m Irving Stringham rics, »ls £ I received ishers to ional ap. ) unique, ^ f calling asses of Each volume in the senes is so consirucxea tnai it may be used with equal ease by the youngest and least disciplined who should be pursuing its theme, and by those who in more mature years and with more ample preparation enter upon the study. Sheldon & Company^s 2'ext-liooks, HilVs Elements of Rhetoric and Cotnposition By D. J. Hill, A.M., President Lewisburg University, author of the Science of Rhetoric. Beginning with the selection of a theme, this book conducts the learner through every process of composition, including the accumulation of material, its arrangement, the choice of words, the construction of sentences, the variation of expression, the use of figures, the formation of paragraphs, the preparation of manuscript, and the criticism of the completed composition. MilVs Science of JRhetoric, An introduction to the Laws of Effective Discourse. By D. J. Hill, A.M., President of the University at Lewisburg. 13mo, 300 pages. This is a thoroughly scientific work on Rhetoric for advanced classes. Intellectual Philosophy (Elements of). 426 pages By Francis Wayland, late President of Brown Univer- sity. The Elements of Moral Science By Francis Waxland, D.D., President of Brown Univer- sity, and Professor of Moral Philosophy. Fiftieth thousand. 12mo, cloth. Elements of Political Economy By Francis Wayland, D.D., late President of Brown Uni- versity. 12mo, cloth, 403 pages. Recast by Aaron L. Chapin, D.D., President of Beloit College. No text-book on the subject has gained such general accept- ance, and been so extensively and continuously used, as Dr. Wayland's. Dr. Chapin has had chiefiy in mind the wants of the dms-TOom, as suggested by an experience of many years. His aim has been to give in full and proportioned, yet clear and compact statement, the elements of this important branch of science, in their latest aspects and applications. Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsoftrigonOOolnerich OLNEY'S MATHEMATICAL SERIES. ELEMENTS or TEIGOiTOMETKY PLANE AND SPHERICAL By EDWARD OLNET, // PRorKssoa op mathbxatiob in thb uinviBBaiTY gb mtchioan. NEW YORK: SHELDON & COMPANY, No. 8 MURRAY STREET. 1885. a^:^ 5^ Entered ^according to Act of Congress in the year 1870. by <. ^ »/" i i SHELDON & COMPANY, ^|l| ^$h© Oflloe 9t the Librarian of Congress at Washington. PROF. OLNEY'S MATHEMATICAL COURSE. INTRODUCTION TO ALGEBRA - , COMPLETE ALGEBRA KEY TO COMPLETE ALGEBRA --.---■ UNIVERSITY ALGEBRA KEY TO UNIVERSITY ALGEBRA A VOLUME OF TEST EXAMPLES IN ALGEBRA - - - ELEMENTS OF GEOMETRY AND TRIGONOMETRY ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University Edition - - ELEMENTS OF GEOMETRY, separate ELEMENTS OF TRIGONOMETRY, separate GENERAL GEOMETRY AND CALCULUS BELLOWS' TRIGONOMETRY PROF. OLNEY'S SERIES OF ARITHMETICS. PRIMARY ARITHMETIC - ELEMENTS OF ARITHMETIC --.----- PRACTICAL ARITHMETIC SCIENCE OF ARITHMETIC - - - - :- - - E. &. Trig. us> CONTENTS. FART IV.— TRIGONOMETRY. CHAPTER I. PLANE TRIGONOMETRY. SECTION I. Definitions and Fundamental Relations between the Trigonometri- LCAL Functions op an Angle (or Arc). page. Definitions 1-5 Fundamental Relations of the Trigonometrical Functions of an Angle 5-7 Signs of the Functions 7-10 Limiting Values of the Functions 10-14 Functions of Negative Arcs 15-lG Circular Functions 10 Exercises 17-20 SECTION II. Relations between the Trigonometrical Functions of different An- gles (or Arcs). Functions of the Sum or Difference 20-25 Functions of Double and Half Angles 20 Exercises 20-29 SECTION HI. Formula for rendering Calculable by Logarithms the Algebraic Sum of Functions 29-30 Exercises 30-31 SECTION IV. Construction and Use of Tables. Definitions 31-32 To Compute a Table of Natural Functions 32-33 To Compute a Table of Logarithmic Functions 33-34 Exercises in the Use of the Tables 34-39 Functions of Angles near the Limits of the Quadrant 39-42 Exercises »-^ w '^^ 800561 IV CONTENTS. SECTION V. Solution of Plane Triangles. pAax. Of Right Angled Triangles 44-45 Exercises and Examples 45-48 Practical Applications 48-49 Of Oblique Angled Plane Triangles 49-51 Exercises 51-55 Oblique Triangles Solved by means of Right Angled Triangles. . . 55-56 Exercises 56-57 functions of the Angles in Terms of the Sides 58-59 Exercises 59-00 Area of Plane Triangles 60-61 Practical Applications 61-64 CHAPTER 11. SPHERICAL TRIGONOMETRY. INTRODUCTION". Projection of Spherical Triangles. Definitions and Fundamental Propositions 65-66 Projection of Right Angled Spherical Triangles 66-71 Projection of Oblique Angled Spherical Triangles 71-73 SECTION I. Solution of Right Angled Spherical Triangles. Definitions 73-75 Exercises 75-76 Napier's Rules 76-78 Determination of Species 79-81 Exercises in Solution of Right Angled Spherical Triangles 81-84 Quadrantal Triangles 84-85 SECTION II. Of Oblique Angled Spherical Triangles. To find the Segments of a Side made by a Perpendicular let fall from the opposite angle 85-86 The Relation of the Sides and Opposite Angles 87 Solution of Oblique Spherical Triangles by Napier's Rules, in Three Problems 87-90 Exercises 90-95 CONTENTS. V SECTION III. General Formula. paob. Angles as Functions of Sides, and Sides as Functions of Angles. 96-100 Gauss's Equations 100-101 Napier's Analogies 10.2 Exercises in tlie Use of tliese Formulae 102-107_ SECTION IV. Aiea of Spherical Triangles 108-110 Practical Applications of Spherical Trigonometry 110-113 TABLES. Introduction to the Table of Logarithms 1-10 Table of Logarithms of Numbers 11-28 Table of Natural Sines and Cosines, and of Logarithmic Sines, Cosines, Tangents, and Cotangents 29-74 Table for Precise Calculation of Functions near their Limits 75-78 Table of Tangents and Cotangents 79^8 PART IV. TRIGONOMETRY, OHAPTEK L PLANE TRIGONOMETRY. SECTION L DEFINITIONS ANl) FUNDAMENTAL RELATIONS BETWEEN THB TRKiONOMETRICAJ. FUNCTIONS OF AN ANGLE (OR ARC). 1, Trigonometry is a part of Geometry whicli has for its sub- ject-matter, Angles. It is chiefly occupied in presenting a scheme for measuring and comparing angles, by means of certain auxiliary lines called Trigonometrical Functions, in investigating the relations between these functions, and in the solution of triangles by means of the relations between tlieir sides and the trigonometrical functions of their angles. 2, JPlane Trigonoinetry treats of plane angles and triangles, in distinction from Spherical Trigonometry^ which treats of spherical angles and triangles. 3, A Function is a quantity, or a mathemaiieal expression, conceived as depending upon some other quantity or quantities for its value. Ill's. — A man's wages /(9r a given time is a function of the amount received per day ; or, in general, his wages is a function of loth the time of service and the amount received per day. Again, in the expressions y = 2a«', y = x^ — 2bx + 5,y = 21ogaa;, y =a^,y ma. function of a; ; since, the numbers 3, 5, a anti b being considered fixed or constant, the value of y depends upon the value we assign to x. For a like reason such expressions as Va^ — a;*, and Scuc' — 2v/~> may be spoken of as functions of a;. Once more, the area of a triangle is a func- •ion of its base and altitude. 4, Angles as Functions of Arcs* — We have learned in Geometry (Part IL, Sec. VI.), that angles and arcs may be treated as functions of each other ; and that, if the angles be taken at the ; ■>^nic ^^^-^ ,.,---.; .^ ;, PLANE TRIGONOMETRY. centre of the same or equal circles, the arcs intercepted nave the game ratio as the angles themselves, and hence may be taken as theii measures or representatives. For trigonometrical purposes, an angle is consiaered as measured by an arc struck with a radius 1, from the angular point as a centre. 5. A Degree being the yj-^- part of the circumference of a circle, becomes the measure of -5^ of a right angle; and, for convenience, it is customary to speak of such an angle as an angle of one degree, of four times as large an angle as an angle of four degrees, etc., apply- ing the term directly to the angle. A small circle written at the right and a little above a number indicates degrees (°). S, A Mimite is -^ part of a degree. Minutes are designated by an accent ( ' ). A Second is -^ part of a minute. Seconds are indicated by a double accent ("). Smaller divisions of angles (or arcs) are most conveniently represented as decimals of a second thougn the designations ^/i2V6Z5,/owr^/iS, etc., are sometimes met with, and signify further subdivisions into 60ths. 5° 12' 16" 13'" is read, " 5 degrees, 12 minutes, 16 seconds, and 13 thirds." Ill's. — In 1 AOB F». 1. an angle of 35°, because the measuring arc ab contains 85 of the 360 equal parts into which the circumference whose radius is Oa, could be divided. In like manner BOC is an angle of 7°. BOC = iAOB = iAOC. Hence, it becomes evi- dent that we may use the numljers 35, 7, and 42 to represent the respective angles AOB BOC and AOC, or the correspondmg arcs ab, be, and 7. A Quadrant is an arc of 90°, and is the measure of a right angle; hence, a right angle is called an angle of 90°. Thus arc ad, Fig. 1, = 90°, or angle AOD = 90°. 8. The Complement of an angle or arc is what remains aftei subtracting the angle or arc from 90°. The Su])pleme?it of an anglu or arc is what remains after subtracting th& angle or arc from 180° Ill's. — In Fig. 1, the angle BOD is the complement of AOB, and the arc bd is the complement of arc ab. The complement of 35° is 90° — 35°- 55°. Thf supplement of 35° is 180°- 35° = 145°. DEFINITIONS AND FUNDAMENTAL RELATIONS. 3 ,9, A Quadrant is often represented by i*, since -n is the semi- cir'jumference when the radius is unity. When this notation is used, 180° the Unit ^^c becomes -^^ = 57°.29578 nearly, or 57° 17' 44 ".8 +, If which is an arc equal in length to the radius. 10, For trigonometrical purposes, an angle is conceived as ger _ erated by the revolution of a line about the angular point, ana hence may have any value lohatever, not only from 0° to 180°, but from 0° to 360°, and even to any number of degrees greater than 300°, as 1280°, etc. An angle of 45° is generated by J of a revolu- tion, 90° by J of a revolution, 180° by i a revolution, 270° by | of a revolution, 360° by one revolution, 450° by IJ revolutions, 1280° by 3|- revolutions, etc., etc. 11, In accordance with the conception of an angle as generated by a revolving line, the measuring arc is considered as originating at the first position of the revolving line {i. e., with one side of the angle), and terminating in the line after it has generated the angle under consideration {i. e., with the other side of the angle). The fi)*st extremity is called the Origin of the arc, and the other the Termination, Ill's.— In Fig. 1, let the angle AOB be considered as generated by a line starting from the position OA, and revolving around the point O, from right to left,* till it reaches the position OB. Oa being taken as unity, the arc ab isthe measuring arc of the angle AOB ; a is its origin, and b its termination. 12, In the generation of angles by means of a revolving line, the normal motion is considered to be from right to left, and the quad- rants are numbered 1st, 2d, 3d, and 4th, in the order in which they are generated. 13, The Trigonometrical Functions are eight in num- ber; viz., sine, cosine, tangent, cotangent, secant, cosecant, versed- si?ie, and cover sed-si7ie. These lines are functions of angles, or. what amounts to the same thing, of arcs considered as measures of angles, and are the characteristic quantities of trigonometry. 14:, Tlie Sine of an angle (or arc) is a perpendicular let fall from the termination of the measuring arc upon the diameter passing through the origin of the arc. Thus in Fig. 2, M is in each case the sine of the angle AOB, or of the arc axh. * The pupil will understand that, if he iraaErines himeelf standing at the centre of moti.:u as the moving body or point passes before him, the distinctions " from right to left." and " from left to right," are easily made. PLANE TKIGONOMETllY. 15, The Trigonometrical TutKjent of an angle (or arc) is a tangent drawn to the measnring arc at its origin, and limited Pie. 2. by the produced diameter passing through the termination of the arc. Thus in Fig. 2, ac is in each case the tangent of the angle AOB, or of the arc axb. 10, The Secant of an angle (or arc) is the distance from the angular point, or centre of the measnring circle, to the extremity of the tangent of the same angle (or arc). Thus in Fig. 2, Oc is in each case the secant of the angle AOB, or of the arc axb. 17 » The Versed-Sine of an angle (or arc) is the distance from the foot of the sine of the same angle (or arc) to the origin of the measuring arc. Thus, in Fig. 2, da is in each case the versed- sine of the angle AOB, or of the arc axb, 18. The prefix co, in the names of the four trigonometrical func- tions in which it occurs, is an abbreviation for the word complement. Thus cosine means complement-sine, i. e., the sine of the com])]e- ment; cotangent means tangent of the complement; etc. The co- sine of 40° is the sine of 90° - 40°, or 50° ; the cosine of 110° is the sine of 90° — 110°, or — 20°; the cotangent of 30° is the tangent of 60° ; the cosecant of 200° is tlie secant of - 110°. ±9. Construction of the Conij^lefnentary Functions. —Let us now see bow the complementary functions are constructed with refer- »uce to their primitives, premising that all arcs in Fig. 3, reckoned fiom A, are DEFINITIONS AND FUNDAMENTAL RELATIONS. 1st. Let AP be Now consideriug to he reckoned around from right to left in this discussion, any arc less than 90* ; then 90' — AP = aPis its complement. a as the origin and P the termination of this complementary arc, Pd is its sine, at its tangent, Ot its secant, and ad its versed- sine. Hence, Prf, at, 0^, and ad are respect- ively the cosine, cotangent, cosecant, and coversed-sine of the arc AP, or the angle AOP. 2d. Letting APP' be any arc between 90° and 180°, its complement is 90° — APP' or — aP', the — sign signifying that the arc is reckoned backward from P' to a. But as the values of the functions will be the same whether the origin be taken at P' or at a, we may take a as the origin of this comple- mentary arc, and P' as its termination, whence p' d' becomes its sine, at' its tan- gen t, Of its secant, and ad' its versed-sine. Therefore P'd\ ai\ Oi\ and ad\ are respect- ively the cosine, cotangent, cosecant, and Fia. 3. coversed-sine of the arc APP', or the angle AOP'. 3d. In like manner, aP' is shown to be the complement of arc APP'P" ; and as P"d", at, Ot, and ad!" are respectively the sine, tangent, secant, and versed-sine of this complement, they are the corresponding cofunctions of the arc APP'P", or the salient angle AOP"« 4th, In the same way, it appears that P"'d"\ at', Ot\ and ad'" are the cosine, cotangent, cosecant, and coversed-sine of the arc APP'P"P'", or the salient angle AOP'". Observe that a.^ a point on the measuring arc 90° from the pnm£tive origin, is the ongin of all the complementary functions. SCH. — It will readily appear from the figure that the cosine of an angle (or arc) is always equal to ffie distance from the foot of the sine to the vertex of the angle (or the centre of the measuring arc). This is the more convenient prac- tical definition. Thus the cosine of AP is Pd = OQ\ the cosine of APP' ia P'd' - D'O, etc. 20, Notation. — Letting x represent any angle (or arc), the several trigonometrical functions of it are writteL sin a;, cos a;, tana;, cot a;, sec a;, coseca;, versa;, and covers a;. They are read "sine a;," ■ " cosine x" " tangent x" " cotangent a;," etc. FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF AN ANGLE (OR ARC). [Note.— Thei?e fundamental relations mn«t be made perfectly familiar They must be memorized, and be as familiar as the Multiplication Table. The etudent can do nothlug in trigonometry without them.] The discussions in this treatise all proceed upon one general plan ; viz., — First obtain the particular property of the e PLANE TlUGONOMETllY. sine and cosine^ and from this deduce all the others, according to the dejjendencies shown in the follow- ing proposition. 21. JProp. — The Fundamental Relations which the Trigono- metrical Function} ^ sustain to each other are: (1) sin'' a; + cos'ic = 1; sin a; (2) tana; cos a; ,^. , cos a; (3) cot X = - — : ^ ' sin a; (4) cota;=:: tana;' (5) sec a; cos a; (6) coseca; = -: — ; ' sma; (7) sec^ a; = 1 •+ tan' x ; (8) cosec'^a; = 1 + cofaj; (9) vers x = 1 — cos a; ; (10) covers a; = 1 — sin a;. (The forms sin'a;, sec'a;, etc., signify the square of the sine, the square of the secant of a;, etc., and are read " sine square x" " secant square a;," etc. The student should distinguish between sin'a;, and sin a;'.) Dem. — In Fig. 4, let x represent any arc as AP, less than 90°. Then PD = sin «, OD or Pd = cos.'T, AT = tana;, OT = seca;, at = cota;, 0^ = cosec«, AD = versin «, and ad = coversin x. (1). In the right-angled triangle POD pd' + OD^ = 0P^ f^i" sin'' a; + cos'' a; = 1, since OP = radius = 1. (2). From the similar triangles POD and TOA, AT PD sma; ^A =rr7=:,ortana; = ^ OA OD' cos a;. (8). From the similar triangles POd and tOa, at Pd ^ cos a; rr- = TT-j, or cota; — -; — . Oa Od sm x (4). Multiplying (2) and (3) together, sinrucosa; . , 1 tana;cota!= : — = 1, oi tan;? = — — cos a; sma; cotst Fig. 4. OT_OP (5). From the similar triangles OTA and OPD, 1 = —: ; but OP and OA each = 1, .•. sec a; OA ~0D DEFINITIONS AND FUNDAMENTAL RELATIONS. (6). From the similar triangles Ota and OP^, Ot OP 1 7^- = 7^, orco8ec«= -; — . Oa Od sm ST (7). From the right-angled triangle OAT, Ot' = oa' + at", or s (8). From the right-angled triangle Oaty Ot^ = Oa + af,0T cosec"^^ = 1 + cot'a;. (9). AD = AO - OD, or vers a; = 1 - cos «. (10). adz=aO — Od, or covers a; = 1 — sin «. Thus the fundamental relations of the functions are established for an ar* less than 90°. But it will readily appear that the relations are the same for any other arc. For example, let x = AP' be any arc between 90° and 1«0°. Then the triangle P'D'O gives sin'^a? + cos'^ic = 1, since P'D' = sin a?, and OD' = cos x. The similar triangles P'D'O and OAT give ^ = %7%, or tan x = ^2^. and the ® OA D'O' cos«' similar triangles P'^^'O and faO give cot x = . In like manner let the sm X student observe the relations when x = APP'P", or an arc between 180° and 270°. So also when x = APPP"P'", or an arc between 270° and 360°. 22. Cor. 1. — T7ie tangent and cotangent of the same angle are reciprocals of each other ; so also are the secant and cosine, and the cosecant and the sine. Thus, if tana; = 3, cota; = J ; since cotic = : . 11 seca; = 2, cosa; = -A-; sinceseca; = , or cosa; = . tana; ''^ cosa; seca; 23, Cor. 2. — Sines and cosines cannot exceed 1. Tangents and cotangents can have any values from. ^o^oo. Secants and cosecants can have any values between^^l and^cxi . Versed-sines and cover sed- gines can have any values hetiveen and 2. These conclusions * will readily appear from the definitions, and an inspection of Fig. 4. SIGNS OF THE TRIGONOMETRICAL FUNCTIONS. 24:, JProp, — Angles (or arcs) considered as generated from right to left being called positive * and marked H-, those considered as gen- crated from left to right are to be called negative and marked —. * ThiB Ib purely an arbitrary convention. W^e might equally well reverse It 8 PLANE TRIGONOMETRY. DfiM. — Tliis is a direct application of the significance of the + and — signs. .e«e Complete School Algebra, pp. 20-23.) Thus, in Fig. 5, if the angle AOP, considered as generated by the revolution of a line from the position OA in the direc- tion of the arrow-head (from right to left), is called positive and marked + , an angle generated by the motion of a line from the position OA in the opposite direction (from left to right), as the angle AOP" thus gen- erated, is to be considered negative and marked — . Let it be carefully observed that it is the assumed direction of the motion of the generatrix that determines the sign of the angle (or arc). Two lines meeting at a common point may be con- sidered as designating either a ^ or a — angle, according to the direction of motion assumed. Thus the lines OAand OP', Mrj. 5, may form the positive angle measured Fia. 5. by the arc APP', or the negative, salient angle measured by the negative arc AP"'P"P'. q. e. d. 23, JProp, — Radius being considered as always extending in the same direction, viz., from the centre toiuard the circumference, is alioays j^ositive. 26. JProp, — The sign of the sine of an angle letiueen 0° and 180° hei7ig -^, that of an angle between 180° aoid 360° is — . Dem.— In Fig. 5, we observe that the sines of all angles terminating in the 1st and 2d quadrants, i. e., between 0° and 180°, may be considered as measured from the primary diameter AB, upward, while those of angles terminating in the 3d and 4th quadrants, i. e., between 180° and 360°, are reckoned downward from the same line; hence, the former being called +, the latter should be — , u (lie two species are estimated in opposite directions, q. e. d. A more elegant conception is to consider the sine as projected upon the diam- eter vertical to that passing through the origin, as aC ; whence Od is the sine of AOP (or arc AP). Now this line evidently is when the angle is ; and as the angle increases, the sine increases, being generated from upward, and hence is called +. This is the same conception as we use in the case of the cosine. Adopting it, we see that sines reckoned from upward are -i-, and downward — . Cosines reckoned from to the right, are -f- , and to the left, — . 27* OoR. — The cosecant of an arc has the same sign as its sine, since coseca; = -. — ; and as 1, being the radius, is +, the sign of sm X -. — is the same as the siffn of sin «, Bin a: ^ DEFINITIONS AND FUNDAMENTAL RELATIONS. 9 28, JProp, — The sign of the cosine of an angle between 0° and 90°, and between 270° and 360°, is +, while that of an angle between 90° cmd 270° is ~. Dem.— In Fig. 5, we observe that the cosines of all angles terminating in the 1st and 4th quadrants, may be considered as estimated from the centre towarc. the right, as OD, OD'"; while correspcmdingly, the cosines of angles terminating in the 2d and 3d quadrants will be estimated from the centre toward the left, aa OD', OD". Hence, by reason of this opposition of direction, the former aie called +, and the latter — . q. e. d. 29, Cor. — The secant of an angle has the same sign as its cosine, eince these functions are reciprocals of each other. (See 27 •) 30* JProp. — The sign of the tangent of an angle between 0° and 90°, and also between 180° and 270°, is + ; while that of an angle between 90° and 180°, and between 270° and 3l)0°, is — . Dem. — Since tan x = , when sinar and cos x have like signs, tan a; is + , by cosa;' . the rules of division; and when sin a; and cos a* have different signs,* tan a; is — . Now, in the 1st and 3d quadrants* the signs of sin a; and cos x are alike, hence in these quadrants tana; is plus; but in 2d and 4th quadrants sin a; and cos a have unlike signs, and consequently in these tan x is — . q. e. d. SI, Cor. — The sign of the cotangent is the same as the sign of the tangent of the same angle, since cot x = . ran ac 32, JProp. — Versed-sine and coversed-sine are always +. Dem. — Vers a; = 1 — cos x ; and as cos x cannot exceed 1, 1 — cos a? is al ways +. In like manner, covers aj = 1 — sin a?; and as sin a; cannot exceed 1, 1 — sin a; is always + . q. e. d. ScH. 1.— It is essential that the law of the signs, as explained above, be well understood, and the facts fixed in memory. Fig. 6 will aid the student in fixing the law in the memory. Having this constantly be- fore the mind, and remembering that tan and cot are + when sin and cos have like signs, and — when they have unlike, and that cos and sec have like Bigns, as also sin and cosec, or, more simply, that tan = sm ^ 1 1 , 1 cot = :: — , sec = — , and cosec = -:— , cos' tan cos sm' the student cannot fail to know the sign of a func- tion at a glance. It will be of service to remember that versed-sine and coversed-sine, and all the functions of angles nf the 1st quadrant, are + ; but that of the other functions than the veraed-sine and coversed-sine, of ^^' " angles terminating in the other quadrants, but Uco are -i- in each quadrant ♦This is a convenient elliptical form for "an angle whose meaooring arc terminateB in the iBt quadrant," etc 10 PLANE TRIGONOMETRY. ScH. 2. — The signs of functions of angles greater than 360* are readily ileter mined by observing in what quadrant the measuring arc terminates. Thus, Bin 570' is — , since an arc of 570° terminates in the 3d quadrant. In any given case, the sign of the function is the same as the sign of the same function of the remainder left after dividing the arc by 360°, or 27t. Thus tan 1180° is the same as tan 100° ; i. i)ictio}i are comprised hetioeen 1 and ± oo , and — 1 and z^ oo. Dem. — Let the student demonstrate and illustrate as in the preceding article. Do not neglect to go through the whole in detail ; it is an important and excellent exercise. 39. JProp.— Versin 0° = 0, versin 90° = 1, versin 180° = 2, ver- sin 270° = 1, versin 360° = 0, and the real limits of the function are and 2. dp:finitions and fundamental relations. 13 t Dem. — The student will readily deduce these results from tlie relation vers a 1 — cos 25. Thus when « = 0, vers = 1— cos = 1 — 1=0, etc. 40, Prop. — Covers = 1, covers 90 = 0, covers 180 = 1, covers^ 270° = 2, covers 360° = 1, and the limits of the real values of this function are and 2. Dbm.— The student should be able to give it 41, General Scholium. — It is important to observe that in the case ol each of the above functions it cMnges its sign hy passing through or co. In fact, it is assumed, in mathematics, that a varying quantity which passes from + to — , does so by passing through or oo . The converse, however, is by no means tiTie ; viz., that whenever a varying quantity passes through or co , its sign necessarily changes.* * The Co-ordinate Ge- ometry affords elegant il- lustrations of the theory of the change in value and sign of these functions. (See Gen. Geom., 23, etc.) The annexed Fig- ure represents a curve, (or as the student may be dii?posed to considei it, a series of curves), constructed as follows : On the indefinite line AE. ^ circumference is developed (as it were straightened out), the origin being at Ai ^^^^ AE l>eing the length ol the circumference. The jurves mn, mW, m"n/' ire drawn by erecting at every few degrees from Ai ^ liQ3 equal to the tangent of the same number of de- grees, above the line AE when the tAngent is +,and below AE when the tangent is — . Thus Aa = 45° in length, and ab = tan 4.5° ; J^c = 135°, and cd = tan 135°. Such lines as aft, cd, etc., are called ordinates of the curve. The law of change in these ordinates is manifestly the same as the law of change in tangents. We see that as we pass from 0° to 90°, the ordinate (tan- gent) passes from to + oo. A; 90° the ordinate (tangent) in both + and — , i. «., ± oo. So also at 270°, and at other similar points. A similar device illustrates the changes in the other trigono- metrical functions. Some may see the propriety of distinguishing oo as + and — , who, never- theless, do not see why it is necessary to make the same distinction in the case of 0. Bui a moment's reflection will show that one distinction involves the other, since oo and are matp- »Uy reciprocals of each other. Pio. 8. 14 PLANE TllIGONOMETRT. 42, ScH. The results of the preceding discussion of the signs and limits of the ti-igonometrical functions, 24 to 41, are exhibited in the annexed .-§ ^ ^ OQ s - a = a = a - ^ \^ 3 ;3 a 1 i 1 1 1 ^ § ^ 1 i 1 N 1 1 o TH + + + + a tH + 8 tH 1 8 ^ ■H 1 H- g B + o + B 1 o 1 % ■*-» •J3 " © 8 lH 8 tH « H- + -H 1 8 tH 8 tH ^ -H 1 H- + 1 B + 3 1 B 1 3 + % ■IH 8 tH 8 + -H 1 Ff- o 8 o 8 ?i -H H- -H H- 1 S + S 1 2 + 3 1 9 8 o 8 © H- -H H- -H 8 o 8 © 5i -H H- -H H- B + o B 1 8 s + © 2 1 8 »+■ -H H- ■H © tH © tH s -H 1 H- + 1 B + B 1 B 1 3 + tH o tH © + -H 1 H- T-( © iH © ij + ■H 1 K- K s + 2 + 3 3 1 S o ■tH o tH H- + -H 1 Viiei fe ^h « ^ s 3 S © ¥il=-^^if-=:_^^ — tan X tan x sm {—x) — sin x sin a = — cot X. ScH. — The proper sign of a functton of a negative angle can always be ascer- tained by observing in what quadrant the measuring arc ends, in a mannei altogether similar to that in which the signs of the functions of positive angles are determined. CIRCULAR FUNCTIONS. 46. Circular Functions are angles (or arcs) expressed as functions of sines, cosines, tangents, or other trigonometrical lines. Ill's. — In the expression sin x, we designate a sine, i. e., a right line, merely using the x to tell wJiat sine, as the sine of 20°, of 135°, etc. But we often wish to speak of an angle (or arc) which has a particular sine, tangent, or other trigo- nometrical line. Thus, we say, " the angle (or arc) whose sine is i,'* " the angle (or arc) whose tangent is 3," etc. In this form of expression, it is evidently the angle (or arc) which is the thing mainly thought of; and it is conceived as de- pendent upon its trigonometrical line. 47. dotation. — The circular functions are written sm~% cos~^a;, tan~^2;, etc. ; and are read "tlie angle (or arc) whose sine is y" "the angle whose tangent is z" etc. Ill's. — The expressions x = sin-^y, and y = sin x, are ultimately equivalent, since the first is " a; = angle whose sine is y," and the second, " 2/ = the sine of «." The only diflFerence is, that in the first form the migle {x) is the thing thought of, and the sine (y) is used merely to tell what angle ; but in the second form, the sine (y) is the prominent thing, and the ajigle {x) is used simply to tell what sine. This mutual relation has caused the circular functions to be called also Inverse Func- tions. ScH. — This notation is rather an unfortunate one, inasmuch as it is the same as has been already adopted in the theory of exponents. The student will how- ever observe that the signification in this instance is altogether difierent from the former. Thus, since we write *' the square of sine a;," sin^a; ; according to the theory of exponents, sin-'sr would be -;— -• ISo also sin-^a; should mean — —. sm^a? sin x Now, the former of these expressions would actually signify as indicated (though it were belter to write it (sin a;)-^), while the latter does not mean at all what the theory of exponents would make it. Unfortunate as the notation is, it is probably best to retain it. It, doubtless, was suggested thus : If we have y = a^x, we may write x = ar-^y, so also y = ax, may \ e written x = o-'y. This afibrds a parallelism hi form, but not in signification. DEFINITIONS AND FUNDAMENTAL llELATIONS. EXERCISES. 17 1. What plement ? IS the complement of 150° 21' 13".5 ? What the sup. Give the complements and also the supplements of 125° 15', 283° 21' 11", 36° 05' 02", and 89° 00' 12". •/I [or arc) -« ? In f-r ? In the 2. How many degrees in the angL urc27r? In J*? InlJ-'r? 3. How many times is the radius contained in 108°? How many times in 27r ? How many in 460° ? How many in 210° ? 4. Radius being taken as the measure of the arc, by what are 45" represented? By what 90°? By what 180°? By what 225°? How many degrees does 1 represent, radius being the measure ? 5. Find the length of a degree of the meridian upon a globe of 18 mches diameter. 6. Express 12° 22' 13" 11'" 05'' in °, ', ", and decimals of a second. So also express 53°.51 in °, ', ", etc. 7. How many degrees, minutes, and seconds in an arc equal to twice radius ? three times radius ? Show that 27° is equivalent to ^*. That 10° to radius 10 ft. = 1.75 ft. What radius gives 1° == 1 inch ? 8. Draw any angle, and construct its sine, cosine, tangent, or any other trigonometrical function, and then determine as nearly as prac- ticable the numerical values of the function by actual measurement. Solution. — Given the angle MON, to find the numerical value of the tangent as near as pi'acticable by measurement. Taking any convenient unit, as OA, for a radius, and striking the arc Aa, draw AT tangent ^o «A at A. Now apply OA to AT and find their ratio (Part I., 56*). In thia case AT = 1^ approximately, .'. tan MON = 1^. [Note. — The student should practice upon such exam- ples, finding the values of all the functions until the process, and the meaning of the numerical value of any function of an angle, are clearly seen.] 9. Construct an angle whose sine is |, ^. e., sin" Solution. — Let O be the required angular point, and OA one side of the angle. Lay off from on OA 3 measures of any convenient length, making On. Using Qa as a radius, describe the indefinite arc «M. Erect 00 perpendicular to OA and take 00 = t of Qa. Through C diuw CP parallel to OA. Finally, draw OB through P. AOB is the angle required, since Oct being 1, the sine cf AOB, PD.is i AOB =siii-if. % Fio.9. FiG.ia 2 18 PLANE TRIGONOMETRI. IC. Coustruct an angle whose cosine is |. That is, construct cos-^|. Sua. — The construction is the same as in the last example, except that instead uf OC being drawn to limit the arc, a perpendicular is erected to Oa at | (the second point of division) from 0, and the point P located where this perpeudiculai intersects the arc. 11. Construct an angle whose tangent is 2. That is, construct tan-^2. SuG. — To construct this at on the line OA, BYg. 10, take any convenient ra- dius, Oa, and strike the indefinite arc. Then erect at a a tangent, and make it equal to twice the radius used. Through the extremity of this tangent and draw a line, and the angle between this and OA is tan-^S. 12. Construct sec~^2; cot~'3; cosec'^lj-; an obtuse angle sin~^^; tan-^-S). SuG. — To construct sin-^^, see Mg. 7. Let OA be one side of the angle, and O the vertex. With any convenient radius draw the semicircumference AaB, and draw the perpendicular Oa. Bisect this perpendicular, and through the point of bisection draw a parallel to AB, intersecting the arc in the 2d quad- rant. Through this intersection draw a line, as OP'. Then AOP' (assuming the constiMiction as specified, and not as in the figure) = sin-'^. 13. Construct the following: tan~'l ; tan~'( — 1); tan~' J ; tan-'(-2); tan-'(— i) ; cos-^— J); sec-*( — 2); cosec-'(— 3); versin-4 ; versin~'lj. 14. From the fundamental relations (21) deduce the following : sm .r=\/l — cos'.t; cosoj^a/I -sin^s:; tana;cota;=l; tan a; cos a; = since; sin X . cos 2; . ^ . ^r.c^ 1 coaa;=:- :sma;= — ^^i sina;= .====, cos a; = — - . tanic cota; yl + cot'a; Vi + tan^' , . , ^ seca; . tanic = sma; sec a;: cot a; = cos.'c coseca;: tan x = ; sma; = coseca; ^08x seca;—l_ coseca;— cot a: ^ _ Vl— sm^ vers X = = ; cotic sma; . _ J y\y±a ^ ycosec^a;— 1 sec a; coseca; 15. Given tana; = j, to find the other trigonometrical functions of a;. Results : Sec a; = f ; cos x = ^; sin a; = -f ; cosec a; = | ; cot a = I ; vers x = ^; covers a; = |. 16. Given sin a; = f , to find the other trigonometrical functions of X. 17. Given sec a; = 2, to find th« other trigonometrical functicns of X. 18. Given tan .7; = — 1, to find the other trigonometrical functionb w DEFINITIONS AND FUNDAMENTAL RELATIONS. 19 Results. — Cot a; = — 1; secx = ^^\^', cosec x = ±^2"; sin x ^z rfc I a/2; CGS^• = rp J a/2 ; versin a; = 1 ± iA/2"; coversm x = 1 qpi V2: [Note. — Observe closely the signs of the functions in Ex. 18.] 19. In the preceding examples the constructions required have " been limited to angles less than 180°, but it is evident that an infi- nite number of angles (or arcs) according to the more comprehensive trigonometrical view, correspond to the same function. What angles or arcs have their tangents each 1 ? What, each — 1 ? Construct an angle between 180° and 270°, whose sine is —J. What other angle less than 360° has the same sine? 20. Having given a sine, how many angles less than 180° corre- spond to it ? Construct the angle or angles less than 180° whose sine is |. How many angles less than 180° have the same cosine ? tan- gent? cotangent ? (In each of the last three cases only one.) Con- struct ?/ = COS"' J; y = cos~^(— J). How are these angles related to each other? Construct^ = tan~^3; y = tan~^(— 3). How are these angles related to each other ? (Restrict the constructions and questions in this example to angles less than 180°.) 21. Given y = sin~X show that cosy = /^/l— .r'; tany = a/T^ ' '^^^ = Vi^T^' y = cosec-^- ; vers y = l- V^^^; 22. What are trigonometrical functions of 450° ? Of 1350° ? Of 900°? 23. Show that the following are true for all integral values of n, including : sin 47i- = ; sin (4^ + 1) - = 1 ; sin Un + 2) - = ; ^ Z Z Bin (471 4- 3) - = — 1 ; cos 4^ - = 1 ; cos (47^ -f 1) ^ = ; cos (47j + 2) ^ ^ /i lii Ji = - 1 ; cos (4?i + 3) - = ; tan ^n\ = ; tan {^2n + 1) ^ =co . 24. What are the signs of the several trigonometrical functions of -110°? Of -35°? Of -500°? Of -2000? 25. Prove that sin 30°= i, cos 30° = i a/3, tan 30°= J a/3, and cot 30°= a/3, sec 30°= I a/3, and cosec 30°= 2. SuG.— Observe that the chord of fiO° = 1, and that the sine of 30°= \ tht chord of 60°. Make the figure. so PLANE TRIGONOMETllV. 26. Prove that sin 45° = ^\/2, cos 45°= ^\/2, tau 45°= 1, cot 45* = 1, sec 45°= V2, and cosec 45°= ^2. SuG. — Observe that sin 45' — cos 45° ; hence sin" 45" + cos" 45° = 1, becomes 2sin'» 45° = 1. Sen. — The viilues obtained in the last two examples should be retained in the memory, as they are of frequent use The functions of 30° and of 45° ait always assumed to be known in any trigonometrical operation. SECTION IL RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF DIFFERENT ANGLES (OR ARCS). {a) FUlfCTIONS OF THE SUM OR DIFFEREN'CE OF AiJ^GLES (or arcs). 4:8, ^voj), — The sine of the sum of two angles {or arcs) is equal to the sine of the first into the cosine of the secondy2^^us the cosine of the first into the sine of the second. Thus letting x and y represent any two angles (or arcs), sin {x ■\- y) = sin x co&y + cos x sin y. Dem. — Let AOB and BOC be the two angles represented respectively by x and y. Draw the measuring arc aP\ and the sines PD, and P'E of the angles. AOC = AOB + BOC, is the mm of the two angles. Draw P'D', the sine of tlie sum of the two angles. Then PD = siuic, P'E = sin y, CD = cos a;, OE = cos y, P'D' = sin {x + y), and OD' = cos(a; + y). Now sin (a; + y) = P'D' = EF + P'L. But from the similar triangles EOFJ" ana POD, we have |? = P^?, or -1^ = '-^. ' OE OP cosy 1 EF = sin X cosy. Also, from P'L op, or P= OP smy cos a; the similar triangles P'EL and POD, we have -^: •*. P'L = cos a; sin y. Substituting these values of EF and P'L, we have sin {x + y) = sinaj cosy + cos a; sin y. q. e. d.* * This demonstration may seem defective, eince the sum of the angles x and y, as represented \n the diagram, is less than 90° ; nevertheless, in the General (Analytical) Geometry, we con- etantly proceed in a manner entirely analo- cot y — tan x osition, bv substituting in it - — for cot, and reducing. Notice that the form in the Prop, is in terms ol the tangents. Also observe why dividing by a certain term gives a particular form. an«i by which to divide to get a required form. FUNCTIONS OF THE SUM OR DIFFERENCE OF ANGLES. 23 !to»80;±fl = :iiilf = tan..etc.,etc. COS (180 + X) — cosaj SS. I^rop* — Tlie tangent of the difference of two angles (or arcs) is eqtial to the difference of their tangents, divided by 1 plus the rect- angle of their tangents. Thus, x and y being the angles, _ tan X — tan y tan X tan y tan(a;-2/) = j^ _ m / N Sin (05 — y) sin a; cos y — cos x sin y Demonstration. Tan {x — y) = — 7 '^ = ^ r ~ — cos {x —y) cos X. cos y + sin a; sin y Bin X cos y _ cos x sin y sin a; _^ sin y cos X cos y cos a; cos y _ cos x cos y _ tan x — tan y © k d fS cos a; cos y sin a; sin y . sin a? sin y 1 + tan x tan y* cos ar Cos y cos a; cos y cos aj cos y foot-notes to preceding proposition.) This proposition is also readily deduced from the preceding by substituting in the formula tan [x + y) = 2L) 1 — tan aj tan J — y for y, and remembering that tan (— y) = — tan y. Cor. Tan{^^°—x)=cotx.Tan{l%0°-x)=z-tanx, Tani^'i^f-x) = co^ X. Tan (360° - a;) = - ^Ja?^ x. Dem. Tan (90^ - ai) = ^''' ^^^I " 1 = ?^ = cot x. Tan (180' - «) ^ cos (90 — a;) sm a; ^ ' sin (180° -^)_. sin a. ^ _ta^^, Tan(370- - a;) =-'^°^^^^" " ^> - " ^°«^ cos (180*— a;) —cos a? cos (270°— a;) —sin a; „, /«««« X sin(360<'— a?) -sin a; cot^ Tan(360»-x) = ^3jL-g^=-__ = _tan«. • These and the kindred formuUe maybe produced by a direct application of the propositioi *mw i. m«, X tan 90" + tan a? tan 90* 1 ^ ^^^ Thus, tan (98'.^x) = ^_^^^^Q,^^^^ = :^^^^g^^-^ = -^^^=:-cotar. (The reason fo, dropping tan a; and 1 is that they are finite terms connected with infinities, as tan 90° = oo . Or, tanoj ^ tana; ^u • * m«n X tan TO" + tan a; ■*■ tan 90° ^ oo 1 )therwisc, tan (90° + ar) — — >-»"■'" ^ ,.• — . 1 — tan90° tnna; 1 ~1 ~ — tana?' r--^^ — tan X tJin a: tan 90° 00 finite divided by an infinite equal* 0. Again, Un (180° + a?) = i£I!i80^±J^!I5_ = !?!E£ ^ ^ ^ I — tan 180° tan « 1 ton«, since tan 180° = 0, etc.. etc. 24 PLANE TRIGONOMETRY. 54:, Prop. — TJie cotangent of the sum of two angles {or arcs) it^ equal to the rectangle of their cotangents minus 1, divided by their sum. Thus, X and y being the angles, , , . cot X cot V — 1 cot (x -\- y) = — 7 ^^— - — . ^' coty + cot a; co&g cosy sina; sinji -^ ^ ^, . cos(a5 + y) cosa; cosy — sin.'C siny sinrz; siny sin.c sint DeM. Cotix +y)= -:-^, —^= —- = -. ' r-^ sin (a; 4- y) smajcosy +co&csiny sin^cosy cos.csin^ sina; smy sina; s'mt cos X cos y sin a; sin y cot a; cot y — 1 _ , , .^ ^- — -. Q. E. D. Or we may deduce it tlius, cos y cos X cot y + cot a; sin y sin a; 1- ^ , . 1 1 — tan X tany cot x coty cot x cot y — 1 cot (i* + v) = ; = = = tan (a; + y) tan x + tany 1 1 coty + cotx cot X cot y Q. E. D. Cor. Cot(90° -h x) = - tarix, Cot{lSO° -{- x) =cot x. Cot(270° + x) — — tan X, Cot (360°+ x) = cot x. Dkm. Divide cosine by sine, or take the reciprocals of llin corresponding ♦.angents Tims, cot (90* + x) = - — j^rjr^ r = — = — tan x, etc. ^ > V / tan (90 + a;) — cota; ' S5. Prop, — Tlie cotangent of the difference of two angles (or arcs) is equal to the rectangle of their cotangents plus 1, divided by their difference. Thus, x and y being the angles, f / \ _ ^^^ ^ ^^^ y + 1 ^ ^^ ~ coty — cot X Dkm. Substitute — y for y, in the preceding formula ; or, divide cos (a? — y) by sin {x — y) and reduce ; or, take the reciprocal of tan {x — y), and substitute —7 for tan. eat Cor. Cot (90° - x) = tan .r. Cot (180°-^;) = - cot x. Cot (370°- x) = tan X. Cot(SCiO°—x) = —cotx. Dem. Same as above. Sen. 1. The formula for the secant and cosecant of the sum and of the difference of two angles (or arcs) are not of sufficient importance to warrant their introduction here ; some of them will be given in the exercises, as also the extension of those already given to the case of the sum of three or more angles, p FUNCTIONS OP' THH: SUM OH DIFFERENCE OF ANGLES. 25 flCH. 2. The results reached iu this discussion are so important that we will eollect them into a (A) sin (x + y) = (B) sin (x -y) = (C) ccw {x -^ y) = VD) cos (a; -y) = (E) tan {x + y) = (F) tan (a; -2/) = (G) cot (« + y) = (H) cot {x-y) = TABLE, sin a; cos y + cos x sin y. sin X cos y — cos x sin y. cos x cos y — sin x sin y. cos a; cos y + sin x sin y. tan X + tan ^ 1— tanaj tany* tana; — tan y ♦ 1 + tan a; tan y* cot a; coty — 1 coty + cota; ' cot a; cot y + 1 ^ coty — cotaj (T) X 90°- » 90»+a; 180°^ a; 180° -»- a; 270°- X 270°+ a? 360°- a; 860°+« ilae cosaj cos a: Binx — Bin a; — cos a; — cos a; — sin a? sin a; ooeine sin a; — Binar — cos a? — cos X — sin X sin a; cos a; COS a; tangent cotaj — cotaj — tana; tan X cot a; — cot a; — tana; tanx cotangent tana; — tanaj — cot X cot X tana; — tan a; — cot a; cot a; It will not be found difficult to memorize and extend set (I), if the studer.! observes, that, when the number of whole quadrants is odd (as 90°, 270°, etc.), the function changes name (as from sin to cos, from cos to sin, etc.) ; but, when the number of whole quadrants is even, the function retains the same name. The sign of the sine and cosine is readily determined according to fundamental principles by observing where the arc ends, assuming x < 90°. Thus 180°+ x ends in the third quadrant ; hence its sine (which in numerical value is sin x) is — , and its cosine is also — . As the signs of tlie tangent and cotangent of tlie same arc are alike, we have only to observe whether the sine and cosine, in any ^iven case, have like or unlike signs, in order to determine the sign of the tan- gent and cotangent. For example, what is cot (630° + a;) equal to? The num ber of quadrants being odd (7), the function changes name, and since the arc ends in the fourth quadrant, its sine is — , and its cosine + ; therefore cot (630° + x) — — tana;. If in any given case x > 90", determine the character of the function as above, on the hypothesis a; < 90°, and then modify the result for the partic- ular value of X. Thus in the last case, if x was between 90° and 180", its taa-J gent would be — , and for such a value cot (630" + a;) = — tan x. Or, wc may consider at first where the arc ends, taking into consideration the given value of a;. 26 PLANE TlilGOxNOMETRY. (b) FUlfCTION^S OF DOUBLE AND HALF ANGLES. SO* JProp,— Leitifig x represent any angle (or arc), 2tan z (K) sin 2a; = 2sin a: cos a: ; (L) cos 2a; = cos" a; — sin" a; = 2cos' a; — 1, or 1 — 2sin" x ; (M) tan 2a; = (K) cot 2a; = 1 -tan' a;' cot' x — 1 2cota; Dem. These results are readily dedi^^ed from (A), (C), (E), and (G). Thus, in sm {x •\- y) = sin x cos y + cos x sin y, if we make y —x^ we have sin 2x — sin X cos X + cos a; sin a; = 2sin x cos x. (In like manner produce the others. ) 57. JProp, — Letting x represent any angle {or arc), we have. (0) sinja; = ±Vi (1— cosa;) ; (P) cosja; = rtViCl + cosa;); ^ 1 + cos a; (E) cotig=±|/l-+°°"^. J. — cos X Dem. From (L), 2sin'a; = 1 — cos 2a;, or sin a; = ± ^\ (1 — cos 2a;). Putting \t for a;, this becomes sin \x— ± ^\ (1 — cos a;). In like manner, from the same .ormula (L), 2cos''a; = 1 + cos 2a; ; whence, cos \x = ± -y/^ (1 + cos a;). Again, , sin ia; . /l — cos a; , . , 1 . . /l + cos a; tania; = — ±i/ ; and cotia; = ;: — — = ±4/ . Q. e. d. cos-Ja; f 1 + cosa; tan^a; r 1 — cosa; ScH. The sign of the function in the case of each of these is + if a; < 180' ; but can only be determined by the value of a; in any given case. EXERCISES. 1. Prove from Fig. {a) that sin {x 4- y) = sin x cosy -{- cos x sin y when X and y are each < 90**, but x + y > 90°. 2. Same as in Ex, 1, from (b), when X < 90°, X -{- y> 90°, and < 180°, and y > 90° and < 180°. Bug. In this case, sin (a; -»- y) = P'D' ^ P'L - EF. Ill other respects the demonstration Is identicai with the preceding. This gives 8iii(« + y)= cos a; sin y — sin a; cosy. But the O F D flf-A («. FUNCTIONS OF THE SUM Oil DIFFEIIENCE OF ANGLES. — sign is accounted for in the general formula, sin {x + y) = sin x cos y + cos x sin y, by notic- ing that cosy is — , when y > 90° and < 180°. 3. Same as in the preceding, when X > 90°, y < 90°, and (x + y) < 180°. Sua. Here sin(a^ + y) = P'D' = EF - P'L. In .all other respects the demonstration is identical with the otJier cases. The — sign in this case arises fron*. x being between 90° and 180", whence cos a? is — . [Note. A number of other cases may be devised, but the Bx + cosy" cos iix + y) cos ^{x — tani(a; + y) tan i{x — y). q. e. d. (Observe the opposition in signs.) CCS a; — cos y _ — sin ijx + y) sin ^{x — y) _ _ sin i{x + y) sin|(a; — y) cos a? + cos y ~ cos i{x + y) cos i{x — y) ~ cos i{x + y) cos i{x — y) 62, I^roh* — To render tan x ± tany calculable by logarithms. ^ ^ , sin a; siny sina; cosy ± cosa; sin y sin (x± y) Dem.— Tan a? ± tan y = ± — - = — ^ — ^ cosaj cosy cos a; cosy cos a; cosy EXERCISES. Let the student deduce the following relations : . ^ i i sin (ic + «/) 1. Cot X + cotv = . ^ : ^\ '^ sin X sm y 2. Sec ic + sec V = — — -^ — -* ^ cos X cos y ^ r, 2 sin Ux 4- y) sin Ux — y) I. Sec « — secy = ^ ^ ^ —. ' cos ic cosy CONfeTRUCTION AND USE OP TRIGONOMETRIOAL TABLOiS. 31 4. 1 + COS ic =2 cos' ^x, (See 57.) 5. 1 — cos a; = 2sin' Ja;. 6. s^^^ + ^^"y = tan Ux + y). (Divide A' by 0', S9.) COSiC + COS^ S\ JJ \ J y J -, sin a; — sin V , w x 7. ; = tan i(a;- y). COS X + COS y » V ^ / ^ ^ - sin cc + sin 2/ i. i / \ 8. -= — cotUx — y). ■ COS X — COS ?/ » \ c/ / ^ sin ic — sin y . . , . 9. ^ = — cot i(a; + y), COS a;— cosy »\ ;// SECTION IV, CONSTRUCTION AND USE OP TRIGONOMETRICAL TABLES. [Note. — In order to read this and the subsequent sections, the student needs a knowledge of the nature of logarithms, and the method of using commou logarithmic tables. If he is familiar with the last chapter in The Complete School Algebra of this series, he is prepared to go on. If he has not this knowledge, he should read the inti'oduction preceding the table of Logarithms before reading this section.] 63, A Table of Trigonometrical Functions is a table containing the values of these functions corresponding to angles of all different values. In consequence of the incommensurability of an arc and its functions, these results can be given only approxi- mately; yet it is possible to attain any degree of accuracy which practical science requires. 64:, There are two tables of trigonometrical functions in common use, the Table of Natural Functions^ and the TaUe of Logarithmic Functions. 65. A Table of Natural Trigonometrical Functions is a table in which are written the values of these functions for angles of various values, the radius of the circle being taken as the measuring unit, and the function being expressed in natural num- bers extended to as many decimal places as the proposed degree of accuracy requires. 66. A Table of Liogarithmic Trigonometrical Func- tions is the same as a table of natural functions, except that the logarithms of the values of the functions are written instead of the functions themselv3S, and to avoid tbe frequent occurrence of nega- 32 PLANE TRIGONOMETRY. tive characteristics, the characteristic of each logarithm is increased by 10. For example, sines and cosines being always less than unity, excei)t at the limit (S3), and tangents of angles less than 45° and cotangents of angles greater than 45° being also less than unity, the logarithms of all such functions have negative characteristics. To obviate the necessity of writing these with their sign, the charac- teristic of each logarithm is increased by 10. 67. JProb, — To compute a taUe of natural trigonometrical func- tions for every degree and minute of the quadrant. SoLUTiON.-^It is evident that an arc is longer than its sine, but that this disparity diminishes as the arc grows less. Thus, in a circle whose radius is 1 inch, the length of the sine of an arc of 1° would not differ appreciably from the arc. Much less should we be able to distinguish between the sine of V and the arc. Now, since when the radius is 1, a semicircumference = tt = 3.1415936, and also = 180°, or 180 X 60 = 10800', we have the length of an arc of V = 3 14159'^ \i\or\t\^ — 0.0002908882 approximately. Assuming this as the sine of 1', we lOoOO obtain the cosine thus, cos V =\/l-sinM' =\/(l + sml')X(l-sinr) = v/l.0002908882 X .9997091118 = 0.9999999577. Having thus obtained sufficiently accurate values of sin 1' and cos 1', we can continue the operation as follows : from the formulm sin (« + y) + sin {x —y) = 2 sin a; cos y, and cos {x + y) ■\- cos {x — y) = 2 cos x cos y, we have sin (« + y) = 2 sin a; cos y — sin {x — y\ cos (a; + 2^) = 2 cos x cos y — cos {x — y). Now letting y remain constantly equal to 1', and letting x take successively the values 1', 2', 3', etc., we have _ j sm 2' = 2cosl' sin 1'- sin 0' = 0.0005817764 For a; _ 1 , j ^os 2' = 2 cos 1' cos 1' - cos 0' = 0.9999998308 sin 3' =r 2 cos 1' sin 2' - sin 1' = 0.0008726646 cos 3' = 2 cos 1' cos 2' - cos V = 0.9999996193 For a? = 2', ] _ j sm 4' = 2 cos r sin 3' - sin 2' = 0.0011635526 For a? _ S , ^ ^^^ ^, ^ ^ ^^^ ^ ^^^ g, _ ^^^ ^, ^ 0.9999993232 _ j sin 5' = 2 cos 1' sin 4' - sin 3' = 0.0014544407 For a; _ 4 , -j ^os 5' = 2 cos 1' cos 4' - cos 3' = 0.9999989425 etc, etc. These operations present no difficulties except the labor of performing the aumerical operations. Of course 60 operations are required for every degree, and for 30°, 1800. But having computed the sines and cosines for every degree and minute up to 30° II CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES. dii we can complete the work by simple subtraction of values already found. For example, letting x = 30°, the first formula used above becomes sin (30° + y) = cosy— sin (30° - y), and from cos [x + y) — cos {x — y) = — 2 sin x sin y, we have cos (30°+ y) — cos (30° — y) — sin y Now making y successively =^ 1', 2', 3', etc., these give j sin 30° 1' = cos 1' — sin 29° 59' ^-- I cos 30' V = cos 29° 59' - sin 1' j sin 30° 2' = cos 2' - sin 29° 58' ( cos 30° 2' = cos 29° 58' - sin 2' j sin 30° 3' = cos 3' - sin 29° 57 ] cos 30° 3' r=: cos 29° 57' - sin 3' etc., etc. All of these values which occur in the second members having been deter- mined in reaching sin 30° and cos 30**, those in the first members can be found by performing the requisite subtractions. Proceeding in this way till we reach 45°, the numerical values oi all sines and cosines become known, since the sine of any angle between 45° and 90", being the cosine of the complementary angle, will have been computed in reaching 45°. And so also the cosines of angles between 45° and 90° will have been computed as sines of the complementary angles below 45°. The sines and cosines being computed, the corresponding tangents, cotan- gents, and, if need be, the secants, cosecants, versed-sines, and coversed- sin X 1 cos X smes, can be calculated from the relations tan x = , cot x = - — or -; — , cos a;' tana; sin.'?! sec a; = , coseca; = - — , vers .a; = 1 — cos a;, and coverea; = 1 — sin x. cos X sin a; 68, ScH. — If it is desired to obtain the natural functions of angles esti- mated to seconds, it is necessary that the values in the tables computed as above be extended to 7 decimals at least From such a table we may make interpo- lations for seconds with suflacient accuracy for most practical ends, except for values near the limits, where the disparity between the variation of the arc and that of the function changes very rapidly. For example, let it be required to find sin 34° 24' 12" from the data sin 34° 24' = .5649670, and sin 34° 25' = .5652070. We observe that an increase of 1' upon the angle of 34° 24' makes' an increase of .5652070 - .5649670 = .0002400 in the sine. Hence an increase of 12", or i of 1', makes an increase of i of .0002400, or .0000480, npp-oxi- mately Adding, we have sin 34° 24' 12" = .5650150. The student must be careful to notice whether an increase of the angle makes a numerical increase or a de crease of the function, and add or subtract as the case may require. 69, JProb, — To construct a talle of logarithmic trigonometrical functions. SoiiUTiON. — Compute the natural sines and cosines as in the preceding prob- lem. Take Uie logarithms of the values thus obtained^ and add 10 to each 3 34 PLANE TKIGONOMETRY. <;Laracteristic. The results are the ordinary tabular logarithmic sines and co Bines. For example, we find from the table of natural functions that sin M° 24' r= .5649670. The logarithm of this number is 1.752023. Adding 10 to the char- acteristic, we have log sin 34" 25' = 9.752023, as usually given in the tables. In like manner the cosines are obtained. sin X To obtain the tabular logarithmic tangents, we have from tan« = »og tan X = log sin x — log cos x. If we now take the log sin x from the table as computed by the preceding part of this solution, and from it subtract the cor- responding log cos X, the result is the true log tan x, since the extra 10 in the tabular log sin and log cos is destroyed by the subtraction. Therefore, to this diflFerence we must add 10 to get the tabular log tan, as above explained. For example, the tabular log sin 34° 24' = 9.752023, and log cos 34° 24' = 9.916514. Hence, the tabular log tan 34° 24' = 9.752023 - 9.916514 + 10 = 9.835509. In like mapuer the tabular log cot a; = log cos x — log sin a? + 10. If thb logarithmic secants are required they can be obtained from the relation sec X = , which gives log sec x = — log cos x. In applying this by means of the tabular functions, it must be observed that the log cos x, as we get It from the table, is 10 too great ; hence, the true log secic — — log cos x + 10. In tabulating log secants and cosecants, it is not necessary to add 10, since, as these functions are never less than 1, their logarithms are never negative. 70. SCH. — The interpolations for seconds are usually made in the same way when using the logarithmic functions, as explained above for the natural func- tions. But to facilitate the operation, the approximate change of the logarithm for a change of 1" of the angle is commonly written in the table, in a column called Tabula/r Differences, and marked D. EXERCISES. 1. Find from the tables at the close of the volume the naturai trigonometrical functions of 25° 18'. Solution. — To find the sine and cosine w look in Table II., and find 25° at Uie top of the page. In the extreme left-hand column we find the minutes, and passing down to 18, find opposite, in the column headed N. sin (natural sine) 42730 ; also in the column N. cos, we find 90408. Now, as these are the lengths of the sine and cosine as compared with radius, we know they are fractions. .-. Sin 25° 18' = .42736, and cos 25° 18' = .90408. To find Vie tangent we turn to Table IV., and finding 25° at the top of the page, cass down the column of minutes, on the left-hand of the page, to 18, opposite which, and under the column headed 25°, we find 2698. To this we prefix the figures 47, which stand in the same column, opposite 11', and belong to the tan- gents of all the angles from 25° 10' to 25° 19', and are omitted in the table shn- ply lo relieve the eye and to economize space. Thus we find tan 25° 18' = .4726981 the number being known to be a fraction because the angle is less than 45° IB To find CONSTRUCTION AND USE OF TRIOONOMETllICAL TABLES. I'd find the cotangent we look at the bottom of the page in the same table till we find 25*, and then passing up the minutes column at the right hand, find col25° 18' = 2.11552. If tite secant were required we should be obliged to obtain it by dividing 1 h^ the cosine, as our tables do not include this function. Thus sec 25° 18' = 1 1 cos 25° 18' .90408 = 1.1061. I I lNote. — Tables of secants and cosecants are sometimes given, but they are not of sufiicient importance to justify their introduction into an elementary text-book.] 3. Show that sin 37° 43' = .61176 ; cos 37° 43' = .79105 ; tan 37° 43' ^^.773353; cot 37° 43' = 1.29307; sec 37° 43' = 1.264142; cosec37° 43' = 1.634628 ; vers 37° 43' = .20895 ; covers 37° 43' = .38824. 3. Find that sin 64° 36' = .90334; cos 64° 36' = .42894 ; tan 64° 36' = 2.10600 ; cot 64° 36' = .474835 ; sec 64° 36' = 2.331328 ; coseo 64° 36' = 1.107003 ; vers 64° 36' = .57106; covers 64° 36' = .09666. SuG. — In looking for sines and cosines of angles above 45°, seek the degrees at the bottom of the page, and be careful to observe that the columns of sines and cosines, as named at the top, change names when read from the bottom. The foundation of this arrangement will be readily perceived. Thus, turning in Table II. to 24° 32', we find sin 24° 32' = .41522. But sin 24° 32' = cos (90 - 24° 32') = cos 65° 28' = .41522. Thus the degrees and minutes read from the bottom of the page are the complements of those read from the top. 4. Find that sin 42° 27' 12" = .67499; cos 42° 27' 12" = .73783; tan 42° 27' 12" = .914834; cot 42° 27' 12" = 1.09309. SuG.— Sin 42° 27' = .67495, and sin 42° 28' = .67516. .-. An mcrease of 1' in the angle makes an increase of 21 (hundred-thousandths) in the sine, and 12" will make ^^ or i as great an increase, approximately. Observe that in the case of cosine an increase of the arc makes a decrease of the function. 5. Find that sin 143° 24' = 0.596225; cos 151° 23' = .877844; tan 132° 36' = 1.08749 ; and cot 116° 7' = .490256. SuG.— Sin 143" 24' = sin (180° —143° 24') = sin 36° 36'. Also the trigone- aielrical function of any angle is numerically equal to the same function of its supplement (56*). 6. Find the logarithmic trigonometrical functions of 32° 16' 32' from the tables at the end of the volume. 36 PLANE TRIGONOMETRY. SOLUTION. — Turning :o Table II. we find 32° at the top of the page, and jpposite 15', and in the column L. sin (logarithmic sine), we get 9.727228 ; i, e. log sm 32° 15' = 9.727228. Now from the column of differences, D. 1", we lejfru that an increase of 1" of the arc at this point makes, approximately, an increase ol 3.34(uiilliontlis) in the logarithm of its sine. Hence, we assume thai an in- crease of 22" makes 22 x 3.34 = 73 (millionths). .*. log sin 32° 15' 22" = 9.727226 -f- .000073 = 9.727301. In a similar manner we have log cos 32° 15' = 9.927231. An increase of l"iu the arc makes a decrease of 1.33 (millionths) in the log cos. .'. an increase of 22" makes 29 (millionths) decrease in the log r,os, and log co? 82° 15' 22" = 9.927202. Log tan 32° 15' 32" = 9.800100 ; and log cot 32* 15 22" = 10199901. 7. Find that log sin 24° 27' 34" = 9.617051 ; log cos 26° 12' 20" = 9.952897; log tan 26° 12' 20" = 9.692125 ; log cot 126° 23' 60" = 9^67579. SuG.— Observe cot (126° 23' 50") = cot (180°- 126° 23' 50") = cot (53° 36' 10"). Also that angles above 45° are found at the bottom of the table ; and remember to subtract the correction for co-functions, if an increase of arc is assumed. 8. Given the natural sine .45621, to find the angle from the tables. Solution. — Looking for this sine in the table of natural sines, we find the next less sine to be .45606, and the angle corresponding, 27° 8'. Now, at thij point, an increase of 1' in the arc makes an increase of 26 (hundred thousandths) in the natural sine. But the given sine .45621 is only 15 (hundred thou- sandths) greater than .45606, the sine of 27' 8'. Hence the required angle is but i^ of 1' or 60" = 35", greater than 27° 8'. .'. sin-\45621 = 27° 8' 35", and its supplement 152° 51' 25", which has the same sign, and these arcs in- creased by every multiple of 27r. 9. Find sin-\62583; cos-\34268; tan-\468531; cot-\876434. Results. Sin-\62583 = 38° 44' 35", and 141° 15' 25"; and these arcs increased by every multiple of 2'7r. cos-\34268 = 69° 57' 36", and 360° - 69° 57' 36" = 290° 2' 24'', and these arcs increased by every multiple of 2;r. tan -'.468531 = 25° 6' 16", and 180° + 25° 6' 16" = 205° 06' 16", and these arcs increased by every multiple of 2;r. cot-\876434 = 48° 46' 3", and 180° -f 48° 46' 3" = 228° 46' 03", and these arcs increased by everj multiple of %Tt, CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES. 37 SuG. — Observe that an increase of the arc makes a decrease of its co-functi'^-ns. In the table of tangents as given, Table IV., the proportional parts given al the bottom of each column are the approximate changes which the functions imdergo for a change of 1" in the function. Thus, in finding cot-'.876434, we find cot-^ .876462 = 48° 46' ; and at the bottom we find that a change of 8.64 (millionths) in the function makes a change of 1" in the angle. Hence, as tlie given cotangent is 28 (millionths) less than the cotangent of 48° 46'. the angle required is 28 -r- 8.64 = 3 (seconds), greater than 48° 46'. 71. ScH. — It is usually best to take from the table that function which is nearest in value to the given function, and then increase or diminish the corresponding arc as the case may require. If we always take from the table the next less function than that given, in the example for sine, tangent, and secant, and the next greater for the cosine, cotangent, and cosecant, corrections for seconds will require always Xo he added. If we always take from the tables the func- tions next less than the one given, the corrections for seconds must be added for sine, tangent, and secant, and subtracted for the co-functions. If we were always to take from the tables the next greater function than the one given, the seconds corrections would be added for the co-functions, and subtracted for the others. [Note.— It is very important that the pupil become so familiar with the nature of these tables as to use them intelligently, and not mechanically. For tills reason we refrain from giving the usual specific, mechanical directions ff)r their use, and substitute illustrations showing how they are used in accordance with the principles upon which they are constructed.] 10. Find sin-^- .34256); cos-^(- .62584); tan-^(- 3.41621) ; cot-^- 1.21648). Results. siii-^(-.34256) = 200° 1' 58", and 339° 58' 02", and these arcs increased by every multiple of ^if. COS-X-.02584) =128°44'38",and231°15'22",and these arcs increased by every multiple of 2-^'. tan-^- 3.41621) = 106° 18' 57", and 286° 18' 57", and these arcs increased by every multiple of 2'7r. cot->(- 1.21648) = 140° 34' 42", and 320° 34' 42", and these arcs increased by every multiple of 2'7r. Bug. — To obtain these results the pupil will need to recall the principles in the corollaries to {48—55). Thus, to find cot- \ - 1.21648), we find from the table that cot-'(1.21648) = 39° 25' 18" ; and from (55) Cor., we learn that cot (180°- X) := - cot X. .'. Cot-»(- 1.21648) = 180° - 39° 25' 18" = 140° 34' 42" Again, from the same corollary, we learn that cot (360° — x) = — cot « •. Cot-»(- 1.21468) = 360° - 39° 25' 18" = 320° 34' 42". 11. Given the logarithmic sine 9.45X234, to find the corresponding angle. 38 PLANE TRIGONOMETRY. Solution.— The next nearest log sin found in Table II., is 9.451204 = lo^' sin 16° 25'. Now we learn from the table that an increase of 1" in the angle al this point, makes an increase in its log sin of 7.14 (million ths). But the given log sin, 9.451234, is 30 (millionths) greater than log sin W 25'. .-. The required angle is 30 -i- 7.14 = 4 (seconds) greater than 1G° 25' ; and we have sin 16° 25' 4" = 9.451234. Again, as sin 16° 25' 4" = sin (180° - 16° 25' 4") = sin 163° 34' 50' the latter angle has for its log sin 9.451234. Finally, either of these angles in. creased by any multiple of 27r has the same logarithmic sine. 12. Sliow from the table that the angle whose log cos is 9.778151, IS 53^ r 49", and also 306° 52' 11", and each of these angles increased by any multiple of 2-^. 13. What angles correspond to the logarithmic cosines 9.246831 and 9.889372? 14. Eind froin the table what angles have for their logarithmic tangents 9.895760, 10.531054, and 11.216313. Results. The first two are the log tans of 38° 11' 20", and 73° 35' 43", and also of 180° + either of these angles, and each increased by any multiple of 2'r. 15. Find the angles corresponding to the logarithmic cotangents 10.008688, 9.638336, and 9.436811. Results. The first two are the log cots of 44° 25' 37", and 66° 29' 54", and also of 180° + either of these angles, and each increased by any multiple of 2"^. 72, ScH. — Strictly speaking, negative numoers have no logarithms; since no base can be assumed, such that all negative numbera can be represented by said base affected with exponents. It is therefore customary to say tliat nega- tive numbers have no logarithms. Nevertheless, we do apply logarithms to nega- tive trigonometrical functions. Thus, if we have — cos a;, the — sign is inter- preted as simply telling in what quadrants z may end ; while, in other respects the function is treated exactly like -f cos x. 16. Given log (- cos a;) = 9.346251, to find x. Solution.— The logarithmic cosine 9.346261, considered independently of ite sign, corresponds to 77° 10' 35". But the — sign requires that the arc shall end in the 2d or 3d quadrant, for such angles, and such only, have negative cosines. •, The angles required are 180° T 77° 10' 35" = 102° 49' 25", and 257° 10' 35", and these increased by entire circumferences, as all these angles have jaga- rithmic cosines, which are numerically equal to 9.346261, and the cosines them- selves are negative. 17. What angle less than 180° has a negative cosine whose tabu lar logarithmic value is 9.653825 ? Ans. 116° 47' 4" OONSTEtJCTION AND tJSE OF TRIGONOMETRICAL TABLES. 39 i8. What angle le&j than 180° has a negative tangent whose tabu nv logarithmic value is 9.884130? Ans. 142" 33' 15" 19. What angle less than 180° has a negative sine whose tabulai logarithmic value is 9.341627 ? 20. What angle less than 180° has a negative cotangent whose tabular logarithmic value is 9.564299 ? Aiis. 110° 8' 15". 21. Find values of a; < 180° wliich fulfil the following conditions: log (- cos x) = 9.562468 ; log (- tan x) = 10.764215 ; log (- s:n x) = 8.886432; log (- cot x) = 11.152161. EesiiUs. 111° 25'; 99° 45' 54"; none; 175° 58' 14". 22. Having at hand only the common logarithmic tables of trig- onometrical functions, and the table of logarithms of numbers, I wish to find the number of degrees, minutes, and seconds corre- sponding to the natural tangent 2.16145. How is it done, and what is the result? Anstuer : Find the logarithm of 2.16145, to this add 10, and find the angle corresponding to this tabular logarithmic tangent. The angle is 65° 10' 20". 23. From the same tables as above find the natural cosine of 35° 23'. Also what angle corresponds to natural tangent 2. 24. From the same tables as above find the angle corresponding to natural tangent — 1.82645. Also to natural cosine — .42536. 25. Why is it in the table of logarithmic functions that the sine of an angle minus its cosine + 10 gives the tangent ? Why that cosine — the sine + 10 gives the cotangent ? Why that the sum of the tan- gent and cotangent of any angle = 20 ? Why is but one column of tabular differences needed for tangents and cotangents, while the sines and cosines require each a separate column ? FUNCTIONS OF ANGLES NEAR THE LIMITS OF THE QUADRANT. TABLE III. [Note. — This subject may be omitted in an elementary course, the first time going over, if thought best.] 73. Failure of Table II, — The method which has been given in the preceding pages for finding the logarithmic functions of anglea involving seconds, by means of the Tabular Differences, Table II., is sufliciently accurate in most cases for practical purposes, but is 40 PLANE TRIGONOMETRY. entirely too rude for the sines, tangents, and cotangents of angles near the beginning of the quadrant (those less than 2° or 3°), and for cosines, tangents, and cotangents of angles near the close of the quadrant (those between 87° or 88° and 90°). An example will render this clear. Suppose we wish to find log sin 1' 12". We find from Table IL, log sin 1' = 6.463726; and also that the average increase of the log sin between 1' and 2' is 5017.17 (million tLs) for every second increase of the angle. But this average rate of increase ©f the function during the minute is much less than its real rate of increase in tlie first part of the minute^ as from I'to 1'12", and much greater than the real rate of increase in the latter part ^ as the angle approaches 2'. In fact, we see from this table, that we should use 2934.85 as the increase of log sin 2' for 1" increase of the arc. Now, in our proposed example, we want the increase of the log sin while the angle is passing from 1' to 1' 12". This, as shown above, is con- siderably more than 5017.17 (millionths) for every second. The cosi7ie being the sine of the complement is subject to the same law of change near the close of the quadrant, that governs the sine at the beginning. The case of the ta7ige7it of a small angle is similar to that of the sine ; and since the cotangent is the reciprocal of the tangent, it has the same laio of change, only that the one increases as the other decreases. Thus, since doubling a small arc, as 1", doubles its tan- gent (approximately), it divides its cotangent by 2. Finally, while the law of change in the sine is very different neai the close of the quadrant from what it is near the beginning, the sine changing very rapidly at the beginning and very slowly at the close, and the cosine is just the opposite, the tangent, and cotangent have the same law of change at both extremities of the quadrant. Thus, if near the beginning of the quadrant a certain small increase of the arc increases the tangent at a particular rate, it decreases the cotangent at the same rate, since these functions are reciprocals of each other. Moreover, since tan 1° = cot (90° — 1°) = cot 89°, cot 89° changes according to the same law as tan 1° ; and tan 89 changes reciprocally with cot 89°. 74, Description of Table HI, — The first page of the table enables us to find the sines of angles less than 2° 36' 15" (and con- sequently the cosines of angles between 87° 23' 45" and 90°) with vt-ry great accuracy. The columns headed Angles contain the degrees, minutes, and seconds of the proposed angles, and the columns at their right give the same angles in seconds. The columns headed CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES.- 41 Diff^ contain the corrections to be used according to the following problems. The second and third pages answer a similar purpose with reference to tangents and cotangents of arcs within 2° 36' 20" of the limits of the quadrant. 75, JPvop* — Letting x represent any number of seconds less than 2° 36' 15", tue have, log sin x" = 4.685575 + log a; - Diff. Dem.— The length of 1" of an arc to radius unity is 3.14159265358979 (the length of the semicircumferenoe) -4- 648000 (the number of seconds in 180**), and = .00000484812. For practical purposes this fraction may also be taken as the sine of 1". Though, actually, the sine is less than the arc, the expressions for arc 1" and sine 1" agree to as many places of decimals as we have here. Again, for these small arcs the sine increases at nearly/ the same rate as the arc, so that gin 8" = .00000484812 x 3 marly; sin 102" = .00000484812 x 102 nearly ; these results being slightly in excess of the true values. It is the correction for this excess that is furnished by Table III. in the columns marked Diff. But this table is adapted to logarithmic computation ; hence we have log sin x" = log sin 1" + logic — Diff. In this expression log sin a;" is the logarithm of the natu- ral sine of x" (not increased by 10,- as each function in Table II. is) ; log sin 1" + log a;, that is, the logarithmic sine of 1" plus the logarithm o^ the number of seconds, corresponds to multiplying the sin 1" by the number of seconds, and gives the Irgarithm of the product, or strictly, the logarithm of the length of the arc of a;". Now, the sine oix" being less than the ai'c, its logarithm is less than the logarithm of the length of the arc. Just how much less the table tells. This difference, therefore, between the logarithm of the arc and the logarithm of its Bine, which is given in the table, is to be subtracted. Finally, to make this result agree with Table II. we must add 10 to the result. Now, log sin 1" = log .00000484813 = 6.085575, and adding 10, we have 4.685575. .-. log sinic" = 4.685575 + log a; - Diff., a result which agrees with the logarithmic functions in Table II. q. e. d. 76, Cor. 1.— 7b oUain the log cos of an angle between 87° 23' 45' ind ^O°,from this tahle^ take the log sin of its complement. 77* Cor. 2. — To obtain the log tan of an angle less than 2° 36' 20", from this table, use the formula, log tan X" = 4.685575 -{- logx + Diff. Dem. — For as small an arc as 1", sine, arc, and tangent are practically equal; hence, log tan 1" = log sin 1'' = 4.685575 (10 being added). Moreover, for these small arcs the tangents increase (like the sines) in nearly the same ratio as the arcs; Jience, we add logx. Finally, the tangent is a little in excess >f the arc, which excess is given iu the table, and is to be added, q. e. d. rj PLANE TRIGONOMETRY. 78, Cor. 3. — To obtain the log cot of an angle less than 2° 36 IS' , f'lwn this table, use the formula, log cot a;" = 15.314425 - logo; - Diff. Demonstration.— Since cot x" = ^^^ ,^, > log cot «" = log 1 — log tan r" ^ 20 - (4685575 + log a; + Diff.) = 15.314425 - log x - Diff. Tlie 20 arises from adding 10 twice to log 1 (= 0). One 10 is added because 4685575 is 10 in excess of the true log tan 1" ; and the other 10 is added in order to make the log cot x' agree with the ordinary tabulated value, as in Table II. q. e. d. 79, Cor. 4. — To obtain the log tan of an angle between 87° 23' 45" and 90°, take the log cot of its complement ; and to obtaiJi the log cot of an angle between the sa/tne values, take the log tan of its complement. SO. Froh. — Having given a log sin less than 8.65')'397 {the log htn 2° 36' 15"), to find the corresponding angle. Solution.— From log sin x" = 4.685575 + log a; — Diff., we have, log x = log sm af' — 4.685575 + Diff. Hence, if from the given log sin, we subtract 4685575, and then add the proper correction as furnished by Table III., we have the logarithm of the number of seconds sought. But we cannot tell what Diff. to take till we know the number of seconds. To meet this difficulty, find the angle corresponding to the given log sin from Table II., and reduce it to seconds. This will be sufficiently accui'ate to furnish the required Diff. 81, Cor. 1. — Having given a log cos less than 8.65 73 9 7 {the log cos of 87° 23' 45"), to find the corresponding angle, treat it as if it were a log sin, and having found the corresponding angle, take its comple- ment. 82. Cor. 2. — For log tan and log cot, the formulce in {77y 78). give, logx = log tan re" - 4.685575 - Diff., and,loga; = 15.314425 — log cot a;" - Diff. These are applied as in {80) ; that is, the Diff. to be subtractad is found by getting from Table II. the required angle in seconds, as near as may be, and then take from Table III. the corresponding DHL CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES. 411 EXAMPLES. 1. Find the log sin, tan, and cot of 1° 11' 15". Solution.— Log sin 1° 11' 15" = 4.685575 + log 4275 - .000031 = 8.316480. r ir 15" = 4275". Since 4275" is between 4230" and 4300, the Diflf. is 31 fmillionths). ^ ~ Log tan 1° 11' 15" = 4.685575 + log 4275 + .000062= 8.316573. Log cot 1° 11' 15" = 15.314425 - log 4275 - .000062 = 11.683427. Or, log cot can be found by subtracting log tan from 20. 2. Verify the following by using Table III. : log sin 56' 26" = 8.215242 ; log tan 56' 26" 8.215301 ; logcot56' 26" =11.784699. 3. Verify the following by using Table III. : log cos 88° 17' 44" = 8.473396 ; log tan 88° 17' 44" = 11.526412 ; log cot 88° 17' 44" = 8.473588. 4. Having given the logarithmic sine 7.246481 to find the angle. Solution.— From Table II. we find 6' 5" =:: 365" as the angle. But tliis is subject to the inaccurac}' exhibited in {73). To obtain the correct result from Table III., we have {80), log X = 7.246481 - 4.685575 + = 2.500906. .-. X = 363.8; or the angle is 6' 3" .8. 5. Given the logarithmic tangent 7.805487, to find the correspond- ing angle. Solution.— Table IL gives 21' 58".3=1318".3 as the angle. Froni Table IIL the Dlff. con*esponding to this is 6 (millionths). Hence, log X = log tan a;" - 4.685575 - Dill'. [82) becomes, log x = 7.805487 - 4.685575 - .000006 = 3.119906. .-. X = 1318; and the angle is 1318" = 21' 58". 6. Given the logarithmic cotangent 12.197148, to find the corre- sponding arc. Table IL gives the angle 21' 49". 8, but the true angle as given by Table III. is 21' 50". i4 PLANE TRIGONOMETBY. SUCTION V: TRIGONOMETRICAL SOLUTION OF PLAIVE TRIANGLES. 83. There are six parts in evei^ plane triangle : three sides anil three angles ; one side and any other two of which being given, the remaining parts can be found by means of the relations which exist l^etween the sides and tabulated trigonometrical functions. To exhibit these highly important practical operations is the object of this section. We shall treat first of right angled plane triangles, and then of oblique angled plane triangles. OF RIGHT ANGLED TRIANGLES. I^rop, 84. — The relations between the sides and the trigonomet- rical functions of the oblique atigles of a right angled triangle are as follows : ,^ . . side opposite (1) sine =-T P ; ^ ' hypotenuse ,„. . side adjacent (2) cosine = -:; -r ; ^ ' hypotenuse . . , side opposite ^ ^ ' ^ ~" side adjacent ' cosecant = hypotenuse side opposite ' hypotenuse secant = — — cotangent side adjacent' side adjacent side opposite* Fig. 12. by cosine, we cave tan B = T7EM.— Let CAB, Fig, 12, be a triangle, right angled at A. Let aM be tlie measuring arc of the angle B, PD = sin B, and BD r= cos B. From the PD similar triangles PDB and CAB, we have □= = CA • • o side opposite — ; I. e., sm B — ^ , BC hypotenuse BP since BP = 1. From the same triangles ^ BA. BC /. «., cos B = Tangent being equal to sine divided side adjacent hypotenuse side opposite _^ side adjacent _8ide opposite hypotenuse The hypotenuse hypotenuse side adjacent other ftuu'Lious being the reciprocals of these three, are as given in the proposi- tion. Finally, as a similar construction could be made about the other oblique angle, C, this demonstration may be considered general, q. k. d. I I SOLUTION OP HlGfiT ANGLED l>LANfi TRIANGLES. 45 IScH. 1.— These formula are so important that it is well to have them fixed in the memory, not only as written above, but also as follows: liy side-adj (1). Side-opp = hy x sin, or , or tan x side-adj, or — ; COSGC COl ,„ O.J T 1 ^y . . , side-opp (2). Side-adi — hy x cos, or — =^, or cot x side-opp, or — ; ^ ' J ^ 'sec tan ' of which the relations side-opp = hy x sin, and side-adj = hy x cos are of the most frequent use. ScH. 2. — The six ratios given in this proposition are frequently made the def Initions of the trigonometrical functions. Thus, referring to a right angled tri- angle, a sine of an angle may be defined to be the ratio of the side opposite to the hypotenuse ; the cosine as the ratio of the side adjacent to the hypotenuse, etc. ScH. 3. — The student will be aided in remembering these important relatit)ns by observing tliat the side opposite the angle is analogous to the sine, and the its analogue side adjacent to the cosine. Now, the sme = r , and so also the co- sine. Tangent equals sine divided by cosine, and in this case it is the pari analogous to the sine, divided by the part analogous to the cosine. One should hy not make the blunder of s&ying that sin = . . , since tliat would make side-opp the sine always more than 1 ; but we have sec n that it never can exceed 1. Similar checks against error may be made in the case of the other relations. EXERCISES. [Note. — The first five of these exercises are mainly designed to illustrate tho proposition, and familiarize the mind with the relations.] 1. In a right angled triangle whose sides are 3, 4, and 5, wliat are the trigonometrical functions of the angles ? What are the functions when the sides are 6, 8, and 10 ? 2. In a right angled triangle the hjrpotenuse is 12, and the angl i at the base sin~^|^. What are the sides ? What is the sine of th/^ other angle ? Sua.— Represent the angles by B, A, and C, A being the right angle; and b h the sides opposite by J, a, and c. Then sin B = -, or ^ = — ; whence, ft = 6 Sin C = cos B = ^/^^ c = 6v^ 3. In a right angled triangle whose hypotenuse is 12, and the tingle at the base tan~^2, what are the other parts ? Am. Cos-^fVB; J^A/57and^\/5; 46 t>LANE TillGONOMETEY. 4. The sides of a right angled triangle are 20 and 32. What are the angles and hypotenuse? Obtain the hypotenuse by means of tlie secant. A71S. Tan-^, tan-^ f , and 4\/89. 5. The hypotenuse of a right angled triangle is 120, and one side 100. Show that the angle opposite the latter is tan~YYVll> the adjacent angle cosec'^^^VH? ^lud the remaining side 20 VH- Obtain tnese results in the order given. Use a trigonometrical function to obtain the last. EXAMPLES. (a) BY MEANS OF THE TAI.LE OF NATURAL FUNCTIONS. 1. In a right angled triangle ABC, tlie liypotenusc BC is 235, and the angle B is 43° 25'. Find the angle C, and the sides AB and AC. C = 46° 35' ; AB = 170.7 ; AC = 161.52. Solution — 7o find 0, we have but to remember tliat the angles of a right angled triangle are complements of each other ; whence, C = 90° — B = 4G^ 35'. AB To find AB, we have cos B = g^' or AB = 235 x cos 43° 25'. Now, from the table of natural functions we find cos 43° 25' = .72637 ; whence AB = 235 x .72637 - 170.7. To find AC, we have sin B = || ; whence AC = 235 x .6873 - 161.52. 2. In a right angled triangle ABC, the hypotenuse AC is 94.6, and the angle C is 56° 30'. Find the angle A, and the sides AB and BC A =33° 30'; AB =78.88; BC = 52.21. 3. In a right angled triangle BDF, the h3rpotenuse BF is 127.9, and the angle B is 40° 10' 30". Find the angle F, and the sides BD and DF. F = 49° 49' 30"; BD = 97.72; DF = 82.51. 4. In the triangle CDE, right angled at E, given the side DE 75, the side CE 50.59, to find the other parts. Hypotenuse = 90.47. '' 5. In the right angled triangle CDE, given the hypotenuse CD 264, the side CE 135.97, to find the other parts. DE = 226.28. 6. Given the hypotenuse 435, and one of the acute angles 44°, to find the other parts. 7. Given the hypotenuse 64, and the base 51.778, to find the other parts. I I I SOLUTION OP lllGHT ANGLED PLANE TRL^GLES. 47 8. Given the hypotenuse 749 feet, and the base 548.255 feet, to find the other parts. 9. Given the hypotenuse 125.7 yards, and one of the acute angles 75° 12 23", to find the other parts."^ 10. Given one side 388.875, and the adjacent angle 27° 38' 50", to find the other parts of a right angled triangle. (b) BY MEANS OF A TABLE OF LOGARITHMIC FUNCTIONS. 11. In a right angled triangle, given an oblique angle 54° 27' 39", rind the side opposite 56.293, to find the other parts. BoLUTiON.— The other oblique angle is 90° - 54° 27' 39" = 35° 33' 21". rr ^ ^.r 7, . u • rr.o n«/ onr, 56.293 , 56.293 To find the hypotemise^ we have sm 54 27 39 = -^ — , ovfiy = -t- hy ' " sin 54° 27' 89" Applying logarithms to facilitate computation, log liy = log 56.293 — log sin 54° 27' 39" + 10. The 10 is added since the log sin 54° 27' 39" found in the table is 10 too great. Now, looking in Table L, we find log 56.293 = 1.750454 and in Table XL, log sin 54° 27' 39" = 9.910473. Hence, log hy = 1.750454 - 9.910474 + 10 = 1.839980, and hy = 69.18. To find the other side we have, tan angle = ^./ ^ , or tan 54° 27' 39" - '' ' *= side adj 66.293 , .^ ^. 56.293 . ^ - ^ -^u i -^ ^■ i^^^ ; whence ^de adj = ^^^ 540 ^y. g^. . Applymg loganthms, log ^adj = log 56.293 - log tan 54° 27' 39" + 10 = 1.750454 - 10.146104 + 10 = 1.604350. .-. SideadS- 40.2115. 12. In a right angled triangle ABC, the hypotenuse AC is 340, and the side AB is 200. Find the acute angles A and c, and the perpen- dicular BC. A = 53° 58' 6" ; C = 36° V 54' ; BC = 274.95. 13. In a right angled triangle ABC, the perpendicular AB is 736.3 and BC 500. Find the acute angles A and C, and the hypotenuse AC, A = 34° 10' 45" ; C = 55° 49' 15" ; AC = 890.02. 14. In a right angled triangle BDF, the perpendicular BD is 246.32, and DF 380.07. Find the acute angles E and F, and the hypote- nuse BF. B = 57° 3' 11"; F = 32° 56' 49" ; BF = 452.91. 15. In a right angled triangle ABC, the side AB is 249, and the angle A is 29° 14'. Find the perpendicular BC, and the hypotenuse AC 80 = 139.35; AC =285.341. 48 PLANE TRIGONOMETRY. 16. Ill a right angled triangle ABC, the hypotenuse AC is 95.76, and t'ae side BC 60. Find the acute angles A and c, and the pei*pen- dicular AB. A = 38° 48' 7" ; C = 51° 11' 53" ; AB = 74.62. 17. In a right angled triangle ABC, the side BC is 364.3, and the angle A is 50° 45'. Find the perpendicular AB, and the hypotenusfl AC. AB = 297.645 ;. AC = 470.433 [Note. — The first ten examples may be solved by logarithms if additional exercises are needed, or these by means of the natural functions. Also any one of the examples will afford several others by giving and requiring different parts. Thus, from Ex. 17, we can give AB = 297.645, A = 50° 45', and require the other parts, etc.]. GENERAL APPLICATIONS. 1. Find the area of a parallelogram whose adjacent sides are 28 and 30 feet, and the included angle 75°. SuG.— First find the altitude. 2. A railroad track 463 feet 3 inches in length has a uniform grade of 3°. Show that the vertical rise is 24 feet 3 inches, nearly. 3. A railroad track makes a vertical rise of 150 feet, by uniform grade, in 3,000 feet of track. What is the grade ? 4. Find the apothem and radius of the circumscribed circle of a regular heptagon one of whose sides is 12 feet. 5. Find the area of a regular dodecagon inscribed in a circle whose radius is 12. 6. The angle of elevation to the top of a steeple is 47° 30', ae measured from a point in the same horizontal plane as its base, and at a distance of 200 feet from it. What is the height of the steeple ? A71S. 218.26. ft. 7. A tower 103 feet high throws a shadow 51.5 feet long, upon the horizontal plane of its base : what is the angle of elevation of the sun 8. The angle at the vertex of a right cone is 52° 23', and the slant height 126 feet : what is the diameter of the base, and what the altitude ? SOLUTION OF OBLIQUE ANGLED I'LANE rillANGLKS. 4\) 9. In Mg. 13, letting EO rep- resent the earth and M the moon, the radius of the earth EO = 3956.2, and the angle EMO * = 57', required to fiud the distance OM, E being a right angle. Fig. 13. The distance of the moon from the earth, as giveii hy this covipu' tation, is 238,613 miles. 10. In Fig. 13, letting ON represent a tangent to the moon's disc at N, the angle NOM is readily measured, being half the moon's apparent diameter. The apparent diameter of the moon being 31' 20", and its distance from the earth as found in the last example, tthe diameter ? A71S, 2176 miles. OF OBLIQUE ANGLED PLANE TRIANGLES. IMPORTAN^T RELATIONS RXISTIKG BETWEEK THE SIDES AND TKIGO- NOMETRICAL FUNCTIONS OF THE ANGLES OF OBLIQUE AKQLED PLANE TRIANGLES. SS, JProp, — Tfie Sides of any plane triangle are proportional to the sines of the angles opposite. Dem. — Let ABC be any plane triangle. Let fall from either angle, as C, a perpendicular upon the opposite side, or upon that side produced. Designate the angles by A, B, and C, the sidea opposite by a, Z»,and c, and the per- pendicular by P. Now, from the right angled triangle ADC we have P = & sin A ; also from CDB, P = a sin B ; sin ABC in the second figure being = sinCBD. Hence, equating the values of P, & sin A = a sin B, or a : & : : sin A : sin B. Fie. 14. Q. B. D. ♦ This angle is called the moon's horizontal parallax, and is readily measured. Some ruvle notion of the manner in which parallax becomes apparent, may be got from considering ihc difference in direction from two observers to the moon, one observer standing directly under the moon, as at F, and the other at E, Boeing the moon in his horizon. The angular displace inept of the moon due to these different points of observation is horizontal parallajc. 4 50 PLANE TRIGONOMETRY. 86* I^vop, — T]ie sum of any two sides of a pliDie triangle is lo their difference^ as the tangent of half the stem oj the angles oj)posite if to the tangent of half their difference. Dem. — Letting a and b represent any two sides of a plane triangle, and A and B llie angles opposite, we have a : 6 : : sin A : sin B. Taking this both by composition and division, we have a + b : a — b : : sin A + sin B : sin A— sin B. But from (6*0), sin A + sin B : sin A — sin B : : tan i(A + B) : tan i(A — B). .-. a + b : a — b :: tani(A + B) : tan i(A — B). q. b. d. 87 > JPvop. — TJie tangent of half of any angle of a plane triangle equals k, divided hy half the jierimeter of the triangle minus the side opposite the angle ; in which Jc is the radius of the inscribed circle, and equals the square root of the continued product of half the peri- meter minus each side separately, divided by half the perimeter* Dem. — Let ABC be any plane triangle. Represent the angles by A, B, and C, the sides opposite by a, b, and c, the perimeter i)y p, and the segments of the sides made ny the radii of the inscribed circle, by «, y^ and 2, as in the figure. Then a + 6 + c = 2aj + 2y + 2s = ^, or r. + y + z=^. Whence x =\p — a, y =ip —b, and z = ^—C] since y + z = a,x + z = b, and x + y = c. Fio. 15. k k Now from the tiiangles AOD, DOB, and COE, tan ^A=-z=- , tan X ip — a iB = - = -A_ andtan ^0 = - = -^-^. y iP-b . z ip — e To find k. iA + ^B + iC = 90°, or ^A = 90" - (^B + ^C) ; whence, tan ^A = ^ = tan [90° - (iB + iC)] = cot (iB -i ^C) = l-taniBtaniC ^^^^ ^^^^ tan^,B + tan IC 1 ^' Substituting for tan ^B, and tan iC, ~, and -, we have - = E y z X k k V z ; wht-nce , , TUZ — xW , , , , , / xuz ^ = ~nr;r ; (^ + ^ + 2)^ = ^s ; and ^ = |/ ^ , ^ -. in this value of A;, sub « -r y v X + y + z ' sUtuting for x, y, and z their values, we have * = jA^ ~ ^^ ^^^ -^)iiP- c) ' ip I [SOLUTION OF OBLIQUE ANGLED PLANE TRIANGLES. 51 JSS, 8cH. 1. — These three propositions {85, 80^ 87), furnish the most elegant and expeditious means for finding tlie unknown parts of an obli(jue angled plane triangle, when a sufficient number of parts are given or known {S-'i) 89, ScH. 2.— Important Practical Suggestions, 1st Two angles of a triangle being given, the third is known by implication, It being the supplement of the sum of the other two. _ 2nd. When two of the known, or given, parts are opposite each other, the ftrst proposition (85) effects the solution. 3rd. When two sides and the included angle are given, the solution is effecTed by means of the second proposition {86). 4th. When the three sides are given, the angles are found by the third proposition {87)- EXERCISES. 1. In the plane triangle CDE, given the angle D = 15" 19' 51", C = 72° 44' 05", and the side c, opposite c, 250.4, to find the other parts. r-^^^c Solution.— i^r«^, E = 180°- (D + C) = 91° 56' 04" L ^ ^^^ {89, 1st). ^ Second, To find side d opposite angle D {89^ 2nd), sin C : sin D : : c : cf, or sm 72° 44' 05" : sin 15"^ 19' 51" : : 250.4 ; d. This proposition may be solved ford:, by taking the natural sine of 15° 19' 51" multiplying it by 250.4, and dividing the product by the natural sine of 72° 44' 05"; or, more expeditiously, by logarithms, as follows : log 250.4 = 2.398634 log sin 15° 19' 51" = 9.422249 logsin73°44'05"(ar. comp.)*= 0.020024 \ogd= t 1.840907 .•.d= 69.328. Third, To find side e opposite angle E {89^ 2nd), sin C : sin E '.-. c : e,ov sin 72° 44' 05" : sin 91° 56' 04" : : 250.4 : e. Maidng the computation by logarithms, log 250.04 = 2.898634 log sin 91° 56' 04" t 9.999752 log sin 72° 44' 05" (ar. corap.) = 0.020024 log e = 2.418410. .-. e = 262.066. * See Introduction to Table I. (17). + The student must bear in mind the fact that all the log. trig, funcs. are 10 too large, and mast see exactly what corrections to make in hia results, on thia account. In this case the ar. comp. is 10 Loo small, since the logarithm we took from the table for log sin 72" 44' 0.5" was 10 too large. But our log sin 15° 1!)' 51" is 10 too large, and just corrects the latter. Hence, we have to reject only 10 from the entire stm 11.840U07, and this on account of the use of ar. comp. X T*ike the sine of the supplement, or the cosine of the given angle miaas 90» 52 PLANE TRIiaONOMETRT. 2. Ill the plane triangle ABC, A = 35° 42', B = 76° 27', an I AB = 142. What are the other parts ? Ans. c = 67° 51'; AC = 149.05; BC = 80.47. 3. Given two angles of a plane triangle, 23° 40' 32" and G9° 39' 51 ', and the included side 100, to find the other parts. 4. In a plane triangle ABC, the side AB is 254.3, the side AC 396.8, and the angle B 94° 29'. Find the angles A and C, and the side BC. A = 45° 48' 21"; c = 39° 42' 39"; BC = 285.37. Suq's. — To find the angle C, we liave 396.8 : 254.3 : : sin 94" 29' : sin C. From this proportion we get log sin C = 9.805443. Now, as we have seen before, there are an infinite number of angles corresponding to any given sine, how shall we know what one to take in this case ? First, no angle of a triangle can exceed 180° ; hence, there arc but two angles, one an acute angle, and the other its supplement, which can come into consideration in the solution of plane triangles. But which of t/iese two are we to take ? Thus, in this case, both the angles 39° 42' 39" and 140° 17' 21" correspond to log sin 9.805443. In this ex- ample the ambiguity is resolved by observing that the given angle B is obtuse, and a plane triangle can have but one obtuse angle. .*. C = 39" 42' 39". 5. In a plane triangle BDF, the angle B is 40°, the side BD is 400, and the side DF 350. Find the angles D and F, and the side BF. Sug's. — To find F, we have 350 : 400 : : sin 40° : sin F, from which log sin F = 9.866059, and F = 47° 16' 28 ", and its supplement 132° 43' 32". How are we to determine which of these to take? The given angle is 40° ; hence, as far as that is concerned, either of the two will meet the conditions. There are, therefore, two angles, F = 47° 16' 28" and F = 132° 43' 32", which fulfill the conditions. We therefore solve two triangles, one having two of its sides 400 and 350, and the angles 40', 47° 16' 28", and 92° 43' 32" ; and another triangle with the same given parts and the two required angles 132° 43' 32", and 7° 16' 28". This is readily illustrated geometrically. Thus, lay off DBF' = 40°. Take BD = 400. Then from D as a centre, with radius 350 describe an arc cutting BF', It is evident that if B is an acute angle the following cases may arise depending upon the value of DF: 1st. If DF is less than the perpendicular p, the problem is impossible. 2iul. If DF = j9, the triangle is right angled at F. 3rd. If DF > j) and DB there is but one triangle which fulfills the conditions, viz.. lUe one with an acute angle at F'. Fia. 16. SOLUTION OF OBLIQUE ANGLED PLANE TRIANGLES. 63 The results in the above examples are, for triangle DF'B, angle DF'B = 47° 16' 28", BDF' = 93° 43' 33", and side BF' = 543.89; for triangle DFB, angle DFB = 133° 43' 33", angle BDF = 7° 16' 28", and side BF = 68.94. 90, Cor. — I?i applying trigonometrical formulce to the solution of triangles, if the jMrt sought is found in terms of its sine, the result is ambiguous, and we are to detennine whether there really are two solutions to the problem in a geometrical sense, by certain geometrical considerations, or else by trying both values for the angle determined by its sine. This ambiguity arises only when an angle is determined by its sine, as will appear hereafter. 6. Given two sides of a plane triangle 201 and 140, and tHe angle opposite the latter 36° 44'. Find the other parts. Results. — There are two triangles. Parts of the first, 130° 49' 49", 22° 36' 11", and 89.34; Parts of the second, 59° 10' 11", 84° 5' 49", and 232.84. 7. Given two sides of a plane triangle 180, 100, and the angle op- posite the former 127° 33', to find the other parts. There is but one triangle, and the parts are 26° 7' 59", 26° 19' 1", and 100.65. 8. Given two sides of a plane triangle 30.8 and 54.12, and the angle opposite the latter 36° 42' 11", to find the other parts. Why but one triangle ? 9. Given two sides of a plane triangle 600 and 250, and the angle opposite the latter 42° 12'. Find the other parts. SuG. — Attempting to get the angle opposite 600, we find log sin = 10.207400, which is impossible. It is in some such way that a trigonometrical solution shows a geometrical absurdity. 10. Given two sides of a plane triangle 1337.6 and 493, and the angle opposite the former 69° 46'. Find the other parts. 11. In a plane triangle, given two sides 1686 and 960, and the in- cluded angle 128° 04', to find the other parts. c SonjTiON.-Let a = 1686, b = 960, and C = 138° 04'. (See 89, 3rd.) The sum of the angles A and B is 180° - 128° 04' = 51^ 56', and ^{^ + B) = 25° 58'. From (86) A , whence AD, P, and angle ACD, can be computed. Then passing to triangle CDB, we know P, and DB (since we have c given and ^ have computed AD). Hence, we can compute B, Fi«. 18. rt, and DCB. Thus, the parts of ACB become known .... When the given parts are two angles and the included side, find the third angle by taking the supplement of the two given. Let fall a perpendicular from one extremity of the given side upon the opposite side. The two right angled triangles thus formed can then be computed. Thus, if A, 5, and C are given, having found B, let fall CD. The triangle ACD has the angle A and side h known, whence its parts can be computed. Having computed P we can pass to the triangle CDB, md knowing P and B, can compute it. Thus the parts of ACB become known. 66 PLANE TRIGONOMETRY. 2nd. When two of the given parts are adjacent and one separated ; i. e , when two angles and a side opposite one are given ; or two sides and an angle oppo- site one are given. The first of these cases is virtually the same as the last given. To solve the other, let fall a perpendicular from the angle between the given sides, and two right angled triangles will be formed which can readily be computed. Thus a, 6, and A being given, and CD let fall from C, the triangle ACD can first be computed, and then CDB. This is the ambiguous case, but it is easily determined. Having computed P, if the given side a is less than P there is no solution ; if = to P, one solution (a right angled triangle) ; if a > P and < Z), there are two solutions, i. e., it will go in between CD and AC, and also beyond CD ; if a> P and also > b there is only one solution, as it will not go in between CD and AC. 3rd. W^ien the three given parts are all separated from each other. This is the case in which the three sides are given to find the angles. It is readily solved by letting fall a perpendicular from the angle opposite the greatest side, upon that side, as CD upon AB. Then compute the segments AD (which call m), and DB {n\ from the following relation (Part II, Ex. 13, page 162) : m + 7i (or c) : 6 + a : : b — a : m-~n. Kjiowing m and n., the angles of the two right angled triangles ACD and CDB can be computed, and these make known the angles of ACB. q. e d [Note. — A few additional examples are here given which the pupil can use to illustrate the theory presented in {91). If more are needed the preceding can be used : these may also be used to apply the methods before given. Again, a very great variety and number of examples may be made from these by as- signing different parts as known.] EXERCISES. 1. In a plane triangle BDF, the side BF is 123.75, DP 500, and the included angle F 120°. Find the angles B and D, and the side BD. B = 49° 12' 4"; D = 10° 47' 56"; BD = 572.006. 2. In a plane triangle ABC, the angle A is 70° 21', the angle B 54° 22', and the side BC 125. Find the angle C, and the sides AB and AC. C = 55° 17' ; AB = 109.1 ; AC = 107.88. 3. In a plane triangle ABC, the side AB is 98, the side BC 95.12. and the angle C 33° 21'. Find the angles A and B, and the side AC A = 32° 14' 55" ; B =:: 114° 24' 5" ; AC = 162.33. 4. In a plane triangle DAC, giyen AD = 450, AC = 309, and D = 27° 50', to find the other parts. C = 137° 9' 36", or 42° 60' 24" ; A = 15° 0'24", or 109° 19' 36"; OC^- 171.36,01 624 5. I SOLUTION OF OBLIQUE ANGLED PLANE TRIANGLES. COMPUTATION. It -will give definiteness to the stu- dent's Ihouglit, if he first sketch the 5gure geometrically. Thus, lay off D .- 27° 50', and taking AD = 450, let fall the perpeDdicular AP. 1st. To compute p. 9 = c sin D = 450 sin 27" 50', log 450 = 3.653213 log sin 27° 50^ = 9.669225 log p = 2.322438. .-. p = 210.106. 57 Fie. 19. Knowing p, we see by inspection that AC can lie in both the positions AL »nd AC, and hence that there are two solutions. 2nd. To compute C, from the triangle ACP, m which d and p are now inown. Sin C = ^ = 210.106 309 log 210.106 = 2.322438 log 309 = 2.489958 log sin C = 9.832480. .'. C ■= 42" 50' 24", and C = 137° 9' 36" 8rd. To find the angU A. DAC = 180° - (D + ACD) = 180° - 164° 59' 36" = 15° 0' 24". DAC = 180° - (D + ACD) = 180° - 70° 40' 24" = 109° 19' 36". 4th. To find DC. Compute DP and CP from the triangles APD and APC. DP - CP = DC, and DP + CP = DC. 5. In a plane triangle ABC, the side AB is 460, BC is 340, and AC 280. Find the angles A, B, and C. A = 47° 23' 16"; B = 37° 18' 31"; C = 95° 18' 13". 6. The sides of a plane triangle are 40, 34, and 25 feet respect- ively ; required the angles. 38° 25' 20", 57° 41' 24", 83' 53' 16". 7. The sides of a plane triangle are 390, 350, and 270 feet respect- >ely; required the angles. 42° 22' 06", 60° 52' 33", and 76° 45' 21". 8. Given two sides of a plane triangle 450 and 540, and the in- cluded angle 80°, to find the remaining parts. Angles, 56° 11', 43° 49'; and the side, 640.08. 9. Given two sides of a plane triangle 76 and 109, and the in- cluded angle 101° 30', to find the remaining parts. Angles, 30° 57' 30", 47° 32' 30"; and the side, 144.8. 58 PLANE TTIIGONOMETUY. FUNCTIONS OF THE ANGLES OF A TRIANGLE IN TERMS OF THE SIDES. 92, Prop. — A7iy side of a plane triangle equals the sum of tlie products of each of the other sides into the cosine of the angle which it makes ivith the first side. Dkm.— In the first figure AB = AD + DB. But AD = 6 cos A, and DB = a cos B. .*. c = b cosA + a cos B. In the second figure AB = AD — DB. But AD = b cos A, and DB = a cos CBD = a (— cos CBA) = —a cos B. .*. = b cos A — (—a cos B) = & cos A + a cos B. In like manner, we have a = b cos C + c cos- B, and b = a cos C + c cos A, Collecting and arranging. (1) a = 6 cos C + c cos B ; (2) & = a cos C + <5 cos A ; (3) c = a cos B + 6 cos A. q. b. d. 93 • Cor. — The square of any side of a plane triangle equals the sum of the squares of the other two, minus twice their rectangle into the cosine of their included angle. Dem.— From (3) {92), we have by transposing and squaring, a' cos" B = c' + &■■* COS'' A — 2bc cos A ; and from {85) a" sin" B = 6" sin" A. Adding, a' = c* + b* — 2bc cos A. In like manner, 6" =0* + c* — 2ac cos B ; and c" =«" + &"— 2ab cos C. Q. e. d. 94, ScH. — These farmulcR aflforJ another means for finding the angles of a plane triangle when the sides are given. Thus, 5« + c« - a» (li) cosA = (2i) cos B = (3i) cos C = 2Ac ga 4. C' - y 2ac ' g' + 5» - c« 2db These formula give directly the natural cosines of the angles in terms of Uie sides. To adapt them to logarithmic computation, we transform them as follows : i" + c" — a' Subtracting each member of (li) from unity, 1 — cos A = 1 g« - y-c» + 25c _ a»-(& - ef _ {a -K^ - c)} {a ~{b - c)} _ {g + b-c){a + e- b] SOLUTION OF OBLIQUE ANGLED PLANE TIlIANilLES. 59 But 1 - cos A = 28in*iA (,57, 0) ; and letting p = a + b + Cyi(a + b - c) = |^_ c, and ^{a + e — b) = ip — b. Whence, substituting, 2siu*^A =r Wc ' I (1.) sin i* = a/SMs-A^-I — 9.. la like manner, (2,)siniB = |/i^Hl(^^;aad (3,)siniC=/(lZ|FE3 In a manner altogether similar, by adding each member of (!') to unity, and reducing, we get (1.) cos iA = i/^ii^Zl^; and from (2,) and (3,), (2.) cosiB = i/SiZEI); ' ac (3,)eosiC=/iMEl. Dividing (la) by (la), (2^) by (2,), and (3^) by (3,), we have. n ^ for. lA - i/SZfHl^"-^. (1.) taniA-^/-^^^---^; (2.) taniB = f-^^^-^, (3,) tanlC = |/^^-*)^^^-^) iPiiP-c) EXERCISES. [Note. — In order to render these /(?rmt^to familiar, and to give the student exercise in applying fat'mulcB, a few examples are appended. If necessary, any which precede can be used.] 1. The sides of a plane triangle being 40, 34, and 25, finl the angles. Solution. — By natural funettons. Let the sides be represented by a, &, and c in order, and the angles opposite by A, B, and C ; then Cos A = &•-' + c" -a'' 1156 +625 - 1600 2bc 1700 = .10647. .-. A = 83* 53' 16". 60 PLANE TRIGONOMETRY. There is no ambiguity in this case, since the cosme is +, and lience the aagle The same angle is found by logarithmic computation, thus : « = 40 ... log = 1.602060 6 = 34 . . .log = 1.531479 = 25 ... log = 1.397940 p=99 ... log = 1.995635 ip = 49.5 . . log = 1.694605 ip-a= 9.5 ... log = 0.977724 ip - 6 = 15.5 ... log = 1.190332 ^ _ c = 24.5 ... log = 1.389166 log {^p -c) = 1.389166 log i^P -b)= 1.190332 a. c. log b = 8.468521 a. c. log c = 8.602060 2)1.650079 1.835039 log sin iA = 9.825039. /. iA = 41° 56' 38" and A = 83° 53' 16". In like manner the other angles may be found. 2. The sides of a plane triangle being 6, 5, and 4, find the angles. The angles are 82° 49' 09", 65° 46' 16", and 41° 24' 35". 3. The sides of a plane triangle being 8601.5, 4082, and 7068, find the angles. The angles are 54° 35' 12", 28° 4' 44", and 97° 20' 4". 4. The sides of a plane triangle being .5123864, .3538971, and 3090507, find the angles. The angle opposite the last side is 36° 18' 10".2. AEEA OF PLANE TRIANGLES. 05, Prop. — The area of a plane triangle is equal to half the l)roduct of any two sides into the sine of the included angle, Dem. — Let ABC be any plane triangle, and h and c any two sides with A as the included angle. From the extremity of one of these two sides roiiiou! from A, let fall a perpendicular p, upon the otbor side Now, Area ACB = ^m. Pie. 21. But, from ACD, \hc sin A. q. b. d. 7? = i sin A. Area ACB OG, Cor. — TJie area of a plane triangle is equal to the square root of the rojitinued product of half its perirteter into half its perinietei minus each side separately. I I I AEEA OF PLANE TKIANGLES. 61 'Dem. — From tlie proposition, and since sin A = 2sin ^A cos JA. we have, A.rea = i5csinA = ^siniAcosiA=Jcj/^SE£lMZI) x >|/iI^EfF=. ^ip{ip-a){ip-b){ip-e). Q. B. D. EXERCISES. ^ 1. Given two sides of a plane triangle 125.81 and 57.65, and tlie included angle 57° 25'. Find the area. Area = 3055.7. 2. Given the sides of a plane triangle 103.5 and 90, and the included angle 100°, to find the area. Area = 4586.74 3. How many square yards are there in a triangle whose sides are 30, 40, and 50 feet? ^rear.-66}. 4. Find the area of a triangle whose sides are 20, 30, and 40. Area = 290.4737. 6. What is the area of a triangle whose sides are 30 and 40, and their included angle 28° 57' ? Area = 290.427. 6. What is the number of square yards in a triangle, of which the sides are 25 feet and 21.25 feet, and their included angle 45° ? Area =^ 20.8694. 7. Find the area of a triangle in which two of the angles are 80° and 60° respectively, and the included side 32 feet. Area = 679.33 square feet. 8. Find the area of a triangular field having one of its sides 45 poles in length, and the two adjacent angles, respectively, 70° and 69° 40'. Area = 1378.411 square poles. 9. Find the jirea of a triangular piece of ground having two angles respectively 73° 10' and 90° 50', and the side opposite the latter 75.3 poles. Area = 748.03 square poles. PRACTICAL APPLICATIONS. [Note. — The following i)roblems arc inserted, not as any part of a treatise upon the subject of trigonometry as pure science, but as affording the student pood mental exercise, aud valuable and interesting information.] 62 PLANE TRIGONOMETRY. 1. To find the length (in miles) of a degree of longitude at Ann Arbor, Mich. Solution.— Let NESQ be a meridian section of the earth, EQ the equatorial diameter, and EL the lati- tude of Ann Arbor, 42° 16' 48".3. A degree of longi tude at L is 3^0 of the circumference of the circle whose radius is LD. CL the radius of the earth at this point = 3957.* Now in the right angled triangle LCD, we have CLD = ECL = 42° 16' 48".3, and CL = 3957; whence, LD = CL x cos 42° 16' 48".3, and LD = 2927.6. .'. A degree = 51.1 miles. As a degree iii longitude makes 4 minutes difference in time, 51.1 miles east or west on this parallel is equivalent to 4 minutes difference in time. Query. — How does it appear from the above solution that the length of a degree of longitude varies as the cosine of the latitude ? 2. To find the distance of a planet from the earth at any par- ticular time. Solution — To render the problem as simple as possible, we will suppose two observatories on the same meridian, at N,and N'; and that when the planet P is on the same meridian, the angles ZNP, and Z'N'P (tlie zenith distances) are measured. With these data and the radius of the earth, CN, CN', known, the problem comes quite within the scope of the present study. The process is as follows: The arc NN' being known, the angle NCN' is known. Then in the triangle NCN', two sides and the included angle are known, whence the other parts can be found. Now, knowing the angles PNC, PN'C, and CNN', CN'N, we can find the angles PNN', PN'N. This affords suflQcient parts of the triangle PNN' to determine the triangle. and we find PN, or PN'. Finally, in the triangle PNC, we know PN, NC, and the included angle ; whence the other parts can be computed. But PC is the distance sought Fig. 23. 3. Suppose in case of the moon, the angles PNZ, and PN'Z', being measured, are found to be respectively 44° 54' 21", and 48° 42' 57", the distance between the points of observation N and N' is 92° 14', and the radius of the earth is 3956.2 miles: find the distance to the moon. Distance = 237,954.098 miles. * The equatorial radius of the earth is 39G2.8 miles ; but in consequence of the flattening in tbe direction of the polar diameter it is less here. PRACTICAL PROBLEMS. 63 4. Required the height of a hill D above a horizontal plane AB, the distance between A and B being equal to 975 yards, and the angles of elevation at a and B l)eing respectively 15° 36' and a^ W ^9'. ...-'--""""";i-::^^^ ...^^^^ DC = 587.61 yards 5. Find the area of a regular hexagon, and also of a regular octagon, whose sides are each 10 feet. Areas, 259.8, and 482.84 square feet 6. Find the area of a regular pentagon, and also of a regular dec- agon, whose sides are each 12 feet. Areas, 247.74, and 1107.96 square feet. I 7 Wishing to know the length of a certain pond of water, 1 measured a line 100 yards in length, and at each of its extremities observed the angles subtended by the other extremity and a couple of trees at the extremities of the pond. These angles were, at one end of the line, 32° and 98°, and at the other, 37° and 118° ; what was the length of the pond ? Draw the horizontal line AB equal to 100; make the angle BAD 32°, BAC 98°, ABC 37% and ABD 118°. The intersections of the linevS AC and BC, AD and BD, determine the extremi- ties of tlie pond; the straight line CD is the length of the pond. CD = 161.868 yards. 8. The distances AB, AC, and BC, between the points A, B, and C, are known ; viz., AB = 800 yds., AC = COO yds., and BC = 400 yds. From a fourth point P, the angles APC and BPC are measured ; viz., APC = 33° 45', and BPC = 22° 30'. Required the distances AP, BP, and CP. r AP = 710.193 yds. Distances, i BP = 934.291 yds. (^ CP = 1042.522 yds. 64 fLANE TRIGONOMETRY. SuG*s. — Conceive tlie circumference passed through A, B, and P, and AD ;ind DB drawn. In the triangle ADB, angle DAB = the given angle DPB, and DBA = APD. Hence, all the parts ol" triangle ADB can be found. Again, since th*.- sides of the triangle ACB are given, its angles can be found. Then, since angU' CAB — DAB = CAD, there are two sides and the included angle known in triangle ACD ; whence angle ACD can be found. Thus we reach the triangle ACP , in wliich there are now known AC and the angles. 9. From the top of a mountain, three miles high, the angle of depression of a line tangent to the earth's surface is taken, and found to be 2° 13' 27". What is the diameter of the earth, considered as a sphere ? Ans. 7958.45 miles. 10. Taking the sun's mean apparent diameter as 32' 3".4, and his distance from the earth 91,430,000 miles, show that, if his centre were coincident with the earth's, his body would extend in all direc- tions nearly 200,000 miles beyond the moon. (See Ex. 3.) Sun's diameter = 852,574 miles. 11. Assuming the height of the Great Pyramid to be 480 feet, how far off may it be seen across the desert ? Aiis.f 27 miles. 12. What was the perpendicular height of a balloon, when its angles of elevation were 35° and 04°, as taken by two observers on the same level, at the same time, both on the same side of it, and in the same vertical plane ; the distance between the two observers being 880 yards ? Ans.f 935.757 yards. 13. Given two sides of a parallelogram 60 and 80, and a diagonal 100. Is this the longer or shorter diagonal ? What is the other ? What are the angles of the parallelogram ? 14. A balloon being directly over one of two towns standing on i\lie same horizontal plane, at a distance of eight miles from each ether, the angle of depression to the more remote town was observed by the aeronaut to be 10°. What was the height of the balloon ? Ans., 1.41 miles. 15. The most recent observations make the sun's horizontal pai-- allax 8''.1)4, and the earth's equatorial radius 3902.8 miles. Show that the distance of the sun from the earth is nearly as given in Ex. 10, instead of 95 millions of miles, as it has been heretofore cod. eidered. OHAPTEE. IL 8PHEBICAL TJRIGONOMETBT. INTRODUCTION, PROJECTION OF SPHERICAL TRIANGLES. 97, To JProject a Spherical Triangle on a plane surface is to draw the triangle on that surface so that it will present the same appearance to the eye, situated at a particular point, as when drawn on the surface of a sphere. 08. The Simplest JKethod of projecting a spherical triangle is to project it on the plane of one of its sides, the eye being supposed situated in the axis of the sphere perpendicular to this plane, and at an infinite distance from it. The plane is called the Plane of Pro- jection; and its intersection with the sphere is called the Primitive Circle, and is the base of the hemisphere on which the triangle is conceived as situated. li 99. Fundamental Propositions, — 1st. Whenthe parts of a spherical triangle ai^e each conceived as less than 180°, amj such ria7igle can he rep)rese7ited on a hemisphere. 2d. The primitive Circle has its axis, and consequently its pole, projected at its centre, 3d. The semi-circumference of any circle of the sphere, perpendic- ular to the Primitive Circle, is projected in the chord representing the intersection of the circles ; and, if the perpendicular circle he a great circle, its semi-circumference is projected in a diameter of the Primitive Circle, (5(5 SPHERICAL TRIGONOMETRY. Ill's.— These propositions are direct consequences of the fundamental con- ception. Thus, let ABA'B' represent the base of the hemisphere on which the triangle is conceived as situated. This is the Primitive Circle, and the eye is supposed situated at an infinite distance, and in a line i^erpendicular to the plane of the paper at P. The pole of the Primitive Circle being in this line is projected (seen as) at P. As all great circles perpendicular to the Primitive Circle pass through its pole and include its axis, the eye is in all such planes, and any lines of these planes, as the semi-circumfer- ences of the great circles in which they intersect the sphere, are projected (appear to the eye) as diameters of the Primitive Circle. Moreover, since the eye is at infinity, it is to be conceived as in the plane of any small circle which is perpendicular to the primitive, and which is therefore projected in a chord, as CC. PROJECTION OF RIGHT ANGLED SPHERICAL TRIANGLES. 100* JProb. 1. — To project a right angled spherical triangle on the plane of one of its sides, when the two sides about the right angle are given.* Solution. — Let the angles of the triangle be represented by A, B, and C, A being the right angle. Let the sides opposite these angles respectively be rep- resented by a, b, and c ; whence b and c are the given sides. Draw the primitive circle and the diameters BB', NN' at right angles to each other. Prom B lay off BA = c.\ Let the right angle be at A ; whence the side b is perpendicular to the primitive circle, and projected in the diameter AA'. To project the vertex C, conceive the semi-circum- ference, of which AA' is the projection, to revolve on AA' until it falls upon the semi-circumference A'BA, then will the point C tsdl at d. Hence make Ad! = b. In like manner revolving the semi-cir- cumference, of which AA' is 'he projection, until it falls upon A'B'A, the point C will fall at d'. Hence make Ad' = b. The point C will de- scribe the semi-circumference of a small circle perpendicular to the primitive circle, and whose projection is dd'. Now, as the Fig. 25. * In this treatise the discussions embrace only such triangles as have each part less than 180°. t For the purposes for which we shall use these projections, an arc can be laid off with sufficient accuracy by observing its relation to 90", 60°, 30°, or some aliquot part of the circum- ference, which is readily obtained on geometrical principles. PROJECTION OF RIGHT ANGLED SPHERICAL TRIANGLES. 67 projection of the vertex of the triangle C is at the same time in AA' and dd\ it must be at their intersecticm. Finally, the liypotenuse a is projected in a curve passing through B, C, and B', since two great circles intersect at the extremities of a diameter. This curve is really an ellipse, but for our present purpose it may be considered as the arc of a circle passing through B, C, and B'. There- fore, BAC is the projection required. Queries. — Will a solution of this problem be possible for all values of h and^ c ? How does it appear from the projection ? EXAMPLES. I 1. Having given h = 110°, and c = 60°, to project the triangle. See Fig, 26. 2. Having given h = 50°, and c = 130°, to project the triangle. 3. Having given 5 = 90°, and c = 30°, to project the triangle. 4. Having given b = project the triangle. \ and c = 90°, to I 101, JProb. 2, — To project a right angled spherical triangle when the hypotenuse and 07ie side are given. Solution. — Using the common notation, let c represent the known side. Drawing the primitive circle and the conjugate* diameters BB', NN', layoff BA = c, and draw the diameter AA'. The projec- tion of b will lie in A A', and the projection of the vertex C will fall somewhere in this line. Now the arc a lies in a semi-circumference passing through B and B'. Conceive this semi-circumfer- ence to revolve on BB', as an axis, till it coincides, first, with BNB', and then with BN'B'. The point C will trace the semi-circumference of a small circle perpendicular to the primitive circle, and whose projection is dd'. Hence, making Bd = Ba = a, and drawing dd', the projection of the vertex C lies at the same time in AA' and dd', and is there- fore at their intersection. Passing an arc of a circle (strictly an ellipse) through BCB', we have ABC, the projection desired. Fia. 27. * Two diameters of a circle which are at right angles to each other are called Conjugate Diameters, m Spherical TRtGONOMETRY, Queries. — Will a solution of this problem be possible for all values of a and (J? If a had been greater than e in the above case, would the solution have been possible ? Will dd' and AA' always intersect, whatever may be the relative values of a and c ? EXAMPLES. 1. Having given c = 75°, and a = 64°, to project the triangle. 2. Having given c = 45°, and a = 136°, to project the triangle. 3. Having given b = 110°, and a = 85°, to project the triangle. See Fig. 28. 4. Having given b = 110°, and a = 120°, to project the triangle. 5. Having given c = 90°, and a = 75*^, or a = 120°, to project the triangle. Pio. 28. Query. — If one side of a right angled spherical ti-iangle is 90°, what must the hypotenuse be ? Why? 102, I^rob, 3, — To project a right angled spherical triangle when an oblique angle and the hypotenuse, or the oblique angle and the adjacent side are given. Solution. — Draw the primitive circle and the conjugate diameters BB', NN' as usual. To construct the given angle B, we observe that this angle is measured by an arc of the great circle which is projected in NN'. Hence, lay ofFNt? = N^' = B , and draw dd! ; then is NO the projection of the arc which measures B, and the projection of a lies in the arc passing through BO 3'.* Having the angle B projected, if the hypotenuse a is the other given part, find the projection of C by taking Be = Be' = a, and drawing ee'. Com- plete the projection by drawing AC through P. When the adjacent side c is given, take BA = c, and draw AC as before. [Tlie student will be able to give the reasons.] Fig. 29. 103, ScH.— As OP is the cosine of the angle ABC, the point O may be found by measuring * As has been remarked, this arc is really a eemi-ellipse. This fact, together with the method of constructing the gemi-ellipse, and thus getting the correct projection of the hypot PROJECTION OF RlGHf ANGLED Sl>HERiOAL TRIANGLES. 6^ from P (towards N if B < 90°, and towards N' if B > 90°) a distance equal to the natural cosine of B. Query.— Is the solution of this problem possible for all values of the hypot- enuse or adjacent side, and the augle? EXAMPLES. ^ — 1. Having given B = 65°, and the hypotenuse a = 120°, to proiect the triangle. See Fig. 30. 2. Having given C = 45°, and the adjacent side b = 50°, to project the triangle. Sug's.— Project the angle C as the angle B of the preceding, and lay oS b = 50° from its vertex on the circumference of the primitive circle. 3. Having given C = 170°, and hypote- nuse a = 160°, to project the triangle. Fig. 30. 4. Having given B = 150°, and c = 40°, to project the triangle. 104, JProb. 4, — To project a right angled spherical tria7igle when an oUique angle and side opposite are given. Solution. — Project the given angle at B, Figs. 31, 32, as in the last problem. Then, from any point in the circumference of the primitive circle, as N, take NO', in the diameter passing through that point, equal to the projection of the given side. (This is done by taking N^ = Hd' = b, as in Prob. 1, and drawing dd'). Now, with P as a centre, and radius PO', describe a cir- cumference cutting BOB', One extremity of the given side b will be projected in this circumference, since this circumference contains the projections of all the points in the surface of the hemisphere which are at a distance b from the circumference BNB'N'. But the vertex is also projected in the enuse, belong to a treatise on Conic Sections. In this case, BB' and 20 P are tiie axes of the ellipse, and the cnrve can be constructed by taking BP as a radius, and striking arc:* from Q as a centre, cutting BB'- These intersections are the foci of the required ellipse. Then take a string equal in length to BB'i and, fastening its ends to the foci, place a pencil against the string, and keeping the string tight, carry the pencil around the curve. TO SPfiEBICAL tlilGONOMETRt. arc BOB'; hence, it must be at the intersections C, C Drawing tlie projections of the perpendic- ulars CA, and C'A', the projection is complete. Queries. — When will there be two triangles fultilUng tlie given conditions? When but one? When none ? When there is but one triangle what kind of a triangle is it? If B > 90°, must h be greater or less than B in order to have two solu- tions ? If B < 90°, how is it ? If B > 90°, can h be less ? If B < 90°, can 6 > 90° ? EXAMPLES. 1. Given C = 120^, c = 150, to project the triangle. See Fig. 33. 2 Given c = 80°, c = 60°, to project tlie triangle. See Fig. 34. Pig. 38. Fig. 34. 3. Given B = 70°, h = 70°, to project the triangle. See Fig. 35. 4. Given B = 64°, b = 75°, to project the triangle. See Fig, 36. Fig. 35. 5. Given b = 160°, d = 110', to project the triangle. PROJECTION OF OBLIQUE ANGLED SPHERICAL TRIANGLES. 71 105, ScH. — When the given parts are the two oblique angles, the projection is most readily effected by first computing one of tlie sides. The projection in this case will be considered in connection with the numerical solution of the case, in the next section. PROJECTION OF OBLIQUE ANGLED SPHERICAL TRIANGLES. — ^ 106. Proh, 1. — To project a spherical triangle when two sides and the included angle are given. Solution. — Let a, e, and B denote the given parts. B at some point in tlie circumference of the primi- tive circle, as B. Lay off one of the given sides, as c, from B on BNB', Let BA = c. Determine the extremity Cof the projection of the other given side a, as in Prob. 2, etc., and drawing the diameter AA', pass the arc through ACA' ; BCA is the pro- jection souglit. Query. — Is this projection always possible, what- ever the relative magnitude of tlie given parts ? Project the given angle Fig. 37. EXAMPLES. 1. Given a = 130°, c — 85°, and h = 100°, to project the triangle. 2. Giyen c = 40°, a = 37°, and b = 80°, to proiect the triangle. 107. I^rob. 2, — To project a spherical triangle when ttvo sides and an angle opposite one of them are given. Solution. — Let the given parts be a, 6, and B. Make the projection upon the plane of the unknown side c. Thus, drawing the primitive circle and the conjugate diameters BB', NN', conceive c as projected from B on the arc BNB', and project the given angle B as in preceding problems. On tlie arc BB', take BC = the pro- jection of the given adjacent side a. To deter- mine the projection of the opposite side h, describe *a circle about P, as a centre, with a radius PC. Through C draw PD, and taking Dd = Dd' = b draw dd'. Tlirough the intersections o, o' of dd! with tlie circumference of the small circle, draw the radii PA, PA'. Finally, passing arcs through the points A, C, A", and A', C, A'", BAC, and BA'C are the projections of triangles which fulfill the given conditions. The projections of B an4 72 SPHERICAL TRIGONOMETRY. a were made upon principles previously established ; and it only remains to show that AC, and A'C are projections of 6. Since by construction DL is the projection of an arc equal to h, the projections of arcs of great circles connect- ing D witii o and o' are projections of arcs equal to b. But the figure DoP — ACP, and Do'P = A'CP; therefore AC = A'C = Do = the projection of b. 108. ScH. — It is evident that this problem has one solution, two solutions, or no solution, according to the value of b as compared with a and B, Thus, if the projection of 6 = DC, o and o' coincide, there is but one solution, and the tri- angle is right angled at A (which in this case falls at D). Also, if the projection of b is intermediate in value between DC and only one of the arcs BC, B'C, there is on\y one solution. If, however, as in the figure given, the projection of b is intermediate in value between DC and both BC and B'C, there are ^w)£> solutions. Finally, if b is given of such value that the chord dd' does not touch the small circle, there is no solution. The latter case occurs when B < 90°, if the projec- tion of 6 < DC ; and when B > 90°, if the projection of ^> > DC, as will appear from Figii. 38, 39. We may observe, also, that there are two solutions when o and o' both fall on the same side of BB' as c ; one solution when o and o' coincide, and when they fall on opposite sides of BB' ; and no solution when <> and o' are imaginary, i. c, when dd' does not touch the small circle, or when both fall on the opposite side of BB' from c. EXAMPLES. Fis. 39. 1. Given b = 110°, a = 120°, and b = 83°, to project the triangle. See Fig. 39. 2. Given B = 110°, a =^ 120°, and b = 130°, ° to project the triangle. ,' 3. Given c = 64°, a = 120°, and c = 75°, to project the triangle. 4 Given C = 80°, b = 60°, and c = 40°, ^ to project the triangle. 5. Given c = 112°, b = 75°, and c = 150°, to project the triangle. 109. I^rob. 3.— To project a spherical triangle when the three are given. SOLUTION OF RIGHT ANGLED SPHERICAL TRIANGLES. 73 Solution. — Drawing the primitive circle and the conjugate diameters, as nsual, take BA = c, the side on the plane of which it is proposed to project the triangle. Take Be = Be' = a and draw ee^ ; then as before shown, the projection of the vertex C lies in ee'. In like manner taking ^d = A sin c = [from (2)J, cos b sin a sin C. Substituting this value of ED, we have _ cos & sin « sin C t . #% /ox cos B = r = cos b sm C. (3) sin a We may write (4) from (3) by analogy, as (2) was from (1) ; or, better, let the student produce it from Fig. 45, as (3) was produced from Fig. 44. Finally, to produce (5), consider the triangle ODE, in either figure, right angled at E. This gives OE=»OD x cos DOE, or cos a = cos 6 cos c, (5) 119. Hule 2. JProp. — In any right angled spher- ical triangle, the sine of the middle part equals the ^jro- duct of the tangents of the adjacent extremes. Dem. — In the spherical triangle BAC, right angled at A, taking b, c, comp B, comp C, and comp a, in succession as middle parts, we are to prove that, 78 SPHERICAL TRIGONOMETRY. sin h = tan c x tan (comp C), or sin b = tan c cot C ; (1) sin c = tan b x tan (comp B), or sin c — tan b cot B ; (2) sin (comp B) = tan c x tan (comp a), or cos B = cot a tan c ; (3) sin (comp C) = tan 6 x tan (comp a), or cos C = cot a tan h ; (4) sin (comp a) = tan (comp B) x tan (comp C), or cos a = cot B cot C • (5) Taking the formulw of Rule 1st, and in the second member of each substitu- ting the value of each factor as found in some other of the set, we readily write the following sin 6 i;^ sin a sin B sin c cos C sin c "~ sin C COSC ~ COSC X cosC smC sin c = sina sin C sin b cos B sin b ~~ sin B cos 6 ~ cos b X cosB sin B cos B = cos 6 sin C cos a sin c cos a X sine cos c sm a sm a cose cos C — cos c sin B cos a sin b cos a ~ cos 6 sin a ~~ sin a X sin b cos 6 cos a = cos & cos c cos B cos C cos B ~ sin C sin B "" sin B X cosC sinC = tan c cot C ; = tan b cot B ; = cot a tan c ; = cot a tan 6 ; = cot B cot C. (1) (2) (3) (4) (5) Q. E. D. 120, ScH. 1. — It will be a good exercise for the student to demonstrate Rule 2d from the an- q £ nexed figures, as Rule 1st was from Figs. 44 and 45. The 5th is not as readily- obtained from the figure as the others. The stu- dent may trace the fol- lowing relations, some in one figure, and some in Fig. 47. Fia. 48. the Other. ^ OD cos b . -o 4. 4. ^ AD sin 5 , ^ _ AD Cos a = ^ — = cos b cos c. But, cot C = ^ = , and cot B = ^ OE sec c AE tanc Ah sin c , ^ _ ^ _ sin 5 sin c , .. o , /-> = ^ — ; : whence, cot B cot C = ; = cos b cos c. .*. cos a = cot B cot C. tan 6' tan* tanc 121, ScH. 2. — It is of much importance, especially for the purposes of Spherical Astronomy, that the student observe that the relations expressed in the above formulae, and in fact all the relations between the sides and angles of spherical triangles, are also the relations between the facial and diedral angles of triedrals. Thus, if a, b, and c represent the facial angles, and A, B, and C the opposite diedrals, all these relations can be established, and in exactly the same manner as above, without any allusion to the spJterical triangle. [The student ghould do it.j I DETEBMINATION OF SPECIES. 79 DETERMINATION OF SPECIES. 122. In the solution of spherical triangles the determination of the species of a part sought becomes of essential importance, since any part of such a triangle may have any value between 0° and 180°. Hence, when we have learned the numerical value of any function erf a part, we have yet to determine whether the part itself is less or greater than 90°, i. e., the species of the part. This may always be effected by some one of the following propositions. 123. I*rop, — //' the part sought is found in terms of its cos, tan, or cot, its sjjecies can be detertnined by the algebraic signs of the functions in the formula used. Dem. — In each oi \hQ formulm arising from the application of Napier's rules, there are three functions, the arcs corresponding to two of which are always known, hence the algebraic signs of theii- functions are known, and the signs of these two determine the sign of the third or unknown function. Now, when a cos, tan, or cot is + , the corresponding angle is less than 90° (if less than 180°) ; and when one of these functions is — , the corresponding angle is greater than 90° ; i. d., in a triangle, it is between 90° and 180°. 124:, When the part sought is found in terms of its sine, the species cannot be determined by the signs of the formula, since the part being less than 180° its sine is always +. The three following propositions dispose of such cases. 125. ^rop. — An oblique angle of a right angled spherical tri- angle and its opposite side are always of the same species. Dem.— From Napier's first rule we have, cos B = cos 5 sin C. But sin C is necessarily + ; therefore, cos B and cos b always have the same sign, and B and h are of the same species. In like manner, we see from cos C = cos c sin B, that C and c are of the same speeies. q. b. d. 126, Prop, — When the hypotenuse of a right angled spherical triangle is less than 90°, the other two sides (and consequently the 80 SPHERICAL TRIGONOMETRY. oUique angles) are of the same species loitli each other. But when the hypotenuse is greater than 90°, the other two sides {and consequently the oUique angles) are of different species from each other. Dem. — From Napier's first rule we have, cos a = cos b cos c. Now, if a < 90°, cos « is + ; lience cos h and cos c must have like signs, and h and c be both less or both greater than 90° But if « > 90° (and less tlian 180°, as it is), cos rt is — ; hence, cos b and cos c must have different signs, and b and c be one greater and the other less than 90°. Finally, since the oblique angles are of the same species as their opposite sides, they are of the same species with each other when a < 90°, and of different species from each other when a > 90°. 127 > JProp, — When a side and its opposite angle are given in a right angled spherical triangle, there is no solution if the sine of the side is greater than the sine of its opposite angle ; there is one solution and the triangle is M-rectangular if the sine of the side equals the sine of its opposite angle ; and there are two solutions if the sine of the side is less than the sine of its opposite angle. Uem. — We have sin& = sin a sm B, or sin a = -: — = . Now, sin 6 > sin B sm B makes 8uia> 1, which is manifestly impossible. Sin & = sin B makes b = B, since they are of the same species. But when the arc included by the sides of an oblique angle of a right angled spherical triangle is equal to the angle, the vertex of the angle is the pole of the arc. Hence, in this case the otlier sides of the triangle are each 90°. Finally, if sin b < sin B, a has two values, one greater and the other less than 90°. Hence there are two triangles. 128, ScH. — These relations between an angle and its opposite side may be observed directly from a figure. When B < 90°, the measure of it, that is b= B, is the greatest included perpendicular which can be dmwn to one side of the lune BAB'C. Hence, in this case, b cannot exceed B, which implies that sin b cannot be > sin B, as when the arcs are less than 90°, the greater arc has the greater sine. If sin 6 = sin B, b = B, and BA = BC = 90°, and the side b can occupy but one position in the lune, thus giving rise to but one triangle BAC, which satisfies the conditions (or two equal triangles BAC and B'AC). If ^> < B, which, when B is less than 90" implies that sin b < sin B, the side can occupy two positions in the lune, b' and b" giving rise to two triangles, BA'C and BA"C", both of which fulfill the conditions. EXERCISES IN SOLVING RIGHT ANGLED TRIANGLES. 81 Again, when B > 90°, the measure of it, i. e., d = B, is the least included perpend icuhir that can be drawn to one side of the lune. Hence, in this case we cannot have 6 < B, which implies that sin h cannot be > sin B, since the greater arc has the less sine. If sin b =r sin B, * = B, and BA r= BC = 90°, and the side b can oc- cupy but one position in the lune, thus giving rise to but one triangle B AC, which satisfies the conditions (or two equal triangles BAC and B'AC). If 6 > B, whicli implies that sin b < sin B, the side b can occupy two positions in the lune, as b' and b'\ thus giving rise to two triangles BA'C, and BA"C", both of which satisfy the conditions. EXERCISES. . In a right angled spherical triangle BACj A heing the right angle, B = 80° 40', and a = 105° 34', to project the triangle and compute the other parts. Projection.*— Projecting the triangle upon the plane of the side c {102), we have, BCA, Fig. 51. [The student should give the process.] Solution.— It is immaterial which of the re- quired parts we seek tiret. We will seek c. Now the three circular parts under consideration are c, compa, and comp B. Comp B is middle part, and the extremes are adjacent ; hence, by Napier's second rule we have, cos B = tan c cota. cos B cos 80° 40' or tan c = cot a cot 105° 34'' Now cos 80° 40' is + , and cot 105° 34' is Fio. 51. Therefore tan c is — , and c > 90". Computing by logarithms, log cos 80° 40' = 9.209993 - log cot 105° 34' = 9.444947 = log tan c = 1.765045 Add 10 for tab. tan 10. 9.765045. 149° 47' 37". * It IS recommended that the projection be given always before the trigonometrical solution. It is an excellent exercise, and gives clearness of perception. 82 SPHERICAL TRIGONOMETRY. To find b, sin & = sin a sin B = sin 105° 34' x sin 80° 40'. Tiiis makes b known by means of its sine, whence the signs of the formula do not determine the species of 5. But b is of the same species as B {124), and therefore less than 90°. Computing by logarithms, log sin 105° 34' = 9.983770 + log sin 80° 40' = 9.994212 Deductmg 10 = 9.977982 = log sin ft. .-. b = 71° 54' 33". To find C, cos « = cot B cot C, or cotC = -^ = ^— iTf5 . Whence cotB cot 80° 40' cot C is — , and C > 90°. Computing by logarithms, log cos 105° 34' = 9.428717 - log cot 80° 40' = 9 .215780 Adding 10 = 10.212937 = log cot C. .-. C = 148° 30' 54 '. ScH.— It is expedient to find each part directly from the parts given in the example, in order that an error in finding one may not extend itself through the whole solution. 2. Giyen a = 86° 51', and B = 18° 03' 32", to project the triangle and compute the other parts. c = 86° 41' 14", b = 18° 01' 50", C = 88° 58' 25". 3. Given b = 155° 27' 54", and c = 29° 46' 08", to project the triangle and compute the other parts. See Fig, 52. a = 142° 09' 13", C = 54° 01' 16", B = 137° 24' 21". 4. Given c = 73° 41' 35", and B = 99° 17' 33", to project the triangle and compute the other parts. C = 73° 54' 46", h = 99° 40' 30", Fre. 52. « = 9^° ^2' 17". 5. Given B - 47° 13' 43", and c = 126° 40' 24", to project the triangle and compute the other parts. b = 32« 08' 56", a = 133° 32' 26", c = 144° 27' 03". EXERCISES IN SOLVING BIGHT ANGLED TRIANGLES. Projection. — In order to project this case, i. e., when the two oblique angles are given {105), it is most convenient to compute the base before pro- jecting. It is also expedient, when two angles are given, to project the larger at a point in the cir- cumference of the primitive circle, as at C, espe- cially if the smaller be quite small. In this case, projecting the angle C at C, Fig. 53, conceive BA as drawn through P (or, if desired, sketch it hypo- thetically), and then compute ^>, from the relation cos B _ . Havmg 83 cos B = sin C cos h. or cos h sin C Pig. 53. found h = 32° 08' 56", take CA = 6, and draw AB through P. 6. Given B = 100°, and h = 112°, to project the triangle and com- pute the other parts. Projection.— See Fig. 54. Numerical Solution. — To find c, we have sin c = tan Z> cot B = tan 112° cot 100°. Computing by logarithms, log tan 112° = 10.393590 -f log cot 100° = 9.246319 Rejecting 10 = 9.639909 = log sin c. .-. c z= 25° 52' 33".4, or 154° 07' 26".6, i. e., BA, or BA'. Fig. 54. To find a, we have sin 5 = sin « sin B ; whence sin a = sin h __ sin 112° sin B ~* sin 100°* I Computing by logarithms, log sin 112° = 9.967166 - log sin 100° = 9.993351 Adding 10 = 9.973815 = log sin a. .'. a = 70° 18' 10".7, or 109° 41' 49".3, i. e., BC, or BC. To find C, we have cos B = cos 6 sin ; whence sin C = Computing by logarithms, log cos 100° = 9.239670 - log cos 112° = 9.573575 cosJB _ cos 100° cos b ~ cos 113°' Adding 10 ».e., BCA, or BC'A'. 9.666095 = log sin C. .-. C = 27° 36' 58".8, or 152° 23' 01". 84 SPHEKICAL TRIGONOMETRY. Thus we see that each of the two triangles BCA and BC'A' fulfills the con- ditions of the problem. 7. Given one side of a right angled spherical triangle 160°, and the opposite angle 150°, to project the triangle and compute the other parts. Results. — There are two triangles. The other sides of the first are 136° 50' 23", and 39° 04' 51"; and the angle opposite the latter side is 67° 09' 43". The corresponding parts of the other triangle are 43° 09' 37", 140° 55' 09", and 112° 50' 17". 8. In the spherical triangle def, right angled at E, given an oblique angle 58°, and the side opposite 64°, to project the triangle and com- pute the other parts. 9. In a right angled spherical triangle given an oblique angle 165°, and the opposite side 112°, to project the triangle and com- pute the other parts. 10. In a right angled spherical triangle given one side 65° 23' 12", and the opposite angle 65° 23' 12", to project the triangle and com- pute the other parts. 11. Given c = 60° 47' 24".3, B = 57° 16' 20".2, and A = 90°, to project and compute. a = 68° 56' 28".9, c = 54° 32' 32".l, and h = 51° 43' 36".l. 12. Given c = 116°, b = 16°, and the included angle 90°, to pro- ject and compute. QUADRANTAL TRIAJVGLES. 129, A Quadrantal Triangle is a spherical triangle one of whose sides is a quadrant, or 90°. Such a triangle is readily solved by passing to its polar, solving it, and then passing back. The polar triangle to a quadrantal triangle, being right angled, is solved by Napier's rules. Ex. 1.- Given a = 90°, B = 75° 42', and c = 18° 37', to compute the other parts. Sug's. — Representing the supplemental parts of the polar triangle by A', B', C, a', b', and c', we have A' = 180" - a = 90°, b' = 180° - B = 104° 18', and OF OBLIQUE ANGLED SrHERICAL TRIANGLES. 85 C - 180' - c = 161° 23', from which to find B', a', and c'. This being ris^^ht ant^led, we find, by applying Napier's rules, B' = 94° 31' 21", a' = 76° 25' 11", and c' = 161° 55' 20". Hence in the primitive triangle we have b = 180° — B' = 85° 28' 39", A = 180° - a' = 103° 34' 49", and C = 180° - c' = 18° 04' 40". Ex. 2. Given a = 90°, C = 42° 10', and A = 115° 20', to find the other parts. __ B = 54:° 44' 24", b = 64° 36' 40", c = 47° 57' 47". SECTION II. OF OBLIQUE ANGLED SPHERICAL TRIANGLES. 130> All cases of oblique angled spherical triangles can be solved by Napier's rules and the following proposition. 131, JProp, — In any spherical triangle, if a perpendicular he let fall from either vertex upon the opposite side or side produced, the tangent of half the sum of the segments^ of that side is to the tangent of half the sum of the other two sides, as the tangent of half the difference of those sides is to the tangent of half the difference of the segments. Dem.— In the triangle BAC let fall the perpen- dicular ^, from C upon the opposite side. Let BD = s, and DA = s'. By Napier's first rule, cos a = cos p cos s, and cos b = cos p cos s'. cos a cos 8 ^ cos s' ' Dividing the former by the latter, , _ * •' ' cos 6 whence, by composition and division, cos b — cos a _ cos s' — cos s cos a + cos b But by {61), cos S + COS s' Pig. 55. COS b COS a and COS a + COS cos «' — COS s cos s + cos « V = tan i (a + b) tan i (<:» — b), tan i (« + «') tan ^ (« — s^* * When the perpendicular falls without the base, as in Fig. 50, this term is to be understood as meaning the distances from the foot of the perpendicular to each extremity of the base, as BD and AD- This, in fact, is the general statement— applying as well to the case when the perpendicular Mia on the base. 86 SPHERICAL TllIGONOMETRY. .'. tan i(a + Z>) tani(a — J) = tan i(8 + «') tan ^(8 — s'); or, tan ^{s + s') : tan ^{a +b) : : tan \{a — b) : tan ^{s — s'). q. e. d. 132» ScH. 1. — Since from a point in the surface of a hemispliere two perpendiculars can always be drawn to the circumference of the great circle which forms its base, and since the feet of these perpendiculars are 180° apart, and no side of a spherical triangle can equal 180°, the foot of one perpendicular will always fall within the base or upon one extremity of it, and the other without the base; or both will fall without the base. If we take the foot of the perpendicular which falls within the base, or the nearer one when both fall without, the sum of the distances from the foot of the perpendicular to the extremities of the base is always less than 180°^ i. e,, 8 + s' < 180°. "When the perpendicular falls within, s + s' makes up one side of the triangle, and hence is less than 180°. If both perpendiculars fall without, let D, Fig. 56, be the foot of the nearer one. Now DB + BD' = 180° ; but by hypothesis DA < BD', .. DB + DA < 180°. When DA = BD', DB + DA = 180°. 133, ScH. 2. — As in spherical triangles the greater segment is not always adjacent to the greater side, it becomes necessary to determine the position of the segments. This can be done by the signs of the proportion tan i (s + s') : tan ^ {a + b) : : tan ^ {a — b) : tan \ {s — s'). 1st. Tan ^ (. 180°. tan ^ {a + b) is — . 3d. When a > b, a — b is a positive arc less than 180°, hence tan ^(a — 6) is + ; and when a < 5, (a — &) is a negative arc and less than 180°, hence tan i {a — b) is — . 4th. The signs of these terms being determined, that of tan ^{s — s') becomes known. Now, as ^{s — s') cannot be numerically greater than 90°, tan i{s — s') is + when s > s', and — when s < s'. 5th. When s + s' = 180°, tan i (s + »')= <». Now as a - 5 < 180°, tan ^ (a-b) cannot be oo, nor can tan ^{s — s') = when the perpendicular falls without. Hence to make the proportion possible, tan ^{a + b) must be e, it is evident that tlie perpendicular lalls without the triangle, w^hicli will agree with the piojection. Passing to the triangle ADC, right angled at D, we now know CD and DA ; whence the other parts can be found. Finally, the angle BCA of the required triangle = BCD + DCA when the perpendicular falls within the triangle, and BCD — DCA when the perpendicular falls without. Case 2d.— Given two angles and the included side. Solution.— The solution of this case is effected by passing to the polar triangle, projecting and solving it by Case 1st, and then passing back, 138, ScH.— A slight saving of labor is effected by using {135) in the solu- tion. Thus, in the triangle BCD, compute CD and BD as before, and (not com- puting angle BCD) then passing to the triangle DCA, compute b and A. Finally, compute C (the entire angle) from the proportion sin 5 : sin c : : sin B : sin C. 1S9. I^rob. 2,— In an oblique angled spherical triangle, given two parts adjacent to each other and one separated from both of them, to solve the triangle. Case 1st. — Given tioo sides and an angle opposite one of them. Solution. — Project the triangle on the plane of the unknown side, with the given angle at B ; and let fall the perpendicular from the angle C opposite the unknown side. Compute the tiiangle BDC. Having computed this triangle, Fig. compare the side opposite the given angle, as 5, with the perpendicular and tlie arcs BC and CB', i. e., with ^, a, and 180° —a. It b = p there is but r OBLIQUE ANGLED TRLiNaLES SOLVED Bt NAPIER* S RULES. 60 I one solution and the triangle is right angled, A falling at D. If 6 is inter- mediate in value between p and both a and 180° — «, it can occupy two positions in Fig. 59, and there are two solutions. If b is intermediate in value between p and only one of the arcs a or 180° — a, there is but one solution. When B < 90° the perpendicular is less than any oblique arc; hence in this case, if Z> <^, there is 110 solution. But if B > OO", the perpendicular is greater than the oblique arcs; hence in this case, \^b> p, there is no solution. [These results should be obtained independently of the results given by the projection, and one be made a check upon the other.] The solution is now completed by computing the parts of DCA, and adding or subtracting the segments BD and AD, and the angles BCD and AC D, as the case may require. Case 2d. — Given tioo angles and a side opposite one of ihem. Solution. — Pass to the polar triangle ; solve it, and then pass back.* 14z0, ScH. — The relation established in {135) may also be used in the solution of this j^roblem. Thus, having projected the triangle, computed J9, and determined whether there are one or two solutions, to find A, when a, b, and B are given, we have, sin b : sin a : : sin B : sin A. Then computing the third side c (or sides), by means of the right angled triangles BCD and DCA as before, we may use the proportion {135) to find the angle BCA and BCA'. But the use of this proportion gives no advantage except in cases in which there is only one solution. I 14=1, JProb. 3, — In an oUique angled spherical triangle, given three parts all separated from each other, to solve the triaiigle. Case 1st. — Given the sides to find the angles. Solution. — Project the triangle on the plane of one of its sides, as c. From the proportion, tan|(s 4- s') : tan \{a ■\-b) \\ tani(« — b) : tan \{s—^\ * This case can be projected and solved in a manner ogether eimilar to the first, without passing to the polar triangle. Thus, let B> «i and A l>e the given parts. Project the triangle on the plane of c, as in the figure. Project B in the usual way, and make BC = the projec- tion of a. Through C draw DD', and make BDO = the projection of A- Drawing the small circle with radius PC, draw diameters through the intersections and 0'* Then will A and A' be the vertices of the triangle re- quired. The student may prove that the figures POD and PC A are equal, and also PQ'D and RCA', and hence that angle BDO = A = A'« FIQ. 61. 90 SPHERICAL TRIGONOMETRY. Fig. 62, find |(s — s'). ■ Then half the sum of the segments + half the difference gives the greater segment, and half the sum — half the difference gives the less. Determine from the signs of the terms whether s is greater or less than s' : and also determine whether the perpendicular lies within or without the tri- angle {134). Observe that these i-esults correspond to those given by the projection. Finally, in each of the two right angled triangles BCD and DC A, there are two sides given ; whence the angles can be found by Napier's rules. If the perpendicular falls within, C = BCD + DCA, and A, of the re- quired triangle := DAC. If the perpendicular falls without, C = BCD — DCA, and A of the triangle = 180° - DAC. Case 2d. — Given the angles to find the Solution. — Pass to the polar triangle ; solve it, and then pass back. 142, ScH. — Here, again, {135) affords a slightly Fig. 63. more expeditious solution. Having projected the triangle, found and located the segments, and com- puted one angle, as B, by the methods given above, the other angles may be found from the proportions, sin 6 : sin a : : sin B : sin A, and sin 5 : sin c : : sin B : sin C. EXERCISES. 1. Given h = 120° 30' 30", c -- to project and solve the triangle. 70° 20' 20", and A = 50° 10' 10", Projection.— See Fig. 64. Trigonometrical Solution. — 1st. To solve the triangle ABD, in which the two known parts are situated. (a) To find p, sin ^ =: sin c sin A. log sin 70' 20' 20" = 9.973912 + log sin 50° 10' 10" = 9.885329 Rejecting 10 = 9.859241 = log sin p .*. p = 46° 19' 01", the species being determined by the opposite angle {125). [Observe that the re- sult corresponds with the projection]. OBLIQUE ANGLED TltlANGLES BOLVED BY KAPIER's EULES. 91 . -^ » r> COS A (6) To find AD, cos A = cot c tan AD, or tan AD = — - . log cos 50° 10' 10" = 9.806532 - log cot 70° 20' 20" = 9.5530 16 Adding 10 = 10.253516 = log tan AD. .-. AD = 60' 50' 49", the species being determined by the signs of the formula. (c) To find angle ABD, cos c = cot A cot ABD, or cot ABD = ~zt • log cos 70° 20' 20" = 9.526929 - log cot 50° 10' 10" = 9.921204 Adding 10 = 9.605725 = log cot ABD. .-. ABD = 68° 01' 53" the species being determined by the signs of the formula. 2d. To solve the triangle DBC. (a) To find DC. Since AD < b, the foot of the perpendicular falls in the base, and DC := AC - AD = 5 - AD = 120° 30' 30" - 60° 50' 49" = 59° 39' 41". (b) To find a, cos a = cosp cos DC log cos 46° 19' 01" = 9.839270 + log cos 59° 39' 41" = 9 .703386 _ Rejecting 10 = 9.542656 = log cos a. /. a = 69'* 34' 56", the species being determined by the signs of the formula. _ sin p (c) To find C, sin p = sma sin C, or sin C = -t— . log sin p = 9.859241 - log sin 69° 34' 56" = 9.971820 Adding 10 = 9.887421 = log sin C. .'. C = 50° 30' 08", the species being determined by the side opposite. {d) To find angle DBC, sin p = tan DC cot DBC, or cot DBC = /^"^ . log sin ^ = 9.859241 - log tan 59° 39' 41" = 10.232653 Adding 10 = 9.626588 = log cot DBC. .'. DBC = 67° 03' 36", the species being determined by the signs of the formula. Finally, B = ABD + DBC =: 68° 01' 53" + 67° 03' 36" = 135° 05' 29" ScH. — We might have omitted the computation of angle ABD in the first part, and DBC in the second, and have found instead the entire angle B from sin a : sin Z> : : sin A : sin B. To compute this requires the looking out of but two logarithms, since sin a is given in the second pai*t (c), and sin A in the first part (a). 2. Given a = 97° 35', J = 27° 08' 22", and A = 40° 51' 18", to project and compute the triangle. Between what limits must the 92 SPHEBICAL TRIGONOMETRY. value of a be assigned in order that there may be two solutions ? Be- yond what limiting values of « is a solution impossible ? ^^.— --~^,^^ Projection. See Fig. 65. /y\ 7n\ ' Trigonometrical Solution.— To find p, sin p Z'' V^ / y\ = sin b sin A, // ^^^^^^-^'')C^ )\ log sin 27° 08' 22" = 9.659115 i ^^^->^ n + ^^S ^^^ ^^° ^^' ^^" = 9-815675 \\ yy \ ^^^^%.// Rejecting 10 = 9.474790 = log sin p. k^/ \y^y^ '''P — •^'^° ^^' ^^ "• [Reason for the species.] ^^::zzl^^^^^f^^ ^^ ^^^ observe that there is one and only one — ""^^ solution, since the arc a (97° 35') cannot lie between Fig. 65. CD (17° 21' 40") and b (27° 08' 22"), but can lie bo tween CD and CA' (180° -h = 152° 51' 38"). cos A To find AD, cos A = cot & tan AD or tan AD = — -r. ' cot 6 log cos 40° 51' 18" = 9.878733 - log cot 27° 08' 22" = 10.290226 Adding 10 = 9.588507 = log tan AD. .-. AD = 21° 11' 30 ".6. cos b To find angle AC D, cos b = cotA cot ACD, or cot ACD =r — — -. * ' cot A log cos 27° 08' 22" = 9.949340 - log cot 40° 51' 18" = 10.0630575 Adding 10 = 9.8862825 = log cot ACD. /. ACD = 52° 25' 01". [Reason for the species. ] To find B, sin p = sina sin B, or sin B = - — -, ' sin a log sin p (as above) = 9.474790 - log sm 97° 35' = 9.996185 Adding 10 = 9.478605 = log sin B. .-. B = 17° 31' 09". [Reason tor the species.] To find DB, cos « = cos -» cos DB, or cos DB = . cosp log cos 97° 35' = 9.120469 - log cos 17° 21' 40" = 9.979750 Adding 10 = 9.140719 = log cos DB. .-. DB = 97° 56' 51".3. [Reason for the species.] AB = AD + DB = 21° 11' 30".6 + 97° 56' 51".3 = 119° 08' 21".9. To find DCB, s'm p = tan DB cot DCB, or cot DCB = 7^^^— • ' tan DB log sin p (as above) = 9.474790 - log tan 97° 56' 51 ".3 = 10.855090 Adding 10 = 8.619700 = log cot DCB. .-. DCB = 92° 23' 7".7. [Reason for the species.] ACB = C = ACD + DCB = 53° 35' 01" + 92° 23' r'.7 = 144° 48' 8".7. OBLIQUE ANGLED TEIANGLES SOLVED BY NAPIER'S RULE. 03 Finally, we observe that, if any value were assigned to a between h (27° 08' 22") and CD (17° 21' 40") there would be two solutions ; since for such values the side a could lie on both sides of CD. But, for any value of a less than CD (17° 21' 40"), there would be no solution ; since CD is the shortest distance from C to the arc ABA'. Also, for any value of a greater than the arc CA' (152° 51' 38"), there would be no solution, as such an arc would fall between CA' and CD' (if not > CD'), and consequently would make c > 180°. ScH. — Such examples as this and the preceding can be more expeditiously solved by using p in each equation in solving the triangles ACD and DCB. By this means and using {135) to determine the side c, the solution can be effected with only 12 logarithms. Thus in Ex. 2. l?t. To find p, 8inj9 = sin 6 ein A. require? 3 logarithms 2d. TofindACDi cos ACD = cot 6 tan;?, requires 3 " 3d. To find DCB» cos DCB = cot a tan ^, requires 2 " (log tan p being known). 4th. To find Bi sin^ = sin a sin Bi requires 2 (log sin ;> being known). 5th. To find c, sin A : sin C : : sin a : sin c, requires 2_ " Gog's of sin A and sin a Total 12 logarithms being known.) 3. Given a = 76° 35' 36'', i = 50° 10' 30", and c = 40° 00' 10", to project and solve the triangle. Projection. — See Fig. 66. Trigonometrical Solution. — 1st. To find the segments CD and DB, we have, tan i{s + 8% or tan ^a : tan ^{b + c) : : tan i{b — c) : tan i(s — s'). Computing by logarithms. a. c. log tan i« = log tan 38° 17' 48" = 0.102561 + log tan \{b + c) = log tan 45° 05' 20" = 10.001347 + log tan i{b -c) = log tan 5° 05' 10" = 8.949406 Rejectmg 10 = 9.053314 = log tan ^(s - s'). .'. i{8 - s') = 0° 27' 02". In order to determine whether 5 or s' is the greater,* we observe the signs of the proportion, and finding tan ^s — s') positive, know that s > s'. Hence, s = K« + «') + i(« - «') = 38° 17' 48" + 6° 27' 02" = 44° 44' 50", and s' = K« + «') - K^ - s') = 38* 17' 48" - 6° 27' 02" = 31° 50' 46". The angles sought are now readily found by computing the two right angled triangles ADC and ADB. * Though the projection generally determines such facts as this, the species of parts, and the number of solutions in ambiguous cases, the student should not rely upon it, but determine each such fact upon purely trigonometrical considerations, merely using the projection to give clearness to the couceptiou, and as a rough check against errors. 94 SPHEllICAL TllIGONOMEl^RY. Or, liaving computed C from the triangle ACD, we may find A and B more expeditiously by using the proportions, sin c : sin 6 : : sin C : sin B, and sin c : sin « : : sin C : sin A. The angles are C = 34° 15' 03", A = 121° 36' 12", and B = 42° 15' 13". 4. Given A = 128° 45', C = 30° 35', and a = 68° 50', to solve the triangle. What values of A give two solutions ? What none ? c = 37° 28' 20", b = 40° 09' 04", and B = 32° 37' 58". 5. Given A = 129° 05' 28", B = 142° 12' 42", and C = 105° 08' 10", to solve the triangle. a = 135° 49' 20", b = 146° 37' 15", and c = 60° 04' 54". 6. Given a = 68° 46' 02", b = 37° 10', and C = 43° 37' 38", to project and solve the triangle. A = 116° 22' 22", B = 35° 29' 54", and e = 45° 52' 34". 7. Given a = 40° 16', h = 47° 44', and A = 52° 34', to project and solve the triangle. What values of a give but one solution? What none ? There are two triangles.— 1\\ the 1st, c = 53° 12' 22", B = 65° 23' 16", and C z= 79° 40' 26". In the 2d, c — 14° 20' 32", B = 114° 36' 44", and C = 17° 43' 06". 8. Given a = 62° 38', b = 10° 13' 19", and C = 150° 24' 12", to project and solve the triangle. A = 27° 31' 44", B =: 5° 17' 58", and c = 71° 37' 06". 9. Given a = 56° 40', b = 83° 13', and c = 114° 30', to project and solve the triangle. A = 48° 31' 18", B = 62° 55' 44", and C = 125° 18" 56". 10. Given A = 50° 12', B = 58° 08', and a = 62° 42', to solve the triangle. What values of A give but one solution ? What none ? There are two solutions.— 1st, b = 79° 12' 10", c = 119° 03' 26", and C = 130° 54' 28". 2d, b = 100° 47' 50", c = 152° 14' 18", and C = 156° 15' 06". 11. Given A = 36° 25', B = 42° 17' 10", and C = 95° 10' 05", to project and solve the triangle. OBLIQUE ANGLED TRIANGLES SOLVED BY NAPIEll's RULES. 95 12. Given a = 124° 53', b = 31° 19', and c = 171° 48' 22", to solve the triangle. 13. Given a = 150° 17' 23", b = 43° 12', and c= 82° 50' 12," to solve the triangle. 14. Given a = 115° 20' 10", b = 57° 30' 06", and a = 126° 37' 30^ to solve the triangle. 15. Given A = 109° 55' 42", B = 116° 38' 33", and C = 120° 43' 37", to solve the triangle. 16. Given A = 50°, b — 60°, and a — 40°, to solve the triangle. 17. Given a = 50° 45' 20", b = 69° 12' 40", and A = 44° 22' 10", \o project and solve the triangle. There are two solutions,— 1st, B = 57° 34' 51", c = 115° 57' 51", and c = 95° 18' 16". 2d, B = 122° 25' 09", C = 25° 44' 32", and c = 28° 45' 05". 18. Given b = 99° 40' 48", c = 100° 49' 30", and A = 65° 33' 10", to project and solve the triangle. a = 64° 23' 15'^, B = 95° 38' 04", and c = 97° 26' 29". 19. Given A = 48° 30', B = 125° 20', and c = 62° 54', to solve the triangle. a = 56° 39' 30", b = 114° 29' 58", and c = 83° 12' 06". 20. Given C = 54° 15' 03", B = 40° 18' 13", and« = 70° 30' 30", to solve the triangle. 21. Given A = 47° 54' 21", c = 61° 04' 56", and a = 40° 31' 20", to project and solve the triangle. 22. Given a = 50° 10' 10", b = 69° 34' 35", and a = 120° 30' 30", to project and solve the triangle. 96 SPHElllCAL TlUGO^sUMETRY. SECTION III. GENERAL FORMUIlE. [Note. — This section is designed for such as make mathematics a specialty. The preceding sections are thought sufficient for the general student] 14z3» Prop. — In a Spherical Tri- angle the cosine of any side is equal to the jjroduct of the cosines of the other two sides, plus the product of the sines of those sides into the cosine of their included a7igle ; that is, (1) cos a = cos h cos c + sin Z> sin c cos A ; (2) cos i = cosa cos c + sina sin c cos B ; (3) cos c = cos aco8b_+ sin a sin b cos C. Dem.— From Fig. 67, we have, cos a = cos {c — x) cos p \ A, cos (c — x) cos b [s. cos 6" cos X L cos X cos b cos c cos X + cos J sin c sin x smce cos p = I cos X J [expanding cos {c — a?)] sm X ^ ~\ smce = tan x I COSiC = cos b cos c + COS b sin c tan x s L cos = cos b cos c + sinb sin c cos A since cos A = cot b tan x In a similar manner (3) and (3) may be produced. cos b tan x sinb 144, Cor. 1. — From set A, h^ passing to the polar triangle, we (1) cos A = — COS B COS c + sin b sin c cos a ; ^ (2) cos B = — cos A cos c + sin A yin c cos b ; V B, (3) cos C = — cos A cos B + sin A sin b cos c. ) Dem.— Letting a', 6', c', A', B', and C represent the parts of the polar triangle, I GENERAL FORMULA. 97 we have a = 180° - A',b = 180° — B', c = 180" - C, A = 180" - a\ B = 180° - b\ and C = 180° - c'. Whence, substituting in (1) A, we have, cos (180° - AO = cos (180° - B') cos (180° - C) + sin (180° - B') sin (180° — C) cos (180° - a'), or, cos A' = — cos B' cos C + sin B' sin C cos a', since cos (180° — A') = — cos A', etc. ; and sin (180° — B) = sin S', etc. Finally, dropping the accents, since the results are general, and treating (2) and (8) of set A in the same way, we have set B. 145. Cor. 2. — From A and B we readily find the angles in terms of the sides, and the sides i7i terms of the angles. Thus, from A, ,. . cos « — cos h cos c (1) cosA = r— ^— . — ; ^ ' sin sm c .^v cos 5 — coso^ cose (2) cos B = : . : ^ ^ ma. a sm c ^ .„. cos c — cos 05 cos 5 (3) cos C = ; : — % . ^ ^ sm a sm & J A'. From B, ,^. cos A + cosB cose ^ (1) COS a =. . : ; ^ ' sm B sm C ,^. , cos B 4- COS A cos C . (2) cos h = :- ^ . — J )■ B'. (3) cos c = smA smc cos C + cos A cos B sin A sin B 146. JProp. — FormulcB A', and B', adapted to logarithmic com- putation, become, (1) 8miA = A/- (2)8in|B = |/- sm {y — b) sin (^s — c) . sin J sin c sin (y — a) sin (^s — c) . sin a sin c ' (3) am^c V' sin { ^s — a) sin {^s — l) sin a mxb V A". SPHERICAL TRIGONOMETUY. And (1) sin ia= a/ (2) sin i^* = |/ ^ COS ^S cos (^S - A). sin B sin c ? cos JS cos (is — B). sin A sin c > (3) sin ^c -V- cos ^S cos (|-S — C ) sin A sin B B". Ubm.— Subtracting each member of (1) A' from 1, we have, cos a — cos b cos c cos h cos c + sin & sin c — cos a 1 — cos A = 1 sin b sin c sin b sin c n, . o , * cos (b — c) — cos a . ^ . « . » , * i^n. ^^ .'. 2sm2 M = —T-r^. , smce 1 — cos A = 2sm* -iA (62, 5), sm&smc a \ ^ /» and cos 5 cos c + sin & sin c = cos (5 — c) {55 f D). Now letting y = b — c, and x = a, we see from {59, D') that cos (6 — e) — cos a = 2 sin ^(a + b — c) sin ^(a + c — 5). Hence, 2 sin* ^A = _ 2 sin i(a 4- & — c) sin i{a + c — J) sin J sin c ... , /sin ^{a + b — c) sm i(a + c — 6) or, sm ^ A = 4/ ^^^ . , . - ' '' r sm 6 sm c Finally, putting « = « + S + c, whence Ka + & — c) = i« — #, and we have, ^{a + c^b) = {8 — 6, sm lA = 4 / sin (^^ - ^) si" (i^ - g) r sin 6 sin c sin 6 sin c In like manner, (2) and (3) of set A' reduce to (2) and (3) of set A". Again, subtracting each member of (1) set B', from 1, we have, cos A + cos B cos C sin B sin C — cos B cos C — cos A 98 1 — cos a == 1 — sin B sin C sin B sill C <. . 2 , — cos (B + C) — cos A _ cos (B + C) + cos A Sin B sm C sm B sm C 2 cos i(A + B + C) cos i(B + C + A) ~ sin B sin C GENERAL FORMULA. 99 Now putting S = A + B + C, whence ^B + C — A) = ^S — A, we have, . . J cosiS cos(iS - A) sin i« = 4/ - . p r -^ ' r sin B sin C In the same manner, (2) and (3) of B" are deduced from (3) and (3) of b . i47« Cor. 1. — Passing to the polar triangle. A" and B" become tA\ 1 i Aos(-|-S — b) cos(-Js (1) cos ia = 4/ — ^^ '- ^-^- C): sm B sill C (2) cosJJ V cos ( ^S — a) cos(-|S — C ). sin A sin C (3) cos Jo = . Aos(iS-A)cos(is-B) . ^ ' ^ 1/ sm A sm B B' And _ ^ /sin |5 sin( |-5 — «)^ sin b sin c ' (1) cosiA = j/' ,-. , . /sin l.s sin (is — d) (2) cos ^B=A/ ^ ^^ ^ sm « sm c (3) cos ^C /sini s sin (Js — c). sin a sin b \A.' 148» ScH. — Formulae A'" and B" can be obtained directly from A and B, in a manner altogether similar to that in which A" and B" were deduced, by addinfj each member of the equations in sets A and B \a 1, instead of subtract- ing, and observing that 1 + cos re — 2cos^ \x. 149, Cor. 2. — Dividi^ig the forrmdce of set K." by the correspond- ing ones of set A.'" ; and, in a similar manner, those of W" hj tUoso of B", and putting ^ ^^^.(is - a) .in(is-b),m(is - c) ^ ^^ and / cob (is - A) cos(ii--Trcos (}S -"^ ^ ^^ ^ — COS is . (1) tan^A (2) tanjB -V sin {^s — b) sin {^s — c) sin y sin L (is - a) sin -«)' sin {\s — a) sin (i* — r) _ sin ^s sin (y - ^) sin -b)' sin {^s —a) sin {\s — ^)_ n) tan 4C - i Am(i-^-^)sin(is- (,3) tan iC - y g.^ ^^ g.j^ ^^^ ___ ^j sin (is — 6')' A" 100 SPHERICAL TRIGONOMETRY. (1) cotia = (2) cotib = (3) cot ic = (is - B) COS (jS - C) cos (^S — A) cos (^S — C) _ cos ^S cos (^S — a) — COS-J^S COS (Js — B) (is - A) COS (is - B) ^ COS is COS (is — C) COS (is — C). K COS (is - -A)' K COS (is - -B)' K B- ScH.— In these farmulcB k is the tangent of the arc with which the inscribed circle is described, and K is the cotangent of the arc with which the circum- FiG. 68. Fig. 69. scribed circle is described. Thus, using the common notation, we have in Mg. 68, AD =: AD' = ^8 — a, and angle PAD = ^A; whence tan PD sin AD = cot PAD x tan PD = tan PAD' tan ^A = — tanPD ,, [(1) A-]. k = tan PD. sin {^8 — a) sin {^s — ay From Mg. 69, we have, AD = ^c, and angle PAD = |S — C. cos (^S - C), Hence, cos (iS — C) z= cot AP x tan^c, or tan Ic = cot AP K or cot ic = cos(iS-C) cos(iS— C)' cotAP [(3)Bi^]. .-. K = cot A P. GAUSS'S EQUATIONS. ISO, J*rob, — To deduce Omissus Equations, which are sin i(A + b) _ cosi(rt — h) ^ (1) (2) COS ic COS-A^C sin i(A — B) _ sin i(« — b) ^ cosic sin ic gauss's equations. IDL . (3) (4) cosi(A + B) _ co8i(a 4- b) . sin ^ C ~ cos ic ' cos ^(A— B) _ sin i(a + b) sin JC sin ^G Solution.— From A, page 25, we have, ^ sin (iA + iB), or sin i{A + B) = sin ^A cos iB + cos iA sin iB. Substituting in tlie second member the vahies of sin ^A, cos ^B, cos ^A, and sin iS , from A" and A'", there results, • i(A B^ — sJQ i¥ — b) . / sin ¥ sin (jg - o) sin (jg — a) . / sin jg sin(ig — C) sm i(. 4- ) — . ^ y g.j^ ^ gjj^ ^ sin c r sin a sin 6 (^s — b) + sin (^g — a) . / sin jg sin {js — c ) sin c y sin a sin b sm c sin (is — b) + sin (|s — a) [See (3) A'"]. cos iC. But sin (is — b) + sin (^s — a) = sin (^a + |6 + ^c — 5) + sin {^a + ib + ^c — a) = sin [ic + i((2 — 6)] + sin [Ic — i(a — b)] = 2 sin ic cos i(a — 5). {59, A'). Also, sin c = 2sin ^c cos ie. Substituting these values, the preceding becomes, 2 sin ^c cos ^{a — b) sin i(A + B) == „ . , '^ 2 sm ^c cos ^ sin i(A + B) _ cos jja — b) cos^C " cos^c ' cos^C (1) In like manner starting with sin (|A — |B ), or sin |(A — B) = sin ^A cos ^B — cos ^A sin ^B, sin -KA — B) Bin ^{a — b) there results, cos iC sin ^c (3) Starting with cos (iA + ^B), or cos i(A + B) =r cos |A cos ^B — sin ^A sin i B, there results, cos i(A + B) _ cos i(« + b) (3) sin |C cos ^c Starting with cos (iA — ^B), or cos^A — B) = cosiAcos^B + sin^AsLn^Bi cos i(A — B) sin \{a + b) there results, siuiC sin^c (4) 10^ , ^ - SPHEIIICAL TltlGONOMETliY. NAPIER'S ANALOGIES. 151, !Pr6h, — To deduce Napier's Analogies, cbhich an (I) tan_J(A_+B) _ cos i(a — h) ^ cot |-c "~ cos \(a + h) ' ^ ^ cot |C "~ sin ^{a + "^ ' (3) (4) tan i (^ -f ^) _ cos ^( A — b). tan ^c ~ cos ^(a + B)' tan \(a—'b) _ sin ^(A — b) tan Jc " sin J(a + b)* Solution.— To deduce (1), divide the 1st of Gauss's Equations by the 3d. To deduce (2), divide the 3d of Gauss's by the 4th. To deduce (3\ divide the 4th of Gauss's by the 3d. To deduce (4), divide the 2d of Gauss's by the 1st. 152, ScH. — In using these formulae the species must be carefully attended to. Thus in (1), cot iC and cos \{a— h) are necessarily + ; hence tan \{k + B) and cos-|(a + h) are of the same sign with each other. In (2), cot ^C and sin \{a + b) are both + ; lience, tan ^(A — B) and sin i(a — b) are of the same sign with each other. And similar inspections may be made upon (3) and (4). EXERCISES. 153 • The proposition that " The sines of the angles are to each other as the sines of their opposite sides" {135), Napier's Analogies {151), and formulae A^"", B'"" {149) are sufficient, in themselves, to effect the solution of all cases of oblique spherical triangles; and for practical purposes they generally require less labor than Napier's Rules. We give a few solutions and refer the student to the pre- ceding Exercises for further practice. 1. Given a = 100°; c = 5° and 5 = 10°, to solve the triangle. {Prod. 1, Case 1st, 137.) 1st. To find A and B we have, cos ^{a + b) : cos i(a — b) : : cot ^C : tan i(A + B) ; and sm K» + b) : sin i{a - 6) : : cot ^C : tan i(A - B) [150 (1) (2)] Napier's analogies. 103 Computing by logarithms, we have, ar. CO. log cos [i{a + b) = 55°] = 0.241409 + log cos [i{a -b) = 45*1 = 9.849485 + log cot [i C = 3° 30'] = 11.35990 7 Rejecting 10 = 11.450801 = log tan KA + B). .-. -J(A + B) = 87° 58' 18". ar. CO. log sin [^{a + b) = 55°] = 0.086635 — + log sin [Ka - b) = 45°] = 9.849485 + log cot [iC = 2° 30'] =r 11 .359907 Rejecting 10 = 11.296027 = log tan KA - B). .-. KA - B) = 87° 06' 16" The signs of all the terms being + , ^A, + B) and ^(A — B) are botli less than 90°. i(A + B) + i(A - B) = A = 87° 58' 18" + 87° 06' 16" = 175° 04' 34" i(A + B) - i(A - B) = B = 87° 58' 18" - 87° 06' 16" = 0° 53' 03". 2d. To find c. This may be found from the proportion, sin A : sin C : : sin « : sin c, or from the 3d or 4th of Napier's Analogies. We use the last, though the first is equally expeditious. sin KA — B) : sin ^(A + B) : : tan i(a — &) : tan ic ar. CO. log sin [KA - B) = 87° 06' 16"] = 0.000555 + log sin [KA + B) = 87° 58" 18"] = 9.999738 + log tan [K« -b) = 45°] = 10.00000 Rejecting 10 = 10.000383 = log tan ic .-. c = 90° 03' 14", 2. Given a = 135° 05' 28".6, c = 50° 30' 08".6, and i = 69° 34' 5 6 ".2, to solve the triangle. 1st. To find a and c. The 3d and 4th of Napier's Analogies give, cos KA + C) : cos KA — C) : : tan ^b : tan K» + <^) ; and sin KA + C) : sin KA — C) : : tan ^5 : tan K« — ^^ Computing by logarithms, we have ar. CO. log cos [KA + C) = 93° 47' 48".6] = 1.3116386 * + log cos [KA - C) = 43° 17' 40"] = 9.8690535 + log tan [kb = 34° 47' 38".l] = 98418537 Rejecting 10 = 11.0335348 = log tan K« + c). .-. Ka + c) = 95° 35' 35". |(a + c) > 90°, since cos KA + C) is — , cos KA — C) is +, and tan i6 is +, making tan K« + c) —. * These logarithms are taken from 7-place tables, in order to obtain the tenths of secondB accurately. 104 SPHERICAL TRIGONOMETRY. ar. CO. log sin [KA + C) = 92° 47' 48".6] = 0.0005176 + log sin [KA - C) = 42° 17' 40 " ] = 9.8279768 + log tan [\b = 34° 47' 28". 1] = 9.8418527 Rejecting 10 = 9.6703471 = log tan \{a - c). .'. i{a -c) = 25° 05' 05". iia — c) < 90°, since the signs of the terms are all + . K« + c) + ^a- c) = a = 120° 30' 30", and }{a + c) — ^a- c) = c = 70° 20' 20 '. 2cl. To find B. Either of the 1st two of Napier's Analogies will give B. Thus (1) becomes, cos i{a — c) : coa i{a + c) : : tan i(A + C) : cot -JB ; and (2) sin i{a — c) : sin ^a + c) : : tan KA — C) : cot -JB- But as ^{a + c) is so near 90°, it will be better to use the second of these than the first. Or we may with equal accuracy use, sin c : sin & : : sin C : sin B. ar. CO. log sin {c = 70° 20' 20" ) = 0.0260878 + log sin (6 = 69° 34' 56".2) = 9.9718202 + log sin (C = 50° 30' 08".6) = 9.887421 Rejecting 10 = 9.8853290= log sm B. .-. B = 50° 10' 10". 3. Given a = 50° 45' 20", b to solve the triangle. 69° 12' 40", and A = 44° 22' 10", 1st. To find B. a i ainb * * 8in A : sin B. ar. CO. log sin {a = 50° 45' 20'') = 0.1110044 + log sin {b = 69° 12' 40") = 9.9707626 + log sm(A=: 44° 22' 10") = 9.8446525 Rejecting 10 z= 9.9264195 = log sin B. .-. B = 57° 84' 51".4, and 122° 25' 08". 6. There are two solutions, since a is intermediate in value between 2> and both b and 180° — b* * The determination of the species of B, or what is the same thing, the number of solutions, can usually be effected by a simple inspection without any computa^ tion whatever. Thus, sin jj = sin 6 sin Ai the loga- rithms of which are given above, as ie log sin a. Now, as both a and p are < 90°, and log sinp < log sin a, p <:a. But a < 6, and aleo less than 180° —b. All this can be seen at a glance. Napier's analogies. 105 To find C and c of the larger triangle in which B = 57° 34' 51".4 Napier's 1st gives ar. CO. log cos [\{b - a) = 9" 13' 40"] - 0.0056570 + log cos [K& + n) = 59° 59'] = 9.6991887 + log tan [KB + A) = 50° 58' 30".7] = 10.0912464 Rejecting 10 = 9.7960931 = log cot iC .-. C = 115° 57' 50^'.T Napier's 3d gives ar. CO. log cos [KB - A) = 6° 36' 20".7] = 0.0028938 + log cos [KB + A) = 50° 58' 30".7] = 9.7991039 + log tan [K& + «) = 59° 59'J = 10.38826 89 Rejecting 10 = 10.0402656 = log tan ^. .'. c = 95° 18' 16".4, 3d. To find C and c of the smaller triangle in which B = 122° 35' 08".6. Using the same formulce as before. ar. CO. log cos [i{b - a) - 9° 13' 40"] = 0.0056570 + log cos [K& + a) = 59° 59'] = 9.6991887 + log tan [KB + A) = 83° 23' 39".3] = 10.93627 03 Rejecting 10 = 10.6411160 = log cot iC- .-. C = 25° 44' 31".6. ar. CO. log cos [KB - A) = 39° 01' 29".3] r= 0.1096506 + log cos [KB + A) = 83° 23' 39".3] = 9.0608369 + log tan [i{b + a) = 59° 59'] = 10.2382 689 Rejecting 10 = 9.4087564 = log tan ic .-. c = 28° 45' 05".2. 154. ScH. — When Napier's Analogies are used for solving Pro5.27Mf(J?59), the most expeditious and elegant method of resolving the ambiguity, is by means of the analogies themselves. Thus, in the above example, after having found that B = 57° 34' 51".4, or 122° 25' 08".6, or both, a simple inspection of the anal- ogy next used will determine the number of solutions. Napier's 1st may be written k ^ .rs COS K^ + Cl) ^ ,,„ .X Now iC < 90°, hence cot iC is +. If, therefore, neither of the values of B renders cot iC —, there are two solutions. If one value renders cot|C +, and the other — , there is one solution and it corresponds to the value of B which makes cot iC +. If both values of B render cot ^C — , there is no solu- tion. In the last example, we see that cos [K& + a) = 59° 59], and cos [K^ — a) = 9° 13' 40"] are both +. Also tan [KB + A) = 50° 58' 30". 7, or 83° 23' 39".3, or both] is + for both values of B. Therefore there are two solutions. 106 SPHEBICAL TRIGONOMETRY. 4. Given A = 95° 16', B = 80° 42' 10", and a = bT 38', to solve the triangle. 1st. To find &, sin A : sin B : : sin a : sin 6. ar. CO. log sin (A = 95° 16') = 0.001837 + log sin (B = 80" 42' 10") = 9.994257 + log sin {a - 57° 38') = 9.926671 Rejecting 10 = 97922765 — log sin h. •. & = 56° 49' 57", or 123° 10' 03", or both. 2d. To find <•, tan \c = ^^ f^ ^ ^ tan i(a + b). Now for b = 56° 49' 57", COS 2\ — '^J tan |c is + ; but for b = 123° 10' 03" tan ic is — ; hence there is but one solu- tion, and that corresponds to the smaller value of b. ar. CO. log cos [i(A - B) = 7° 16' 55"] = 0.003517 + log cos [KA + B) = 87° 59' 05"] = 8.546124 + log tan lHa + b) = 57° 13' 58"] = 10.191352 Rejecting 10 = 8.740993 = log tan ic .-. c = 6°18'19". 3d. To find C, we may use (1) or (2) of Napier's Analogies, or sin a : sin c : : sin A : sin C, the last of whicn is the most expeditious. ar. CO. log sin {a = 57° 38') = 0.073329 + log sin {c = 6° 18' 19") = 9.040705 + log sin (A = 95° 16') = 9.998163 Rejecting 10 = 9.112197 = log sin C .-. C = 7° 26' 22' This value is taken for C instead of its supplement, since C is opposite the smallest side c. 5. Given a = 70° 14' 20", b = 49° 24' 10", and c = 38° 46' 10"; to solve tlie triangle. COMPUTATION-. a = 70° 14' 20" b = 49° 24' 10" c = 38° 46' 10" 8 = 158° 24' 40" is = 79° 12' 20" ar. CO. log sin = 0.007753 ^s- a = 8° 58' 00" " " = 9.192734 ^s- b = 29° 48' 10" " " = 9.696370 is- c = 40° 26' 10" " " = 9.8119T7 2)18.708834 .-. log k = 9.354417 NAPIER S ANALOGIES. 107 log tan iA = \ogk- log sin (is - a) + 10 = 10.161683. .'. A = 110° 51 16". log tan iB = log A; - log sin {^s -^) + 10 = 9.658047. .. B = 48* 56' 04". log tan iC = log A: - log sin (is - c) + 10 = 9.542440. .-. C = 38° 26' 48". 6. Given A = 109' to solve the triangle. 55' 42", B = 116° 38' 33", andc = 120°43' 37", COMPUTATION-. A = 109° 55' 42" B = 116° 38' 33" C = 120° 43' 37'^ S =: 347° 17' 52" iS = 173° 38' 56" ar. co. log iS- - A = 63° 43' 14" iS - B = 57° 00' 23" iS - C= 52° 55' 19" .-. log K 0.002683 9.646158 9.736035 9.780247 2)1 9.165123 9.583561 log cot ia = log K - log cos (iS - A) + 10 = 9.936403. .-. a = 98° 21' 38". log cot i& = log K - log cos (iS - B) + 10 = 9.846526. .-. b = 109° 50' 20". log cot ic = log K - log cos (iS - C) + 10 = 9.802314 .: c = 115° 13' 28". ScH. 1.— The student can use the exercises in the preceding section to famil- iarize the methods here given. In doing so, it will be well for him to seek the most expeditious solutions. He will find that Examples under Prob. 1 require 11 logarithms by Napier's Analogies and {135), and 12 logarithms by Napier's Rules and {135). Examples nnder Prob. 2, when there is but one solution, require 10 loga- ritlmis by Napier's Analogies and {135), and 12 logarithms by Napier's Rules and {135). When there are two solutions, 15 logarithms are required by Napier's Analogies and {135), and only 14 by Napier's Rules alone, or by these rules and {135). Examples under Prob. 3 require but 7 logarithms by the method given in this section and 13 by the previous method. ScH. 2. — In cases in which the angles or sides are near the limits 0°, 90°, or 180°, so that the functions used in the particular solution change very rapidly in proportion to the arc, it is often possible to select one among the several methods which will give more accurate results than the others. Tliere are also other methods which are better adapted to such cases than those here given. For these, as well as for much other interesting matter, and especially for the discussion of the General Spherical Triangle, American students liave an excellent resource in the treatise of Professor Chauvenet of Washington Univer- sity, St. Louis. 108 SPHERICAL TEIGONOMETEY. SECTION IV. AREA OF SPHERICAL TRIANGLES. 155, I^roh. — Having the angles of a spherical triangle given, to find the area. Solution. — [The solution is given in Part II. {613), and we simply re- produce the result in order to give completeness to this section.] The area is equal to the ratio of the spherical excess to 90°, or |;r, into the trirectangular tri- angle. That is, letting the sum of the angles be S°, the area K, and the radius of the sphere 1, whence the area of the trirectangular triangle is ^tt, we have In the latter expression S is the sum of the angles in terms of the radius, i. e., CO QO ^ = 5r:29578' °' ^PP^o^i^^tely, S = ^^ {9). EXERCISES. 1. What is the area of a spherical triangle whose angles are 100°, 58°, and 62°, on a sphere whose diameter is 6 feet ? 220° Solution. K = S — tt = z^r^^ — 3.14159 = .698, the area of a similar tri- angle on a sphere whose radius is 1. Hence, the area of the required triangle is .698 X 32 = 6.282. [The method given in Part II. (613) is more expedi- tious, but it is our purpose to illustrate the form here given.] 2. What is the area of a spherical triangle whose angles are 170°, 135°, and 115°, on a sphere whose radius is 10 feet? Ans. 418.875 square feet. 3. What is the area of a spherical triangle whose angles are 150°, 110°, and 60°, on a sphere whose radius is 3 feet ? 156. Prob. — Having the sides of a spherical triangle given, to find the area. Soj^yTjON. — The angles may be fomid by {148\ and then the area by {155). AREA OF SPHERICAL TRIANGLES. 100 But a more direct method is to find the spherical excess by means of LhuiU Iter's formula, which we will now produce. iK=i{^ + B + C-7t) Whence tan iK = tan i[A + B + C - ^] = tan [KA + B) - ^(;r - C)] _ sin i (A + B) - sin ^{tv — C) ~ cos UK + B) + cos i{7t - C) (7, page 31] _ sin KA + B) - cos jC ~ cos i(A + B) + sin iC ^ [cosi(a-5)-cosic]cosiC ^^^^^ jg^ ^^^ 3^^ [cos Ma + b) + cos ic] sin ^C _ cos i{a - b) - cos jc / sin js sin {js — c) ^^^^ ^^^. ~ cos ^{a + b) + cos ic r sin {is — a) sin {^s — b) _ sin i (a + e - b) sin :|(5 + c — «) / sin js sin (jg — c) ^, j^,. ~ cos i(a + 6 + c) cos i(a + 6 — c) r sin (is — «) sin (is - 6) ^ ' ' s in'-^ i(is - 6 ) sin'^ jljs - ft) sin js sin (js - c) (as s = a + b + c) cos^ is cos=^ ids — c) sin (is — a) sin (is — 6) sin^ U^s — b) sin^ i(is — 90', K woul4 be > 560' which }8 iropos- eible. (Part HI., 256), 110 SPHRIIICAL TllIGONOMETRY. Whence area = 15?!^^" . ^tt (4000)^ = ^-~ . ^it (4000)^ = 46,450,440, nearly. 2. Giyen a = 70° 14' 20", b = 49° 24' 10", and c = 38° 46' 10", to find the area of a spherical triangle on a sphere whose diameter is 8 feet. Ans. 6.1, nearly. Id7, JProb. — Having tivo sides and their included angle given in a spherical tria7iglef to find the area. Solution. — Compute the other two angles by Napier's Analogies, and find .1 ^ ., 1 rrru P 1 * 1 1^ cot |« cot ^b + COS C . the area from the angles. [The formula cot ^K = ^ — gives the spherical excess in terms of two sides and their included angle ; but it is of no practical value for finding the area, as it is not adapted to logarithmic compu- tation. For the manner of producing it and several other forms for K, see Todhunter's Spherical Trigonometry, {103)]. PRACTICAL APPLICATIONS. [Note. — The three following problems are given merely to indicate to the student some departments of investigation in which Spherical Trigonometry is of essential service. The two sciences to which this branch of Pure Mathe- matics is indispensable, are Geodesy, or the mathematical measurement of the earth, and Astronomy.] I^rob. 1, — To find the shortest distance on the earth^s surface be- JP__ tiuee7i two points whose latitudes and longi- tudes are known. Sue's. — The shortest distance on the surface be- tween two points is the arc of a great circle joining the points. Hence, the Problem is : Given two sides (the co-latitudes) and the included angle (the differ- ence in longitude), to find the third side. Fig. 71. Ex. 1. Berlin is situated in Lat. 52° 31' 13" N., Lon. 13° 23' 52" E., and Alexandria, Egypt, in Lat. 31° 13' X., Lon. 29° 55' E. What is the shortest distance in miles on the earth's surface between them, the earth being considered a sphere whose radius is 3962 miles ? Ans. 1691.96 miles. PRACTICAL APPLICATIONS. Ill Ex. 2. A ship starts from Valparaiso, Chili, Lat.33° 02' S., Lon. 71° 43' W., and sails on the arc of a great circle in a northwesterly direc- tion 3S-iO miles, when her longitude is found to be 120° \V. What is her latitude ? Ans. 51' 16" S. Ex. 3. A ship starts from Kio Janeiro, Brazil, Lat. 22° 54' S., Lon. 42° 45' W., and sails in a northeasterly direction on the arc of a great_ circle 5624.4 miles, when her latitude is found to be 50° N. What is her longitude ? Ans. 2° 01' 18" W. JProb, 2, — To find the time of day from the altitude of the sun. Sug's.— Let NESQ represent the projection of the concave of the heavens upon the plane of the meridian of observation. The equator of this concave sphere is simply the intei-sectiou of tlie plane of the earth's equator with this imaginar}' concave sphere, audits axis is the prolongation of the earth's axis, the poles being the points N and S where the axis pierces the imaginary concave. EQ is the projection of this celestial equator, NS the axis or the projection of a great circle perpendicular to the meridian of the observer (NESQ) and to the equator, HO the projec- tion of the horizon, and ZZ' the projection of the prime vertical (that is, a great circle of the heavens passing through the zenith of the observer and the east and west points in his horizon). Now let S' be the place of the sun at the time of observation. RS', the sun's declination, is known from the almanac ; LS', the sun's altitude, is measured with the sextant (or other instrument) ; and EZ is the latitude of the observer. Hence, in the spherical triangle ZNS' we know the three sides, viz., NS' = the co-declination of the sun, ZS' = the co-altitude of the sun, and ZN = the co- latitude of the observer. We may therefore compute the angle ZNS', which reduced to time gives the time before or after noon as the case may be. Ex. 1. On April 21st the master of a ship at sea in latitude 39° 06' 20" N., observed the altitude of the sun's centre at a certain time in the forenoon and found it to be 30° 10' 40", and looking in the almanac found the sun's declination at that time to be 12° 03' 10" N. What was the time of day ? COMPUTATION. 90" - 30° 10' 40" = 59° 49' 20" 90° - 12° 03' 10" = 77° 56' 50" 90° — 39° 06' 20" =: 50° 53' 40" 2 )188° 39' 50" W 19' 55" a. c. log sin 94° 19' 55" = 0.001242 a. c. log sin 34° 30' 35" = 0.24676") log sin 16° 23' 05" = 9.450381 log sin 43° 26' 15" = 9837312 2)19.5357011 log tan 30° 22' 03".3 = 9.767850 112 SPHEIUCAL llUGONOMETRY. Therefore i the hour angle NNS' = 30° 22' 08" .3, and the hour angle is CO' 44' 07". This reduced to time at 4 minutes to a degree, gives 4 h. 2 m. 56 s. be- fore noon, or 7 h. 57 m. 4 s. a. m. Ex. 2. In latitude 40° 21' N., when the declination of the sun is 3° 20' S., and its altitude 36° 12', what is the time of day ? Ans. 9 h. 42 m. 40 s. a. m. Ex. 3. In latitude 21° 02' S., when the sun's declination was 18° 32' K, and the altitude in the afternoon 40° 08', what was the time of day? Ans. 2 h. 3 m. 57" p. m. I^rob. 3» — To find the time of simrisi7ig and sunsetting at any given place on a given day. Sdg's.— The projection being the same as before, let M'RS'M represent the ap* parent diurnal path of the sun. Since S'M is described in 6 hours, the time taken to describe RS' is the time before 6 o'clock, at which the sun rises, i. e.^ passes the horizon HO. But the time requisite to describe RS', is the same part of 24 hours (360° angular measure) that the angle CNL (=: arc CL) is of 360°.' Hence, the arc CL, in time, is the time before 6 o'clock at which the sun rises. In a similar manner, C/, in time, is seen to be the time after 6 o'clock when the sun is south of the equator. The solution of the prob- lem, therefore, consists in finding CL. Now, in Fig. 73. the triangle RLC, right angled at L, LR = the sun's declination at the time, and angle RCL — ECH = the co-latitude of the place.* From these data CL is readily found. Ex.l. Required the time of sunrise at latitude 42° 33' N., when the sun's declination is 13° 28' N. COMPUTATION. cot 47° 27' = 9.962813 tan 13° 28' = 9.379239 sin 12° 41' 52" = 9.342052 (12" 41' 52") X 4 gives the time before 6 o'clock as 50' 47". .*. The sun rises at 5 h. 09 m. 13 s. * This may be eeen thus : Suppose a person to start from the equator at Hand travel north. When he is at E, the south point of his horizon (H) is at S ; ^^^^ for every degree lie goes north, the south pole (S) sinks a degree below his horizon. Hence, HCS = bis latitude, and ECH = co-latitude. PRACTICAL APPLICATIONS. 113 I Ex. 2. Required the time of sunrise at latitude 57° 02' 54" X., when the sun's declination is 23° 28' N. Sun rises at 3 h. 11 m. 49 s. Ex. 3. How long is the sun above the horizon in latitude 58° 12' N., when the sun's dechnation is 18" 41' 8., that is about January Soth ? Ans. 7 h. 35 m. 36 s. Ex. 4. What is the length of the longest day at Ann Arbor, Mich., Lat. 42° 16' 48".3, the sun's greatest declination being 23° 27' ? Ans. 15 h. 05 m. 50 s. [Note. — Tn such problems as the foregoing, several small coiTections have to be made in order to entire accuracy, such, for example, as that for refraction in taking the altitude, and for tlie time required for the sun's disk to pass the hori- zon. But they would be out of place here.] 8 THE END. INTRODUCTION TO THE TABLE OF LOGARITHMS, [Note. — If the student understands the nature and use of logarithms so as to be able to use the common tables, it will not be necessary that he should read this introduction. Otherwise a careful study of it will be needful before reading Section 4 of the Plane Trigonometry.] 1, A Logarithm is the exponent by which a fixed number ia to be affected in order to produce any required number. The fixed number is called the Base of the System. III. Let the Base be 3 : then the logarithm of 9 is 2 ; of 27, 3 ; of 81, 4 ; o( 19633, 9 ; for 3' = 9 ; S" = 27 ; 3* = 81 ; and 3* = 19683. Again, if 64 is the base, the logarithm of 8 is ^, or .5, since 64 , or 64* = 8 ; i. e., |, or .5 is the exponent by which 64, the base, is to be affected in order to produce the num* ber 8. So also, 64 being the base, J, or .333 + , is the logarithm of 4, since 64 , or 54.333 + _ 4. J- g^ ^^ Qj. 333 ^^ ig jijg exponent by which 64, the base, is to be § affected in order to produce the number 4. Once more, since 64 , or 64 ''^^ + = -i, 16, §, or .666 +,is the logarithm of 16, if the base is 64. Finally, 64 or 64—' = I, or .125 ; hence — i, or — .5, is the logarithm of J, or .125, when the base is 64. In like manner, with the same base, — ^, or — .333 + , is the loga- rithm of i, or .25. EXAMPLES. 1. If 2 is the base what is the logarithm of 4 ? of 8 ? of 32 ? of 128? of 1024? Solution. 7 is the logarithm of 128, if 2 is the base, since 7 is the exponent by which 2 is to be affected in order to produce the number 128. 2. If 5 is the base, what is the logarithm of 625? of 15625? of 126? of 25? 2 iNl^RODUCTION TO THE TABLE OF iX)GARtTHMS. 3. If 10 is the base, what is the logarithm of 100 ? of 1000 ? of 10,000? of 10,000,000 ? 4. If 2 is the base, what is the logarithm of J, or . 25 ? of |, or .1 25 ? of ^, or .03125 ? Ans. to the last, - 5. 5. If 8 is tlie base, of what number is -f, or .666 + the logarithm ? of what number is |, or 1.333 +, the logarithm ? of what number is 2 the logarithm ? of what number is 2J, or 2.333 + ? of what num- ber 3f, or 3.066 + ? Ans, to the last, 2048. ScH. Since any number with for its exponent is 1, the logarithm of 1 is in all systems. Thus 10' = 1, whence is the logarithm of 1, in a system in ''which the base is 10. 2. A System of LogaritJiTHS is a scheme by which all num- bers can be represented, either exactly or approximately, by expo- nents by which a fixed number (the base) can be affected. 5. There are Tiuo Systems of Logarithms in common use, called, respectively, the Briggean or Commo7i System, and the Napierian or Hyperholic System. The base of the former is 10, and of the latter 2.71828 -f • In speaking of logarithms of numbers, the com- mon logarithm is always signified, if no specification is made. 4, The logarithms of all numbers, except the exact powers of the base, indicate a power of a root, and are consequently fractional and usually only approximations. It is customary to write them in the form of decimal fractions. The integral part is called the ChdV- acteristiCf and the fractional part the Mantissa, The charac- teristic can always be told by a simple inspection of the number itself; hence only the mantissa is commonly given in the table. 5, JProp. — The characteristic of the common logarithm of any numher greater than unity, is one less than the number of integral figures in the given number. ' III. The logarithm of 4685 is more than 3, because 10' = 1000, and less than 4, l)ecause 10* = 10,000; hence it is 3 + a fraction. The same method may be pursued to determine the characteristic of the logarithm of any other number greater than unity, and the truth of the proposition be observed. Thus the logarithm of 25645.827 is 4, since the number lies between the 4th and 5tli powera ot the base, 10. 6* J^rop. — Tlie mantissa of a decimal fraction, or of a mixed number, is the same as the mantissa of the number considered as inifigral IKTUODtJCTlON TO THE tk^lM 01P LOGARITHMS. 3 Dem. Suppose it is known log 2845673 = 6.454185. This means thai jQ3.«t4m __ 2845672. Dividing by 10 successively, we have IO»-«"i86 ^ 284567.2, or log 284567.2* = 5.454185, 10*'""" _ 28456.72, or log 28456.72 = 4.454185, 10»-""" = 2845.672, or log 2845.672 = 3.454185, 10'-*"'" = 284.5672, or log 284.5672 = 2.454185, ;l()i.454i8» _ 28.45672, or log 28.45672 = 1.454185, 10o-«M»«» - 2.845672, or log 2.845672 = 0.454185. Now if we continue the operation of division, only writing 0.454185 - 1. i. 454185, meaning by this that the characteristic is negative and the mantissa positive, and the subtraction not performed, wc have ! iQ-*tAm _ 0.2845672, or log 0.2845672 = T.454185, 1QT454186 ^ 0.02845672, or log 0.02845672 = 2:454185, lQi:4»4i86 ^ 0.002845672, or log 0.002845672 = 3.454185, etc., etc. Q. B. D. 7, Cor. The characteristic of a number coiisistiiig entirely of a decimal fraction, is negative, and one more than the number of 0' 9 immediately following the decimal point, as appears from the last demonstration, or from the fact that 1"* = ^ = '.1 ; 10~' = ^hr = .01 ; 10-' = ttjVtt = -001 ; etc., etc. 8. One of the most important uses of logarithms is to facilitate the multiplication, division, involution, and extraction of roots of large numbers. These processes are performed upon the following principles : 9» JProp. 1. — The sum of the logarithms of two numbers is the logarithm of their product. Dem. Let a be the base of the system. Let m and n be any two numbers whose logarithms are x and y respectively. Then by definition a* = w, and d' — n. Multiplying these equations together we have a^-^^ =.mn. Whence a + y is the logarithm of mn. q. e. d. 10, JProp. 2. — The logarithm of the quotient of two numbers is the logarithn of the dividend 7ninu$ the logarithm of the divisor. Dem Let a be the base of the system, and m and n any two numbers whose kgarithms an;, respectively, x and y. Then by definition we have a"^ — m, and «» =: n. Dividing, we have o*-" = — . Whence a? — y is the logarithm of -. q. E. D. • This is tlie common abbreviation indicating the logarithm of a number, and should w read "logarithm of 284567.2," not "/oc 284567.2,"' which is grossly inelegant. i IKTllODtCTION TO THE TABLE OF LOGARITHMS. 11, JProp, 3, — The logarithm of a power of a member is th6 logarithm of the number multiplied by the index of the poiver. Dem. Let a be the base, and x the logarithm of m. Then of rz m ; and raising l>oth to any power, as the 2th, we have a^" = m". Wlience xz is the logarithm of the 2th power of m. q. e. d. 12. Prop, 4. — The logarithm of any root of a number is the logarithm of the number divided by the number expressing the degree of the root, Dem. Let a be the base, and x the logarithm of m. Then a^ = m. Extract- Ing the 2th root we have a^= y/m. Whence - is the logarithm of v^m. q. e. d. TABLE OF LOGARITHMS. 13. In order to apply the above principles practically, we need what is called a TaUe of Logarithms, That is, a table from which Wti can readily obtain the logarithm of any number, or the number corresponding to any logarithm. Such a table is given on pages 11 to 28. For methods of computing it, the student is referred to algebra. Its nature and manner of use will be learned from the two following problems : 14i. 'Proh,— To find the logarithm of a number from the table. Solution. The logarithm of any number between 1 and 100 is seen directly fioni the table on page 11. The column marked N contains the numbere, and the adjacent column to the right gives the logarithm of the corresponding number to C places of fractions. Thus, log 7 = 0.845098 ; log 33 = 1.518514. The mantissa of the logarithm of any number exjwessed with three figures vs. found in the column headed 0, on some one of the pages from 12 to 26 inclusive. The given number being found in the column marked N, the mantissa of it? logarithm stands opposite. Tlie characteristic can be determined by (5), (6*), {7) When but four figures are found opposite the number in the column, the two left hand figures of the mantissa are the same as in the next mantissa above, which lias six. Thus, log 443 = 2.646404. To find the logarithm of a number consisting of four figures. Let it be required to find the logarithm of 2936. Looking for 293 (the first three figures) in the column of numbers, and then passing to the right until reaching the column headed 6, the fourth figure, we find 7756, to which prefixing the figures 46, which belong to all the logarithms following them till some others ai-e indicated, we have for the mantissa of the logarithm of 2936, .467756. But, as 3 is the I INTRODUCTION TO THE TABLE OF LOGARITHMS. logarithm of 1000, and 4 of 10,000, log 2936 is 3 and this decimal, or log 2936 = 3.467756. 2b find the logarithm of a number consisUng of mxyre than four figures. Let it be required to find the logarithm of 2845672. Finding the decimal part of logarithm of the first four figures 2845, as before, we find it to be .454082. Now tlie logarithm of 2346 is 153 (million ths, really) more than that of 2845, as found in the right-hand colunm, marked D. Hence, assuming that if an increase of the- number by 1000 makes an increase in its logarithm of 153, an increase of 672 in the number will make an increase in the logarithm of 1^0%, f>r .672 of 153, or 103, omitting lower orders, and adding this to .454082, we have .454185 as the mantissa of log 2845672. The integral part is 6, since 2845672 lies between the 6th and 7th powers of 10. Hence, log 2845672 = 6.454185. Q. e. d. SCH. 1. If in seeking the logarithm of any number, any of the heavy dots noticed in the table are passed, their places are to be filled with O's, and the first two figures of the decimal of the logarithm taken from the column in the line below. Thus, log 3166 is 3.500511. This arrangement of the table is merely a cx)nvenient method of saving space. ScH. 2. The column marked D is called the column of Tabular Differences ; and any number in it is the difference between the mantissas found in columns 4 and 5, which is usually the same as between any two consecutive logarithms m the same horizontal line. The assumption made in using this difierence, viz., tliat the logarithms increase in the same ratio as the numbers, is only approxi- mately ti'ue, but still is accurate enough for ordinary use. EXERCISES. 1. Find the logarithm of 2200. ....... Logarithm, 3.342423. 2. Find the logarithm of 24.36 Logarithm, 1.386677. 3. Find the logarithm of 2.698 Logarithm, 0.431042. 4. Find the logarithm of 3585.9 Logarithm, 3.554598. 5. Find the logarithm of 42.6634 Logarithm, 1.630056. 6. Find the logarithm of 331.957 Logarithm, 2.521082. 7. Find the logarithm of 2519.38 Logarithm, 3.401294. 8. Find the logarithm of .538329 Logarithm, 1.731047. 9. Find the logarithm of .087346 Logarithm, 2.941243. 10. Find the logarithm of .007389 Logarithm, 3.86858a 15, ScH. 3. It will be observed that the tabular difference varies rapidly at the beginning of the table ; hence, for numbers between 10000 and 11000 it ia better to use the last two pages of the table. 16. Prob 2. — To find a number corresponding to a given logarithm. INTKODUCTION TO THE TABLE OF LOGARITHMS. Solution. Let it be required to find the number corresponding to thf loo:aritlim 5.515264 Looking in tlie table for tlie riext less mantissa, we find .515211, tlie number corresponding to which is 3275 (no account now being !aken as to whether it is integral, fractional, or mixed ; as in any case the figures vvill be the same). Now from the tabular difference, in column D, we find that an increase of 133 (millionths, really) upon this logarithm (.515211), would make an increase of 1 in the number, making it 3276. But the given logarithm is (mly 53 greater than this, hence it is assumed (th iUgh only approximately correct) that the increase of the number is -^^ of 1, or 53 -i- 133 = .3984 + . riiis added (the figures annexed) to 3275, gives 32753984 + . The characteristic, being 5, indicates that the number lies between the 5th and 6th powers jf 10, and hence has 6 integral places. .'. 5.515264 = log. 327539.84 + . q. e. d. EXERCISES. 1. Find the number whose logarithm is 1.240050. Number, 17.38. 2. Find the number whose logarithm is 2.431203. Number, 269.9. 3. Find the number whose logarithm is 3.503780. Number, 3189.91. 4. Find the number whose logarithm is 0.138934. Number, 1.377. 5. Find the number whose logarithm is 1.368730. _ Number, .233738. 6. Find the number whose logarithm is 2.448375. ^Number, .028078. 7. Find the number whose logarithm is 3.538630. Number, .003456. 8. Find the number whose logarithm is . 843970. _ Number, 6.98184. 9. Find the number whose logarithm is 1.867372. iVi^wJer, .736837. 10. Find the number whose logarithm is .003985. _ Number, 1.00921. 11. Find the number whose logarithm is 3.723460. Number, .005290. APPLICATIONS 1. Find, by means of logarithms, the product of 57.98 by 18. Solution. As the logarithm of the product equals the sum of the logarithms q[ the factors [9), we find the logarithms of 57-98, and 18 fi-om the table, and INTRODUCTION TO THE TABLE OF LOGARITHMS. 7 adding tliein together, find the number corresponding to the sum. The latter number is the proJuct required. Thus, log 57.98 = 1.763278 log 18 = 1.225273 3.018551 = log 1043.64. Multiply 23.14 by 5.062. Prod. 117.1347: Multiply 0.00563 by 17. Prod, 0.09571. Multiply 397.65 by 43.78. Prod. 17409.117. Multiply 0.3854 by 0.0576. Prod. 0.022199. Find the continued product of 3.902, 597.16, and 0.0314728. Prod. 73.3354. 7. Multiply 832403 by 30243. Prod. 25174363929.* 8. Multiply 9703407 by 90807. Prod. 881137279449. 9. Multiply 3.47 by 9.83. Prod. 34.1101. 10. Multiply 12.763 by 10.976. Prod. 140.086688.* [Note. The examples in division below will offer additional exercise, il accessary.] 1. Divide 24163 by 4567. Solution. Since the logajithm of the quotient equals the logarithm of the dividend minus the logarithm of the divisor, we have the following operation : log 24163 = 4.383151 log 4567 = 3.659631 0.723520 = log 5.89078, which number is the quotient. 2. Divide 56.4 by 0.00015. Operation, log 56.4=1751279 log 0.00015 = 4176091 Difference of log's = 6.575188, .*. The quotient is 376000. Bug. Observe that only the characteristic of the logarithm is negative, and* that in subtracting we are to regai'd the nature of the logarithmic numbei-s »* positive or negative. 3. Divide 461.02876 by 21.4. 4. Divide 25.49052 by 2.46. 5. Divide 17610.8248 by 37.6. 6. Divide .00144 by 1.2. 7. Divide .0000025 by .005. Arts. 21.5434. Ans. 10.362. Ans. 468.373. Ans. .0012. Ans, .0005. ♦ Qrdtoary 6 or 7 place logarithms will not ^ive these products correct. Wby ? 8 INTRODUCTION TO THE TABLE OF LOGAKITHMS. 8. Divide 43.2 by .24. Ans. 180 9. Divide 59.74514 by 1.36. Ans. 43.93025. 10. Divide .0001728 by 2.4. Ans, .000072. [Note. The examples in multiplication given above will aflPord additional jccrcise, if necessary.] 17* ^CYL.— Arithmetical Complement.— T\iq arithmetical complement of a number is simply tiie remainder after subtracting the number from some particular fixed number. Thus, the a. c. of 5 with reference to 9 is 4 ; of 3, 6; of 7, 2; etc. The a. c. of 7 with reference to 10 is 3 ; of 4, 6 ; of 2, 8 ; etc. When required to subtract one number from another, we may, if we choose, add its a. c. and then subti-act the number with reference to which the a. c. is taken. This process will give the true difference. Thus, if we are to subtract 6 from 9, we may add to 9 what 6 lacks of being 10 (10 — 6 = 4, the a. c. of 6 with reference to 10) and then subtract 10. 9 — 6 = 9 + 4 — 10. A few such questions as the following will render this simple process familiar. What number must I add to 576, in order that I may subtract 100 from the sum, and get the same remainder as if I had subtracted 58 in the first instance ? Again, if I wish to take 37 from 160, what must I add to the latter, in order that I may subtract 40 from the result, and get the difference sought ? This principle is sometimes used in computing by means of logarithms. It is especially convenient when there are several multipliers and several divisors involved in the same computation. An example or two will make the process familiar. The complements of logarithms are usually taken with reference to 10. If the logarithm exceeds 10, 20 may be used, etc. IS. Required the result of tbe following combinations : 346 X 27.8 H- 1156 X 3426 -^ 2.004 X 27 -i- 11.05. Opbration. log 346 = 2.539076 log 27.8 = 1.444045 a. c. log 1156 = 6.937042 log 3426 = 3.534787 a.c. Iog2.004 = 9.698102 log 27 = 1.431364 a. c. log 11.05 = 8.956638 34541054 Rejecting 30.000000 as three complements are used. 4.541054 is the logarithm of the re quired result .*. As 4.541054 = log 34757.92, the latter is the result sought. [Note. The preceding examples can be used to familiarize this principle if fliought desirable.] SuG. The a. c. of 2.468216 is 11.531784, since 2 is negative. An a. c. can b« written directly from the table with nearly the same ease as the logarithn" Itself, by writing from left to right, and taking each figure from 9, except the INTRODUCTION TO THE TABLE OF LOGAKITHMS. 9 nght-band one, which is to be taken from 10. Thus, if the characteristic is 8^ we write 6 • the next figure being 2, write 7 ; for 4, write 5, etc. 1. What is the cube of 32 ? Soi.DTiON. Since the logarithm of the cube of a number is three times tb«- logarithm of the number itself {11), we have log (32)' = 3 log 32 = 4.515450 = log 32767.97, which number is the cube of 32, as accurately as the process gives it (32)' by multiplication = 32768. 2. What is the cube root of 7896.34? SuG. Log (7896.34)^ = i log 7896.34 = 1.299142 = log 19.913. .*. 17896.34)^ = 19.913. {See 12.) 3. What is the 20th power of 1.06 ? Ans. 3.2071. 4. What is the 5th power of 2.846 ? 5. What is the 5th root of 31152784.1 ? Ans, 31.52+. 6. What is the cube root of 30 ? Ans. 3.107 + . 7. What is the cube root of .03 ? SuG. Log .03 ="2.477121. Now to divide this_by 3, we have to bear in mind that the characteristic alone is negative ; i. e., 2.477121 = —2 +_.477121, or — 1.522879. This divided by 3 gives - .507626, or - .507626 = 1.492374. But a more convenient method of effecting this division is to write for the — 2, -3 +_1, whence we have for 2.477121,-3 + 1.477121, which divided by 3 gives 1.492374, nearly. 8. Divide 3^261453 by 2, by 4, by 5. Last quotient, 1.4522906. 9. What is the 4th root of .00000081 ? Ans. .03. 10. What is the 7th root of 0.005846? Ans. .4797. 1. If 28.035 : 3.2781 : : 3114.27 : x, what logarithmic operations will fiud X? SuG. The logarithm of the product of the means is the sum of their loga- rithms ; and the logarithm of the quotient of this product divided by the first extreme, is the logarithm of said product minus the logarithm of the other extreme. .-. log x = log 3.2781 + log 311427 - log 28.035 = 0.515622 + 3.493356 - 1.447700 = 2.561278. Hence, x = 364.1478 + . 2. Given 72.34 : 2.519 : : 357.48 : x, to find x, by logarithms. X = 12.448. a Given 6853 '. 489 : : 38750 : x, to find a, by logarithms. X = 2765. 10 INTRODUCTION TO THE TABLE OF LOGAlilTHMS. Sua. The most elegant way to solve such propositions by loganthms is tc take the sum of the logarithms of the means and the a. c. of the given extreme Ecd reject 10. The result is log x. 4. Givsn 497 : 1891 : : 376 : x, to find x, usin^ the a. c. log. Operation. log 1891 = 3.276692 log 370 = 2.575188 a. c. log 497 = 7.303644 Sum, less 10 = 3.155524 = log 1430.62. /. x = 1430.62. INoTK. Solve the preceding in a similai- manner, by usmg a. c. log.] Let the student give the reasons for the following: 1. Given (|)2 -^ (■^)^ = x, we have 2 log 2 = 0.602060 i log 16 = 0.903090 a. c. 2 log 3 = 9.045758 a. c. f log 5 = 9.475773 Sum, less 20 = 0.026681. /. x = 1.0033+ . 2. Given a/| : a; : : (3J)2 : y^P", to find x. Log X = 2 log 3 + i log 2 + I log 6 + a. c. 2 log 10 + a. c. 4 log 5 — 20 = 374039. .'. a; = 0.1879 + . 3.. Given Vll5 X V^i016 : (0.0051)' : : x :4|^V Log 35 = i log 115 + i log .016 + i log .32 + a. c. } log 1146 + a e. S log 0051 - 20 = 2.729701. /. x = 536.66 +. TABLE I, OONTAINIKa LOGARITHMS OF NUMBERS Feom 1 TO 11,000. N. Log. N. Log. N. Log. N. Ln. 1 o-oooooo 26 1-414973 51 1.707570 76 1.880814 2 o-3oio3o 27 1-431364 52 1.716003 77 I. 88649 I 3 0-477I2I 28 1-447158 53 1-724276 78 1.892095 4 0. 602060 29 1.462398 54 1.732394 79 1.897627 1.903090 5 0-698970 80 1.477121 65 1.740363 80 6 0.778151 0.845098 81 1.491362 56 1.748188 81 1.908485 7 82 i-5o5i5o 57 1.755875 82 1-913814 8 0.903090 83 i.5i85i4 58 1-763428 83 1-919078 9 0.954243 34 I -531479 I • 544068 59 1.770852 84 1-924279 10 I . 000000 35 60 1.778151 85 1-929419 11 1-041393 86 i-5563o3 61 1.785330 86l 1.934498 12 1-079181 37 1-568202 62 1.792392 87 1.939519 1.944483 13 1-113943 38 1-579784 63 'Mil!, 88 14 I. 146128 39 1-591065 64 89 1.949390 15 1-176091 40 1 -602060 65 1.812913 90 1.954243 ir> I- 2041 20 1 41 1-612784 66 1-819544 91 1.959041 1.963788 17 1-230449 1-255273 42 1-623249 67 1-826075 92 18 43 1-633468 68 I -832509 1'3 1.968483 19 1-278754 44 1-643453 69 1-838849 94 1.973128 20 i-3oio3o 45 1-653213 70 1-845098 95 1.977724 21 1-322219 46 1-662758 . 71 1-851258 96 1.982271 22 1.342423 47 1-672098 72 1-857333 97 1.986772 23 1-361728 48 1-681241 73 1-863323 98 I. 991226 24 1-380211 49 1-690196 74 1-869232 99 1-995635 25 1-397940 50 1-698970 75 1-875061 100 2 • 000000 Remark. — In the following Table, the Jirst two figures^ in the first column of Logarithms, are to be prefixed to each of the numbers, in the same horizontal line, in the next nine columns; but when a point (•) occurs, a is to be put in its place, and the two initial Jigures are to be taken from the next line below. 12 LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 7 8 9 D. 100 101 102 103 104 105 106 107 108 109 000000 4321 86oo 012837 7o33 021189 53o6 9384 033424 7426 0434 4751 9026 3259 745i i6o3 57.5 9789 3826 7825 0868 5i8i 0451 368o 7868 2016 6125 •195 4227 8223 i3oi 5609 9876 4100 8284 2428 6533 •600 4628 8620 1734 6o38 •3oo 4521 8700 2841 6942 1004 5029 9017 2166 6466 •724 4940 9116 3252 7350 1408 5430 9414 2598 6894 1147 536o 9532 3664 7757 1812 583o 9811 3029 7321 1070 5779 9947 4075 S164 2216 623o •207 3461 7748 1993 6197 e3oi 4486 8571 2619 6629 •602 3891 8174 24i5 6616 8978 3o2i 7028 •998 432 428 424 4.9 416 412 408 404 400 396 110 111 112 113 114 115 116 117 118 119 041393 5323 9218 053078 6905 060698 4458 8186 071882 5547 1787 5714 9606 3463 7286 1075 4832 8557 225o 59.2 2182 oio5 9993 3846 7666 1452 5206 8928 2617 6276 2576 6495 •38o 423o 8046 1829 558o 9298 2985 6640 6885 •766 46i3 8426 2206 5953 9668 3352 7004. 3362 2582 6326 ••38 3755 7664 1538 5378 9185 2958 6699 •407 4o85 7731 4148 8o53 1924 5760 9563 3333 7071 •776 445i 8094 4540 8442 2309 6142 9942 3709 7443 1145 4816 8457 4932 8b3o 2694 6524 •320 4o83 7bi5 i5i4 5i82 8819 393 389 386 382 l]l 372 363 36o 357 355 35i 349 346 343 340 338 335 120 121 122 123 124 125 126 127 128 129 079181 082785 6360 9905 093422 6910 100371 38o4 7210 1 1 0590 9543 3i44 6716 •258 % 0715 4146 7549 0926 !L1 7071 •611 4122 7604 io5q 4487 7888 1263 •266 386i 7426 •963 4471 7951 l4o3 4828 8227 1599 •626 4219 n^\ i3i5 4820 8298 5r69 8565 1934 8i36 1667 5169 8644 2091 55io 8903 2270 i347 4934 8490 2018 55i8 8990 2434 585i 9241 26o5 1707 5291 8845 2370 5866 9335 2777 6191 9579 2940 2067 5647 9198 2721 62i5 0681 3119 653 1 r4 2426 6004 9552 3071 6562 ••26 3462 6871 •253 3609 130 131 132 133 134 135 136 137 138 139 113943 7271 120574 3852 Tio5 i3o334 3539 6721 i4Joi5 7603 0903 4178 7429 o65d 3858 7037 •194 3327 4611 7934 I23l 4504 7753 0977 4177 7354 •5o8 3639 4944 8265 i56o 483o 8076 1298 4496 7671 •822 3951 5278 8595 1888 5i56 8399 1619 4814 VSL 4263 56ii 8926 2216 5481 8722 1939 5i33 83o3 i45o 4574 5943 9206 2544 58o6 9045 2260 5451 8618 1763 4885 6276 9586 287I 6i3i 9368 258o 5769 8934 2076 5196 6608 6456 9690 2900 6086 9249 2389 5507 6940 •245 3525 6781 ••12 3219 64o3 9564 2702 58i8 333 33o 328 325 323 321 3i8 3i5 3i4 3ii 140 141 142 143 144 145 146 147 14S 149 150 151 152 153 154 155 156 157 158 159 146128 i5?28l 5336 8362 i6i3b8 4353 7317 170262 3i86 6438 9527 li^^ 5640 8664 1667 465o 7613 o555 3478 6748 9835 2900 5943 8965 1967 4947 3769 7o58 •142 32o5 6246 9266 2266 5244 8203 1141 4060 7367 •449 35io 6549 9567 2564 5541 8497 1434 435i 7676 •756 38i5 6852 9868 2863 5838 8792 1726 4641 7985 io63 4120 7154 •168 3i6i 6i34 9086 2019 4932 82J94 1370 4424 7457 •469 3460 643o 9380 23ll 5222 86o3 1676 4728 7759 V,^ 6726 9674 2603 55i2 8911 1982 5o32 8061 1068 4o55 7022 ^5 5802 309 3o7 3o5 3o3 3oi 299 297 295 293 291 176091 i8i§44 4691 7521 igo332 3i25 5900 8657 201397 638i 9264 2129 % 0612 34o3 6176 8932 1670 6670 9552 24i5 5259 8084 0892 368i 6453 9206 1943 6959 9839 2700 5542 8366 1171 3959 6729 9481 2216 7248 •126 2985 5825 8647 i45i 4237 7oo5 9755 2488 7536 •4i3 3270 6108 8928 1730 45i4 7281 ••29 2761 7825 6391 9209 2010 4792 7556 •3o3 3o33 8ii3 5069 7832 •577 33o5 8401 1272 4123 6956 9771 2567 5346 8107 •85o 3577 8689 1558 4407 7239 ••5i 2846 5623 8382 1124 3848 289 287 285 283 281 -I 276 274 272 N. 1 2 8 4 6 6 7 8 9 D. LOGARITHMS OF NUMBERS. 13 N. 1 2 8 4 5 6 7 8 9 D. 160 204120 4391 4663 4934 5204 5475 5746 6016 6286 6556 271 161 6826 5?^^ 7365 7634 7904 8173 8441 8710 8979 9247 269 162 95i5 ••5i •319 •586 •853 1121 1 388 1654 1921 4579 267 163 212188 2454 2720 2986 3252 35i8 3783 4o4q 43.4 266 164 4844 1 5 109 5373 5638 5902 6166 6430 j 6694 6957 7221 264 165 7484 7747 8010 8273 8D36 8798 9060 9323 9585 9846 262 166 220108 0370 o63i 0892 ii53 1414 1675 ; 1936 2196 2456 1 261 167 2716 2976 3236 3496 3755 4oi5 j 4274 4533 4792 5o5i 259 168 5309 5D68 5826 6084 1 6342 ! 6600 6858 7ii5 7372 763o 25o 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 •193 256 170 230449 2996 5528 0704 0960 12l5 1470 1724 1979 4517 2234 2488 2742 254 171 3250 35o4 3757 4011 4264 4770 5o23 5276 253 172 5781 6o33 6285 6537 6789 7041 7292 7544 7795 252 173 8046 8297 8548 8799 9049 9299 9550 9800 ••5o •3oo 25o 174 240549 3o38 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 175 3286 3534 3782 4o3o 4277 4525 4772 5019 5266 248 176 55i3 5759 6006 6252 8^?! 6745 6991 7237 7482 7728 246 177 7973 8219 8464 8709 9198 9443 9687 2125 ^11 •176 245 178 250420 0664 0908 3338 ii5i 1395 1638 1881 2610 243 179 2853 3096 358o 3822 4064 43o6 4548 4790 5o3i 242 180 255273 5514 5755 5996 6237 6477 6718 6953 9355 7198 l$'i 241 181 7679 7018 o3io 8.58 83q8 0787 8637 8877 1263 9116 9594 239 182 260071 0548 1025 i5oi 1739 1976 4346 2214 23J 183 245i 2688 2925 3.62 3399 3636 3873 4109 4582 237 184 4818 5o54 5290 5525 5761 8344 6232 6467 6702 6937 235 185 7172 7406 7641 7875 8110 8578 8812 9046 9279 234 186 95i3 9746 9980 •2l3 •446 •679 •912 1 144 1377 1609 233 187 271842 2074 23o6 2538 2770 3ooi 3233 3464 3696 3927 232 188 41 58 4389 4620 485o 5o8i 53ti 5542 5772 6002 6232 23o 189 6462 6692 6921 7i5i 7380 7609 7838 8067 8296 8525 229 ll'O 278754 8982 9211 9439 1715 9667 9895 •123 •35i •578 •806 228 191 281033 1261 1488 1942 2169 46?6 2622 2849 3075 227 192 33oi 3527 3753 3979 42o5 443 1 4882 5107 5332 226 193 5557 5782 6007 6232 6456 668 1 6905 7.30 7354 7578 225 194 7802 8026 8249 8473 8696 8920 9143 9366 9580 ibi3 9812 223 195 290035 0257 0480 0702 0925 1147 3363 1369 1591 2o34 222 196 2256 2478 2699 2920 3i4i 3584 38o4 4025 4246 221 197 4466 4687 4907 5.27 5347 5567 5787 6007 8198 6226 6446 220 198 6665 6884 7104 7323 7542 7761 7979 8416 8635 219 199 200 8853 9071 9289 9507 9725 9943 •161 •378 •595 •8i3 218 3oio3o 1247 1464 1681 1898 2114 233i 2547 2764 2980 217 i 201 It 3412 3628 3844 4o5g 4275 4491 4706 4921 5i36 216 ; 202 5566 5781 5996 6211 6425 6639 8778 •6854 7068 72S2 2l5 203 '4t 7710 7924 8137 835i 8564 8991 9204 9417 2l3 204 9843 ••56 •268 •481 •693 •906 1118 i33o 1542 2t2 205 311754 1966 4289 2389 2600 2812 3o23 3234 3445 3656 211 206 3867 4078 4499 4710 4920 5i3o 5340 555i 5760 7854 210 207 5970 6180 6390 6599 6809 8898 7018 7227 7436 7646 209 203 8o63 8272 8481 86S9 9106 9314 9522 9730 9938 208 209 210 320146 o354 o562 0769 0977 1184 1391 1598 i8o5 2012 207 322319 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 4282 44B8 4694 ilfo 5io5 53io 55i6 5721 5926 6i3i 205 £12 6336 6541 6745 7i55 7359 9398 7563 9805 7972 8176 204 213 8380 8583 8787 8991 9194 9601 •••8 •211 203 214 330414 0617 0819 1022 1225 1427 i63o i832 2o34 2236 202 215 2438 2640 2842 3o44 3246 3447 3649 5658 3850 4o5i 4253 202 216 4454 4655 4856 5o57 5257 5458 5859 6059 8o58 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 8257 •246 200 218 8456 8656 8855 9054 9253 945i 965o 9849 ••47 2028 ]^ 219 340444 0642 0841 1039 1237 i4£5 1632 i83o 2225 N. 1 2 8 4 5 6 7 8 9 D. K )* 14 LOGARITHMS OF NUMBERS. m 1 2 8 4 5 6 7 8 9 D. 220 221 222 223 224 225 226 227 228 229 342423 4302 63d3 83o5 350248 ai83 , 4io8 6026 7035 9835 2620 4589 6549 85oo 0442 2375 43oi 6217 8125 ••25 2817 4785 6744 8694 0636 2568 4493 6408 83i6 •2l5 3oi4 4981 ^^ 0829 2761 4685 tl^ •404 3212 5178 7135 9083 1023 2o54 4876 6790 8696 •593 3409 5374 7330 9278 1216 3i47 6981 8886 •783 36o6 5570 7525 9472 1410 3339 5260 7172 9076 •972 38oa 5766 7720 9666 i6o3 3532 5452 7363 9266 1161 7915 9860 1796 3724 5643 7554 9456 i35o 4196 6157 8110 ••54 5834 7744 9646 1539 195 194 193 193 192 191 280 281 282 233 284 285 236 237 238 239 361728 36i2 5488 7356 9216 371068 2912 4748 5n 5675 7542 9401 1253 3096 4932 6759 8580 2.05 7729 9587 1437 3280 5ii5 6942 8761 2294 4176 6049 79i5 9772 1622 3464 5298 7124 8943 2482 4363 6236 8101 ??S 3647 5481 7306 9124 2671 455i 6423 8287 •143 5664 7488 9306 2859 4739 6610 8473 •328 2175 401 5 5846 7610 9487 3048 4926 6796 86D9 •5i3 236o 4198 6029 7832 9668 3236 5ii3 6983 8845 •698 2544 4382 6212 8o34 9849 3424 5301 7169 9o3o •883 2728 4565 6394 8216 ••3o 188 188 187 186 i85 184 184 1 83 182 181 240 241 242 243 244 245 246 247 248 249 3802 1 1 2017 38i5 56o6 390935 2697 4452 6199 0392 5785 7568 9343 1112 2873 4627 6374 0573 2377 4174 5964 7746 9520 1288 3048 4802 6548 0754 2557 4353 6142 7923 9698 1464 3224 4977 6722 0934 2ll 6321 8101 9875 1641 3400 5i52 6896 iii5 2917 4712 6499 8279 ••5 1 1817 3575 5326 7071 1296 3097 4891 6677 8456 •228 1993 3731 55oi 7245 1476 3277 5070 6856 8634 •4o5 3926 5676 7419 i656 3456 5249 7034 881 1 •582 2345 4101 585o 7592 1837 3636 5428 7212 6989 2521 4277 6025 7766 181 180 178 177 176 176 175 174 250 251 252 253 254 255 256 257 258 259 397940 9674 401401 3l2I 4834 6540 8240 9933 41 1620 33oo 8114 9847 1573 3292 5oo5 6710 8410 •102 1788 3467 8287 ••20 1745 3464 5176 6881 8579 •271 1956 3635 8461 •192 5346- 7o5i 8749 •440 2124 38o3 8634 •365 2089 3807 55i7 7221 8918 2293 3970 8808 •538 2261 l^l 7391 9087 •777 2461 4137 8981 •711 2433 4149 5858 756i 9257 •946 2629 43o5 9154 •883 26o5 4320 6029 7731 9426 1114 2796 4472 9328 io56 2777 4492 6199 7901 fil 2964 4639 9501 1228 2949 4663 6370 8070 9764 i45i 3i32 4806 173 173 172 171 171 X 167 260 261 262 268 264 265 265 287 268 269 414973 6641 83oi 9956 421604 3246 4882 65ii 8i35 9752 5i4o 6807 8467 •121 1768 3410 5o45 6674 8297 9914 5307 6973 8633 •286 3?74 5208 6836 8459 ••75 5474 S Si se •236 5641 7306 8964 •616 2261 3901 5534 7161 8783 •398 58o8 7472 9129 •781 2426 4o65 5697 7324 8944 •d59 5974 7638 9295 X. 4228 5860 7486 9106 •720 6141 7804 9460 1110 2754 4392 6023 7648 9268 •881 63o8 7970 9625 1275 2918 4555 6186 7811 9429 1042 6474 8i35 9701 14^9 3082 4718 6349 7973 9591 1203 167 166 i65 i65 164 164 i63 162 162 161 270 271 £72 278 274 275 276 277 278 279 43 1 364 2969 4369 6i63 440909 2480 4045 56o4 i525 3i3o 4729 6322 7909 tz 2637 4201 5760 1685 32go 4888 6481 8067 9648 1224 2793 4357 59,5 1846 3450 5o48 6640 8226 9B06 i38i It, 6071 2007 36io 5207 tin 3 1 06 4669 6226 2167 3770 5367 & •122 1695 3263 4825 6382 2328 3930 5526 7II6 8701 •279 i852 3419 4981 6o37 2488 4090 5685 7275 8839 •437 2009 3576 5i37 6692 2649 4249 5844 7433 9017 •394 2166 3732 5293 6848 2809 4409 6004 7592 9175 •732 2323 3889 5449 7003 161 160 ;59 1 58 \ll i55 N. 1 2 8 4 5 6 7 8 9 D. LOGAKfrHMS OF NUMBERS. 16 .. 1 2 8 4 5 6 7 8 "I-^ D. 280 447158 73iS 8861 7468 7623 7778 7933 8088 8242 8397 8552 1 55 231 8706 9015 9170 9324 9478 9633 9787 9941 ••95 i633 i54 2S-2 450249 0403 0557 071 1 0865 1018 1172 i326 1479 ' 1 54 283 1786 1940 2093 2247 2400 2553 2706 2859 3oi2 3i63 i53 284 33.8 3471 3624 3777 3930 4082 4235 4387 4540 4692 i53 285 4845 i 4997 1 5i5o 1 53o2 1 5454 56o6 5758 5910 { 6062 6214 I 52 286 6366 65i8 6670 6821 6973 7125 8638 7276 7428 7579 7731 l52 287 7882 8o33 8i84 8336 8487 8789 8940 9091 9242 i5i 288 9392 9543 9694 9845 9995 •146 •296 •447 1948 ^ •748 i5i 289 460893 1048 1198 1348 1499 1649 1799 2248 i5o 290 462398 2548 2697 2847 2997 3i46 3296 3445 3594 3744 i5o 291 3893 4042 4191 4340 4490 4639 4788 4936 5o85 5234 149 292 5383 5532 5680 5829 5977 6126 6274 6423 i 6571 6719 \% 293 6868 7016 7164 8643 23.2' 8790 7460 7608 7706 7904 i 8o52 9380 9527 8200 294 8347 8495 8938 9085 9233 9675 143 295 9822 VM •116 •263 •410 •557 •704 •85 1 •998 1145 147 146 296 471292 1 585 1732 1878 2025 2171 23i8 2464 2610 297 2736 2903 3o49 45o8 3195 4653 3341 3487 3633 377? 5235 8925 538i 4071 146 298 4216 4362 4799 4944 6397 6090 5526 146 299 5671 58i6 5962 6107 6232 6542 6687 6832 6976 145 800 4,11121 7266 7411 7555 0438 7700 7844 7989 8i33 8278 8422 145 801 8566 8711 8855 9143 9287 9431 9575 9719 9863 144 302 480007 oi5i 0294 o582 0725 0869 1012 ii56 1299 144 803 1443 1 586 1729 1872 2016 2159 2302 2445 2588 2731 143 804 2874 3oi6 3i59 33o2 3445 3587 3730 3872 4oi5 4157 143 805 43oo 4442 4583 4727 4869 Soil 5i53 5295 5437 5579 142 806 5721 5863 6oo5 6147 6289 643o 6572 6714 6855 6997 142 ' 807 7i38 855i 7280 7421 7563 7704 7845 It 8127 8269 8410 141 308 8692 8833 '.Vst 9II4 9255 9537 9677 9818 141 809 9958 ••99 •239 •520 •661 •801 •941 1081 1222 140 810 491362 l502 1642 1782 1922 3319 2062 2201 2341 2481 2621 140 811 2760 2900 3o4o 3i79 3458 3597 3737 3876 401 5 139 812 4i55 4294 4433 4572 4711 485o 4989 6376 5i28 5267 6653 5406 139 813 5544 5683 5822 5960 6099 6238 65i5 6791 139 814 6930 83ii 7068 7206 8586 7^44 8724 7483 7621 7759 7897 8o35 8173 9550 i33 815 8448 8862 9137 9275 9412 i33 816 9687 9824 9962 ••99 •236 •5n •643 •785 •922 137 817 5oio59 1196 1333 1470 1607 1744 1880 2017 2i54 2291 137 818 2427 2564 2700 2837 2573 4335 3109 3246 3382 35i8 3655 i36 819 3791 3927 4o63 4199 4471 46©7 4743 4878 5oi4 i36 820 5o5i5o 5286 5421 5557 5693 5828 7316 8664 0099 745 1 8799 •143 6234 6370 1 36 821 65o5 6640 6776 691 1 7046 8395 7181 7586 7721 i35 822 7856 7991 8126 8260 853o 8934 9068 i35 823 9203 9337 9471 9606 9740 9874 •••9 •277 •411 i3.i 324 5io545 0679 o8i3 0947 io8i 12l5 i349 1482 1616 1750 i34 825 1 883 2017 2.5l 2284 2418 255i 2684 2818 2951 3o84 i33 826 3218 335i 3484 3617 3750 3883 4016 4149 5476 4282 44i5 i33 827 4548 4681 4813 4946 5o79 64o3 52II 5344 5609 5741 i33 828 5874 6006 6139 6271 6535 6668 6800 6932 7064 8382 l32 829 7196 7328 7460 7592 7724 7855 7987 8119 825i l32 830 5i85i4 8646 8777 8909 9040 '-Z 93o3 9434 9566 9697 i3i 3;31 9828 9959 ••90 •221 •353 •6i5 •745 •876 2i83 1007 i3i C32 52II38 1269 2575 1400 i53o 1661 1792 1922 2o53 23i4 i3i 833 2444 2705 2835 2966 3096 3226 3356 3486 36i6 i3o 334 3746 3876 4006 4i36 4266 4396 4526 4656 4785 4915 i3o 835 5o45 5i74 53o4 5434 5563 5693 6985 5822 5951 6081 6210 129 836 6339 6469 6598 7888 6727 6856 8402 7243 8531 7372 7501 129 337 763o 7759 9045 8016 8145 8274 8660 8788 129 338 8917 9174 9302 943o 9559 ^l 9815 9943 ••72 128 839 N. 53o2oo o328 0456 0584 0712 0840 1096 1223 i35i 128 1 2 3 4 5 6 7 8 9 D. 16 LOGARITHMS OF NUMBERS. N. 1 2 8 4 5 6 7 8 9 D. 840 841 842 843 844 845 846 847 848 819 850 851 852 853 854 855 856 357 858 359 531479 2704 4026 5294 6558 7819 9076 540329 nil 1607 2882 41 53 5421 6685 7945 9202 0455 1704 2900 •1734 3009 4280 5547 681 1 8071 9327 o58o 1829 3074 1862 3i36 4407 5674 6937 8197 9432 0705 1953 3199 1990 3264 4534 5800 7063 8322 9578 o83o 2078 3323 2117 3391 4661 5927 7189 8448 9703 0955 22o3 3447 2245 35i8 4787 6o33 73i5 8374 9829 1080 2327 3371 2372 3645 4914 6180 7441 6609 99% 1205 2452 3696 25oo 3772 5o4i 63o6 & 2576 3820 2627 6432 •204 1454 2701 3944 128 127 III 126 126 125 125 125 124 544068 5307 6543 7775 9003 550228 i45o 2668 3883 5094 i'X 6666 7898 9126 o35i 1572 2790 4004 52i5 43 16 5555 6789 8021 9249 0473 1694 2911 4126 5336 4440 5678 6913 8144 9371 0095 1816 3o33 4247 5457 4564 58o2 7o36 8267 9494 0717 1938 3i55 4368 5578 4688 5925 9616 0840 2060 3276 4489 5699 4812 6049 7282 85 1 2 9739 0962 2181 3398 4610 5820 4936 6172 74o5 8635 9861 1084 23o3 3519 4731 5940 5o6o 6296 2425 3640 4852 6061 5i83 6419 7652 8881 •106 1328 2547 3762 4973 6182 124 124 123 123 123 122 122 121 121 121 860 861 862 863 864 865 866 867 868 869 5563o3 8709 9907 56II0I 2293 3481 4666 5848 7026 6423 7627 8829 ••26 1221 2412 36oo 4784 5966 7144 6544 lit •146 1 340 253i 3718 4903 6084 7262 6664 7868 9068 •265 1459 265o 3837 502I 6202 7379 6785 7988 9188 •385 1578 2769 3955 5i39 6320 7497 6905 8108 9308 •5o4 1698 2887 4074 5237 6437 7614 7026 8228 9428 •624 18x7 3oo6 4192 5376 6335 7732 7146 8349 43ii 5494 6673 7849 2o55 3244 4429 5612 6791 7967 7387 8589 9787 •982 2174 3362 4548 5730 6909 8084 120 120 120 119 119 119 \\l 118 118 870 871 872 873 874 875 876 877 878 879 568202 9374 570543 1709 2872 4o3i 5i88 6341 7492 8639 83 19 9491 0660 1825 2988 4147 53o3 6457 7607 8754 8436 9608 0776 1942 3io4 4263 5419 6572 2S 8554 9725 0893 2o58 3220 4379 5534 6687 7836 8983 8671 9842 1010 2174 3336 4494 5650 6802 7951 9097 8788 9939 1126 2291 3432 4610 5765 6917 8066 9212 8905 ••76 1243 2407 3568 4726 5880 7032 8i8i 9326 9023 i339 2323 3684 4841 5996 7147 8295 9441 9140 •309 1476 2639 3800 4957 6111 7262 8410 9555 nil 39.5 5072 6226 ^'5^5 9669 »'7 117 117 116 116 116 ii5 ii5 ii5 114 880 881 882 883 884 885 886 887 888 889 579784 580925 2o63 5461 6587 mi 9950 9898 1039 l',]l 4444 5574 6700 7823 8944 ••6 1 ••12 ii53 2291 3426 4557 5686 6812 7935 9036 •173 •126 1267 2404 3539 4670 llU 8047 9167 •284 •241 i38i 25i8 3652 4783 59,2 7037 8160 •355 i4o5 263 1 3765 4896 6024 7149 8272 9391 •507 •469 1608 2745 3879 5009 6137 7262 8384 95o3 •619 •583 1722 2858 3992 5l22 625o 7374 8496 90i5 •730 :p2 2972 4io5 5235 6362 7486 8608 lilt •811 1950 3o85 4218 5348 6475 7599 8720 9838 •953 114 114 114 ii3 ii3 ii3 112 112 112 112 890 891 892 393 394 395 896 397 398 899 591065 nil 4393 5496 p 8791 9883 600973 Ills 56o6 6707 7805 8900 1287 2399 35o8 4614 5717 6817 7914 9009 •lOI 1191 1 399 25lO 36i8 4724 5827 6927 8024 9119 •210 1299 i5io 2621 3729 4834 7^037 8i34 9228 1621 2732 3840 4945 6047 7146 8243 9337 •428 I5i7 1732 2843 3950 5o55 6157 7256 8353 9446 •537 1625 1843 2954 4061 5i65 6267 7366 8462 9556 •646 1734 1955 3o64 4171 5276 6377 7476 8572 9663 •755 1843 2066 3175 4282 5386 6487 7586 8681 9774 •864 1951 III 111 III no no no no 109 109 109 N. 1 2 3 4 5 6 ■ 7 8 9 D. LOGARITHMS OF NUMBERS. 17 N. 1 2 3 4 5 6 7 8 9 D. 400 401 402 403 404 405 406 407 408 409 602060 3i44 4226 53o5 6381 7455 b526 9594 610660 1723 2169 3253 4334 54i3 6489 7562 8633 9701 0767 1829 2277 3361 444:* 5521 6596 7669 87^0 9808 0873 1936 2386 3469 455o 5628 6704 6847 9914 0979 2042 2494 3577 4658 5736 6811 7884 8954 ••21 1086 2148 2603 3686 4766 5844 6919 7991 9061 •128 1192 2254 2711 3794 4874 5951 7026 8098 9167 •234 1298 2360 2819 3902 4982 82o5 9274 •341 i4o5 2466 2928 4010 5089 6166 7241 g3l2 9381 •447 i5ii 2572 3o36 4118 5197 6274 7348 8419 9488 •554 1617 2678 108 108 108 108 107 107 107 107 106 106 410 411 412 413 414 415 416 417 418 419 612784 3842 4897 5900 7000 8048 90q3 62oi36 1176 2214 2890 Itil 6o55 7io5 8i53 9198 0240 1280 23i8 5io8 6160 7210 8257 9302 o344 1 384 2421 3l02 4169 52i3 6265 73i5 8362 9406 0448 1488 2525 3207 4264 5319 6370 7420 8466 9311 o552 1592 2628 33i3 4370 5424 6476 7525 8571 9615 0656 1695 2732 3419 4475 5529 658i t]l 9719 0760 1799 2835 3525 458 1 5634 6686 8780 9824 0864 1903 2939 3630 4686 5740 6790 & 2007 3o42 3736 4792 5843 6895 7943 8989 ••32 1072 2IIO 3146 106 106 io5 io5 io5 io5 104 104 104 104 420 421 422 423 424 425 426 427 4S8 429 623249 4282 53i2 6340 7366 8389 9410 630428 1444 2457 3353 4385 541 5 6443 7468 8491 9512 o53o 1545 2559 3456 4488 55i8 6546 'ill 9613 o63i 1647 2660 3559 4591 5621 6648 7673 8695 9715 0733 1748 2761 3663 4695 5724 675, 7775 8797 9817 o835 1849 2862 3766 4798 5827 6853 7878 8900 1951 2963 3869 4901 5929 6956 7980 io38 2o52 3o64 3973 5oo4 6o32 7o58 8082 9104 •I23 1.39 2i53 3i65 4076 5107 6i35 7i6i 8i85 9206 •224 1241 2255 3266 4179 52IO 6238 7263 8287 9308 •326 i342 2356 3367 io3 io3 io3 io3 102 102 102 102 lOI 101 430 431 432 433 434 435 436 437 438 439 633468 5484 6488 7490 8489 9486 640481 1474 2465 3569 4578 5584 6588 7590 8589 9586 o58i 1573 2563 3670 4679 5685 6688 7690 8689 9686 0680 1672 2662 3771 t]ll 6789 7790 B789 9783 0779 1771 2761 3872 4880 5886 6889 8888 9885 0879 1871 2860 3973 4981 5986 6989 7990 8958 9984 0978 1970 2959 4074 5o8i 6087 & 9088 ••84 1077 3o58 4175 5i82 6187 7189 8190 9188 •i83 2168 3i56 4276 5283 6287 7290 8290 V2 1276 2267 3255 4376 5383 6388 7390 8389 9387 •382 1375 2366 3354 100 100 100 100 99 99 99 99 99 99 440 441 442 443 444 445 446 447 448 449 643453 4439 5422 6404 7383 8360 9335 65o3o8 1278 2246 355i 4537 5521 65o2 7481 8458 9432 o4o5 1375 2343 365o 4636 5619 6600 9530 o5o2 1472 2440 3749 4734 5717 6698 7676 8653 9627 0D99 i569 2536 3847 4832 58i5 6796 ^ro 9724 0696 1666 2633 3946 493 1 59.3 6894 8848 9821 0793 1762 2730 4044 5029 6oii 6992 ? i8?9 2826 4143 5127 6110 7089 8067 9043 ••16 0987 1956 2923 4242 5226 6208 9140 •ii3 1084 2o53 3019 4340 5324 63o6 7285 8262 9237 •210 1181 2i5o 3u6 98 98 98 98 98 97 97 97 97 97 96 96 96 93 93 93 93 93 450 451 452 458 454 455 456 467 458 459 653213 4177 5i38 6098 7o56 801 1 8965 9916 660865 i8i3 3309 4273 5235 6194 71D2 8107 ^, 0960 1907 34o5 4369 533i 6290 7247 8202 9i55 •106 io55 2002 35o2 4465 5427 6386 7343 8298 9200 •201 ii5o 2096 3598 4562 5523 6482 7438 8393 9346 •296 1245 2191 3695 46D8 5619 6577 7534 8488 9441 •391 1339 2206 3791 4754 5715 6673 7629 8584 9536 •486 1434 238o 3888 485o 58io 6769 7725 8679 963 1 •58 1 1 529 2473 3984 4946 5906 6864 7820 8774 1623 2569 4080 5o42 6002 6960 7916 8870 9821 •771 1718 2663 N. 1 2 3 4 5 6 7 8 9 D. 18 LOGARITHMS OF NUMBERS. LOOAEITHMS OF NUMBEE8. 19 N. 1 2 8 4 5 6 7 8 ' ). 520 7i6oo3 6087 6170 6254 6337 6421 65o4 6588 6671 6754 b3 521 6838 6921 7004 7088 7171 7254 7338 8169 7421 8253 7504 8336 7587 83 522 Itll 7754 8585 & 7920 8oo3 8086 8419 83 523 8751 8834 89.7 9000 9083 9.65 9248 83 524 933i 9414 9497 9580 9663 9743 9828 99'' 9994 0821 ••77 83 525 720159 0242 o325- 0407 1233 0490 0573 0655 0738 0903 83 526 . ??u 1068 n5i i3i6 1398 1481 1563 1646 1728 82 527 1893 1975 2o58 2140 2222 23o5 2387 2469 2552 82 5'^8 * 2634 27.6 3538 2798 2881 2963 3045 3gll 3209 329. 3374 82 529 3456 3620 3702 3784 3866 4o3o 4H2 4194 82 680 724276 4358 4440 4522 4604 4685 itl 4849 4931 5oi3 82 531 6095 5176 5258 5340 5422 55o3 5667 5748 583o 82 532 591-2 5993 6075 6.56 6238 6320 6401 6483 6564 6646 82 533 6727 6809 6890 7??5 7053 7134 72.6 7297 8uo 7379 7460 81 534 7541 7623 8435 Ei 7866 8678 7948 8029 8.91 8273 8. 535 8354 8597 9408 8759 8841 8922 9003 9084 81 536 9165 9246 9327 9489 9570 965 1 9732 98.3 9893 81 537 9974 ••55 •i36 •217 •298 •378 •459 •540 •621 •702 81 538 730782 o863 0944 1024 no5 1.86 1266 1347 1428 i5o8 8i 589 1589 1669 1750 i83o 1911 1991 2072 2.52 2233 23.3 81 540 732394 2474 2555 2635 27.5 2796 2876 2956 3o37 3.17 80 541 3197 3278 3358 3438 35.8 3598 3679 3759 3839 3919 80 542 3999 4079 4.6o 4240 4320 4400 4480 436o 4640 4720 55.9 80 543 4800 4880 4960 5o4o 5.20 5200 5279 5359 5439 80 544 5599 5679 5759 5838 59.8 5998 6078 6.57 6237 63.7 80 545 6397 6476 6556 6635 67.5 6795 '6874 6954 7034 7. .3 80 546 7193 7987 8067 7352 743 1 75,1 83o5 ilr, 7670 11^ 7829 7908 8701 79 547 8.46 8225 8463 8622 79 548 878. 9572 8860 8939 90.8 « 9'77 996^ 9256 9335 9414 9493 •284 79 549 9651 9731 98.0 ••47 •126 •205 79 550 740363 0442 052I 0600 0678 0757 o836 09.5 0994 1073 79 551 1.52 1230 i3o9 i388 1467 1546 1624 1703 1782 i860 79 5.-2 1939 2018 2096 2.75 2254 2332 2411 2489 3275 2568 2647 ?^ 553 2725 2804 2882 2961 3o39 3ii8 IX 3353 343 1 554 35io 3588 3667 3745 3823 390? 4o58 4i36 42.5 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5.53 523. 5309 5387 5465 5543 5621 5699 5777 78 557 5855 5933 60.1 6868 6167 6245 6323 6401 6479 6556 78 558 6634 67.2 6790 6945 7023 7.01 V^it 7256 7334 78 559 7412 7489 7567 7645 1122 7800 7878 8o33 8110 78 560 748188 8266 8343 8421 8498 8576 8653 8,3. 8808 8885 77 561 8963 9040 91.8 9195 9272 9350 9427 9504 9582 9659 77 562 9736 98.4 9891 9968 ••45 •123 •200 •277 •354 •43 1 77 563 75o5o8 o586 0663 0740 0817 0894 0971 1048 1125 1202 77 564 i?79 .356 1433 i5io i587 1664 1741 I818 1895 1972 77 565 2048 21^5 2202 2279 2356 2433 2 509 2586 2663 2740 77 566 2816 289] 2970 3o47 3.23 3200 3277 3353 343o 35o6 77 567 3583 366o 3736 38.3 3889 3966 4042 4119 4883 4.95 4272 77 568 4348 4425 45oi 4578 4654 4730 4807 4960 5o36 76 569 5ll2 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 755875 5951 6027 6io3 6180 6256 6332 6408 6484 6560 76 571 6636 67.2 6788 6864 6940 70.6 'X 7168 7244 7320 76 572 '^t 7472 7548 7624 8382 7700 7775 7927 8oo3 8079 76 573 8230 83o6 8458 8533 8609 8685 8761 8836 76 574 B912 8988 9063 9.39 9214 9290 9366 9441 95.7 9592 76 575 9668 9743 9819 9894 9970 ••43 •121 •.96 •272 •347 7? 576 760422 0498 0073 0649 0724 0799 0875 0950 1025 1. 01 75 577 1176 I2DI .326 1402 1477 1 552 1627 .702 1778 1 853 75 578 1928 2003 2078 2i53 2228 23o3 2378 2453 2529 3278 2604 75 579 2679 2754 2829 2904 2978 3o53 3ii8 32o3 3353 75 N. 1 2 3 4 5 6 7 8 9 D. 20 LOGAlilTHMS OF NUMBERS. N. 1 2 3 4 5 6 7 8 9 D. 580 681 582 583 684 585 586. 687 588 589 763428 4176 4923 5669 641 3 7i56 2638 9377 770115 35o3 425i 4998 5743 6487 723o 7972 8712 945 1 0189 3578 4326 5072 58i8 6562 7804 8046 8786 9525 0263 3653 4400 5i47 5892 6636 7379 8120 8860 tin 4475 5221 5966 6710 B 9673 0410 38o2 455o 5296 6041 6785 7527 8268 9008 9746 0484 3877 4624 5370 6ii5 6859 76QI 8342 9082 9820 0557 3952 4699 5445 6190 6933 7675 8416 9i56 2^?f 4027 4774 5D20 6264 7007 7749 8490 9230 9968 0705 4101 4848 5594 6338 7082 7823 8564 lit 0778 i 74 74 74 74 74 74 74 590 591 592 593 594 595 596 597 598 599 770852 i587 2322 3o55 3786 4517 5246 5974 6701 7427 0926 1661 2895 3128 3860 4590 53,9 6047 6774 7499 0999 1734 2468 3201 3933 4663 5392 6120 6846 7572 1073 1808 2542 3274 4006 4736 5465 6193 6919 7644 1 146 1881 26i5 3348 4079 4809 5538 6265 6992 7717 1220 1955 2688 3421 4i52 4882 56io 6338 7064 7789 1293 2028 2762 3494 4223 4955 5683 641 1 1 367 2102 2835 3567 4298 5o28 5756 6483 7209 7934 1440 2.75 2908 3640 4371 5ioo 5829 6556 7282 8006 i5i4 2248 2981 3713 4444 5173 5902 6629 7354 8079 1 73 72 600 601 602 603 604 605 606 607 608 609 778.51 8874 9596 780317 io37 1755 2473 3189 3904 4617 8224 8947 9669 0389 1109 1827 2544 3260 3975 4689 8296 9019 9741 0461 1181 3332 4046 4760 8368 9091 98.3 o533 1253 Ills 34o3 4118 483 1 8441 9163 9885 o6o5 1324 2042 2759 3475 4189 4902 85i3 9236 9957 0677 .396 2114 283 1 3546 4261 4974 8585 9308 ••29 0749 1468 2186 2902 36i8 4332 5045 8658 9380 •lOI 0821 1 540 2258 2974 3689 44o3 5ii6 8730 9452 •173 0893 1612 2329 3o46 3761 4475 5187 8802 9524 •245 0965 1684 2401 3ii7 3832 4546 5259 72 72 72 ■ 72 72 72 72 71 71 71 610 611 612 618 614 615 616 617 618 619 785330 6041 675. 2460 8168 8875 9581 790285 5401 6112 6822 753. 8239 8946 9601 0356 1059 1761 5472 6i83 6893 7602 83io 9016 9722 0426 1 1 29 i83i 5543 6254 6964 7673 838 1 9087 9792 0496 1199 1901 56i5 6325 7035 9157 9863 o567 1269 1971 5686 6396 7106 78.5 8522 9228 9933 0637 i34o 2041 5757 6467 I'll 7885 8593 9299 • 9*4 0707 1410 2111 5828 6538 7248 7956 8663 9369 ••74 0778 1480 2181 6609 73.9 8027 8734 9440 •i44 0848 i55o 2252 III: 7390 8098 8804 9510 •2l5 0918 1620 2322 71 71 71 71 71 71 70 70 70 70 620 621 622 628 624 625 626 627 628 629 792392 3092 3790 4488 5i85 5880 6574 7268 ^5? 2462 3i62 3860 4558 5254 5949 6644 8029 8720 2532 3231 3930 4627 5324 6019 6713 7406 87^9 260a 33oi 4000 4697 53q3 6782 7475 8167 8858 2672 3371 4070 4767 5463 6i58 6852 7545 8236 8927 2742 3441 4139 4836 5532 6227 6921 7614 83o5 8996 2812 35ii 4209 4906 5602 6297 6990 7683 8374 9065 2882 358i 4279 4976 5672 6366 7060 7752 8443 9134 2952 365i 4349 5o45 5741 6436 7129 7821 85i3 9203 3022 372. 4418 5ii5 58ii 65o5 7198 7890 8582 9272 70 70 70 70 F 69 630 631 632 633 634 635 636 637 638 639 799341 800029 0717 1404 2089 2774 3457 4139 4821 55oi 9409 0098 0786 1472 2i58 2842 3525 4208 4889 5569 9478 0167 o854 i54i 2226 2910 3394 4276 4957 5637 9547 0236 0923 2295 2979 3662 4344 5o25 5705 9616 o3o5 Xs 2363 3047 3730 4412 5093 5773 9685 0373 1061 1747 2432 3ii6 3798 4480 5i6i 5841 9754 0442 1 129 i8i5 25oo 3iS4 3867 4548 5229 5908 9823 o5ii 1 198 1884 2568 3252 3935 4616 9892 o58o 1266 1952 2637 3321 4oo3 4685 5365 6044 9961 0648 i335 2021 2705 3389 4071 4753 5433 6112 69 69 69 68 68 68 68 N. 1 2 3 4 5 6 , 7 8 9 D. LOOAEITHMS OF NtJMBEElS. 21 N. 1 2 3 4 5 6 7 8 9 D. 640 806180 6248 63i6 6384 645 1 65i9 6587 6655 6723 6790 68 641 6858 6926 6994 7061 7129 7197 7264 7332 7400 l^^l 68 642 7535 8211 7603 ^6 7738 7806 7873 7941 8008 8076 8143 68 643 ^S 8414 8481 8549 86i6 8684 8751 8818 A^ 644 8886 9021 9088 9i56 9223 9290 9358 9425 9492 67 645 9060 9627 9694 9762 9829 9896 9964 ••3 1 ••98 •l63 67 646 810233 o3oo o367 0434 o5oi o569 0636 0703 0770 0837 67 647 T4 0971 1039 1106 1173 1240 i3o7 1374 1441 i5o8 67 048 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 049 2245 23l2 2379 2445 25l2 2579 2646 2713 2780 2847 67 650 8i2oi3 3d8i 2980 3047 3ii4 3i8i 3247 33i4 338i 3448 35i4 ^7 651 3648 3714 3781 3848 3qi4 45»i 3981 4048 4114 4181 67 652 4248 43i4 438i 4447 45i4 4647 4714 4780 ^?^"' ii 053 4913 5578 4980 5046 5ii3 5i79 5843 5246 53i2 5378 5445 55ii: 66 654 5644 6374 5777 5910 5976 6042 6109 6175 66 655 6241 63o8 6440 65o6 6573 6639 6705 6771 6838 66 656 6904 6970 7o36 7102 7169 7235 7301 lltl 7433 7499 8160 66 657 7565 763 1 8292 89D1 & 7764 7830 7962 8094 66 658 8226 8424 8490 8556 8622 8688 8754 8820 66 659 8885 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 9610 9676 9741 9807 9873 9939 •••4 ••70 •i36 66 661 820201 0267 o333 0399 0464 o53o 0595 0661 0727 0792 66 002 o858 0924 0989 io55 1120 1186 I25l i3i7 i382 1448 66 003 i5i4 1 579 2233 1645 1710 1775 1841 1906 1972 2037 2io3 65 664 2168 2299 2364 243o 2495 256o 2626 2691 2756 65 665 2822 2887 2952 3oi8 3o83 3i48 32i3 3279 3344 3409 65 6G6 3474 3539 36o5 3670 3735 3 800 3865 3930 3996 4061 65 067 4126 4f9i 4256 4321 4386 445r 45i6 458i 4646 471 1 65 663 4776 4841 4906 4971 5o36 5ioi 5i66 523i 5296 536i 65 669 5426 5491 5556 5621 5686 5751 58i5 588o 5945 6010 65 670 826075 6140 6204 6269 6334 6399 6464 6528 6593 6658 65 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 73o5 65 672 7369 801 5 liso 7499 7363 7628 7692 7757 7821 7886 9239 65 673 8144 8209 8273 8333 8402 8467 853i 64 674 8660 8724 8789 8853 8918 9561 8982 9046 9111 9175 64 675 9304 9368 9432 9497 9625 9690 9754 9818 9882 64 676 9947 ••11 ••75 •i39 •204 •268 •332 •396 •460 •525 64 677 83o589 0653 T3^ 0781 0845 0909 i5jo 0973 1037 1102 1166 64 678 I230 1294 1934 1422 i486 1614 1678 1742 1806 64 679 1870 1998 2062 2126 2189 2253 23i7 238i 2445 64 680 832509 2573 2637 2700 2764 2828 2892 2956 3020 3o83 64 681 3i47 32II 3275 3338 3402 3466 3530 3593 423o 3657 3721 64 682 3784 3848 39.2 3975 4039 4io3 4166 4294 4357 64 688 4421 4484 4548 461 1 4675 il',^ 4802 4866 4929 4993 64 684 5o56 5l20 5i83 5247 53io 5437 55oo 5564 5627 63 685 5691 5754 5817 588i 5944 6577 6007 6071 6i34 6197 6fi6i 63 686 6324 6387 645 1 65i4 6641 6704 6767 683o 6894 63 087 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 8i56 63 688 7588 8219 7652 8282 ll\l 7778 7841 8471 zi 7967 8o3o 8093 63 689 8408 8597 8660 8723 8786 63 690 838849 8912 8975 9o38 9101 9164 9227 9280 o545 9352 941 5 63 691 9478 9541 9604 9667 9729 9792 9855 9981 ••43 63 692 840106 0169 0232 0294 0357 0420 0482 0608 0671 63 693 0733 0796 o859 0921 0984 1046 1109 1172 1234 1297 63 694 1359 1422 1485 1547 1610 1672 1735 1797 i860 1922 63 695 1985 2047 2110 2172 2235 2297 236o 2422 2484 2547 62 696 2609 2672 2734 2796 2859 3544 2983 3046 3io8 3170 62 697 3233 3295 3357 3420 3482 36o6 3669 3731 3793 62 698 3855 3918 3o8o 4042 4104 4166 %ll 4291 4353 44i5 62 699 4477 4539 4001 4664 4726 4788 4912 4974 5o36 62 N. 1 2 3 4 5 6 7 8 9 D. 22 LOGARITHMS OF NUMBER^. N. 1 2 8 4 5 6 7 8 9 D. 700 845098 5i6o 5222 5284 5346 5408 5470 5532 5594 5656 62 701 57.8 5780 5842 5904 5966 6028 6090 6i5i 62i3 6275 62 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 i 6894 62 708 8180 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 704 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 705 8231 83.2 8374 8435 8497 8559 8620 8682 8743 62 706- 88o3 8866 8928 8989 905 1 9112 9'74 9235 9297 9358 61 707 9419 85oo33 9481 9542 9604 9665 9726 9788 9849 9911 till 61 708 0095 01 56 0217 0279 o34o 0401 0462 0324 61 709 0646 0707 0769 o83o 0891 0952 1014 1075 ii36 1197 61 710 851258 l320 i38i 1442 i5oJ i564 1625 1686 1747 1809 61 711 1870 1931 2541 1992 2o53 2114 2175 2236 2297 2358 2419 61 712 2480 2602 2663 2724 2785 2846 2907 2968 3029 61 71S 3090 3i5o 32II 3272 3333 3394 3455 35i6 3577 3637 61 714 3698 3759 3820 388i 3941 4002 4o63 4124 4i85 4245 61 715 43o6 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 716 4913 5519 4974 5580 5o34 5095 5i56 5216 5277 5337 5398 5459 61 717 5640 5701 5761 5822 5882 5943 6oo3 6064 61 718 6124 6i85 6245 63o6 6366 6427 6487 6548 6608 6668 60 719 6729 6789 685o 6910 6970 703 1 7091 7152 7212 7272 60 720 857332 7393 7453 75i3 7574 8176 7634 7694 7755 78.5 7875 60 721 7935 7995 8o56 8116 8236 8297 8357 8417 8477 60 722 8537 8597 8657 8718 8778 8838 8898 8958 9559 9018 9078 60 728 9i38 9.98 9258 9318 9379 9978 9439 9499 9619 9679 60 724 9739 mi 985o 0458 9918 ••38 ••98 •i58 •218 •278 60 725 86o338 o5i8 0578 0637 0697 0757 0817 0877 60 726 0937 1 534 0996 io56 1116 1176 i?36 1295 i355 I4i5 1475 60 727 1594 1654 1714 1773 1833 1893 2489 iq52 2012 2072 60 728 2l3l 2191 225l 23io 2370 243o 2549 2^>o8 2668 60 729 2728 2787 2847 2906 2966 3o25 3o85 3144 3204 3263 60 730 863323 3382 3442 35oi 356i 3620 368o 3739 3799 3858 59 731 30,7 45ii 3977 4o36 4096 4i55 4214 4274 4333 4392 4q85 4452 59 732 4570 463o 4689 4748 4808 4867 4926 5045 59 / 733 5io4 5i63 5222 5282 5341 5400 5459 55i9 5578 5637 59 734 5696 6287 5755 58i4 5874 5933 6524 IVsl 6o5. 6110 6169 6228 59 735 6346 6405 6465 6642 6701 6760 6819 5q 736 6878 6937 7526 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7585 7644 7703 7762 7821 7880 7939 7998 59 738 8o56 8n5 8174 8233 8292 83 5o 8409 8468 8527 8586 59 739 8644 8703 8762 8821 8879 8938 8997 9o56 9114 9173 59 740 869232 9290 9349 9408 9466 9525 9584 9642 9701 9760 ^ 741 9818 9877 9935 9994 ••53 •hi •170 •228 •287 •345 59 742 870404 0462 0521 o579 0638 0696 0755 o8i3 0872 0930 58 743 0989 1047 1 106 1164 1223 1281 i33q 1398 1456 i5i5 58 744 1573 i63i 1690 1748 1806 i865 1923 1 98 1 2040 2098 58 745 21 56 22l5 2273 233 1 2389 2448 25o6 2564 2622 2681 58 746 2739 2797 2855 29.3 2972 3o3o 3o88 3i46 3204 3262 ^S 747 3321 3379 3437 3495 3553 36ii 3669 3727 3785 3844 58 748 8902 3960 4540 4018 4076 4134 4192 425o 43o8 4366 4424 58 749 4482 4598 4656 4714 4772 483o 4888 4945 5oo3 58 750 875061 5ii9 5.77 5235 5293 535i 5409 5466 5524 5582 58 751 1 5640 5698 5756 58i3 5871 5929 5q87 6045 6102 6160 58 752 ! 6218 6276 6333 6391 6449 65o7 6564 6622 6680 6737 58 753 j 6795 6853 6910 6g68 7026 7083 7141 7199 7256 73i4 58 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 755 j 7947 8004 8062 8119 8l7T 8234 8292 8349 8407 8464 57 756 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 757 9096 9669 9.53 9211 9268 9^5 9383 9440 1 9497 9555 9612 ^■^ 758 9726 9784 9841 9898 9056 ••i3 ••70 •127 •i85 ^7 759 880242 0209 o356 o4i3 0471 0528 o585 0642 0699 0756 57 N. 1 1 2 3 4 5 6 7 8 9 D. tOGAElTHMg OF NUiVfUEilg. 23 I N. 1 2 3 4 5 6 7 8 9 B. 760 88o8u 0871 0928 rfd 1042 1099 ii56 I2l3 1271 i328 57 761 i385 1442 1499 i6i3 1670 1727 1784 1841 1898 57 7C2 ;?^f 2012 2069 2126 2i83 2240 2297 2354 241 1 2468 57 763 258i 2638 2695 2752 2809 2866 2923 2980 3o37 57 764 3003 3i5o 3207 3264 3321 3377 3434 3491 3548 36o5 57 765 366 1 3718 3775 3832 3888 3o45 4002 4059 4ii5 4172 57 766 4229 4795 4285 4342 4399 4455 43 12 4569 4623 4682 4739 57 767 4852 4909 4960 5022 5o-jS 5i35 5192 5248 53o5 57 763 536i 5418 5474 553 1 5587 5644 5700 5757 58i3 6870 57 769 5926 5983 6039 6096 6i52 6209 626D 6321 6878 6434 56 770 886491 7054 6547 6604 6660 6716 6773 6829 6885 8067 8629 6098 7561 56 771 7m 7167 7223 7280 7336 7392 8573 56 772 773 7617 8179 ni3o 8292 8348 7842 8404 7898 8460 Itl 8123 8685 56 56 774 8741 9302 8797 88d3 8909 8965 9526 9021 9077 9134 9190 9246 56 775 9358 9414 9470 9582 9638 9694 9750 9806 56 776 9862 9918 9974 ••3o ••86 •141 •197 0766 •233 •309 ■ •365 56 777 890421 0477 0533 0589 0645 0700 0812 0868 0924 56 778 0980 i537 io35 1091 1705 I203 1259 1816 i3i4 1370 1426 14S2 56 779 1593 1649 1760 1872 1928 1983 2039 56 780 892095 2i5o 2206 2262 23i7 2373 2429 2484 2540 2595 3i5i 56 781 26D1 2707 2762 33i8 2818 2873 2929 2980 3040 3096 56 782 3207 3262 3373 3429 3484 3340 3595 365i 3706 56 783 3762 3817 3873 3928 3984 4039 4593 4094 4.5o 42o5 4261 55 784 43 16 4371 4427 4482 4538 4648 4704 4759 4814 55 785 4870 4925 4980 5o36 5091 5146 5201 5257 53i2 5367 55 786 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 55 787 6526 6o3o 6o85 6140 6195 625i 63o6 636i 6416 6471 55 788 658i 6636 6692 6747 6802 6857 6912 6067 7517 7022 55 789 7077 7i32 7187 7242 7297 7352 7407 7462 7572 55 790 897627 7682 7737 8286 lit iu S? IfJ. 8012 8067 8122 55 791 8176 823i 856i 861 5 8670 55 792 8725 8780 8835 8890 9437 8944 8999 9547 9054 9109 9164 9218 55 793 9273 932S 9383 9492 9602 9656 971i 9766 55 794 9821 9875 9930 9985 ••39 ••94 •149 •203 •258 •3l2 55 795 900367 0422 0476 053 1 o586 0640 0695 0749 1295 0804 0859 55 796 0Ql3 0963 1022 1077 ii3i 1186 1240 i349 1404 55 797 1458 i5i3 1 567 1622 1676, 1731 1785 1840 1894 1948 54 798 2003 2057 2112 2166 222Z 2275 2329 2873 2384 2438 2492 54 799 2547 2601 2655 2710 2764 2818 2927 2981 3o36 54 800 '16?? 3 1 44 3199 3253 3307 336i 3416 3470 3524 3578 54 801 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 802 4174 4229 4283 4S1I 4391 4932 4445 4499 4553 4607 4661 54 803 4716 4770 4824 49'^6 5o4o 5094 5i48 5202 54 804 5256 53io 5364 5418 5472 5526 558o 5634 56S8 5742 54 805 5796 5850 5904 5958 6012 6066 6119 6658 6173 6227 6281 54 806 6335 6389 6443 6497 655i 6604 6712 6766 6820 54 807 6874 6927 6981 7035 7089 7143 7196 725o 7304 7358 54 808 741 1 7465 7573 8110 7626 7680 8217 7734 7787 7841 IX 54 809 7949 8C02 8i63 8270 8324 8378 54 54 810 908485 8539 8592 8646 ^ 8753 8807 8860 8914 8967 811 9021 9074 9128 9181 9289 9342 9396 9449 93o3 54 812 9556 9610 9663 9716 9770 o3o4 9877 9930 9984 ••37 53 813 910091 0144 0107 0731 025l o358 041 1 0464 o5i8 0571 53 814 0624 0678 0784 o838 0891 0944 0098 io5i 1 104 53 815 ii58 1211 1264 i3i7 1371 1424 1477 i53o i584 1637 53 816 1690 1743 >797 i85o 1903 1956 2009 2063 2116 2169 53 817 2222 2275 2328 238i 2435 2488 2541 2594 2647 2700 53 818 2753 2806 2859 2913 2966 3019 3072 3i25 3178 3231 53 819 3284 3337 3390 3443 3496 3549 36o2 3655 8708 3761 53 N. 1 2 3 4 5 6 7 8 9 D. 24 LOGARITHMS OF NtJMBEltS. N. 82 82t; 8^7 828 829 9i38i4 4343 487'^ 1 :-4uo 5927 6454 6980 7506 8o3o 8555 1 2 8 4 6 6 ' ' 7"' s' 9 D. 3867 4396 49'^5 5453 5980 65o7 7033 7558 8o83 8607 3920 4449 ^977 5oo5 6o33 6559 7085 7611 8i35 8659 3973 45o2 5o3o 5558 6o85 6612 7i38 7663 8188 8712 4026 4555 5o83 56ii 6i38 6664 7190 7716 8240 8764 5i36 5664 6191 6717 7243 7768 8293 8816 4i32 4660 5189 5716 6243 6770 7295 7820 8345 8869 4184 4713 5241 5769 6296 6822 7348 il^ 8921 4237 4766 5294 5822 6349 6875 7400 7925 8450 8973 4290 4819 5347 5875 6401 6927 7433 85o2 9026 53 53 53 53 53 53 53 52 52 52 880 831 832 888 834 885 836 837 888 889 919078 9601 0201 23 0645 1166 1686 2206 2725 3244 3762 9i3o 9653 0176 ISl 1738 2258 2777 3296 38i4 9183 9706 0228 0749 1270 1790 23lO 2829 3348 3865 9235 9758 0280 0801 l322 1842 2362 2881 3399 3917 9287 9810 o332 o853 1374 1894 2414 2933 345i 3969 9340 9862 o384 0906 1426 1946 2466 2985 3oo3 4021 9392 9914 0436 0958 1478 i??8 3o37 3555 4072 9444 9967 0489 lOIO i53o 2o5o 2570 3089 3607 4124 9496 ••19 o54i 1062 1 582 2102 2622 3i4o 3658 4176 2^^? 0593 1114 1 634 2i54 2674 3192 3710 4228 52 52 52 52 52 52 52 52 52 52 840 841 842 843 844 845 846 847 848 S49 924279 4796 53 1 2 5828 6342 6857 7370 7883 b396 8908 433i 4848 5364 5879 6394 6908 8447 8959 4383 i^ 5931 6445 6959 7473 7986 8498 9010 4434 4951 5467 5982 6497 7011 7524 8037 8549 9061 4486 5oo3 55i8 6o34 6548 7062 ^576 8088 8601 9112 4538 5o54 5570 6o85 6600 7H4 7627 8140 8652 9163 4589 5io6 5621 6137 665 1 7165 7678 8191 8703 9215 4641 5i57 5673 6188 6702 7216 7730 8242 8754 9266 4693 5209 5725 6240 6754 7268 8293 88o5 9317 4744 5261 5776 6291 68o5 7319 7832 8345 8857 9368 52 52 52 5i 5i 5i 5i 5i 5i 5i 850 851 852 853 854 855 856 857 858 859 929419 9930 930440 0949 1458 1966 2474 2981 9470 9981 049' 1000 1 509 2017 2524 3o3i 3538 4044 9521 ••32 o542 io5i i56o 2068 2575 3082 3589 4094 0592 1 1 02 1610 2118 2626 3i33 3639 4145 9623 •i34 0643 ii53 1661 2,69 2677 3i83 3690 4195 9674 •i85 0694 1204 1712 2220 2727 3234 3740 4246 9725 •236 0745 1254 1763 2271 2778 3285 3791 4296 9776 •287 0796 i3o5 1814 2322 2829 3335 3841 4347 9827 •338 0847 i356 1 865 2372 lilt 3892 4397 nil 0898 1407 1915 2423 2930 3437 3943 4448 5i 5i 5i 5i 5i 5i 5i 5i 5i 5i 860 861 862 863 864 865 866 867 868 869 934498 5oo3 5507 60U 65i4 7016 75i8 8019 8520 9020 4549 5o54 5558 6061 6564 7066 7568 8069 8570 9070 4599 5io4 56o8 6111 6614 7117 7618 8119 8620 9120 465o 5i54 5658 6162 6665 7167 7668 8169 8670 9170 4700 52o5 5709 6212 6715 7217 7718 8219 8720 9220 475r 5255 5759 6262 6765 7267 7769 8269 8770 9270 4801 53o6 5809 63i3 68i5 7317 8320 8820 9320 4852 5356 586o 6363 6865 7867 7869 8370 8870 9369 4902 5406 5910 64i3 6916 7418 7919 8420 8920 9419 4953 5457 5960 6463 6966 7468 7969 8470 8970 9469 5o 5o 5o 5o 5o 5o 5o 5o 5o 5o 870 871 872 878 874 875 876 877 878 879 93951Q 940018 o5i6 1014 i5n 2008 25o4 3ooo 3495 3989 9569 0068 o566 1064 i56i 2o58 2554 3o49 3544 4o38 961Q 0118 0616 jn4 1611 2107 2603 3099 3593 4088 0168 0666 n63 1660 2i57 2653 3148 3643 4137 97'9 0218 0716 I2l3 1710 2207 2702 3198 9769 0267 0765 1263 1760 2256 2752 3247 3742 4236 9819 o3i7 o8i5 i3i3 1809 23o6 2801 3297 3791 4285 9869 0367 o865 i362 1859 2355 285 1 3346 3841 4335 99.8 0417 0915 1412 1909 24o5 2901 3396 3890 4384 9968 0467 0964 1462 1958 2455 2o5o 3445 5o 5o 5o 5o 5o 5o 5o 49 49 49 N. 1 2 8 4 5 6 7 8 9 D. logAbithms of ntjmbebs as N. 1 2 3 4 5 6 7 8 9 D. 880 881 882 883 884 885 886 887 88S 889 944483 it 5961 6452 6943 7434 7924 8413 8902 4532 5o25 55i8 6010 65oi 6992 74»3 7073 8462 8951 458i 5074 5567 6059 655i 7041 7532 8022 85ii 8999 463i 5i24 56i6 6108 6600 7090 7581 8070 856o 9048 4680 5173 5665 6157 6649 7140 763o 8119 8609 9097 4729 5222 57,5 6207 6698 7189 8657 9146 4779 5764 6256 6747 7238 7728 8217 8706 9195 4828 5321 58i3 63o5 6796 7287 7777 8266 8755 9244 5370 5862 6354 6845 7336 7826 83x5 8804 9292 4927 5419 5912 6403 6894 7385 2875 8364 8853 9341 49 49 49 49 49 49 49 49 49 49 890 891 892 893 8'J4 695 896 897 898 899 949390 9878 95o365 o85i i338 1823 23o8 2792 3276 3760 9439 9926 0414 0900 1386 2356 2841 3325 38o8 9488 9975 0462 1920 24o5 2889 3373 3836 9536 ••24 o5ii 0997 1483 Iti 2938 3421 3905 9585 "ll 1046 i532 2017 2502 2986 3470 3953 9634 •121 0608 1095 i58o 2066 255o 3o34 35i8 4001 9683 •170 0657 1143 1629 2114 2599 3o83 3566 4049 973. •219 0706 1192 1677 2i63 2647 3i3i 36i5 4098 0734 1240 1726 2211 2696 3i8o 3663 4146 o8o3 I 2 fig 1773 2260 2744 3228 37.1 4194 49 49 49 49 it 48 48 48 48 900 901 902 903 904 905 906 907 908 1'09 954243 47^5 5207 5688 6168 7607 8086 8564 4291 4773 5255 5736 6216 6697 7176 7655 8i34 8612 4339 4821 53o3 5784 6265 6745 7224 7703 bi8i 8659 4387 4869 535i 5832 63i3 6793 7272 7751 8229 8707 4435 4918 5J99 5880 6361 6840 7320 7799 fe? 4484 4966 5447 5928 6409 6888 7368 7847 8325 88o3 4532 5oi4 5495 5976 6457 6936 7416 7894 8373 885o 458o 5o62 5543 6024 65o5 6984 7464 IT. 8898 4628 5iio 5592 6072 6553 7032 7512 IZ 8946 4677 5i58 5640 6120 6601 7080 til 85i6 8994 48 48 48 48 48 48 48 48 48 48 910 911 912 913 914 915 916 917 918 919 959041 9018 9995 960471 0946 1421 1895 2369 2843 33i6 9089 9566 ••42 o5i8 0994 1469 1943 2417 2890 3363 9137 9614 ••90 0566 1041 i5i6 1990 2404 2937 3410 9i85 «J o6i3 1089 i563 2038 25ll 2985 3457 9232 0661 ii36 1611 2o85 2559 3o32 35o4 9280 9757 •233 0709 1184 1658 2l32 2606 3?79 3DD2 9328 9804 •280 0756 I23l 1706 2180 2653 3126 3599 9375 9^32 •328 0804 2227 2701 3174 3646 9423 9900 •376 o85i i326 1801 2275 2748 3221 3693 9471 9947 •423 0899 1374 1848 2322 2795 3268 3741 48 48 48 48 47 47 47 47 47 47 920 921 922 923 924 925 926 927 928 929 963788 4260 473 1 5202 5672 6142 661 I 7080 7548 8016 3835 4307 477» 5249 ■5719 6189 6658 7595 8062 3882 4354 4825 5296 5766 6236 6705 7173 7642 8109 3929 4401 4872 5343 58i3 6283 6752 7220 7688 8i56 4448 i%l 5860 6329 6799 7267 .735 8203 4024 4495 4966 5437 lr,i 6845 7314 2782 8249 4071 4542 5oi3 5484 5954 6423 6892 736i 7829 8296 4118 4590 5o6i 5531 6001 6470 6939 7408 7875 8343 4i65 4637 5io8 5578 6048 6986 7434 7922 8390 4212 4684 5i55 5625 6095 6564 7033 7501 7969 8436 47 47 47 47 47 47 47 47 47 47 980 931 932 938 934 935 936 937 938 989 N. 968483 8950 9416 9882 970347 0812 1276 1740 2203 2666 8530 8996 0393 o858 l322 1786 2249 2712 8576 9043 9009 9975 0440 0904 1369 i832 2295 2708 8623 9090 9Do6 ••21 0486 0931 I4i5 1879 2342 2804 8670 9i36 0602 ••68 o533 0997 1461 'i^ 285i 8716 9183 9649 •114 0579 1044 i5o8 1971 2434 2897 8763 9229 9693 •161 0626 1090 1554 2018 2481 2943 8810 9276 0742 •207 0672 ii37 . 1601 2064 2527 2989 8856 9323 9789 •254 0719 ii83 1647 2110 2573 3o35 8903 9369 9835 •3oo 0765 ;ip 2137 2619 3082 47 47 47 46 46 46 46 46 1 2 8 4 5 6 7 8 9 D. 26 LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 ' 8 9 D. 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 973128 35^0 4031 45i2 4972 5432 • 689 1 63oo 6808 7266 3i74 3636 tt^l 5oi8 5478 5937 6396 6854 7312 lltt 8683 9i38 9594 0049 o5o3 0957 1411 1864 3320 3682 4143 4604 5o64 5524 5983 6442 6900 7358 3266 3728 4189 465o 5iio 5570 6029 6488 6946 7403 33i3 3774 4235 5iD6 56i6 6075 6533 6992 7449 3359 3820 4281 4742 5202 5662 6121 6579 7037 7493 34o5 3866 4327 4788 5248 5707 6167 6625 7083 7541 345i 3gi3 4374 4834 5294 5,53 6212 6671 7129 7586 3497 3939 4420 4880 5340 5799 6258 6717 7175 7632 3543 4oo5 4466 4926 53b6 5845 63o4 6763 7220 7678 46 46 46 46 46 46 46 46 46 46 977724 8181 8637 9093 9548 980003 0458 1819 78,5 8272 8728 9184 9639 0094 o54o ioo3 1456 1909 7861 83,7 «774 923o 9685 0140 0594 1048 i5oi 1954 8819 9270 9730 oi85 0640 1093 1 547 2000 8409 8865 9321 9776 023l 0685 1139 1592 2040 8911 9366 9821 0276 0730 1184 1637 2090 8043 85oo 8956 9412 9867 o322 0776 1229 i683 2135 8089 8546 9002 9^57 0821 1275 1728 2l8l 81 3,1 8591 9047 95o3 9958 0412 0867 l320 1773 2226 46 46 46 46 46 45 45 45 45 45 960 961 962 963 964 965 966 967 963 969 982271 2723 3626 4077 4527 nil 5875 6324 23i6 2769 3220 3671 4122 4572 5022 5471 5920 6369 2362 2814 3265 37.6 4167 4617 5067 55.6 5965 64i3 2407 2859 33io 3762 4212 4662 5lI2 5561 6010 6458 2452 2904 3356 3807 4257 5i57 56o6 6o55 65o3 2497 2949 3401 3852 43o2 4752 5202 565 1 6100 6548 2543 2994 3446 3897 4347 4797 5247 5696 6144 6593 2588 3o4o 3491 3942 4392 4842 5292 5741 6189 6637 2633 3u85 3536 3987 4437 4887 5337 5786 6234 6682 2678 3i3o 358i 4o32 4482 4932 5382 583o 6279 6727 45 45 45 45 45 45 45 45 45 45 970 971 972 973 974 975 9:6 977 978 979 986772 7219 7666 8n3 8559 9000 9459 6817 7264 8604 9049 9494 0827 6861 7309 7756 8202 8648 9094 t^M 0428 0871 6906 7353 7800 8247 8693 9138 9583 ••28 0472 0916 6951 7398 7845 8291 till 9628 ••72 o5i6 0960 6996 7443 7890 8336 8782 9227 9672 •117 o56i 1004 7040 7488 8826 9272 9717 •161 o6o5 1049 7085 7532 ^^4^5 8871 9316 9761 •206 o65o 1093 7577 8024 8470 8916 9361 9806 •25o 0694 II37 7175 7622 8068 85i4 8960 94o5 9800 •294 0738 1182 45 45 45 45 45 45 44 44 44 44 980 981 9S2 933 984 935 936 9^7 9^8 989 991226 1669 2III 2554 2995 3436 1270 1713 2156 2598 3o39 3480 3921 436i 4801 5240 i3i5 1758 2200 2642 3o83 3524 3965 44o5 4845 5284 1359 1802 2244 2686 3568 4009 4449 4689 5328 i4o3 1846 2288 2730 3172 36i3 4o53 4493 4933 5372 1448 1890 2333 2774 3216 3657 4097 4537 4977 5416 1492 1935 2377 2819 3260 3701 4141 458i 502I 5460 1536 1979 2421 2863 3304 3745 4i85 4625 5o65 55o4 i58o 2023 2465 3S 3789 4229 4669 0108 5547 1625 2067 25o9 2951 3392 3833 4273 4713 5i52 5591 44 44 44 44 44 44 44 44 44 44 900 ii'.il W2 993 994 995 906 997 9'i8 999 995635 6074 65i2 %l 7823 8259 ^', 9565 5679 6117 6555 6993 743o Ifol 8739 9>74 9609 5723 6161 6599 7037 7474 7910 8J47 8782 9218 9652 6643 7080 7517 79^4 8390 8S26 9261 9696 58ii 6249 6687 7124 756i l^l 8869 93o5 9739 5854 6293 6731 7168 7605 8041 8477 8913 9348 9783 5898 6337 6774 7212 7648 8o85 85a I 8956 9392 9826 5942 638o 6818 7255 7692 8129 8564 9000 9435 9870 5986 6424 6862 7299 7736 8172 8608 9043 9479 9913 6o3o 6468 6906 7343 7779 8216 8652 9087 9522 9957 44 44 44 44 44 44 44 44 44 43 "■1 ' i 2 3 4 5 6 7 8 9 D. LOGARITHMS OF NUMBERS. 27 N. lOOO 1 2 3 4 5 6 ■^ 8 9 D. 0043 0087 oi3o 0174 0217 0260 o3o4 o347 0391 43 lOOI 0434 0477 052I o564 0608 06 n 0694 0738 0781 0824 43 1002 o«68 1344 0954 0908 l43i 1041 iob4 1128 1171 1214 1258 43 ioo3 i3oi 1 388 «474 i5i, i56i 1604 1647 1690 43 1 004 1734 1777 1820 • 863 Zl 1950 2382 1993 2o36 2080 2123 43 ioo5 002166 2209 2252 2296 2420 2468 25l2 2-.55 43 1006 2598 2641 2684 2727 2771 2814 2807 IMf, 2986 43 1007 3029 3073 3ii6 3i59 3202 3245 3288 34.7 43 1008 3461 35o4 3547 3590 3633 3676 3719 3762 1 38o5 3848 43 1009 3891 00432 1 3934 3977 4020 4o63 4106 4149 4192 4235 4278 43 lOIO 4364 4407 445o 4493 4536 4579 4622 4665 4708 43 ion 47^1 4794 4837 4880 4923 5352 4966 5009 5o52 5095 5i38 43 1012 5i8i 5223 5266 5309 5738 5395 5438 5481 5524 5567 43 ioi3 5609 5652 5695 5781 5824 5867 l^s 5952 6380 5995 43 1014 6o38 6081 6i?4 6166 6209 6202 6295 6423 43 lOlD 006466 6 509 6936 7364 6552 6594 6637 7065 6680 6723 6765 6808 685 1 43 1C16 6894 6979 7022 7107 7i5o 7193 7200 7278 43 1017 7321 7406 7449 7492 7534 7577 7620 8046 7662 85i? 7705 8i32 43 50l8 7748 7790 8217 7833 8259 7876 7Qlu 8387 S004 ■43 1019 8174 83o2 8^45 843o 8472 8558 43 1020 008600 8643 8685 8728 8770 88i3 8856 8898 8941 8983 43 1021 9026 9068 9111 9153 9196 923s 9281 9323 9366 94°? 42 1022 9451 9493 9536 9578 9621 9663 9706 9748 9791 9833 42 1023 9876 9918 o342 & •ooo3 •0045 .0088 .oi3o •0173 •02l5 • 0258 42 1024 oio3oo 0427 0470 o5i2 0554 0597 o63o io63 0681 42 1025 010724 0766 0809 o85i 0893 oq36 i3.-)9 0978 1020 iio5 42 1026 1 147 1190 1232 1274 i3i7 1401 1444 i486 i528 42 1027 1028 1570 i6i3 i655 1697 1740 1782 1824 1866 ;s? 1951 2373 42 1993 203") 2078 2120 2162 2204 2247 2289 42 1029 24i5 2458 25oo 2542 2584 2626 2669 2711 2753 2795 42 io3o 012837 2879 2922 2964 3385 3oo6 3048 3090 3i32 3174 3217 42 io3i 3259 33oT 3343 3427 3469 35ii 3553 3596 3638 42 I032 368o 3722 3764 38o6 3848 3890 3932 lV4 4016 4o58 42 io33 4100 4142 4184 4226 4268 43io 4353 4437 4479 4898 42 1034 4321 4563 46o5 4647 t^ 4730 47.72 4814 4856 42 io35 014940 5360 4982 5o24 5o66 5i5o 5,92 5234 5276 53i8 42 io36 5402 5444 5485 5527 598? 0611 5653 5695 5737 42 wil 5779 5821 5863 5904 5o46 6365 6o3o 6072 6114 6i56 42 6197 6239 6281 6323 6407 6448 6490 6532 6574 42 1039 6616 6657 6699 6741 6783 6824 6866 6908 6950 6992 42 1040 017033 7075 7117 7159 7200 7242 7284 7326 7367 7409 42 1041 745 1 7492 7534 7576 7618 8o34 7659 7701 1 7743 7784 7826 42 1042 7868 8284 IT,1 Its 7993 8076 8118 8x59 8201 8243 42 1043 8409 8825 845 1 8492 8534 8576 8617 8659 9075 42 1044 8700 8742 8784 8867 8908 9324 8950 9366 8992 9033 42 1045 019116 9i58 9199 9241 9282 9407 9449 9490 42 1046 9532 9573 9656 9698 9739 9781 9822 9864 9905 •o320 42 1047 9947 9988 • oo3o • 0071 •oii3 • 0154 • 0195 • 0237 •0278 41 1048 o2o36i o4o3 0444 0486 o527 o568 0610 o65i 0693 0734 41 1049 0775 0817 o858 0900 0941 0982 1024 io65 1107 1148 41 io5o 021189 i6o3 123l 1272 i3i3 i355 1396 1437 1479 |520 i56i 41 io5i 1644 1685 1727 1768 1809 i85i 1892 1933 2346 •,W; 41 io52 2016 2057 209H 2140 2181 2222 2263 23o5 41 io53 2428 2470 23l 1 2552 2593 2635 2676 2717 2758 2799 41 io54 2H41 2H82 2923 2964 3oo5 3047' 3o88 3129 3170 32II 41 io55 023232 329', 3335 3376 3417 3458, 3499 3828 3870 1 3911 4239, 4280; 4 521 3541 3582 3623 41 io56 3664 3705 3746 3787 3952 4363 3993 4o34 41 io57 io3S 4075 4116 4157 4198 4404 4445 41 44B6 4527 4568 4609 -i^^^o ! Abqi I 4732 4773 4814 4855 41 ioSq 4896 4937 4978 5oi9 5o6o 5ioi 5i42 5i83 5224 5265 41 r« 1 2 3 4 6 6 7 8 9 D. 28 f.O0ARlTtfM8 OP NUilBEES. N. 1 2 3 4 5 6 1 8 9 D. 1060 o253o6 5347 5388 5420 5838 5470 55ii 5552 5593 5634 5674 4t .061 5715 5756 5797 5879 5920 6329 5o6i 6370 6002 6043 6084 41 J 062 6125 6i65 6206 6247 6288 6411 6452 6492 41 ro63 6533 6574 6082 7390 661 5 6656 6697 6737 7146 6778 6819 6860 6901 41 1064 6942 027350 7023 7064 7io5 7.86 7227 7268 7300 7716 41 io65 1 743 1 7472 75i3 7553 & 7635 8042 7676 41 1066 7757 7798 8205 7839 7«79 7920 8368 8o83 8124 41 1067 8164 8246 8287 8693 8327 8409 88i5 8449 8490 853 1 41 1068 8571 8612 8653 8734 8775 9181 8856 8896 ¥1 41 1069 8978 9018 9059 9100 9140 9221 9262 93o3 9343 41 1070 029384 9424 9465 9506 9546 9587 9^27 • 0033 "^68" 9708 9749 41 1071 9789 9830 9871 To m 9992 .0073 •0114 • oi54 41 1072 03019D 0235 0276 0397 0438 0478 o5i9 0923 i328 0559 40 1073 0600 0640 0681 0721 0762 0802 0843 o883 0964 1 368 40 1074 1004 1045 io85 1126 1 166 1206 1247 1287 40 1075 o3i4o8 1449 i85i 1489 1893 i53o 1570 1610 i65i 1691 .732 1772 40 1076 1812 ;?^^ IV-,', 2014 2o54 2095 2i35 2175 40 1077 2216 2256 2296 2417 2458 2498 2538 2578 40 1078 2619 ?659 2699 2740 2780 2820 2860 2901 33o3 2941 3343 2981 3384 40 1079 302I 3062 3l02 3i42 3582 3223 3263 40 1080 033424 3464 35o4 3544 3585 3625 3665 3705 3745 3786 40 1081 3826 3866 3906 43o8 3946 4348 3986 4388 4027 4067 4107 45o8 45S 4588 40 1082 4227 462B 4267 4428 4468 40 io83 4669 4709 4749 4789 4829 4869 tu 4949 5330 it 40 1084 5029 5069 5,09 5i49 5,90 523o 5270 40 1085 o3543o 5470 55io 555o 5590 563o 5670 5710 5750 5790 40 1086 583o 5870 5910 6309 5950 6349 IVsl 6o3o 6070 6II0 6i5o 6100 698? 7387 40 1087 623o 6269 6429 6828 7227 6469 6868 7267 65o9 6908 7307 6549 40 1088 1089 6629 7028 6669 7068 670Q 7108 6740 7148 6789 7187 6948 7347 40 40 1090 037426 7466 ~^o6 7546 7586 7626 7665 i7o5 81 o3 vS 7785 40 1091 7825 8223 7865 7904 83o2 7944 79«4 8342 8382 8024 8064 8i83 40 1092 8262 8421 8461 85oi 8541 858o 40 1093 8620 8660 8700 8739 8779 8819 9216 8859 8898 8938 9335 8978 9^74 40 1094 90.7 9057 9097 9493 9i36 9176 925d 9295 40 1095 039414 9454 9533 9573 9612 9652 9692 973 1 977' 40 1096 981 1 985o fz 9929 o325 o36? • 0009 o4o5 • 0048 .0088 • 0127 o523 • 0167 40 1097 1098 040207 0246 0444 0484 0563 40 0602 0642 0681 0721 0761 0800 0840 0879 T, 0958 1 353 40 1099 0998 1037 1077 1116 n56 1195 1235 1274 39 N. 1 2 3 4 5 6 7 8 9 D. % OONTAININO KATURAL SINES AND COSINES, . LOGAKITHMIC SmES, COSmES, TAIS^GENTS, ANB COTANGEISTTS, EVERT DEGREE AND MINUTE OF THE QUADRANT. 30 TRIGOis'OMETRlCAL FUNCTIONS. — 0° Nat. Functions. Logarithmic Functions + 10. ~~^ N.sine. N.C08. L. sine. D.l'^ L.CO8. iD.l" L. tang. D. 1" L. cot 1 ooooo Unit. • 000000 10-000000 0-000000 1 Infinite. |60 1 00029 Unit. 6-463726 5oi7 n 000000 •00 6-463726 5oi7 11 13.536274:59 2 ooo58 Unit. 764756 2934 85 000000 •00 764756 2934 235244 oS 3 00087 Unit. 940847 2082 3i 000000 -00 940847 2082 3i 059153 57 4 00116 Unit. 7.065786 i6i5 11 000000 •00 7.065786 i6i5 17 12.934214 56 837304 55 5 00145 Unit. 162696 i3i9 000000 •00 162696 i3i9 69 6 00175 Unit. 241877 iii5 75 9.999999 •01 241878 iii5 78 758122 i 54 7 00204 Unit. 308824 966 53 999999 -Of 308825 t. 53 601 175 53 8 00233 Unit. 366816 852 54 999999 .01 366817 54 633 1 83 52 9 00262 Unit. 417968 762 63 999999 999998 •01 417970 463727 762 63 582o3o 51 10 00291 Unit. 463725 689 88 -01 689 88 536273 i 50 11 00320 99999 7.5o5ii8 629 81 9.999998 .01 7 '505 1 20 629 81 12.494880 49 12 oo34o 00378 99999 542906 579 36 999997 .01 542909 579 33 457091 48 13 99999 577668 536 41 999997 .01 577672 536 42 422328147 14 00407 99999 609853 499 38 999996 •01 609857 499 39 390143 46 15 00436 99999 639816 467 438 14 999996 .01 639820 467 i5 36oi8o 45 16 00465 99999 667845 81 999995 •01 667849 438 82 332i5i 44 17 00495 99999 694173 4i3 11 999995 -01 694179 4i3 73 3o5S2i 43 18 00324 99999 718997 391 999994 -01 719003 39, 36 280997 42 19 00553 99998 742477 371 :i 999993 .0. 742484 371 28 257516 41 20 00582 99998 764754 353 999993 ■01 764761 35i 36 235239 40 "i^l 0061 1 99998 7-785943 336 72 9.999992 -01 7.785951 336 73 12.214049 'W '22 00640 99998 806146 321 75 999991 .01 806 1 55 321 76 193845 88 23 00669 99998 825451 3o8 o5 999990 .01 825460 3o8 06 174540 87 24 00698 99998 843934 \t 47 999980 999988 .02 843944 295 49 l56o56 86 25 00727 99997 861662 ■ 88 .02 861674 2b3 90 138326 85 26 00706 99997 878605 273 17 999988 .02 878708 273 18 121292 84 27 00785 99997 8950^5 263 23 999987 .02 895099 263 25 1 0490 1 83 28 00814 99997 910879 253 ?? 999986 .02 910894 254 01 089106 82 29 00844 99996 926119 245 999985 .02 926134 245 40 073866 059142 12.044900 31 SO 81 00873 99996 940842 7 •956082 237 33 999983 1 .02 940858 7.955100 _237_ 35 80 29 00902 99996 229 80 9.999982 229 81' 82 00931 99996 96S870 222 73 999981 .02 968889 222 75 o3iiii 28 £3 00960 99995 982233 216 08 999980 .02 982253 216 10 017747 27 £4 00980 01018 99995 995198 8- 007787 ^0? 81 999979 .02 995219 209 83 004781 26 85 99995 ?? 999977 .02 8.007809 203 11 11.992191 979955 25 86 01047 99995 020021 198 999976 .02 020045 198 24 87 01076 99994 0319I9 1 88 02 999975 .02 031945 043627 ,93 o5 96S055 23 88 oiio5 99994 o435oi 01 999973 .02 188 o3 956473 22 89 oii34 99994 054781 1 83 25 999972 .02 054809 i83 27 945191 21 40 01 164 99993 065776 178 72 999971 -02 o658o6 178 74 934194 20 41 01 193 99993 8-076500 086965 •74 41 '■& -02 8-076531 066997 174 44 11.923469 9i3oo3 19" 42 01222 99993 170 3i •02 170 34 18 43 OI25l 99992 097183 166 39 999966 -02 097217 166 42 002783 17 44 01280 99992 107167 162 65 999964 -o3 107202 162 68 16 45 01 309 01338 99991 1 16926 'M 08 999963 -03 VMfo 159 10 15 46 99991 126471 66 999961 .o3 i5d 68 873490 14 47 01367 99991 i358io 1 52 38 999959 999958 •03 i3585i l52 41 864149 13 48 01396 99990 144953 146 24 .03 144996 149 27 855004 12 49 01425 99990 153907 22 999956 .o3 153932 146 27 846048 11 50 51 01454 01483 99989 162681 143 33 999954 9.99995a -o3 -o3 162727 143 36 83/273 10 99989 8.171280 140 54 8.171328I 140 57 11.828672 9 52 oi5i3 9998^ 179713 .37 86 999950 .o3 179763 i88o36 i37 I2 820237 8 53 01 542 187985 i35 29 999948 -03 i35 81 1964 7 54 01571 99988 196102 l32 80 999946 .03 196156 l32 84 8o3844 6 55 01600 99987 204070 i3o 41 999944 • 03 204126 i3o 44 795874 5 56 01629 oi658 99987 211895 128 10 999942 .04 211953 21964.' 128 14 788047 4 57 99986 219581 125 87 999940 .04 125 90 780359 3 58 01687 01716 99986 227134 123 72 999938 • 04 227195 123 76 772805 2 59 99985 234557 121 64 999936 .04 234621 121 68 765379 1 60 01745 99985 241855 119-63 999934 -04 241921 119.67 758079 N.C08. N. sine. L. COS. D.l" L. sine. j L.cot. D. 1" L. tang. ' 89° 1 trigonomI':trical functions. — i' 31 Nat. Functions. Logarithmic Functions + 10. ' ! o! N.slne. N.CP3.' L. sine. D.l" L. COS. ] 3.1", L.tang. D.l" L-cot. 60 01745 1 999S5 8-241855 119-63 9-999934 •041 8.241921 119.67 1. -758079 1 01774199984 oi8o3 1 99984 249033 117 68 999932 •041 249102 117.72 750S98 743833 59 2 256094 ii5 80 999929 .04 256i65 115-84 68 ^ 01832 99983 263o42 113 98 999927 .04 263ii5 114-02 736885 57 4 01862 99983 269881 112 21 999925 .04 269956 112-25 730044 66 b 01891 199982 276614 no 5o 999922 .04 276691 110-54 723309 55 6 01920 : 99982 283243 108 83 999920 .04 283323 108-87 716677 64 ^ 01949 99981 01978 99980 289773 107 21 999918 • 04 289836 107-26 7>oi44 63 8 296207 io5 65 999915 .04 296292 io5-7o 703708 62 9 02007 99980 302546 104 i3 9999 '3 .04 302634 104-18 697366 61 10 11 02036 99979 308794 102 66 999910 • 04 308884 102-70 691116 11-684954 50 49 02o65 99979 99978 8^4957 101 22 9-999907 .04 8.3 1 5046 101-26 12 02094 321027 ^ 82 999903 .04 32II22 n 678878 48 13 02123 99977 327016 47 999902 .04 327114 672886 47 14 02l52l 99977 332924 97 14 999''99 .05 333025 97-19 666975 46 15 O2181 99976 338753 95 86 999897 .05 338856 95-90 661 144 45 16 02211 99976 344504 94 60 999894 • 05 344610 94-65 655390 44 17 02240 99975 35oi8i 93 33 999891 .05 350289 355895 93.43 6497'! 43 18 02269 02298 99974 355783 92 '9 999888 .05 92.24 644105 42 19 99974 36i3i5 88 o3 999885 .05 36i43o 91-08 ^9-95 88-85 638570 41 20 21 02327 99973 366777 8-372171 90 80 _9998^ 9-999879 .o5 .05 366895 8-372292 633io5 II -627708 40 89 "^56 99972 22 02385 99972 377499 87 72 999876 .o5 377622 87-77 622378 38 23 02414 99971 382762 86 67 999873 .o5 382889 86-72 617111 37 24 02443 99970 387962 393101 85 64 999870 .o5 388092 393234 85.70 61 1908 36 25 02472 99969 84 64 909867 • o5 84-70 606766 35 26 025oi 99969 398179 83 66 999864 -05 3983.5 83-71 601685 34 27 o253o 99968 4o3i99 82 71 999861 .05 403338 82-76 ■ 596662 33 28 02560 99967 408161 81 H 999858 .05 4o83o4 81.82 59.696 586787 82 29 O258o 02618 02647 02676 99966 4i3o68 80 999854 .o5 4i32i3 80-91 31 80 31 99966 4i79'9 8.422717 79 79 96 It 999851 9-999848 .06 .06 418068 8.422869 80-02 70-14 78.30 581932 11-577131 80 29 99965 32 99964 427462 78 999844 .06 427618 572382 28 33 02705 99963 432156 77 40 999841 •06 43231 5 77-45 567685 27 84 02734 99963 436800 76 •57 999838 -06 436962 441 56o 76-63 563o38 26 35 02763 99962 441394 75 •77 999834 •06 75-83 558440 25 36 02792 99961 445941 74 99 999831 • 06 4461 10 75-05 553890 24 37 02821 99960 450440 74 -22 999827 .06 45o6i3 74-28 549387 23 38 o285o 99959 454893 73 .46 999823 .06 455070 73-52 544930 540319 22 39 02879 99959 459301 72 •73 999820 .06 459481 463849 72-79 72-06 ^21 40 "if 02908 99958 463665 8T467985 72 71 00 .29 999816 9-999812 .06 -06 536i5i 20 02938 99957 8.468172 71-35 11-531828 TvT 42 02967 99956 472263 70 60 999809 999805 .06 472454 70.66 527546 18 43 02996 j 99955 476498 69 91 .06 476693 69-98 523307 17 44 o3o25 I 99954 480693 tl •24 999801 .06 480892 485o5o 69-31 519108 16 45 o3o54 1 99953 484848 -59 999797 .07 68-65 514950 5io83o 15 46 1 o3o83 99952 488963 67 t 999793 .07 489170 68-01 14 47 03lI2 99952 493040 67 999790 999786 .07 493250 67-38 506750 13 48 o3i4i 99951 497078 5oio8o 66 ii •07 497293 66.76 502707 12 49 o3i7o 999DO 66 999782 •07 501298 66.1 5 498702 11 50 51 o3i99 99949 5o5o45 65 -48 999778 9-999774 .07 .07 505267 65-55 494733 10 ~9 03228 99948 8.508974 64 -89 8.509200 513098 516961 64-96 64-39 11-490800 62 o3257l 99947 512867 64, • 3i 999769 .07 486902 8 53 } 03286 : 99946 516726 63' •75 999765 .07 63-82 483o39 7 54 o33i6 99945 52o55i 63 -19 999761 •07 520790 63-26 479210 475414 6 55 ' 03345 ! 99944 524343 62 -64 999757 . .07 524586 62-72 5 56 i; 03374 199943 528102 62 •II 999753 .07 528349 62-18 47i65i 4 57 o34o3 ■ 99942 531828 61 -58 999748 • 07 532080 61 -65 467920 S 5S 03432 ! 99941 535523 61 -06 999744 •07 535779 6i-i3 464221 2 59 ,03461 99940 539.86 60 -55 999740 •07 539447 60-62 460553 1 60 03490 99939 542819 60 -04 999735 ■07 543084 6o-i2 11-456916 N. COS. N. Bine. L. COS. D.l" L.sine. | L.eot D.l" 1 L. tang. 1 ' 88"^ 32 TRIGONOMETRICAL FUNCTIONS. — 2""' Nat. Functions. Logarithmic Functions + 10. 1 f N.Bine. N. cos. L. sine. D. 1" L. COS. D.l" L. tang. 1 D. r< L. cot. ~o" 03490 1 99939 8-542819 60-04 9-999735 .07 8-543084 60-12 11-456916 453309 60 1 o35i9 ! 99938 03548 , 99937 546422 59 55 999731 •07 546691 1 59 62 69 2 549995 06 999726 -07 550268 59 14 449732 68 3 035X7 1 99936 553539 58 999722 ■08 553817 58 66 446 1 83 57 4 o36o6 199935 557054 58 II 999'7n .08 557336 58 it 442664 56 5 o3635 1 99934 56o54o 57 65 999713 .08 560828 57 439172 55 61 03664 99933 563999 57 19 999708 .08 564291 57 27 430709 54 7 s o36q3 I 99932 567431 56 74 999704 .08 567727 56 82 432273 53 03723 1 99931 570830 56 3o 999699 .08 571,37 56 38 428863 52 9 03702 99930 574214 55 87 999694 .08 574520 55 t 425480 51 10 IT 03781 I 99929 577566 55 il_ 999689 9-999685 .08 .08 577877 55 422123 50 o38To 99927 8-580892 55 02 8.581208 55 10 11-418792 415486 49 12 03839 o386a 99926 584193 54 60 999680 .08 584514 54 68 48 13 99925 587469 54 19 999675 .08 587795 54 27 4l22o5 47 14 03897 99924 590721 53 79 999670 -08 591001 53 87 408949 46 15 03926 99923 593948 53 39 999665 .08 594283 53 47 405717 45 16 03955 99922 597152 53 00 999660 .08 597492 53 08 4o25o8i44l 17 03984 99921 6oo332 52 61 999655 .08 600677 52 70 399323 i 43 1 18 040 1 3 99910 99918 603489 52 23 999650 .08 6o3839 52 32 396161 42 19 04042 606623 5i 86 999645 .09 606978 5i 94 393022 41 20 21 04071 99917 609734 5i 49 999640 -09 .09 610094 8-6i3i89 5i 58 40 89 04100 99916 8-612823 5i 12 9-999635 5i 21 11-386811 22 04129 99915 615891 5o 76 999629 -09 616262 5o 85 383738 38 23 04159 04188 99913 618937 5o 41 999624 .09 619313 5o 5o 380687 37 24 99912 621962 5o 06 999619 .09 622343 5o i5 377657136 | 25 04217 99911 624965 49 11 999614 .09 625352 49 81 374648 35 26 04246 99910 627948 49 999608 .09 628340 49 47 371660 34 27 04275 99909 63091 1 il 04 999603 -09 63i3o8 49 48 i3 36S692 83 28 04304 99907 633854 71 999597 -09 634256 80 365744 32 29 04333 99906 636776 639680 48 39 999592 -09 637184 48 48 362816 31 30 31 04362 99905 48 06 999586 9. 99958 I -09 -09 640093 48 16 359907 80 29 04391 99904 8-642563 47 75 8-642982 645853 47 84 11-357018 32 04420 99902 645428 47 43 999575 -09 47 53 354147 28 33 04449 04478 99901 648274 47 12 999570 -09 648704 47 22 351296 27 34 ^ 65iio2 46 82 999564 -09 65i537 46 91 348463 26 35 04507 65391 1 46 52 999558 •10 654352 46 61 340648 25 86 04536 99897 656702 46 22 999553 -10 657149 65992^ 46 3i 342851 24 37 04565 99896 659475 45 tl 999547 -10 46 02 340072 23 88 04594 99894 662230 45 999541 -10 662689 45 73 337311 22 39 04623 99893 664968 45 35 999535 -10 665433 45 44 334067 ; 21 1 40 04653 99892 667689 45 06 999529 •10 668160 8-670870 45 44 26 88 331840 ii-329i3o 20 19 41 04682 99890 8-670393 673080 44 If 9-999524 -10 42 047 1 1 99889 44 999518 •10 673563 44 61 326437 18 43 04740 99888 675751 44 24 999512 -10 676239 44 34 323761 17 44 04769 04798 04827 99886 &78405 43 9? 999506 •10 678900 I 44 68 1 544! 43 17 32II00 j 16 45 99885 681043 43 70 999500 •10 80 3i8456!l5 46 99883 683665 43 44 999493 -10 684172 686784 43 54 3i5S28!l4 47 04856 99882 686272 43 18 999487 -10 43 28 3i32!6il3 48 04885 99R81 688863 42 92 999481 •10 689381 43 o3 310619! 12 49 04914 99879 691438 42 67 999475 •10 691063 42 27 3o8o37 11 50 51 04943 04972 99878 99876 693998 8-696543 42 42 _999469^ 9-999463 -10 694529 i 42 52 3o547i 10 42 17 8-697081] 42 28 II -302919 9 62 o5ooi 99875 699073 41 92 909456 -II, 699617 42 o3 3oo383 8 53 o5o3o 99873 701589 41 68 999450 •II 702139 41 It 297861 ^r 54 o5o59 o5o88 99872 704090 41 44 999443 - II 704646 i 41 295354 6 55 99870 706577 41 21 999437 -II 707140 41 32 292860 5 56 o5ii7 99869 709049 40 97 999431 -III 709618 41 08 290382 4 67 o5i46| 99867 7ii5o7 40 74 999424 -II 712083! 40 85 287917 3 58 o5i75 j 99866 713952 40 5i 999418 •11 714534! 40 62 285465 2 69 o52o5 1 99864 716383 40 29 9994 1 1 -II 716972 40 40 283028 1 60 05234 99863 718800 40-06 999404 •II 719396 40-17 280604 N. COS. N. sine. L. COS. D. 1" L.6ine. L.cot 1 D.l" L.tang. / 87° 1 ■ Nat. TRIGOi^OMETRICAL FUNCTIONS.— 3°. 33 Nat. Functions. LOGABITHMIC FUNCTIONS + 10. 1 ' 'N.sine. N. COS.! L. sine. D.l" L. COS. ] ^.V L. tang. 1 D. 1" L.cot. o5a34 99863 3.718800 40-06 9-999404 -II 8.719396 40-17 721806 39-95 11.280604 60 1 05263 99861 721204 39-84 999398 .11 278194 59 2 05292 I 99S60 723595 39-62 999391 -II 724204 39-74 275796 58 8| o532i 1 99853 725972 39-41 999384 .11 726588 39-52 278412 57 4 o535o 1 99857 V^l \l\l 999378 •II 728959 39-30 271041 5'3 5 05379 99855 999371 999364 •II 73i3i7 ll^ 268683 65 ^ 05408 99854 733027 38-77 •12 733663 266337 54 7 05466 99852 735354 38-57 999357 •12 735996 38-68 264004 53 8 99851 737667 38-36 999350 -12 738317 38-48 261683 52 9 05495 99849 739969 38- 16 999343 •12 740626 38-27 259874 51 10 11 05524 99847 742259 37-96 999336 -12 742922 38-07 257078 50 05553 99846 3-744536 37-76 37-56 9.999329 •12 8-745207 37.81 37.68 11.254793 49 12 05582 99844 746802 999322 •12 747479 252521 43 13 o56n 99842 749055 37-37 9993 I 5 •12 749740 37-49 25o26o 47 U o564o 99841 ]\\VA IV^, 999308 •12 751989 37-29 24801 1 46 15 05669 05698 99839 999801 .12 754227 37-10 36-92 245773 45 16 9983s ?"^^? fa\ 999294 999286 •12 756453 243547 44 17 "^.U 99836 •12 758668 36-73 36-55 241882 43 18 99834 76oi5i 36-42 999279 •12 760872 289128 42 19 05785 o58i4 99833 762337 36-24 999272 •12 763o65 36-36 286935 41 20 21 99831 764311 36 -06 999265 •12 765246 36-i8 284754 40 89 o5844 99829 8-766675 35-88 9-999257 •12 3-767417 36-00 11.232588 22 05873 99827 768828 35-70 999250 -l3 769578 35-83 280422 88 23 05902 99826 770970 35-53 999242 -l3 771727 35-65 228278 226184 87 24 0593 1 99824 773ioi 35-35 999235 •l3 773866 35-48 36 25 05960 99822 775223 35-18 999227 •13 775995 35-31 224005 85 26 a 99821 777333 35-01 999220 .13 778114 35-14 221886 84 27 99819 779434 34.84 999212 .13 780222 tv. 215592 83 28 06047 99817 781524 34-67 999205 • 13 782320 32 29 06076 99815 7836o5 34-5; 999107 999189 .13 784408 34-64 81 30 o6io5 99813 785675 34-31 • i3 7864C6 8-788554 34-47 2i35i4 80 061 34 99812 8-787736 34-18 9-999181 • i3" 34-31 I 1.21 1446 29 32 06 1 63 99810 789787 34-02 999174 • i3 790613 34-15 209387 28 33 06192 99808 791828 33-86 999166 • i3 792662 33-99 207888 2o5299 27 34 06221 99806 793859 33-70 999158 • i3 794701 33-83 26 35 o625o 99804 795881 33.54 9991 5o • i3 796731 33-68 208269 201248 25 36 o63o8 99803 797894 33-39 33-23 999142 • i3 798752 800763 33-52 24 37 99801 799897 999134 • 13 33.37 199287 23 38 06337 99799 801892 33.08 999126 •13 802765 33-22 197285 195242 22 89 06366 99797 803876 32-93 999118 .13 804758 33-07 21 40 06395 9979^ 8o5852 32.78 999110 .13 806742 8^08717" 32^92 198258 20 19 41 06424 99793 8-807819 32-63 9.999102 .13 32.78 32-62 u. 191288 189817 42 06453 99792 809777 32-49 ^^ •14 810683 18 43 06482 99790 811726 32-34 •14 812641 32-48 187859 17 44 o65ii 99788 813667 32-19 32-05 999077 •14 814589 32-33 185411 16 4") 06540 99786 815599 999069 •14 816329 32-19 32-05 188471 15 46 o656q 99784 817522 3i-9i 999061 •14 818461 181539 14 1 13 47 99782 819436 31-77 999053 •14 820384 81-91 179616 48 06627 99780 821343 31-63 999044 •14 822298 31-77 177702 12 49 06656 99778 823240 3i-49 999036 •14 824205 31-63 175795 11 50 06685 99776 825i3o 31-35 999027 j_i4 826103 3i-5o 178897 10 ~9~ •51 .06714 99774 8-827011 31-22 9-999019 •14 8-827992 81-36 11.172008 52 06743 99772 828884 3i-o8 999010 •14 829874 3I-23 170126 8 53 06773 99770 830749 30-95 30-82 •14 831748 3i^io 1682=2 7 54 06802 99768 832607 998993 • 14 8336i3 80.96 30-83 1663C7 6 55 0683 1 99766 834456 30-69 998984 •14 835471 164529 5 56 06860 99764 836297 3o-56 998976 .14 837321 80.70 30^57 80^45 162679 4 57 06889 069 1 S 99762 838 1 3o 3o-43 m .i5 839163 160887 3 58 99760 839956 3o-3o .i5 840998 842825 159002 2 59 06947 99758 841774 30-17 998950 .i5 80.32 157175 1 60 06976 99756 843585 3o-oo 998941 .i5 844644 30^19 155356 ' ' 1 N. COS. iN. sine. L. COS. ! D.l" L.8ine. L. cot D.l" L. tang. 86° 1 34 TRIGONOMETRICAL FUNCTIONS. — 4°. Nat. Functions. Logarithmic Functions + 10. ! i ' N.sine. N. COS. L. sine. D.1" L. COS. D.l"|| L. tang. Dl." L.cot. 1 1 06976 99756 8.843585 3o-o5 9.99^941 .i5 ,8.844644 30.19 11.155356 GO 1 07005 99754 845387 29 92 990932 .13 846455 3o °] 153545 59 2 07034 00752 847183 29 80 998923 • 15 848260 29 93 i5i74o 58 s 07063 I 99750 848971 29 67 9989 '4 .15 85oo57 29 82 149943 57 4 07092 i 99748 85o75i 29 55 99^905 .i5 85 1846 29 70 148154 66 5 071 21 199746 852525 29 43 998896 •13 853623 29 58 146372 55 6 07i5o 99744 854291 29 3i 998887 • 15 855403 29 46 144597 54 7 07179 99742 856049 29 '9 998878 .i5 857171 29 35 142829 53 8 07208 99740 857801 29 07 998869 .i5 858q32 29 23 141068 52 9 07237 99738 839546 28 96 998860 .i5 86o6t:6 29 11 139314 51 10 11 07266 0729T 99736 861283 i 2S 84 998851 9-998841 .i5 • 15 862433 8-864173 29 28 00 "b8~ 137567 11-135827 50 49 99734 8-863014 28 73 12 07324 99731 864738 28 61 998832 .15 865906 28 77 134094 48 13 07333 99729 866455 2S 5o 998823 .16 867632 28 66 132368 47 U 07382 99727 868i65 2-' 39 998813 .16 869351 28 54 1 30649 46 15 0741 1 99725 869868 2 J 28 998804 .16 871064 28 43 128936 45 16 07440 99723 871565 28 11 998795 .16 872770 28 32 127230 125531 44 17 07469 99721 873255 23 998785 .16 874469 28 21 43 18 07498 997 '9 874938 27 95 998776 .16 876162 28 n 123838 42 19 07527 99716 876615 27 86 998766 .16 877849 28 00 I22l5l 41 20 21 07556 99714 878205 27 "27 73 63 998757 9.998747 .16 .16 879529 8.881202 27 89 120471 40 07585 99712 8.879949 27 79 II -118798 "svT 22 07614 99710 881607 27 52 998738 .16 882869 27 68 II7131 115470 88 23 07643 99708 883258 27 42 998728 .16 884530 27 58 37 24 07672 99705 884903 27 3i 998718 .16 8861 85 27 47 ii38i5 86 25 07701 99703 886542 27 21 998708 .16 887833 27 37 1 1 2 1 67 35 26 07730 99701 888174 27 11 ^», .16 889476 27 27 iio524 34 27 07759 99699 889801 27 00 .16 89 1 1 1 2 27 17 108888 83 L8 07788 1 99696 891421 26 90 998679 • 16 892742 27 07 107258 82 29 07817 j 99694 893035 26 80 998669 •17 894366 26 97 105634 81 30 07846 99692 99689 894643 26 70 998659 •17 895984 26 87 104016 u - 102404 80 29 '31 07875 8-896246 26 "6o"~ 9-998649 8.897596 ~2^ 77 32 07904 99687 897842 26 5i 998639 •17 899203 26 67 100797 28 33 07933 996S5 899432 26 41 998629 •17 900803 26 58 099197 27 34 07962 99683 901017 26 3i 998619 •17 &^ 26 48 097602 26 35 07991 99680 902596 26 22 998609 •17 26 38 096013 25 CO 08020 99678 904169 26 12 998599 •17 905570 26 29 094430 24 37 08049 99676 905736 26 o3 998580 998578 •n 907147 26 20 092853 23 33 08078 99673 907297 25 g •n 908719 26 10 (591281 089715 22 SO 08107 99671 908853 25 998568 •17 910285 26 01 21 40 41 081 36 ^8^65 99068 910404 25 75 998558 •17 •'7 91 1846 8.913401 25 25" 92 "83"" 088154 1 1 . 086399 20 I9" 99666 8.911949 913488 23 66 9 . 998548 42 08194 99664 25 56 998537 •17 914951 25 74 oS5o49 18 43 08223 99661 9l5022 25 47 998527 •17 916495 25 65 o835o5 17 44 |oS252 99639 9i655o 25 38 998516 .18 918034 25 56 081966 16 45 !o82Si 99657 5.8073 25 29 998506 ■ 18 919568 25 47 080432 15 46 IETRICAL FUI^-CTIONS.— 7^ 37 Nat. Functions. Logarithmic Functions + 10. N.slne. N. cos. L. sine. D.1" L.C08. ] D.l" L. tang. 9-089144 D.l" L. cot. 12187I 99255 9.085894 I7-I3 9-996751 .26 '7^33" 1.0 -910856 60 i 12216 9925i 086922 17-09 996735 -26 090il]7 17-34 909813 59 2 12245 99248 087947 17-04 996720 -26: 09122J 17-30 900772 58 3 12274:99244 088970 17-00 16-96 996704 -26 092266 17-27 907734 57 4 12302 99240 089990 996688 -26 093302 17-22 906698 56 b\ 12331 99237 091008 16-92 996673 -26 094336 17-19 903664 55 61 12360 99233 092024 16-88 996657 -36 095367 096395 17-13 904633 54 ^i 12389 9923o 093037 16-84 996641 -26 17.11 903 6o5 58 « 12418 99226 094047 16-80 996625 -26 097422 098446 17.07 902578 52 9 12447 I 99222 095o56 16-76 996610 -26 17-03 901554 51 10 11 12476 99_^'_9_ 99215 096062 16.73 996594 9-996578 •27 099468 9-100487 16.97 16.95 900532 10-899513 r.c 4'i 13304 9.097065 16-68 12 12533 99211 098066 16-65 996562 •27 ioi5o4 16.91 898496 48 13 12562 99208 099065 i6-6i 996546 -27 102319 16-87 897481 47 14 12591 99204 100062 16.57 996530 -27 103532 16.84 896468 4P 15 12620 99200 ioio56 16-53 996514 -27 104542 16.80 895458 45 16 12649 99197 102048 16-49 996498 •27 io555o 16.76 894430 44 17 12678 99193 io3o37 i6-4D 996482 •27 106556 16.72 893444 4; 18 12706 99189 104025 16-41 996465 -27 107559 16.69 892441 42 19 12735199186 lODOlO i6-38 tt& •27 io856o i6-65 891440 41 20 21 12793 99182 105992 . 16-34 •27 -27 109559 i6-6i 16-58 890441 10-889444 40 99178 9.106973 16.30 9-996417 9-110556 22 12822; 99175 107951 108927 16.27 996400 -27 iii55i 16-54 888449 3^ 23 12851 1 99171 16.23 996384 -27 112543 i6-5o 887457 37 24 12880 99167 109901 16.19 996368 -27 113533 16-46 886467 30 25 1 2908 99 1 63 110873 16.16 9063 5 I -2T 114521 16-43 885479 884493 35 26 13937 99160 1 1 1842 16-12 996335 •27 11 5507 16-39 34 27 12966 99156 112809 16-08 996318 •27 11 649 1 16-36 883509 33 28 12995 99 1 52 113774 i6-o5 996302 -28 117472 16-32 882528 32 29 i3o24 99148 1 14737 16-01 996285 .28 118452 16-29 881548 31 80 31 i3o53 99144 II 5698 15.97 996269 -28 119429 16-23 16-22 880571 10-879596 30 2y" i3o8i 99141 9. 1 1 6656 15.94 9-996252 .28 9-120404 32 i3iio 99137 117613 \lr, 996235 -28 ,2.377 16-18 878623 28 33 i3i39 99133 1 1 8067 996219 .28 122348 i6-i5 877652 27 34 i3i68 99129 119519 i5-83 996202 .28 123317 16-11 876683 20 35 i3i97 99123 120469 i5-8o 996185 -28 124284 16-07 875716 25 36 13226 99122 121417 15.76 996168 -28 125249 16-04 874751 24 37 13254 99118 122362 15.73 996151 .28 12621 1 16-01 873789 23 38 13283 99114 i233o6 15.69 996134 -28 127172 15.97 872828 22 39 i33i2 99110 124248 i5.66 996117 .28 i28i3o 15.94 871870 21 40 i334i 99106 125187 15-62 996100 .28 129087 ,5.9. 870913 20 41 13370 99102 9-126125 15-59 9-996083 .29 9-i3oo4i 15.87 i'o^699'59 I'.t 42 13399 99098 127060 15-56 996066 •29 130994 15-84 869006 18 43 13427 99094 127993 15.62 996049 •29 i3i944 i5-8i 868o56 17 44 13456 99091 128925 15.49 996032 •29 132893 15-77 867107 16 45 13485 99087 129854 15.43 996015 •29 133839 15-74 866 1 61 15 46 i35i4 99083 130781 15-42 995998 .29 134784 15-71 865216 14 47 13543 99079 131706 15.39 993980 •29 135726 15.67 864274 13 48 13572 99075 13263.J i5-35 995963 •29 136667 15-64 863333 12 49 i36oo 99071 i33J5i i5-32 995946 .29 137605 i5-6i 862395 11 50 "ol 13629 73658 99067 99063 134470 i5-29 _995928; 131 • 29 138542 i5-58 861438 10 9' 9.735387 15-25 9-995911 19-139476 i5-55 io-86o524 52 13687 99059 i363o3 l5-22 995894 •29 140409 i5-5i 85^66^ 8 53 13716 99055 137216 l5-iq 995876 993839 -29 141340 15-48 7 54 13744 9905 1 i38i28 i5-i6 •29 142269 15-45 857731 6 55 13773 99047 139037 l5-I2 993841 • 29 143196 .5-42 856804 5 56 i38o2 99043 139944 15-09 995823 •29 144121 15-39 855879 4 57 1 383 1 199039 i4o85o i5.o6 995806 •29 143044 i5-35 854956 3 58 i386o 1 9903 D 141754 i5-o3 995788 .29 145966 i5-32 854034 2 59 13889 9903 1 142655 15-00 995771 •29 146885 \l:lt 853ii5 1 60 13917 99027 143555 14-96 995753 .29 147803 852197 L. tang. 1 N. COS. N, sine. L. COS. D.l" L.sine. L.cot D.l" 82° 1 38 TRIGONOMETRICAL rUI^CTIONS.-^'§\ Nat. Functions. Logarithmic Functions + 10. |N.sine. 13917 N. COS. L. sine. 9- 143555 D.l" L. COS. D.l" L. tang. Dl." L.cot. 1 99027 14 .96 9.995753 .30 9-147808 15.26 10-852197 60 1 18946 99023 144453 14 93 993783 •30 148718 15-23 85l2o2 59 2 13975 99019 145349 14 90 993717 .3o 149682 l5-20 85o363 53 3 14004 990 ID 146243 14 87 993699 .3o i5o544 15-.7 849436 57 4 14033 99011 i47i36 14 84 990681 .30 i5i454 15-14 843546 56 5 14061 99006 148026 14 8i 995664 •3o 152363 i5-i 1 847637 55 6 14090 99002 148915 14 78 995646 .30 153269 i5-o3 846731 54 7 14119 9S998 149302 14 75 995628 .30 154174 i5-o5 845326 53 8 14143 98994 i5o686 14 72 995610 .30 155077 l5-02 844923 52 9 14177 9S990 i5i569 14 69 995591 .30 155978 14-99 844022 51 10 11 i42o5 14234 989S6 9893^ i5245i 9.153330 14 66 _995573^ • 30 •3o 156877 9.157775 14-96 14-93 843123 10-842225 50 49 14 63 9.995555 12 14263 98978 1 54208 14 60 995537 .30 158671 14-90 841329 43 13 14292 9-'973 i55o83 14 57 993519 .30 159565 14-87 840435 47 U 14320 9'J969 155957 14 54 993301 .31 160457 14-84 839543 46 15 14349 9:960 i5683o 14 5i 995482 •3i 161847 14-81 83tJ653 45 16 i437tJ 90961 157700 14 48 995464 'i' 162286 14-79 837764 44 17 14407 9J9D7 1 53569 14 45 995446 'i' i63i23 14-76 836877 43 18 14436 9S953 1 59433 14 42 995427 .31 164008 14-73 835992 4 -J 19 14464 98948 i6o3oi 14 39 995409 .31 164892 14-70 835 108 41 20 21 I4493_ 14522 98944 98940 i6u64 9.162025 14 14 36 33 995390 9.995372 .31 •31 165774 9-166654 14-67 834226 10-833346 40 14^64 22 14551 98936 162885 14 3o 995353 .31 167532 i4-6i 832468 38 23 14530 98931 168743 14 27 995334 t 168409 14^58 83 1591 37 24 14608 9S927 164600 14 24 995316 169284 14-55 880716 829843 36 2.> 1463] 14666 98923 165454 14 22 995297 .31' 170157 14-53 35 26 98919 166807 14 \l 995278 •3i: 171029 i4-5o 828971 84 27 14695 98914 167159 168008 14- 995260 .31 171899 14-47 828101 83 28 14723 98910 14- i3 995241 .32 172767 14-44 827233 82 29 14752 98906 168856 14- 10 995222 .32' 173634 14-42 826366 31 80 14781 98902 169702 14- 07 995203 •32 174499 14-39 825501 30 "sT i43io 98897 9-170547 14- o5 9.995184 .32 9-175362 14-36 10-824688 2\) 32 14838 98^93 171389 14- 02 995i65 .32 176224 14-33 828776 28 33 14867 98889 172280 i3. 99 995146 •32 177084 i4-3i 822916 27 84 14896 98884 178070 i3. 96 995127 •32 177942 14-28 822058 26 G5 14925 98880 178908 i3. 94 995108 •32 178799 14-25 821201 25 86 14954 98876 174744 i3. n 995089 .321 179635 14-23 820345 24 87 14982 9S871 175578 i3. 995070 •32) i8o5o8 14-20 819492 23 38 iSoii 98867 17641 1 i3. 86 995o5i •32! i8i36o 14-17 818640 22 89 i5o4o 98863 177242 i3 83 995o32 •321 182211 I4-.5 817789 21 40 41 15069 "i5o97 98858 17S072 i3 80 9950 1 3 •32 i83o59 14-12 816941 20 IF 98854 9.178900 i3 77 9.994993 .32 9- 168907 14-09 10-816093 42 i5i26 98849 179726 i3 74 994974 •32! 184732 14-07 8i5248 18 43 i5i55 98843 i8o55i i3 72 994955 .32! 185597 186439 14-04 8i44o3 17 44 i5i84 9S841 181874 i3 994935 .32 14-02 8i356i 16 45 l52I2 98836 182196 i3 66 9949 '6 •33 187280 188120 13-99 812720 15 46 15241 9333 2 i83oi6 i3 64 994896 .33 13.96 811880 14 47 15270 98827 98823 183834 i3 61 994877 994857 .33 188958 13-93 81 1042 13 48 15299 18465 1 i3 59 • 33 189794 18-91 810206 12 49 15327 15356 T5385" 98818 185466 i3 56 994838 •33 190629 13-89 809871 11 50 51 98814 186280 i3 53 994818 •33 •33' 191462 13-86 8o8538 10 9 98809 98805 9.187092 i3 5i 9.994798 9^192294 13-84 10-807706 52 i54i4 187903 i3 48 994779 •33 198124 i3.8i 806S76 8 53 15442 98800 188712 i3 46 994759 •33 198953 \t]t 806047 7 54 15471 98796 1895.9 i3 43 994739 •33 194780 8o5220 6 55 i55oo 98791 190825 i3 41 994719 •33 195606 13.74 804394 5 56 15529 98787 191130 i3 38 994700 •33 196430 13.71 803570 4 57 15557 98782 191933 i3 36 994680 •33 197253 13.69 802747 3 58 15586 98178 192734 i3 33 994660 •33 1 198074 i3.66 801926 2 59 i56i5 98773 193534 i3 3o 994640 • 33 198894 i3-64 801106 1 60 15643 98769 194332 13.28 994620 .33 199713 13.61 800287 N.C08. N. sine. L. COS. D.l" L. sine. j L. cot. D.l" L. tang. ~^ 81° 1 TRIGONOMETRICAL FUNCTIONS. — ^9° 39 Nat. Functions. 1 Logarithmic Functions + 10. 1 1 ; ■ N.8ine> N. COS. L. sine. D.l" 13-28 L.CO8. D.l" L. tang. D.l" L.oot 15643 98769 9- 194332 9.994620 .33 9.199713 i3-6i 10-800287 60 1 n672 9^764 193129 13-26 994600 •33 200529 i3-59 799471 59 iS-oi 9S760 195925 13-23 994380 •33 201843 i3-56 798655 58 ^!i 15730 98755 196719 13-21 994560 •34 202159 .3.54 797841 57 4'' 1 5758 1 98731 197311 i3-i8 994540 •34 202971 13.52 797029 5ij 5i: 15787 j 98746 198302 13.16 994519 •34 208782 13.49 796218 55 6:! i58i6 198741 199091 i3-i3 994499 •34 204592 13.47 795408 5i 7,115845 98737 199879 13-11 994479 •34 205400 18.43 794600 53 8 15873 98732 200666 i3-o8 994439 •34 206207 13.42 798798 52 9! 10902 9S728 20I43I i3-o6 994438 •34 207013 13.40 792987 51 10 i| 1 593 1 n 13959 98723 98718 202234 9-2o3oi7 i3-o4 .994418 •34 •34" 207817 9.208619 13.88 _J?2i8£ 10.791881 50 4J i3-oi 9-994397 13.35 12^ 15988 98714 203797 12-99 994377 •34 209420 13.33 79o58o 789780 4S 13; 16017 14' 16046 98709 204377 12-96 994337 •34 210220 i3.3i 47 Q8704 203354 12-94 994336 •34 211018 13.28 788982 46 15 1 16074198700 1 2061 3 1 12-92 994816 •34 2ii8i5 13.26 788185 45 16 i6io3 19^693 206906 12-89 994295 •34 2 1 26 11 13.24 787389 41 ^^i i6i32 j 98690 207679 12-87 994274 •35 2i34o5 l3-21 786595 43 is' 16160 '98686 20;ii32 12-85 994254 •35 214198 18.19 785802 42 19; 16189:98681 16218198676 16246198671 209222 12-82 994233 .35 214989 13.17 i3.i5 783011 41 20! 21; 209992 12-80 _9942I_2_ •35 •35 215780 9-216368 784220 ro^83432" 40 9-210760 12-78 9.994191, 13.12 2-j! 16275198667 21l526 12-75 994I7I .35 217336 218142 13.10 782644 38 23 i63o4 ! 9S662 16333 I 98657 21229I 12-73 9.74 i5o .35 i3.o8 781838 37 24 2i3o55 12-71 994129 •35 218926 i3^o5 781074 36 25 i636i 1 9S652 2i38i8 12-68 994108 •35 219710 i3.o3 780290 35 26 16390198648 214579 12-66 994087 .35 220492 13.01 779508 04 27 i64i9:9S643 215338 12-64 994066 .35 221272 12-99 778728 83 28' 16447 9' 633 216097 12-61 994045 .35 222o52 12.97 777943 82 29 i 16476 ; 9':633 216834 12-59 994024 .35 222880 12-94 777170 81 30! 311 i65o5 90629 T6533 "9S6"'24~ 217609 9-218363 12-57 12-55 994003 9-998981 ^5 .35 228606 12.92 77^394 SO 9-224882 12-90 10.775618 29 32 16562 1 9S619 219116 12-53 998960 .35 225i56 12.88 774844 23 33' 16391 ' 98614 219868 i?-5o 998989 .35 225929 12-86 774071 27 34: 16620 98609 220618 12.48 998918 -35 226700 12.84 778800 26 35! 16648:98604 221367 12-46 998896 .36 227471 12^8l 772529 25 36 16677 ! 98600 2221l5 12-44 998875 •36 228289 12^79 771761 24 37 16706 98395 222S61 12-42 993854 •36 229007 12-77 770993 23 S3 16734' 98390 223606 12-39 998882 •36 229773 12-75 770227 22 89 16763 , 98585 224349 .2-37 998811 .36 23o539 12-73 769461 21 40 41 16792:98580 16820 98575 223092 9-225833 12-35 993789 .36 .36 281802 9-282065 12.71 768698 20 12.33 9.998768 12-69 10-767985 19 42 16849 98570 16878 198565 226373 12-31 993746 • 36 1 282826 12-67 767174 18 43 227311 12-28 998723 .36 1 233586 12-65 766414 17 44 16906 {98361 228048 12-26 998708 •36 234345 12-62 765655 16 45 16935 I 98556 228784 12-24 998681 • 36 235io3 12-60 764897 15 46 16964 198351 229318 12-22 993660 .36 235859 12-58 764141 14 47 16992 98546 230252 12-20 993638 .36 2366.4 12-56 768886 13 43 17021 [98341 230984 12-18 998616 • 36 287868 12-54 762632 12 49 i7o5o 98536 23i7i5 I2-l6 993594 •37 288120 12-52 761880 11 50 51 170^8! 98531 17107 98526. 232444 12-14 993572 •37 238872 12 -5o 761 128 j 10 1 9-233172 12-12 9.993550 !"-37 9^39622 , 12-48 10-760878 9 L2 I7i36 98521 233899 12-09 993528 .37 i 240871 12-46 759629 8 53 17164! 98516 234625 12-07 12-05 993506 .37 1 241118 12-44 758882 7 .1 17193 I98311 235349 286073 993484 .37 1 241865 12-42 758i35 6 55 17222 198506 I2-03 993462 .37 242610 12-40 757890 5 56 17250:98501 286795 2375i5 12-01 993440 .37 243354 12-38 756646 4 57 17279 98496 11-99 993418 .37 l'.%% 12-36 755908 3 58 17308.98/91 238235 11-97 993396 .37 12-34 755161 2 59 173365984 6 288953 11-95 993374 .37 245579 12-32 754421 1 60 17365,98481 239670 1 11-98 993351 .37 1 246819 I2-3o 753681 N. COS. ;N. sine. L. COS. D.l" L. Bine. | ' L. cot D.l" L. tang. 80° 1 40 TKIGONOMETRICAL FUNCTI0N8. — 10 Nat. Functions. Logarithmic Functions + 10. t 1 2 8 4 • 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 82 33 34 35 86 37 88 89 40 41 42 43 44 45 4ti 47 43 49 50 1a 52 53 54 55 5 ) 57 5S 59 eo N.sine.'N. COS. i L. sine. D. 1" L. COS. D.l" L. tang. D. 1" L. cot. 60 /.9 58 57 50 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 17365 17393 17422 17431 •7.479 17308 17:37 17365 17594 17623 17651 17680 17708 ;^?!? \% 17852 17880 17909 17937 98481 98476 98471 98466 98461 98455 98450 98445 98440 98435 98430 98425 98420 98414 98409 98404 98399 98394 98383 98378 9.239670 240386 241101 241814 242526 243237 243947 244656 245363 246069 246775 9-247478 248181 248883 249583 230282 250980 251677 232373 253067 253761 11.93 ;;:§; 11.87 11.85 11.83 11.81 11-79 11-71 11.69 11.63 11.61 :;:?? 11.56 1J.54 9-993351 993329 998262 998240 993217 998195 998172 998149 998127 9-998104 998081 993059 998086 998013 992990 992967 992944 992921 992698 1 1 .38 .38 .38 .38 .38 .38 • 38 • 38 -38 -38 -38 -38 • 38 -38 9.246819 247057 247794 248530 249264 249998 250780 251461 252191 252920 253648 12-30 12.28 12.26 12.24 12-22 12-20 12-18 12-17 I2-l5 12-13 12-11 io-7.:368i 7 J 2943 732206 751470 730736 730002 749270 740339 747809 7470S0 746352 10-745626 744900 744176 743453 742781 742010 741290 740571 739834 789187 9.254874 255100 255824 256547 257269 257990 258710 259429 260146 260868 12-09 12-07 12-05 12.03 12.01 12.00 11.98 11.96 11-94 11.92 .7966 ':^995 18023 i8o52 18081 18109 i8i38 18166 18195 18224 18232 18281 i83o9 18338 18367 18395 18424 18452 18481 i85o9 98373 98368 98362 98357 ?8352 93347 93341 98336 9S331 98325 98320 983 1 5 98310 98304 93299 93294 98268 98283 98277 98272 98267 98261 98256 98250 98245 98240 98234 tVA 98218 9-234433 253144 255834 256523 257211 257898 258563 259268 259951 260633 9-261314 261994 262673 263331 264027 264703 26J377 266o5i 266723 267893 9.26J065 268734 269402 270069 270735 271400 272064 272726 273388 274049 11.52 11. 5o 11.48 11.46 11-44 11.42 11-41 iil 9-992875 992852 992829 992806 992783 992759 992786 992713 992690 992666 -38 -38 1 9-261578 262292 263oo5 268717 264428 265i38 265847 266555 267261 267967 11.89 11.83 11.81 11-79 11-78 11.76 11-74 10-788422 787708 736995 786283 735572 784862 784153 788445 782789 782033 3y 33 37 36 35 34 33 32 31 30 11.33 II. 3i 11.30 11-28 11-26 11-24 11-22 11-20 11.19 11.17 9-992643 992619 992596 992372 992349 992525 992 5o I 992478 992434 992480 i -39 •40 •40 •40 9.268671 269875 270077 270779 271479 272178 272876 273573 274269 274964 11.72 11.64 11-62 11-60 11-58 11-57 10.731829 780625 729923 729221 726521 727822 727124 726427 723781 725o36 29 23 27 28 25 2i 23 21 21 20 18538 18567 18595 18624 18652 18681 18710 18738 18767 18795 n.i5 11. i3 ll'U II .10 n-08 11 -06 II. o5 11-03 IIOl 10-99 9-992406 992882 992339 992883 992811 992287 992268 992289 992214 992190 .40 .40 .40 .40 .40 .40 -40 -40 -40 .40 .40 .40 -41 -41 .41 -41 -41 -41 •41 -41 9.275658 276351 277043 277734 278424 279113 279801 280488 281174 28i858 11-55 11-53 11-51 ii-5o 11-48 11-47 11-45 11-43 II-4I 11-40 10-724342 728649 722957 722266 721376 720867 720199 719312 718S26 718142 19 18 17 16 15 14 13 12 11 10 18824 '98212 18852*98207 18881198201 18910' 98196 18933198190 18967 1 98185 1 8995. 98 1 79 19024 98174 19052 98168 19081 98163 9.274708 275367 276024 276681 277337 277991 278644 279297 279918 280399 10.98 10-96 10.94 10-92 10.91 10.89 10.87 10-86 10-84 10-82 . 9-992166 902142 992117 992093 992069 992044 992020 991996 991971 991947 9.282542 283^25 288907 284588 285268 285947 286624 287801 287977 288652 11-38 11-36 n-35 11-33 11-31 11-30 11-28 11-26 11-25 11-23 10-717458 716775 716093 7J54I2 714732 714053 718876 713699 712028 711848 9 8 7 6 5 4 3 2 1 N. COS. N. sine. L. COS. D. 1" L. sine. 1 L. cot. D. V L. tang. / 79° 1 TEIGOKOMETRICAL FUNCTIONS.— 11°. 41 Nat. Functions. Logarithmic Functions + 10. 1 N.8itte.iN.cos. L.8ine. D.l" L. COS. D.l" L. tang. D.l" 11-23 L. cot. 10.711848 60 19081 98163 9-280599 10-82 9-991947 •41 '9-288652 1 1 19109! 9^07 281248 10 .81 991922 •41 289826 11 -22 710674 59 2 19133 9S152 281897 10 79 991897 •41 289999 11 20 710001 58 3 19167 98146 282544 10 77 991873 •41 290671 II-18 709829 57 4 19195 : 9S140 283190 10 76 991848 •41 291842 11-17 7ot)658 66 6 19224 98135 283836 10 74 99.823 •41 292018 iii5 707987 55 6 19232 98129 284480 10 72 991799 •41 292682 11-14 707818 64 7 19281 98124 285124 10 71 991774 •42 2933J0 11-12 7o665o 53 8 & 98118 285766 10 69 991749 •42 294017 11-11 705988 7o53i6 52 9 98112 286408 10 67 991724 •42 294684 1.-09 51 10 19866 ^^'i'L 287048 10 66 991699 •42 295849 u-07 704651 50 11 19395 98101 9-287687 288326 10 64 9-99'674 •42" 9-296018 1 1 06' 10.708987 49 12 19423 98096 10 63 99 "049 •42 296677 11-04 708828 48 13 19452 98090 288964 10 61 991624 •42 297339 11-03 702661 47 U 1 948 1 98084 289600 10 59 991599 •42 298001 11-01 701999 46 15 \t^ 98079 290236 10 58 99' 574 •42 298662 11-00 701888 45 16 98073 290870 10 56 991549 •42 299822 10-98 700678 44 17 19566 98067 291504 10 54 991524 •42 299980 10-96 700020 43 18 19595 98061 292137 292768 10 53 991498 •42 3oo638 10-95 699862 42 19 19623 o8o56 10 5i 991473 •42 3012^5 10-98 698705 41 20 19652 98050 293399 10 5o 99 '448 •42 801901 9-802607 1092 10-90 698049 10.697898 40 39" 19680 "gS^T 9-294029 10 48 9-991422 -42 22 19709 & 294658 10 46 991397 •42 808261 10-89 696789 38 23 19737 295286 10 45 991872 •43 808914 10-87 696086 37 24 19766 98027 295913 10 43 991846 •43 804567 3o52i8 10-86 695433 36 2r> 19794 98021 296539 10 42 901821 •43 10-84 694782 35 26 19823 98016 297164 10 40 991295 •48 805869 10-83 694181 34 27 19851 98010 297783 10 39 991270 •43 8o65i9 10-81 693481 33 28 19880 98004 298412 10 37 991244 -43 807168 10-80 692882 32 29 19908 97998 299034 10 36 991218 •48 807815 10-78 692185 31 SO ~8Y J9937 19965 97992 299655 9-300276 10 10 34 32 991198 9-991167 •43 •43 808468 9*309109 10-77 10-75 _^9il32 1 • 69089 1 30 2 J 97987 32 19994 97981 300895 10 3i 991141 •43 809754 10-74 690246 28 33 20022 97975 3oi5i4 10 11 991115 •43 810898 19-73 689602 27 34 2oo5i 97969 97963 302l32 10 991090 •43 311042 10-71 688958 26 35 20079 302748 10 26 99 1 064 •48 3ii685 10-70 6888.5 2") 36 20108 97958 3o3364 10 25 991088 •43 812827 10-68 687678 24 37 2oi36 97952 1:1%"^ 10 23 991012 •48 3I3608 10-67 10-65 687088 23 3=i 2oi65 97946 10 22 990986 •43 686892 22 39 20193 97940 3o5207 10 20 990960 •43 3.4247 10-64 685758 21 40 41 20222 2025o 97934 97928 3o58i9 10 19 990934 •44 314885 10-62 685ii5 20 9-3o643o 10 17 9-990908 •44 9.815528 10-61 10.684477 I'J 42 20279 97932 307041 10 16 990882 •44 3i6i59 10 -60 688841 18 43 2o3o7 97916 307650 10 14 990855 •44 316795 10-58 683205 17 44 2o336 97910 308259 10 i3 990829 •44 3i743o 318064 10.57 682570 16 4.5 2o364 97905 308867 10 11 990808 •44 10-55 68.936 68.3o3 15 46 20393 [ 97899 20421 197893 309474 10 10 990777 •44 318697 10-54 14 47 3 10080 10 08 9907D0 •44 819829 10-53 680671 13 48 20450 1 97887 3io685 10 07 990724 •44 819961 io-5i 680089 12 49 20478 j 97881 311289 311893 10 o5 990697 •44 820592 io-5o 670408 678778 11 61 2o5o7 20535 97875 10 04 990671 •44 •44 821222 10-48 10 9 97869 9-312495 10 o3 9-990644 I9-321851 10-47 10-678149 62 ! 20563 : 97863 3 1 3097 10 01 990618 •44 822479 10.45 677521 8 63 20592 ' 97857 313698 10 00 990591 •44 328106 10.44 676S94 7 64 20620 97851 314297 9 98 990565 •44 323788 10-48 676267 6 55 20649 97845 314897 9 97 990538 •44 324358 10-41 675642 5 56 20677 97839 20706 97833 315495 9 -96 9905 1 1 .45 824988 lo-4o 675017 4 67 316092 316689 9 94 990485 •45 3 2 5607 10.39 674898 3 58 20734 97827 9 93 990458 •45 826281 10-37 678769 2 69 ; 20763 97821 317284 9 91 990431 .45 326853 10. 36 678.47 1 60 i 20791 97815 317879 9.90 990404 •45 327475 10.35 672525 { N. COS. N. sine. L. COS. D.l" L. sine. L.oot D.r'" L. tang. 78° 1 ^2 TR1G0N"0METRICAL FUNCTIONS. — 12°. Nat. Functions. Logarithmic Functions + 10. j / N.8me.|N.co3, L. sine. D. 1'' L.COS. D.l" L. tang. D.l'' L. cot 20791 1 97815 "it]! Its 9-990404 •45 9-327474 10-35 10-672526 fiO 1 20820 : 97809 990378 •45 328095 10-33 671905 59 2 20848 \ 97803 319066 It 99035 I •45 328715} 10-32 671285 58 3 ' 20877 1 97797 319658 990324 •45 329334 io-3o 670666 57 4 20903 ! 97791 320249 9.84 990297 •45 3299331 10-29 330570 10-28 670047 56 5 20933 1 97784 320840 9-83 990270 •45 669430 55 6 20962 j 97778 32i43o 9-82 990243 •45 331187' '0-26 668^13 54 7 20990; 97772 322019 9-80 990215 •45 33i8o3i 10-25 668197 53 8 210191 97766 322607 9-79 990188 •45 332418 10-24 667382 52 9 : 21047 97760 323194 9-77 9.76 990161 •45 333o33 10-23 666967 51 10 : 21076 97754 323780 990134 •45 333646 10-21 666354 50 49 Tf 21104 9774H 9-324366 9.75 9-990107 .46 9-334259 10-20 10-663741 12 2Il32| 97742 324950 9.73 990079 -46 334871 10-19 665129 48 13 21161 1 97735 325534 9.72 990052 -46 335482 10-17 664518 47 14 2I189 21218 97729 q7723 326117 9.70 990025 .46 336093 10-16 663907 46 15 326700 9-69 989997 -46 336702 io-i5 663298 45 16 21246 97717 327281 9-68 989970 .46 3373 II io-i3 662689 44 17 i2'275 977 1 J 327862 9-66 989942 .46 337919 10-12 662081 43 18 :2i3o3 97705 328442 9-65 M .46 338527 339133 lO-II 661473 42 19 2i33i 97698 32g02I 9-64 .46 10-10 660867 41 20 2i36o '21388 97692 97686 _329599 9-330176 9-62 989860 -46 339739 10-08 660261 40 39 9-61 9-989832 -46 9-340344 10-07 10-659656 22 21417 97680 330753 9-6o 9S9804 -46 340948 34i552 io-o6 659052 38 23 1 21445 97673 33i329 331903 9-58 989777 -46 10-04 658448 37 24 ,21474 97667 9-57 989749 •47 342156 10 -03 657845 36 25 I2l502 97661 332478 9-56 989721 •47 342757 10-02 657243 35 26 !2i53o 97655 333o5i 9-54 989693 •47 343358 10-00 656642 34 27 1 21559 97648 333624 9-53 989665 •47 343958 9-99 656042 33 28 21 587 97642 334195 9-52 989637 •47 344558 9.98 655442 32 29 21616 97636 334766 9-50 989609 •47 343157 9-97 654843 31 30 21644 21672 97630 97623 335337 9.49 989382 •47 •47^ 345755 9-346353" _9l9^_ 654245 30 9-335906 9-48 9-989553 9-94 10-653647 29 32 21701 97617 336475 9.46 989325 •47 346949 347545 9-93 653o5i 28 83 21729 21758 97611 337043 9-45 989497 •47 9-92 652455 27 34 97604 337610 9-44 989469 •47 348141 9-91 65i859 661263 26 35 21786 97598 338176 9-43 98.9441 .47 348735 9.90 25 36 21814 97592 338742 9-41 989413 •47 349329 9-88 650671 24 37 21843 97585 339306 9-40 989384 •47 349922 9-87 650078 23 38 J2I87I l]l]^ 33987. 9.39 989356 •47 35o5i4 9-86 649486 22 39 21899 21928 2'r956 340434 9ii 9-36 9-35 989328 •47 35iio6 9-85 648894 21 40 41 _97566 97560 340996 9-341558 989300 9-989271 •47 •47 35,697 9-352287 9-83 6483o3 20 19 9-82 10-647713 42 121985 97553 342119 9-34 989243 •47 352876 9-81 647124 18 48 !220l3 97547 342679 9-32 989214 •47 353465 9-80 646535 17 44 22041 97541 343239 9-31 989186 •47 354053 9^79 645947 16 45 1 22070 97534 343797 9 -30 989157 •47 3546^0 9-77 645360 15 46 22098 97528 344355 9-29 989128 .48 355227 9-76 644773 644187 14 47 ' 22126 97521 344912 9-27 989100 .48 3558i3 9-75 13 48 '22i55| 975i5 345469 9-26 989071 .48 356398 9^74 643602 12 49 |22i83: Q75o8 346024 9-25 989042 .48 356982 9-73 643018 11 50 ~5r , 22212 1 97502 346579 9-24 989014 .48 357566 9-358149 9-71 642434 10 ~9~ T2'24o j 97'496 i 22268 97489 9-347134 9.22 9-988985 .48 9-70 io-64i85i 52 347687 9-21 988956 .48 358731 It 641269 8 5 [22297! 97483 348240 9-20 988898 .48 359313 640687 7 |54 22325 97476 348792 9-19 .48 359H93 9-67 640107 6 . 55 : 22353 97470 349343 9-17 988869 .48 360474 9-66 639526 5 56 =22382 97463 349893 9-16 988840 -48 36io53 9-65 638947 4 57 22410 97457 350443 9-.5 988811 •49' 36i632 9-63 638368 3 •^8:22438 9745o| 350992 35i54o 9-14 988782 •49! 362210 9-62 637790 2 «!M: 22 467 97444 9-,3 988753 •49 362787 9-61 637213 1, 60 2249D 'l N. COS. 97437 352088 9-II 988724 •49 363364 9-60 636636 ^ N. sine. L. COS. D.l" L. sine. L. cot. D.l" L. tang. ^ 77° 1 TRIGONOMETRICAL FUNCTIONS. — 13° 43 Nat. Functions. Logarithmic Functions f 10. N.»ineJN. COS. L. sine. ! D. 1" L. COS. D.l" L. tang. Dl." 1 L.cot 22495197437 9-352088 9-II 9.988724 -49 9-363364 9-60 10-636636 60 1 22323 i 97430 352635 9-10 988695 •49 1 363940 36451 5 9-59 636o6o 59 2 22552 97424 353181 9.09 9-08 988666 •49 9-58 635485 53 8 22580 97417 353726 988636 •49 1 365090 9.57 634910 57 4 22608 97411 354271 9-07 988607 •49 365664 9-55 634336 66 5 22637 22665 97404 354.S15 9-o5 988578 •49 366237 9-54 633763 55 6 97398 355358 9-04 988548 •49 366810 9.53 633190 54 7 22693 97391 355901 9-o3 988519 •49 367382 9.52 682618 63 8 22722 97384 356443 9-02 988489 .49 367953 9-51 682047 52 9' 22750 97378 3569S4 9-01 8-99 8-98 988460 •49 368D24 9.50 681476 51 11 22778 22807 97371 97365 357524 9.350064 988430 9.988401 .49 •49 369094 ^69663^ 9.49 680906 50 49 9.48 10-680337 12 22835 97358 358603 8-97 988371 .49 370232 9.46 629768 43 13 22863 97351 359141 8-96 988342 .49 370799 9-45 629201 47 U 22892 97345 359678 8-95 988312 .50 371367 9-44 628688 46 15 22920 97338 36o2i5 8.93 988282 .50 371933 9^43 628067 45 16 22948 97331 360752 8.92 988252 .50 372499 9^42 627501 44 17 ^!S? 97325 361287 8-91 988223 • So 373064 9-41 626986 43 18 97318 361822 8.90 988193 .5o 373629 374193 9-40 626811 42 19 23o33 973 1 1 362356 8-89 988163 .50 lit III 625807 41 20 21 23062 97304 3628S9 8-87 988133 9-988103 .5o -5o ._^47j6 9-375319 625244 10-624681 40 39" 23090 97298 9-3634?2 22 23ii8 97284 363954 8-85 988073 -50 375881 624119 88 23 23146 3644S5 8-84 9S8043 .5o 376442 9.34 623558 87 24 28175 97278 365oi6 8-83 988013 .50 377003 9-33 622997 622487 36 2o 23203 97271 365546 8-82 987983 -50 377563 9-32 35 26 23231 97264 366075 8-81 987953 -5o 378122 9-31 621878 34 27 23260 97257 366604 8-80 987922 -5o 378681 9-3o 621819 33 2S 23288 97251 367i3i 8-79 987892 -50 379739 9-29 620761 32 29 233i6 97244 367659 8-77 987862 -So 379797 9-28 620208 81 80 81 23345 23373 97237 368i85 8-76 8.75 987832 9^987¥oT .5i .5i 38o334 9-27 61^9646 10-619090 30 29 97230 9-368711 9-380910 9.26 32 23401 97223 369236 8-74 987771 .51 381466 9.25 6 18534 28 33 23429 23458 97217 369761 8.73 987740 • 51 382020 9-24 617980 27 34 97210 370285 8-72 987710 • 5i 382575 9.23 617425 26 3.) 23486 97203 370808 8.71 987679 .51 383129 9.22 616871 25 36 235i4 l^t 37i33o 8.70 987649 •51 383682 9.21 616818 24 87 23542 371802 8-69 987618 • 5i 384234 9.20 615766 23 88 23571 97182 372373 8-67 987588 .5i 384786 9.10 9.18 6r52i4 22 89 23599 97176 372894 8-66 987557 .5i 385337 614668 21 40 41 23627 23656 97169 97162 373414 9-373933 8-65 987526 ^9874^ .51 385888 9.386438 9.17 9-15 614112 10T6 13562 20 19 8-64 4-2 23684 97155 374452 8-63 987465 -51 l%ii 9-14 618018 IS 43 23712 97148 374970 8-62 987434 .51 9.13 612464 17 44 23740 97141 375487 8.61 987403 •52 388084 9-12 611916 61 1869 16 45 23769 97134 376003 8-60 987372 -52 388631 9. 11 15 46 23797 97127 '^fo'^ 8-5? 987341 -52 389178 9-10 610822 14 47 23825 97120 987310 -52 389724 12 610276 13 4S 23853 97113 377549 8-57 987279 -52 390270 609780 12 4.) 23882 97106 378063 8-56 987248 -52 390815 9.07 609185 11 50 Id 23910 97100 378577 8-54 987217 •52 .52 391860 9-06 608640 10-608097 607553 10 "9" 23938 ^^ 9-379089 8-53 9.987186 9.391903 9 -05 52 23966 379601 8-52 987.55 .52 392447 9-04 8 53 23995 97079 38on3 8-51 987124 -52 3929^9 9 -03 607011 7 ^54 24023 97072 380624 8-50 987092 .52 393531 9-02 606469 6 55 24o5i 91065 38u34 8-49 987061 •52 394073 9-01 603927 6 56 24070 24108 97058 381643 8-48 987030 -52 394614 9-00 8-99 6o5386 4 57 97o5i 382152 8-47 986998 -52 395154 604846 8 68 24136 97044 382661 8-46 986967 •52 395694 8-98 604806 2 59 24164 97037 383 1 68 8-45 986936 -52 396233 8.97 608767 1 60 24192 97o3o 383675 8-44 986904 -52 396771 8-96 608229 N. COS. N. sine. L. COS. 1 D.l" Lsine. 1 L. cot. D.l" L. tang. ~^ 76° 1 44 TRIGONOMETRICAL FL'NCTIONS. — 14®. Nat. Functions. L06ARITUMIC Functions + 10. ' N.sine. N. cos. L. sine. D. 1" L. COS. D.l" i L. tang. D.l" L. cot 24192 97o3o 9-383675 8.44 9.986904 •52 9-396771 8.96 io^6o3229 60 1 24220 97023 384182 8 43 986873 •63 i 397309 8 96 602691 59 2 24249 97015 334687 8 42 986841 -63 ; 397846 8 95 602164 58 S 24277 97008 385192 8 41 ,986809 .63 398383 8 94 601617 57 4 243o5 97001 385697 8 40 986778 -63 396919 8 93 601081 56 6 24333 96994 386201 8 ll 986746 -63 I 399455 8 92 600646 55 ' 6 24362 I 96987 386704 8 986714 .63 399990 8 91 600010 54 7 24390 j 96980 387207 8 ll 986683 .53 400624 8 90 699476 63 8 i 24418196973 387709 8 986651 -53 401068 8 89 698942 52 9 24446 : 96966 388210 8 35 986619 -63 401691 8 88 698409 51 10 i^4474 I 9^^939 3S8711 8 34 986587 •63 402124 8 87 597876 50 11 , 245o3 i 96952 9.389211 8 33 9.986555 .53 9-402656 8 86 10-697344 49 12 24331 96945 389711 8 32 986523 • 53 1 403187 403718 8 86 60681 3 48 13 24!>59 j 96937 390210 8 3i 986491 •63 8 84 696282 47! 14 24587 i 96930 24615 96923 390708 8 3o 986459 •53 404249 8 83 695761 46 15 391206 8 28 986427 986395 -63 404778 8 82 696222 45 i 16 24644 96916 391703 8 27 -53 4o53o8 8 81 694692 44 i 17 24672 96909 ^6^? 8 26 986363 •54 405836 8 80 694164 45 i 18 24700 96902 8 25 986331 •54 4o6364 8 ]l 593636 42 ' 19 24728 96894 393191 8 24 986299 •54 406892 8 693108 41 20 21 24756 24784 96887 96880 393685 9-394179 394673 8 23 22 986266 9.986234 •64 •54 407419 9-407945 8 77 592681 40 ^ 8 76 10- 692066 39 ' 22 24813 96873 8 21 986202 •64 408471 8 75 591620 691003 33 23 24841 96866 395166 8 20 986169 •54 408997 8 74 37 24 24869 96858 395658 8 •9 986137 •54 409621 8 74 690479 36 25 24897 96801 396150 8 18 986104 •54 410045 8 73 689955 85 26 24925 96844 396641 8 17 986072 •54 410669 8 72 689431 84 • 27 24954 96837 397132 8 '7 986039 •64 411092 8 71 688908 33 28 24982 96829 397621 8 16 986007 •64 411616 8 70 588385 82 29 25oio 96822 398111 8 i5 985974 •64 412137 8 69 687864 81 30 31 25o38 96815 398600 8 8 14 i3 985942 9.980909 985876 •54 .55 412668 8 68 587342 30 25o66 96807 9T399088 9-4i3i79 8 67 10-586821 29 32 25094 96800 399575 8 12 .55 413699 8 66 586301 2S 83 25l22 96793 400062 8 II 985843 .65 414219 8 65 586781 i'7 34 25l5l 96786 400649 8 10 9S58u • 55 414733 8 64 686262 2'] 85 25i79 96778 4oio35 8 09 985778 • 55 416207 8 64 584743 25 36 25207 25235 96771 4oi520 8 08 985745 .55 416775 8 63 684226 24 37 96764 4o20(>5 8 07 985712 .56 416293 8 62 683707 2 5 38 25263 96756 402489 8 06 985679 .55 416810 8 61 683190 22 39 25291 96749 402972 8 o5 985646 • 65 417326 8 60 682674 21 40 41 25320 25348 96742 403455 8 04 98561 3 9-985580 • 55 .55 417842 9-418358 8 "8^ 59 58 682168 10-681642 20 19 96734 9-403938 8 o3 42 25376 96727 404420 8 02 985547 • 65 418S73 8 57 681127 18 43 25404 96719 404901 8 01 985514 •55 419387 8 56 58o6i3 17 44 25432 96712 405382 8 00 985480 .55 419901 8 66 680099 679583 16 45 25460 96705 4o5862 99 985447 .65 420416 8 65 15 46 25488 96697 406341 98 985414 .66 420927 8 54 670073 578560 14 47 255i6 96690 406820 97 985380 • 66 421440 8 63 13 43 25545 96682 407299 96 985347 • 56 4219^2 8 52 678048 12 40 25573 96675 407777 93 985314 .66 422463 8 61 677537 11 50 256oi 96667 408254 94 985280 .56 422974 9-423484 8 5o 677026 10 51 25629 96660 9-408731 94 9.985247 • 56 ~8^ 1? 10-576616 ~9" 52 "25657 i 96653 409207 93 985213 • 56 1 423993 . 8 576007 8 53 25685 1 96645 409682 92 985 1 80 .56 4245o3 8 48 675497 7 54 25713 '96638 410157 91 986146 .66 426011 8 47 574989 6 55 25741 96630 4io632 985ii3 • 56 426619 8 46 674481 5 56 25769 96623 25798 96615 411106 986079 986045 • 66 426027 8 45 573973 4 57 411579 •66 426534 8 44 573466 3 68 25826 96608 4l2052 87 98601 1 • 56 427041 8 43 672969 2 59 25854 ' 96600 412524 86 984978 • 56 i 427547 8 43 672453 1 60 25882 96593 412996 7-85 984944 .56 428062 8-42 571948 N. COB. N.sine. L. COS. D.l" L. sine. i L. cot D.l" L. tang. t 75° 1 T&lGONOMEtillCAL FtJNCTlONS. — 1$°. 45 I r \ Nat. Functions. LOGABITHMIO FUNCTIONS + 10. N.8ine.'N.cos. L. sine. D.l" L.CO.. D.l^' 1 L.tang. D.l" L.cot. 1 25882 96593 9-412996 7-85 9-984944 •57 '9-428052 8-42 10-571948 160 1 ! 15910 196585 413467 84 984910 •57 428337 8-41 571443 i 59 2 i 23938 1 96378 413938 83 984876 •57 429062 8.40 570938 58 8 23966 96370 414438 83 984842. •57 429366 8-39 570434 57 4 23994196362 414378 82 984808 •57 430070 8-38 569930 56 6- 26022 96355 415347 81 9B4774 •57 430373 8-38 569427 55 6 26o5o 96547 4.5.. 5 7 80 984740 •57 431075 tu 568925 j 54 7 26079 96540 416233 79 984706 •57 43 1577 568423 53 8 26107 96332 416751 7« 984672 •57 432079 8-35 56792 1 52 9 26135 96524 417217 77 984637 •57 432580 8.34 567420 51 10 26163 26191 963r7_ 417684 76 984603 •57 -57 43 3 080 8.33 566920 50 10-566420 1 4U 96309 Q.4i8i5o 75 9.984569 " 9.433580 8.32 "12 26219 96502 418615 74 98453D •57 434080 8.32 565920 48 13 26247 96494 419079 73 984500 •57 434579 8.3i 565421 47 14 26275 96486 419544 73 984466 •57 435078 8-3o 564922 46 15 263o3 96479 420007 72 984432 -58 435576 8-29 564424 45 16 263 3 1 1 96471 420470 71 984397 -58 436073 8-28 563927 44 17 26339 1 96463 420933 70 984363 -58 436570 8-28 563430 43 18 26337196436 421395 tt 984328 -58 437067 8-27 562933 42 19 26415 1 96448 421837 422318 984294 984269 -58 437363 8-26 562437 41 20 21 26443 26471 96440 67 • 58 438059 ^38554 8-25 561941 10-661446 40 S9 96433 9 •42^2773 67 9.984224 -58 8-24 22 265oo 96425 423238 66 984190 984155 -58 439048 8-23 560952 38 23 26323 96417 423697 65 -58 439543 8-23 560457 37 24 26336 96410 4241 56 64 984120 -58 440036 8-22 559964 36 25 26384 96402 424615 63 984085 • 58 440529 8-21 559471 35 26 26612 96394 425073 62 984050 • 58 441022 8-20 558978 558486 34 27 26640- 96386 425530 61 984015 • 58 44i5i4 8-19 33 28 26663 96379 425987 60 983981 -58 442006 8-19 mii 32 29 26696 96371 426443 60 983946 -58 442497 8-i8 31 SO ^31 26724 26752 96363 426S99 59 98391 1 .53 -58 _^42988_ 9-443479 8.17 8.16 557012 10-556521 SO 2y 96335 9-427334 58 9.983875 S2 26780 96347 427809 428263 57 983840 .59 443968 8.16 556o32 28 S3 26808 96340 56 9838o5 -59 444458 8.i5 555542 27 84 1 26336 96332 428717 55 983770 .59 444947 8.14 555o53 26 S5 26,;64 96324 429170 54 983735 -59 445435 8-i3 554565 25 86 26392 96316 429623 53 983700 .59 445923 8-12 554077 24 S7 26920 96308 430075 52 983664 -59 446411 8-12 553589 23 88 1 26948 96301 43o527 430978 52 983629 -59 446898 8-11 553102 22 89 j 26976 96293 5i 983594 -59 447384 8-10 552616 21 40 "4T 27004 27032 96285 96277 431429 5o 983538 9.983523 -59 .59 447870 9-448356 8-09 552i3o 20 19 9.431879 49 8.09 10-551644 42 1 27060 96269 432329 43277a ii 983487 .59 448841 8.08 55ii59 18 43 127088! 9^)261 983452 -59 449326 8.07 550674 17 44 1271 16 1 96233 433226 47 983416 .59 449810 8.06 550190 16 45 127144 [96246 433675 46 983381 .59 450294 8.06 549706 15 46 1 27172 96238 434122 45 983345 .59 450777 8.o5 549223 14 47 27200 1 96230 434569 44 983309 983273 983238 •5o 451260 8-04 548740 13 48 27228 9-3222 435oi6 44 .60 451743 8-03 548257 547775 12 49 27256 i 96214 435462 43 • 60 452225 8.02 11 50 51 27284 96206 435908 42 983i202 9.983x66 -60 -60 452706 97453187" 8.02 8.01 547294 10-546813 10 9 27312 96198 9-436353 41 52 j 27340 itx 436798 40 983i3o .60 453668 8.00 546332 8 53 ! 27368 437242 40 983094 983o58 • 60 454148 7-99 545852 7 54 j 27396 96174 437686 39 -60 454628 7-99 545372 6 55 127424 96166 438129 38 983022 .60 455107 7.98 544893 5 56 1 27452 96158 438572 37 982986 -60 455586 7-97 544414 4 57 27480 96150 439014 36 982950 -60 436064 7.96 543936 3 158 27508 96142 439456 36 982914 982878 .60 456542 7.96 543458 2 69 27536 96134 439897 35 -60 457019 7.95 542981 1 60 27564 96126 440338 7 34 982842 .60 457496 7.94 542504 |; N. COS. N. sine. 1 L. COS. D.l" L.sine. L.<»t D.r- L. tang. ' 1 74° 1 46 TEIGOITOMETRICAL FUNCTIONS.— 16*. Nat. Fukctions. Logarithmic Functions + 10. i / N.sine/N.cos. L. sine. D.l" L. COS. D.l" j L. tang. D.l" L.cot ~^ 27564 96126 9.440333 7-34 9-982842 1 -60 '9.457496 •94 10- 542504 60 1 27092 961 j8 440778 33 982805 .60 437973 •93 542027 59 2 1 27620 961 10 441218 32 982760 . 982733 •61 458449 93 54i55i 58 3 27648 96102 441638 7 3i •61 458925 92 541075 57 4 27676 96094 442096 3i 982696 •61 459400 91 540600 56 5 27704 96086 442535 3o 982660 .61 459875 90 540125 55 6 , 27731 96078 442973 29 982624 -61 460349 460823 90 539651 54 7 27759 96070 443410 28 982587 .61 89 539177 53 8 i 27787 96062 443847 27 982551 .61 461297 88 535703 52 9 27815 96054 444284 11 982014 •61 461770 88 538230 51 10 27843 96046 444720 982477 .61 '•6? 462242 9-462714 -y 87 86 537753 10-537286 50 49" 11 1 27871 96037 9.443155 25 9.982441 121127899 96029 443390 24 982404 .61 463 1 86 85 536814 4S 13 Ij 27927 96021 446025 23 982367 .61 463658 i 85 536342 47 14 !' 27955 96013 446459 23 982331 .61 464129 84 535871 46 15 I 27983 96005 446893 22 982294 .61 464599 83 535401 45 16 ;j 2:ioii 95997 447326 21 982237 .61 465069 83 534931 44 17 j; 28039 95989 447759 20 982220 .62 465539 82 534461 43 18 28067 : 95981 448191 20 982183 .62 466008 81 533992 42 19 28095 95972 20 '1 28123 95964 21| 28i5i 95956 448623 19 982146 .62 466476 80 533324 41 449054 9 •449485 18 982109 .62 466945 80 533o55 40 39 \l 9.982072 • 62 9.467413 79 10-532587 22 28178 95948 449915 982035 .62 467880 78 532120 33 23 28206 95940 430345 16 981998 •62 468347 78 53i653 37 24 28234 95931 450773 i5 981961 .62 468814 ]l 531186 36 25 28262 95923 451204 U 981924 .62 469280 530720 35 26 28290 '95915 45i632 i3 981886 .62 469746 75 530234 34 27 283 18 95907 452060 i3 981849 •02 470211 75 529789 33 28 28346 93898 452488 12 981812 • 62 470676 74 529324 32 29 28374 95890 452913 II 981774 981737 .62 47U4I 73 528859 528395 31 80 31 I 28402 i 95882 284 2'9 95874" 453342 9-453768 10 •62 •63 471605 73 30 ID 9-981699 9.472068 72 10-527932 29 32 28457 95865 454194 09 981662 -63 472532 7 71 527468 28 33 28485 1 95357 434619 08 981625 •63 472995 71 527005 27 34 i285i3i 95849 455044 07 981587 • 63 473437 70 526543 26 35 1 28541 j 95841 455469 07 981549 •63 473919 69 526081 25 36 28569 95832 455S93 06 981512 •63 474381 69 525619 24 37 1 28597 i 95824 4563 16 o5 981474 •63 474842 68 523158 23 38 2S625I95816 456739 04 981436 .63 4753o3 67 524697 22 39 ' 28652 1 95807 457162 04 981399 .63 473763 67 524237 21 40 41 28680 1 95799 '287081 95791 457584 o3 981361 .63 .63 476223 ^476683 66 5.3777 20 19 9.458006 02 9.981323 65 10-523317 42 '28736195782 458427 458848 OI 981285 .63 477142 65 522858 13 43 28764195774 01 981247 •63 477601 64 522399 17 44 2S792 1 95766 459268 7 00 981209 •63 478059 63 521941 16 45 ! 28820195757 45o688 6 99 . 981171 •63 478317 63 521483 15 4() : 28847 ' 9^749 460108 6 98 981133 .64 478975 62 521023 14 47 : 28875 1 93740 460527 6 98 981095 -64 47943^ 61 520368 13 43 1 28903 i 95732 460946 6 97 981037 -64 t& 6i 520111 12 49 1 28931 95724 46 1 364 6 96 981019 -64 60 519655 11 5<:> IT 289391 93715 461782 6 95 980981 .64 .64 480801 9.481257 59 519199 10 28987 1 93707 29015 195698 9.462199 6 95 9.980942 u 10-518743 52 462616 6 94 980904 980S66 •64 481712 518288 8 53 29042 1 95690 463o32 6 93 -64 482167 57 517833 7 54 29070193681 463448 6 93 980827 •64 482621 57 517379 6 5.') ! 29098 1 95673 463864 6 92 980789 •64 483075 56 516925 5 5; ; 29126 9 "664 464279 6 91 980750 -64 483529 55 516471 4 57 '29154 '93656 464694 6 90 980712 -64 483982 55 5i6oi8 3 58 29182 93647 465 1 08 6 % 980673 -64 484435 54 5i5565 2 59 29209 93639 465522 6 980635 -64 484887 53 5i5ii3 1 CO 29237 1 9563o 465935 6-88 980596 • 64 485339 7.53 5i466i N. COS. N. 6ine. L. COS. D. 1" L. sine. Loot. D.l" L. tang. / 73^ 1 TRIGONOMETRICAL FUNCTIONS. — 1*' 47 Nat. Functions. Logarithmic Functions + 10. 1 ' N.slne. N. cos. L. sine. D.l" L.C08. D.l" L. tang. D.l'' L. cot. 29237 95630 9.465935 6.88 9.980596' .64 9^485339 7^55 io^5i466i 60 1 2926J 95622 466348 6.88 980558 .64 485791 7-52 514209 59 2 29293 956i3 466761 6.87 6.86 980519 •63 486242 7-5i 5x3758 58 3 29321 956o5 467173 980480 .65 486693 7-5i 5i33o7 57 4 29348 95596 467585 6-85 950442 ' •65 487143 7^5o 512857 56 5 29376 955b8 467996 6.83 930 .o3 •65 487593 T49 512407 55 6 29404 95579 468407 6.84 980364 •65 488043 7-49 Oij?o8 54 7 29432 95571 468817 6-83 980325 •65 488492 7-48 53 8 29460 95562 469227 6.83 980286 •65 488941 7-47 5iio59 52 9 29487 95554 469637 6.82 980247 •65 489390 - 489838 9-490286 7-47 5io6io 51 10 11 295i5 95545 470046 6.81 980208 •65 .7.46 510162 50 2^5^ 95536 9.470455 6-80 9.980169 •65 7-46 io^5o97i4 49 12 29371 95528 470863 6-80 980130 • 65 490733 7-45 509267 48 13 29599 95519 471271 6.79 980091 • 65! 491 180 7-44 308820 47 14 29626 95511 471679 6-78 980052 • 65 491627 492073 7-44 508373 46 15 29654 955o2 472086 6-78 980012 •65 7-43 507927 45 IG 29682 95493 472492 6.77 979973 • 65 492519 7-43 507481 44 17 29710 95485 472898 6.76 979934 • 66 492963 7-42 507035 43 18 29737 95476 473304 6.76 979895 • 66 493410 7-41 506590 42 19 29765 9-5467 473710 6.75 979855 • 66 493854 7-40 506146 41 20 '21 19791 95459 4741 i5 6-74 979816 • 66 494299 7-40 5o57oi 40 29821 95450 9.474519 474923 6.74 9-979776 .66 9-494743 7-40 io-5o5237 oy 22 29849 95441 6.73 979737 • 66 493186 7-39 504814 38 23 29876 95433 475327 6.72 979697 • 66 495630 7^38 504370 37 24 29904 95424 475730 6.72 979608 • 66 496073 7-37 503927 36 25 29932 93415 476133 6.71 979618 • 66 4965 1 5 7-37 5o3485 35 26 29960 95407 476536 6.70 979579 •66 496957 7-36 5o3o43 34 27 29987 95398 476938 6-69 979539 • 66 497399 7-36 5o26oi 33 28 3ooi5 95389 477340 6.69 979499 •66 497841 7^35 5o2i59 32 2S 30043 95380 477741 6.68 979439 • 66 498282 7-34 501718 31 30 31 30071 30098 95372 95363 478142 6.67 979420 9.979380 .66 •66 498722 9-499«63 7-34 7-33 501278 io-5oo837 30 2J 9.478542 6.67 32 30126 i 95354 478942 6.66 979340 •66 499603 7-33 500397 28 33 3oi54 95345 479342 6.65 979300 • 67 5ooo42 7^32 499938 v7 34 30182 95337 479741 6-65 979260 •67 500481 7-3i 499319 26 85 30209 95328 480140 6.64 979220 •67 500920 7-3. 499080 '25 36 30237 95319 480539 6-63 979180 •67 5oi359 7^30 49'--;64i 24 87 30265 95310 480937 6-63 979 '40 •67 501797 7^30 498203 .'23 88 30292 95301 481334 6-62 979100 •67 502235 ]:?s 497765 22 39 3o32o 95293 481731 6.61 979059 • 67 502672 497328 21 40 3o348 _95^84_ 482128 6.61 979019 •67 5o3io9 7^28 _i96891 10.496454 20 19~ 41 30376 95275 9.482025 6.60 9.978979 .67 9-5o3546 7.27 42 3o4o3 95266 482921 6.59 978939 .67 503982 7-27 496018 13 43 3043 1 95257 4833 16 6.59 978898 .67 504418 7^26 495582 17 44 30459 95248 483712 6.58 978858 •67 504854 7-25 493146 16 45 30486 95240 484107 6.57 978817 .67. 505289 7-25 494711 15 46 3o5i4 9523 1 484501 6.57 978777 .6-^ 505724 7-24 494276 U 47 3o542 95222 484895 6.56 978736 •67 1 5o6i59 7-24 493841 13 48 30570 95213 485289 6.55 978696 .68 506593 7-23 493407 492973 492340 10-492107 12 49 3o597 95204 485682 6.55 978655 •68 507027 7^22 11 50 51 30625 3o653 95195 486075 6.54 978615 • 68 • 68 507460 7.22 7-21 10 ~9" 95186 9.486467 6.53 9.978574 9-507893 52 3o68o 95177 486860 6.53 978533 •681 5o8326 7-21 491674 8 53 30708 95168 487251 6.52 978493 •68 508739 7-20 491 241 7 54 30736 95159 487643 6.5i 978452 • 68 509191 7-'9 490809 6 55 30763 95i5o 488034 6.5i 978411 • 68 509622 7-19 490378 5 56 30791 95142 488424 6.5o 978370 • 68 5ioo54 7.18 489946 4 57 30819 95i33 488814 6-50 978329 • 68 5 1 0485 7.18 489515 3 58 30846 95124 489204 6-49 978288 •68 510916 7-'7 489084 2 59 30874 95ii5 489593 489982 6-48 978247 ■ 68 5ii346 7.16 488634 1 60 3090a 95106 6-48 978206 • 68 511776 7.16 488224 N.CM. N. Bine. L. COS. D.l" L. sine. L.cot. D.l" L. tang. ' 72° 1 48 d:RlGONOMETRtCAL FUKCTIOKS. — 18% Nat. Functions. Logarithmic Functions + 10. 1 N.sine. N. COS. L. sine. D. 1" L. COS. D.l" L. tang. D.l" L. cot 30902 93106 9-489982 6-48 9-978206 .68 9.511776 7.16 10-488224 60 1 30929 93097 490371 6 48 978165 .68 5l22o6 7-16 il]iii 59 2 30937 93088 490739 6 47 978124 -68 512635 7-i5 58 S 30985 95079 491147 6 46 978083 •69 5i3o64 7-14 486936 57 4 ■3lO!2 95070 491535 6 46 978042 .69 513493 7-14 486307 5t) 5 3 1 040 95061 491922 6 45 978001 .69 513921 7-13 486079 55 6 3 1 068 95o52 492308 6 44 977959 977918 .69 514349 7-i3 485651 54 7 31095 93043 492695 6 44 -69 514777 7-12 485223 53 8 31123 95o33 493081 6 43 977877 -69 513204 712 484796 52 9 3ii5i 95024 493466 6 42 977835 .69 5i563i 7-11 484369 483943 51 10 11 31178 3 1206 950 1 5 95006 493831 6 42 977794 .69 5 1 6057 7-10 50 49" 9-494236 6 41 9.977752 .69 9^i'6484' 7-IO 10-483516 12 31233 94997 494621 6 41 97771 1 .69 516910 7-09 483090 48 13 31261 94988 495oo5 6 40 977669 .69 517335 11 482665 47 14 1 31289 94979 495388 6 39 977628 .69 517761 482239 481815 46 15 3i3i6 94970 495772 6 39 977586 .69 5i8i85 7-o8 45 16 3 1 344 94961 496154 6 38 977544 .70 5i86io 7-07 481390 44 17 31372 94952 496537 6 37 9775o3 .70 519034 7-06 480966 43 18 3 1 399 94943 496919 6 37 977401 .70 519458 7-06 480342 42 19 31427 94933 497^01 6 36 977419 .70 519882 7-05 480118 41 20 31454 94924 94915 497632 6 36 977377 .70 52o3o5^ 7-o5 __479695. 10-479272 40- 39 "211314^2 9.498064 6 35 9.977335 .70 9.520728 7 -04 22 3i5io 94906 498444 6 34 977293 .70 52ii5i 7-03 478849 38 23 i'^-il 94S97 498825 6 34 977231 .70 521573 7-03 478427 37 24 3 1 365 94888 499204 6 33 977209 .70 521995 7-03 478005 36 25 3 1593 94S78 499584 6 32 977167 977125 .70 522417 7.02 477583 35 26 31620 94869 499963 6 32 .70 522838 7.02 477162 84 27 31648 94j6o 5oo342 6 3i 977083 .70 523259 7.01 476741 83 28 31675 94851 500721 6 3i 977041 -70 523680 7.01 476320 32 29 31703 94842 501099 6 3o 976999 •70 524100 7.00 475900 31 30 31 31730 94832 5oU7^ 6 29 976937 .70 524520 6.99 475480 30 31758 94823 9-5oi854 6 29 9-976914 •70 9.524939 6.99 io-475o6i 29 32 31786 3i8i3 94814 50223l 6 28 976872 •71 525359 6.98 474641 28 33 94805 502607 6 28 976830 •71 525778 6.98 474222 27 34 31841 94795 502984 6 27 976787 •71 IIIIV5 6-97 473803 26 85 3 1 868 94786 5o336o 6 26 976745 •71 t^^ 473385 25 36 31896 94777 5o3735 6 26 976702 •71 527033 472067 472349 24 87 31923 94768 5o4iio 6 25 976660 •71 527451 6-96 23 38 31951 94758 5o4485 6 25 976617 •71 527868 6-95 472132 22 39 31979 94749 504860 6 24 976574 •7' 528285 6-95 47i7«5 21 40 41 32006 94740 5o5234 6 23 976532 •71 528702 6-94 471298 20 32o34 94730 9-5o56o8 6 23 9.976489 •71 9.529119 529535 6-93 10-470881 19 42 32o6i 94721 505981 6 22 976446 •71 6-93 470465 18 43 32089 32116 94712 5o6354 6 22 976404 •71 529950 6-93 470o5o 17 44 94702 506727 6 21 976361 •71 53o366 6-92 469634 16 45 32144 94693 94684 507099 6 20 976318 •71 530781 6-91 469219 15 46 32171 507471 6 20 976275 976232 •71 531196 6-91 468804 14 47 32199 94674 94665 507843 6 19 .72 53i6ii 6-90 468389 467975 467561 13 48 32227 508214 6 19 976189 1 .72 532025 6-90 12 49 32254 94656 5o8585 6 18 976146 .72 532439 6-89 11 50 32282 94646 508956 6 18 976103 •72 532853 6-89 467147 10 32309 94637 9-509326 6 17 9.976060 •72 9.533266 6-88 10-466734 9 52 32337 94627 509696 6 16 976017 .72 533679 6-88 466321 8 53 32364 94618 5 10065 6 16 975974 .72 534092 5.87 465908 7 54 32392 94609 510434 6 i5 975930 .72 534504 6-87 465496 465o84 6 55 32419 94599 5 10803 6 i5 975887 .72 534916 6-86 5 56 32447 i$ro 511172 6 14 975844 •72 535328 9-86 6-85 46467 J 4 57 32474 5ii54o 6 i3 975800 .72 535739 464261 3 58 32502 94571 511907 512275 6 i3 975757 .72 536i5o 6-85 463850 2 59 32329 94561 6 12 975714 •72 536561 6.84 463439 463028 1 60 32557 94552 512642 6-12 975670 -72 536972 6.84 N. COS. N. sine. L. COS. D.r' L. sine. L. cot D.l" L. tang. » •71° 1 TRIGOKOMETRIOAL FUNCTIONS. — 1»°. 4S Nat. Functions. LOGABITHMIO FUNCTIONS + 10. 1 / 1 N.sine. N. COS. L. sine. D. 1" L.CO8. ] 0.1" L. tang. 9.536972 537382 Dl." L.cot 3a557 94552 9-5i2642 6.12 9.975670 ^ 6.84 10.463028 60 1 32584 94542 5 1 3009 6. II 975627 .73 6.83 462618 59 2 32612 94533 513375 6.11 975583 •73 537791 6-83 462208 58 3 33639 94523 5i374i 6- 10 975539 .73 538201 6-82 461798 57 4 32667 94514 514107 6-09 975496 .73 53861 1 6-82 461389 56 5 32694 94304 514472 6-09 975432 •73 539020 6-81 460980 55 6 32722 94493 514837 6-08 975408 •73 539429 6.81 460571 54 7 32749 94485 5l5202 6.08 975365 •73 539837 6.80 460163 53 8 32777 94476 5 15566 6.07 975321 •73 540245 6-80 459755 52 9 32804 94466 5 1 5930 6.07 975277 •73 540653 6-79 459347 51 10 11 32832 94457 516294 6.06 975233 •73 541061 6-79 458939 10 -458532 50 49" 32859 9443B 9.516657 6.o5 9.975189 975145 •73 9.541468 6-78 12 32887 517020 6.o5 •73 541875 6-78 458120 48 13 32914 94428 517382 6.04 975101 •73 542281 6-77 457719 47 14 32942 94418 518107 6-04 975057 •73 542688 6-77 407312 46 15 32969 94409 6-03 975oi3 •73 543094 6-76 406906 45 16 32997 94399 518468 6.02 974969 974923 •74 543499 6-76 456501 44 17 33024 94390 518829 6-02 •74 543905 6.75 456090 43 18 33o5i 94380 519190 6-01 974880 •74 544310 6-75 450690 42 19 33079 94370 519551 6-01 974836 •74 544715 6-74 455280 41 20 21 33 1 06 94361 519911 6.00 6.00 974792 9-974748 •74 •74 545119 6-74 454881 10-454476 40 3y 33i34 943JI 9.520271 9.545524 6.73 23 33i6i 94342 52063 1 5-99 974703 •74 545928 546331 6.73 454072 38 23 83189 94332 520990 ^99 974659 •74 6.72 453669 37 24 33216 94322 521349 5.98 974614 •74 546735 6.72 453260 36 25 33244 943 1 3 521707 5.98 974570 •74 547138 6.71 452862 35 26 33271 943o3 522066 5.97 974525 •74 547540 6.71 452460 84 27 33298 94293 522424 5.96 974481 •74 547943 6-70 452057 33 28 33326 94284 522781 5.96 974436 •74 548345 6-70 45i65o 32 29 33353 94274 523i38 5.95 974391 •74 548747 6.69 451253 31 80 81 33381 33408' 94264 523495 5.95 974347 •75 549149 6-69 45o85i 10.450450 30 29 94254 9.523852 5.94 9.974302 •75 9.549550 6.68 82 33436 94245 524208 5-94 974257 •75 549951 6.68 450049 28 33 33463 94235 524564 5.93 974212 •75 55o352 6.67 449648 27 34 33490 94225 524920 5.93 974167 •75 550752 6.67 449248 26 35 335i8 9421 5 525275 5.92 974122 •75 55ii52 6.66 448848 25 36 33545 94206 525630 5.91 974077 •75 55i552 6.66 448448 24 37 33573 94196 525984 5.91 974032 •75 551952 552351 6-65 448048 r3 38 33600 94186 526339 526693 5.90 973987 •75 6.65 447649 22 39 33627 94176 iT 973942 •75 552750 6-65 447250 21 40 41 33655 94167 527046 9.527400 5.89 973897 •75 •75 553149 6.64 446851 10.446452 20 19 33682 94157 5.89 9.973852 9.553548 6.64 42 33710 94147 527753 5.88 973807 •75 553946 6.63 446054 18 43 33737 94137 528io5 ■. 5.88 973761 •75 554344 6.63 445656 17 44 1 33764 94127 528458 5.87 973716 .76 554741 6.62 440209 16 45 33792 94118 528810 5.87 973671 .76 555 1 39 6.62 444861 15 46 33819 94108 529161 5.86 973625 .76 555536 6.61 444464 14 47 33846 94098 529513 5.86 973580 .76 555933 6.61 444067 13 48 33874 94088 529864 5-85 973535 .76 556329 6-60 443671 12 49 33901 94078 53021 5 5.85 973489 •76 556725 6.60 443275 11 50 51 33929 94068 53o565 5.84 973444 .76 557121 6.59 442879 10.442483 10 33906 94058 9.530915 5.84 '■V,llt .76 9.007317 6-59 52 33983 94049 531265 5.83 .76 557913 442087 8 53 34011 94039 53i6i4 5.82 973307 .76 5583o8 441692 7 54 34038 94029 531963 5.82 973261 •76 553702 6.53 441298 6 65 34065 94019 532312 5.81 973215 .76 559097 6.57 440903 5 56 34093 94009 532661 5.81 973169 •76 559491 6.57 44o5o9 4401 1 5 4 57 34120 93999 533009 5.80 973124 •76 559885 6.56 3 58 34147 93989 533357 5-80 973078 •76 1 560279 560673 6.56 439721 2 59 34175 93979 533704 i:]? 973o32 •77 6.55 439327 1 60 34202 93969 534052 973986 •77 561066 6.55 438934 N. COS. N. Bine. L. C03. D. 1" L. sine. j L. cot. D.l" L. tang. ' 70° 50 TRIG0J5"0METIIICAL FUKCTIOI^-S. — 30". Nat. Functions. LoGABiTHjuc Functions + 10 1 , N.sine. N. cos. L. sine. 1 D. 1" L. COS. D.l" L. tang. D 1" L. cot 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 IS 19 20 21 22 23 24 25 26 27 28 29 30 34202 34229 34207 34284 343-1 1 34339 34366 34393 34421 34448 34475 93969 93939 93949 93939 93929 93919 93909 93899 93889 93879 93869 g. 534052 534399 534745 535092 535438 535783 536129 536474 536818 537163 537507 5 5 5 5 5 5 5 5 5 5 5 78 77 77 77 ^A ]t 74 i 9.972986 972940 972894 972848 972802 972755 972709 972663 972617 972570 972524 •77 •77 •77 •77 •77 •77 •77 •77 •77 •77 •77 9.561066 561439 56i85i 562244 562636 563028 563419 563811 564202 564592 564983 6 6- 6 6 6 6 6 6 6 6 6 55 54 54 53 53 53 52 52 5i 5i 5o 10.438934 438341 438149 437364 436972 436581 436189 433798 435408 435017 60 59 53 57 56 55 54 53 52 51 50 49 4S 47 46 45 44 43 42 41 40 345o3 34530 34557 34584 34612 34639 34666 34694 34721 34748 93859 93849 93839 93829 93819 93809 93799 93789 93779 93769 9-537851 538194 538538 538880 539223 539365 539907 540249 540590 540931 I 5 5 5 5 5 5 5 72 72 71 71 70 70 A^ tl 68 9.972478 972431 972385 972338 972291 972245 972198 o72i5i 972105 972o58 .78 .78 .78 .78 .78 .78 9-565373 565763 566 1 53 566542 566932 567320 567709 568098 568486 568873 6 6 6 6 6 6 6 6 6 6 5o 49 49 49 48 48 47 % 46 10.434627 434237 433847 433438 433068 432680 432291 431902 43i5i4 431127 34775 34803 34830 34857 34884 349i'-« 34939 34966 34993 35021 93759 93748 9373b 93728 93718 93708 93698 93688 93677 93667 9-541272 54i6i3 541953 542293 542632 542971 543310 543649 543987 544325 5 5 5 5 5 5 5 5 5 5 tl 66 65 65 64 64 63 63 9-972011 971964 971917 971870 971823 971776 971729 971682 971635 971588 .78 •78 •78 .78 .78 -78 •79 •79 •79 •79 9-569261 569648 570035 570422 570809 571196 571381 571967 572352 572738 6 6 6 6 6 6 6 6 6 6 45 45 45 44 44 43 43 42 42 42 10-430739 43o332 429965 429378 429191 428803 428419 428033 427648 427262 89 33 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 40 42 43 44 45 4t5 47 4S 49 50 51 52 53 54 55 56 57 58 59 eo 35048 35075 35102 35i3o ,35i57 35i84 '35211 35239 : 35266 35293 93657 93647 93626 93616 93606 llllt 93575 93565 9-544663 543000 545338 545674 546011 546347 546683 547019 547354 547689 9.548024 548359 548693 549027 549360 549693 550026 55o359 550692 55io24 5 5 5 5 5 5 5 5 5 5 '5 5 5 5 5 5 5 5 5 5 62 62 61 61 60 60 59 ?s 58 J .56 56 55 -55 •54 •54 -53 .53 9.971540 971493 971446 971398 97x331 97i3o3 971256 971208 971161 971113 •79 •79 •79 •79 •79 •79 •79 •79 •79 •79 9-573123 573507 573892 574276 574660 575044 575427 575810 576193 576376 9-576938 57734. 577723 578104 578486 578867 579248 579629 580009 58o389 9-580769 581149 58i528 581907 582286 582665 583043 583422 583800 584177 6 6 6 6 6 6 6 6 6 6 "6" 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6- 41 41 40 40 ? 39 38 38 37 3] 36 36 36 35 35 34 34 34 33 33 32 32 32 3i 3i 3o 3o 29 29 10-426877 426493 426108 425724 425340 424056 424573 424190 423807 423424 10-423041 422659 422277 421896 42i5i4 421133 420752 420371 4 1 9991 419611 10.419231 4i885i 418472 418093 417714 417333 416037 416378 416200 415823 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 ~9" 8 7 6 I 3 2 1 35320 35347 35375 35402 35429 35456 35484 35511 35538 35565 93555 93544 93534 93324 93514 93303 93493 93483 93472 93462 93432 93441 93431 93420 93410 93400 93389 93379 93368 93858 9.971066 971018 970970 970922 970874 970827 970779 970731 970683 970635 -80 -80 -80 .80 -80 -80 -80 -80 .80 .80 35592 35619 35647 35674 35701 35728 35755 35782 358io 35837 9-551356 551687 552018 552349 552680 553010 555341 553670 554000 554329 5 5 5 5 5 5 5 5 5 5 52 52 52 5i 5i -5o 5o 49 49 48 9-970586 970538 970490 970442 970394 970345 970297 970249 970200 970152 .80 .80 .80 .80 .80 .81 .81 .81 -81 .81 N.C08. N.sine. L. COS. D. 1" L.sine. 1 L.cot. D.1" L. tang. 1 69° 1 TRIGONOMETRICAL PUNCtlONS.— 21°. £1 Nat, Functions. Logarithmic Functions + 10 . 1 2 3 4 5 6 7 8 I? 11 12 Vi U 15 16 17 18 19 20 [N.8ine.[N. COS. L. sine. D. 1" L.C08. D.l" L.tang. D 1" L.cot 1 35837 35864 35891 35918 35945 35973 36ooo 36027 36o54 36o8i 36 108 93358 93348 93337 93327 93316 93306 93295 93285 93274 93264 93253 93243 93232 93222 93211 93201 93190 93180 93,69 93i3q 93148 9.554329 5:.46J8 554987 55531 5 555643 55597. 556299 506626 556953 557280 557606 9.557932 558238 558583 558909 559234 559558 559883 560207 56o53i 56o855 5 5 5 5 5 I 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 .48 48 47 47 46 46 45 45 44 44 43 43 4i 42 42 41 41 40 40 ll 33 38 i 36 36 35 35 84 34 33 33 32 32 3i 3i 3i 3o 3o 9-970152 970103 970055 970006 969907 969909 969860 9698 1 1 969762 969714 969665 .81 .81 .81 •81 .81 •81 •81 •8. •8. .81 .81 9-584177 584000 584932 585309 585686 586062 506439 5868.0 587.90 587066 587941 t 6 6 t t t 6 "6" 6 6 6 6 6 6 6 6 6 29 29 28 28 27 27 27 26 26 25 25 10.415823 415445 4.5o68 414691 414314 4.3938 4i3o6i 4i3i85 412810 412434 412059 CO 59 t>a- 55 54 53 52 51 50 36i35 36162 36190 36217 36244 36271 36298 36325 36352 36379 9.969616 969067 969018 969469 969420 969370 969321 969272 969223 969.73 9.969.24 969070 969025 968976 968926 968877 968827 968777 968728 968678 9.96S628 968578 968528 968479 968429 968379 968329 968278 968228 96.3.78 .82 .82 •82 .82 .82 • 82 .82 .82 •82 .82 •62 .82 .82 .82 .83 •83 -83- •83 • 83 -83 •83 •83 •83 •83 .83 .83 •83 .83 .84 •84 -84 •84 •84 .84 • 84 .84 -84 •84 •84 •84 9-5883.6 588691 589066 589440 5898.4 590188 590062 590935 591308 591681 9.592054 592426 592798 593.71 593042 593914 594285 594656 595027 595398 9.595768 596.38 596508 596878 597247 5976.6 597980 598354 598722 599091 23 24 24 23 23 23 22 22 22 21 IO-4U684 41 1 309 410934 410OO0 410186 409812 409438 409065 408692 408319 4 'J 48 47 46 4.-) 44 4i 42 41 40 21 22 23 24 25 2« 27 28 29 30 31 32 33 84 35 36 37 38 39 40 36406 36434 36461 36488 365i5 36542 36569 36596 36623 36650 93.37 93127 93116 93106 93095 93084 93074 93o63 93o52 93042 9.561178 56i5or 561824 562146 562468 562790 563112 563433 563755 564075 6 6 6 6 6 6 6 6 6 6 21 20 20 19 19 .8 18 18 17 17 10-407946 407574 407202 406829 406408 406086 405715 405344 404973 404602 89 38 37 36 35 31 33 32 31 30 2J 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 36677 36704 36731 36758 36785 36812 36839 36867 36894 36921 930J1 93020 93010 92999 92988 92978 92967 92956 92945 92930 9.564396 564716 565o36 565356 565676 566632 566951 567269 6 6 6 6 6 6 6 6 6 6 17 16 16 16 10 i5 i5 14 14 i3 10.404232 403862 403492 403l22 402753 402384 4020.5 40.646 401278 400909 41 42 43 44 45 46 47 48 49 r-,0 TT 52 53 54 55 56 57 53 59 60 36948 36975 37002 37029 37056 ! 37083 37110 j 37137 137164 I37191 92924 92913 92902 92892 92881 92870 92859 92849 92838 92827 9.567587 567904 568222 568539 568856 569172 569488 569804 570120 570435 5 5 5 5 5 5 5 5 5 5 29 29 28 28 28 27 27 26 26 25 9.968.28 968078 960027 967977 967927 967876 967826 967770 967720, 967674 9-599409 599827 600194 6oo562 600929 601296 601662 602029 602395 602761 6 6 6 6 6 6 6 6 6 6^ i3 i3 12 12 11 1 1 11 10 10 10 10.400541 4J0173 399806 399438 399071 39S704 398338 397971 397605 397239 J37218 37245 37272 37299 37326 37353 37380 37407 37434 37461 92816 92805 92794 92184 92773 92762 92751 92740 92729 92718 9.570751 571066 571380 571695 572636 572950 573263 573575 5 5 5 5 5 5 5 5 5 5 25 24 24 23 23 23 22 22 21 21 9.967624 967573 967522 967471 967421 967370 967319 967268 9672.7 967166 -84 .84 •85 .85 -85 • 85 •85 •85 -85 •85 9-603127 603493 6o38o8 604223 604538 604953 6053.7 6o5682 606046 606410 6- 6. 6. 6. 6- 6. 6- 6- 6- 6- 09 09 09 08 o3 07 07 07 06 06 10-396873 396507 396142 390777 395412 395047 394683 394318 393954 393090 7 9 8 6 5 4 3 2 1 N. COS. N. sine. L. COS. D.1" L. sine. L.cot. D.r' L. tang. ' 1 6§° " 1 S2 I'RlGON-OilETfilCAL FUNCTION'S. — SS". Nat. Functions. 1 Logarithmic Functions + 1 \ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 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 N.sine.' N. cos.' L. sine. | D. 1" L. COS. D.l" L. tang. D.l" L.cot 37461 37488 375i5 37542 37622 37649 37676 37703 3773^ 37757 37784 3781 1 37838 37865 37892 37919 37946 37973 37999 38026 38o53 38080 38107 38i34 38i6i 38i88 382 1 5 38241 38268 92718 92707 92697 92686 92675 92664 92653 92642 9263i 92620 92609 92598- 92987 92376 92D6D 92554 92543 92J32 92521 92510 92499 92488 92477 92466 92455 92444 92432 92421 92410 92399 923b8 9.573575 573888 074200 574512 574824 575i36 575447 575758 576069 5"»6379 576689 5-21 5-20 5-20 5.19 5-19 0-19 5-i8 5-18 5-17 5-16 9-967166 i 967113 967064 967013 966961 966910 966859 966808 966756 966705 966653 .80 •85 -85 • 85 • 85 • 85 • 85 -85 .86 • 86 • 86 9.606410 606773 607137 607500 607863 608225 6o8588 608900 6093 1 2 609674 6ioo36 6 6 6 6 6 6 6 6 6 6 6 06 06 00 00 04 04 04 o3 o3 o3 02 10-393090 393227 392863 392000 392137 391775 391412 39io5o 390688 390326 389964 60 69 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 od 38 37 36 35 34 33 32 31 30 9.576999 577309 577618 577927 578236 578545 578853 579162 579470 579777 9-58ooa5 580392 58?oo3 58i3i2 581618 581924 582229 582535 582840 5.16 5-16 5-15 5-13 5.14 5-14 5-13 5-i3 5-i3 0-12 9-966602 966550 966499 966447 966395 966344 966292 966240 966188 966136 .86 • 86 • 86 • 86 .86 .86 .86 .86 .86 .86 .87 .87 .87 .87 .87 .87 .87 .87 .87 .87 9.610397 610759 611120 611 480 611841 612201 6i256i 612921 613281 6i364i 6 6 6 6 6 6 6 6 5 5 02 02 01 01 01 00 00 00 99 99 10-389603 369241 3do6«o 3 8020 38ai59 387799 387439 387079 386719 386309 5-12 5-11 5-n 5-11 5-10 5-10 5-09 0-09 5.09 5-08 9.966065 966033 965981 965928 965876 965824 960772 965720 965668 96561 5 9.614000 614359 614718 6i5o77 615435 615793 616101 616009 616867 617224 5 5 5 5 5 5 5 5 5 5 98 98 98 97 97 96 90 10-386000 385641 380282 384923 384065 384207 383849 383491 383i33 382776 38295 3832 2 38349 38376 38403 38430 38456 38483 1 38510 ! 38537 38564 38591 38617 38644 38671 38698 138-25 13^702 38778 388o5 ^883 2 38859 38886 38912 38939 38966 38993 39020 39046 39073 92377 92366 92355 92343 92332 92321 92310 92299 92287 92276 92265 92254 92243 9223l 92220 92209 92198 92186 92175 92164 9-583145 583449 583754 584o58 584361 584665 584968 585272 585574 585877 5-o8 5-07 5.07 5-06 5-o6 5-06 5 -05 5-o5 5-04 5-04 5-03 5-o3 5-o3 5-02 5-02 5-01 5-01 5-01 5.00 5-00 9-965563 965011 960458 965406 965353 965301 965248 965195 965143 965090 .87 :i .88 .88 -88 .88 -88 9-617082 617939 618290 6i8652 619008 619364 619721 620076 620432 620787 5 5 5 5 5 5 5 5 5 90 95 94 94 94 93 93 92 92 10.382418 382061 381700 38i348 380992 380636 380279 379924 379568 379213 29 28 27 26 25 24 23 22 21 20 ly 18 17 16 15 14 13 12 u ID y 8 7 6 5 4 3 2 1 0, 9-586179 586482 586783 587085 587386 587688 588090 588890 9-965037 964984 964931 964879 964826 964773 964719 964666 964613 964560 -88 -88 -88 -88 -88 .88 .88 .89 .89 .89 .89 .89 .89 .89 .89 .89 .89 .89 .89 .89 9-621142 621497 621802 622207 622561 622915 623269 623623 623976 624330 5 5 5 5 5 5 5 5 5 5 5' 5 5 5 5 5 5 5 5 5 92 91 91 90 90 90 ^9 89 10.378858 3785o3 378148 377793 377439 377080 376731 376377 376024 375670 92l52 92141 92i3o 92119 92107 92096 92085 92073 92062 92o5o 9-589190 589489 589789 5900S8 5903 S7 590686 590984 591282 591580 591878 4-99 4-99 4-99 4.98 4-98 4-97 4-97 4-97 4-96 4-96 9-964507 964454 964400 964347 964294 964240 964187 964133 964080 964026 9.624683 620o36 620388 62 '741 626093 626440 626797 627149 627501 627852 88 88 -87 ?7 87 86 • 86 86 85 85 10.370317 374964 374612 374259 373907 373000 373203 372851 372499 372148 : N. COS. N. sine.j L. cos. D.l" L. sine. L. cot D 1" L. tang. / 67° 1 TRIGONOMETEICAL FUNCTION-S. — 23" 53 Nat. Functions. LooAKiTHMic Functions + 10. 1 / N.sine. N. COS. L. sine. D. 1" L. COS. ] D.l" L. tang. D.l" L.cot 39073 92o5o 9091878 4-96 9-964026 -17i 9.627862 62S203 5-85 10-372148 60 1 39100 192039 692176 4-93 96J972 .89 5-85 371797 59 2 39.27 92U2? 692473 4-95 9639 r 9 -89 628554 5-85 37i4h6 58 8 39153 92016 592770 4-95 963o6J •99 628905 5-84 37102') 57 4 39180 9200 ) 50J007 4-94 96381 1 •90 629255 5-84 370743 ~ij{) 5 39207 9'99i 593363 4-94 963757 •90; 629606 5-83 370394 55 6 39234 91982 593639 4-93 963704 •90 1 629966 5-83 370044 54 7 39260 91971 59395D 4-93 963650 -90 63o3o6 5-83 369694 53 8 39287 91959 594251 4-93 963596 .901 63o656 5-83 369344 52 9 393.4 9.94« 594547 4-92 963542 •90 63ioo5 5-82 368995 51 10 11' 39341 91936 594842 4-92 963488 .90 63i355 5-82 368645 50 1 39367 91925 9095137 4-91 9.963434 .90 9-631704 5-82 10-368296 i 49 1 12 39394 91914 693432 4-91 963379 -90 632053 5-81 367947 48 13 39421 91902 593727 4-91 963325 •90 632401 5-81 367099 47 14 39448 9.89. 596021 4.90 963271 .90 632760 5-81 367230 46 15 39474 91879 5963.5 4.90 963217 -90 633098 5-8o 3669')2 45 16 39501 91868 696609 4.89 963.63 .90 633447 5-80 366553 44 17 39528 91856 596903 4-89 963108 •91 633795 5-80 366205 43 18 39555 91845 597196 963o54 •91 634143 5-79 365857 42 19 39581 9.833 597490 962999 •91 634490 5-79 365510 41 20 39608 91822 597783 4-88 962945 .91 634838 5.79 365i62 40 21 39635 91810 9-698075 4-87 9.962890 -91 9-635185 5.78 io-3648i5 89 22 39661 91799 598368 4-87 962836 .91 635532 5-78 364468 38 23 39688 91787 598660 4-87 962781 .91 635879 5.78 3641 2 1 37 24 397.5 91775 598952 4-86 962727 .91 636226 5-77 363774 36 25 3974. 91764 599244 4-86 962672 .91 636572 5-77 363428 35 26 39768 9175: 699336 4-85 962617 .91 636919 637265 5-77 363o8i 34 '27 IX 9174. 599827 4-85 962562 .91 t]l 362735 33 28 91729 600118 4-85 962608 .91 63761 1 362389 32 29 39848 91718 600409 4.84 962453 •91 637966 5.76 362044 31 30 39875 91706 600700 4.84 962398 .92 638302 5.76 361698 80 29 31 39902 9.694 9.600990 4.84 9-962343 .92 9-638647 5.75 10-36x353 32 39928 91683 601280 4-83 962288 •92 638992 5.75 361008 28 33 39955 91671 601670 4-83 962233 .92 639337 5.75 36o663 27 34 39982 91660 601860 4-82 962178 •92 639682 5-74 36o3i8 26 35 40008 91648 602 1 5o 4-82 962123 •92 640027 5-74 359973 25 36 4oo35 91636 6®2439 602728 4-82 962067 .92 640371 5-74 359629 24 37 40062 91625 4-8i 962012 •92 6407 1 6 5.73 350284 358940 23 38 40088 91613 6o3oi7 4-8i 961967 .92 641060 5.73 22 39 401 1 5 91601 6o33o5 4-8i 961902 •92 641404 5.73 358596 21 40 40141 91590 603594 4-8o 961846 .92 641747 5-72 358253 20 41 40168 91578 9-6o3882 4-8o 9.961791 •92 9-642991 "5.72" 10-357909 19 42 40195 91566 604170 4-79 961735 •92 642434 5-72 357366 18 43 40221 91555 604457 4-79 961680 .92 642777 5-72 357223 17 44 40248 91543 604745 4-79 961624 .93 643120 5-71 356880 16 45 40275 9i53i 6o5o32 4-78 t^ ■93 643463 5.71 356537 15 46 4o3oi 9i5i9 6o53i9 4-78 .93 643806 5-71 -35(?t94 14 47 40328 9i5o8 6o56o6 4-78 961459 .93 644148 5.70 355852 13 48 40355 91496 91484 605892 4-77 961402 •93 644490 5.70 355510 12 49 4o38i 606179 4-77 961346, •93 644832 5-70 355168 11 50 40408 91472 606465 4.76 961290 .93 646174 5-69 354826 10 9 51 40434 91461 9-606751 4-76 9.961235 .93 9.645316 5.69 io-3344^->4 52 40461 91449 607036 4-76 961179 .93 643867 5-69 354.43 8 53 40488 91437 607322 4-75 961123 .93 646199 5.69 353801 7 54 4o5l4 91425 607607 4-75 961067 •93 646540 5-68 353460 6 55 4o54i 91414 607892 4-74 9610-1 .93 646881 5-68 353.19 5 56 40567 j 91402 608177 4-74 960935 -93 647222 5-68 352778 4 57 4059419.390 608461 4-74 960899 •93 647662 5-67 352438 3 68 J4062I 91378 608745 4-73 960843 •94 647903 5.67 352097 351757 2 59 40647 9.366 609029 4-73 960786 •94 648243 5-67 1 60 40674 9.355 609313 4-73 960730 •94 648583 5-66 351417 N. COS. N. sine L. COS. i D. 1" L. sine. L. cot. D.l' L.tang. t 66" 1 54 TRIGONOMETRICAL FL^NCTIOXS. — 24^ Nat. Functions. . LOGABITHMIC FUNCTIONS + 10. 1 ' N.sine.|N. C08. L.8ine. D.l"! L. COS. D.l" L. tang. Dl." L. cot 40674 91355 9.609313 4-73 9-960780 .94 9-648583 5-66 io-35i4i7 60 1 40700 91343 609J07 472 960674 •94 648928 5-66 801077 59 2 40727 9l33i 609880 4-72 960618 •94 649268 5-66 350787 58 3 40753 91819 610164 4-72 960561 •94 649602 5-66 350898 .=>7 4 40780 9i3o7 610447 4-71 96o5o5 •94 649942 5-65 35oo58 56 5 40806 91295 610729 4-7' 960448 •94 65o28i 5-65 3497 '9 55 6 40833 91283 611012 4-70 960892 960885 •94 65o620 5-65 349380 54 7 40860 91272 611294 4-70 •94 650959 5-64 34904 1 53 8 40886 91260 611576 4-70 960279 •94 65i297 5-64 848708 52 9 40913 91248 6ii858 4-69 960222 •94 65i636 5-64 348864 51 10 40939 91286 612140 _4j^69_ 960165 •94 651974 5-63 '5T68 848026 10-347688 50 49 11 40966, 91224 9-612421 4-69 9-960109 .90 9-652812 12 40992 91212 612702 4-63 960052 .90 652630 5 •03 347350 48 13 41019 91200 612983 4-68 939995 .93 652988 5-63 347012 47 U 4I04D 91188 618264 4-67 959988 .93 658826 5-62 346674 46 15 41072 91176 613545 4-67 959882 -93 653663 5-62 346887 45 16 41098 91 164 618825 4-67 959825 • 90 654000 5-62 346000 44 17 4II25 91152 6i4io5 4-66 959768 -93 654337 5.61 345663 43 IS 4n5i 91140 614385 4-66 959711 • 93 654674 5-6i 345326 42 ly 41178 91128 614665 4-66 959654 -95 6550II 5.61 344989 41 ifO 41204 91 1 16 614944 4-65 959596 -95 655848 5-61 5-60" 344632 10.344316 40 39 21 4i23i I91104 9-6i5223 4-65 9-959089 -95 9-655684 22 41237 i 91092 6i55o2 4-65 959482 .95 656020 5-60 343980 33 2:} 41284191080 615781 4.64 959425 -95 656356 5-60 343644 37 24 4i3io 91068 616060 4-64 959868 -95 656692 5-59 343808 36 25 41337 9io56 616888 4-64 959810 .96 657028 5-59 342972 85 26 ! 41 363 91044 616616 4-63 959253 .96 657364 5-59 342636 84 27 41390 [91032 616894 4-63 959195 959188 .96 6576Q9 658o34 n^ 342801 33 2S 4i4i6j9io20 617172 4-62 .96 341966 32 29 41443 91008 617450 4-62 959081 .96 658369 5-58 341681 31 30 41469190996 617727 4-62 959028 -96 658704 5-58 341296 30 IT i 41496 |909y4 9-618004 4-6r 9-958965 .96 9-659089 659878 '5-58 10-340961 ^9~ 32 41522:90972 618281 4-6i 958908 -96 5-57 340627 23 33 j 41549 90960 618553 4-6i 958d5o .96 659708 5-57 340292 27 34 141575 190948 618884 4 -60 958792 .96 660042 5-57 339938 26 35 1 41602 1 90986 6191 10 4-6o 958784 .96 660876 5-57 339624 25 36 ' 41628 90924 619886 4«6o 958677 •96 660710 5-56 33^7 388623 24 37 4i655 90911 619662 4-59 958619 .96 661043 5-56 23 88 41681 90899 619988 4.59 958561 .96 661877 5-56 22 89 41707 90887 620218 4-59 9585o3 •97 661710 5.55 888290 337957 21 40 41734 90875 620488 4-58 4-58 958445 9-958887 •97 662043 5-55 20 19 41 41760 90b68 9'620763 •97 1 9.662876 5.55 10-337624 42 fXl 90801 621088 4.57 958829 •97 662709 5-54 887291 18 43 90889 Q0826 62i3i3 4-57 958271 •97 668042 5.54 886938 17 44 41840 621587 4-57 958218 •97 663375 5.54 386625 16 45 '41866190814 621861 4-56 9581 54 •97 668707 5.54 386298 15 46 41892 90802 622135 4-56 958096 958o38 •97 664039 5-53 335961 14 47 |4i9'9 90790 622409 4-56 •97 664371 5-53 335629 13 48 [41943 90778 622682 4.55 957979 •97 664708 5-53 385297 12 49 ,41972 90766 622956 4.55 957921 •97 665o35 5-53 334965 11 50 41998 907^ 628229 4-55 957868 ^1 665866 5-52 334684 10 51 42024 90741 9-623502 j 4.54 9-957804 '97 -98 9-665697 5.52 io-3843o3 9 52 42o5i 90729 628774 1 4-54 957746 666029 5-52 888971 8 53 42077 90717 624047 4-54 957687 957628 .98 666860 5.51 333640 7 54 42104 90704 624819 4-53 .98 666691 5-5i 333309 6 55 42i3o 90692 624591 4-53 957570 .98 667021 5-51 332979 5 56 421 56 90680 624868 4.53 957511 .98 667832 5-5i 882648 4 57 42183 90668 625 1 35 4-52 957452 .98 667682 5-50 332818 3 58 42209 90655 625406 4-52 937893 -98 668018 5-5o 881987 2 59 ! 42235 90048 625677 4-52 957335 .98 668343 5-5o 381657 1 60 42262 90681 625948 4-5i 957276 .98 668672 5-5o 33i328 |n. cos. N. sine L. COS. 1 D. 1" j L. sine. L,cot. D.l" L. tang. ' 65° 1 TRIGONOMETRICAL FUNCTIONS. — 25°. 55 Nat. Functions. Logarithmic Functions + 10. 1 N.Bine. N.cos. L.sine. D. 1" L.COS. D.l" L. tang. D 1" L.cot. 42262 906 G I 9-625948 4-5i 9-957276 .98 9.668673 5.5o io.33i327 330998 60 i 42288 90618 626219 4-5i 957217 957158 •98 669002 5 49 59 2 423i5 90606 626490 4-5i .98 669332 5 49 330668 58 3 42341 90594 626760 4-5o 957099 -98 669661 5 49 330339 57 4 42367 9o582 627030 4-5o 957040 -98 669991 670320 5 48 330009 56 5 42394 90569 627300 4-5o 956981 -98 5 48 329680 55 6 42420 9055? 90545 627570 4-49 956921 •99 670649 5 48 329351 54 7 42446 627840 4-49 956862 •99 670977 5 48 329023 53 8 42473 90532 628109 4-49 9568o3 •99 671306 5 47 328694 52 9 %tn 90520 628378 4.48 956744 •99 671634 5 47 328366 51 10 90507 628647 4.48 956684 •99 •99 671963 5 5" 47 328037 60 "u 425D2 90495 9.628916 4-47 9-956625 9.672291 47 10.327709 49 12 42378 90483 629185 4-47 956566 •99 672619 5 46 327381 48 13 42604 90470 629453 4-47 9565o6 •99 672947 5 46 327033 47 14 42631 90458 629721 4.46 956447 •99 673274 5 46 326726 46 Id i 42637 42683 90446 629989 4-46 956387 •99 673602 5 46 326398 45 16 1 90433 630237 4.46 956327 •99 673929 5 45 326071 44 17 ; 42709 90421 63o524 4.46 956268 •99 674237 5 45 325743 43 18 42736 90408 630792 4-45 956208 1-00 674584 5 45 325416 42 19 42762 Q0396 63 1039 4-45 956148 I -00 674910 5 44 325090 41 20 21 42788 90383 63i326 4-45 956089 I-OO 675237 5 44 324763 40 42815 90371 9.631593 4-44 9.956029 1. 00 9.675564 5 44 10-324436 39 22 42841 90358 63 1839 4.44 955969 1-00 675890 5 44 324110 38 23 42867 90346 632125 4.44 955909 1-00 676216 5 43 323784 37 24 42894 90334 632392 4-43 955849 1-00 676543 5 43 323457 36 25 42920 90321 632638 4-43 955789 1-00 676869 5 43 323i3i 35 26 42946 90309 632923 4-43 955729 I-OO 677194 5 43 322806 34 27 42972 90296 633189 4-42 955669 1-00 677520 5 42 322480 33 28 42999 90284 633454 4-42 955609 1-00 677846 678171 5 4^ 322154 32 29 43025 90271 90259 633719 4-42 955548 1-00 5 42 321829 31 30 31 43o5i 633984 4-4i 955488 1. 00 678496 5 42 32i5o4 30 43077 90246 Q.634249 4 -'41 9.955428 1-01 9.678821 T 41 10.321179 320854 29 32 43104 90233 634514 4.40 955368 I-Ol 679146 5 41 28 33 43i3o 90221 634778 4.40 955307 l-OI 679471 5 41 320520 320205 27 34 43 1 56 90208 635042 4-40 955247 1-01 679793 680120 5 41 26 33 43182 ! 90196 6353o6 4-39 955186 I-OI 5 40 319880 25 36 43209 90183 635570 4-39 955126 1-01 680444 5 40 319556 24 37 43235! 90171 635834 4-39 955o65 1.01 680768 5 40 319232 23 3S 43261 90 1 58 636097 4-38 955oo5 1-01 681092 5 40 318908 22 39 43287 90146 636360 4-38 954944 I -01 681416 5 39 3x8384 21 40 41 433 1 3 90133 636623 4-38 954883 l-OI 1. 01 681740 ! 9.682063 5 39 318260 20 1^ 43340 90120 9-636886 4.37 9-954823 5^ 39 10-317937 42 43366 90108 637148 4-37 954762 1-01 682387 5 39 317613 18 43 43392 90095 63741 1 4.37 954701 l.OI 682710 5 38 317290 17 44 43418 90082 637673 4-37 954640 1.01 683033 5 38 316967 16 45 43445 90070 637935 4-36 954579 954518 l-OI 683356 5 38 3 1 6644 15 46 43471 90057 638197 638458 4-36 1-02 683679 5 38 3i632i 14 47 tin 90043 4-36 954457 1-02 684001 5 37 3 15999 13 48 90032 638720 4.35 954396 954335 1-02 684324 5 37 3 15676 3 1 5354 12 49 43549 43575 90019 638981 4-35 1-02 684646 5 37 11 50 51 90007 639242 4-35 954274 9-954213 1-02 684968 5 37 3i5o32 10 1 43602 89994 9 -639503 4-34 1-02 j 9.685290 T 36 io-3i47«o 9 52 1 43628 89981 639764 4-34 954152 1-02 1 685612 5 36 314388 8 53 J 43654 89968 640024 4.34 954090 1-02 ! 685934 5 36 3 1 4066 7 54 43680 89956 640284 4-33 954029 953968 1.02 i 686255 5 36 3i3745 6 55 43706 89943 640544 4-33 1-02 686577 5 35 3i3423 5 56 43733 89930 640804 4-33 953906 1-02 686898 5 35 3i3io2 4 57 43759 89918 641064 4-32 953845 1-02 687219 5 35 312781 3 58 43785 1 89905 641324 4-32 953783 1-02 687540 5 35 3 1 2460 2 59 43811I89892 641583 4-32 953722 i-o3 687861 5 34 3i2i39 1 1 60 43837 1 89879 641842 4-3i 953660 1-03 688182 5.34 3u8i8 1 N. COS. N. sine. L. COS. D.l" L. sine. Kcot D.l" L. tang. 1 ' 64° 5Q TRIGONOMETRICAL FUNCTIOXS.— 26°. • Nat. Fukctions. Logarithmic Functions + 10. / N.sine N.cos L. sine. D. 1" L. COS. D.l" L. tang. D.l" j L. cot 43837 89879 9.641842 4-31 9.953660 i.o3 9-688182 5.34 |io-3ii8i8 60 1 43863 89867 642101 4-3i 953599 i-o3 688502 5.34 3.1498 59 2 1 43889 89854 642360 4-3i 953537 1-03 688823 1 5.34 3.1177 58 8 ,43916 89841 642618 4-3o 953475 i-o3 689143 5.33 3 10857 57 4 : 43942 89828 642877 643 1 35 4-3o 953413 i-o3 689463 5.33 3.0537 56 5 43968 89816 4-3o 953352 i-o3 689783 5-33 3.0217 55 6 i 43994 89803 643393 64365o 4-3o 953290 i-o3 690103 5-33 309897 54 / 7 1 44020 89790 4-29 953228 1-03 690423 5.33 309577 309258 53 8 44046 89777 643908 4-29 953166 i-o3 690742 5-32 52 9 i 44072 B9764 644165 4-20 4-28 953104 i-o3 691062 5-32 308938 51 10 : 44098 89752 644423 953042 1-03 691381 5-32 3086.9 50 11 44124 89739 9.644680 4-28 9-952980 I1.04 9-691700 5-3i io.3o83oo 4y 12 44151 89726 644936 4-28 952918 1-04 692019 692338 5-3i 307981 4-! 13 '44177 89713 645193 645450 4-27 952855 1.04 5-3i 307662 47 14 1 44203 89700 4-27 952793 1-04 952731 I1.04 692656 5-31 307344 46 15 44229 44255 89687 645706 4-27 692975 5-3i 307025 45 16 89674 645962 4-26 952669 1.04 693293 5-3o 306707 3o6388 44 17 4i28i 89662 646218 4-26 952606 1-04 693612 5-3o 43 18 ! 44307 89649 646474 4-26 952544 1-04 693930 5-30 306070 3o5752 42 19 144333 89636 646729 4-25 952481 1.04 694248 5-3o 41 20 21 44359 1 44385 89623 646984 4-25 952419 1-04 694566 5-29 3o5434 40 89610 9-647240 4-25 9-952356 1-04 9-694883 5-29 io.3o5ir7 3y 22 : 4441 1 89597 647494 4-24 952294 1-04 695201 5-29 304799 33 23 44437 89584 647749 4-24 952231 1-04 695518 5-29 304482 37 24 44464 89571 648004 4-24 952168 i-o5 695836 lit 3o4.64 36 25 44490 89558 648258 4-24 952106 1-05 696153 3o3847 85 26 445 1 6 89545 89532 648512 4-23 952043 1-05 696470 5-28 3o353o 84 27 44542 648766 4-23 951980 i.o5 696787 5-28 3o32.3 33 28 44568 89519 649020 4-23 951917 |.o5 697103 5-28 302897 32 29 44594 89506 649274 4-22 951 854 i-o5 697420 5-27 3o258o 81 80 44620 89493 649527 4-22 951791 i.o5 697736 5.27 302264 30 31 44646 89480 9.649781 4-22 9.951728 i-o5| 9-698053 5.27 10-30.947 2y' 32 44672 89467 65oo34 4-22 95 I 665 i-o5 698369 5-27 3oi63i 28 33 44698 89454 6502S7 4-21 951602 i-o5 698685 5-26 3o.3i5 27 84 44724 89441 65o539 4-21 951539 i-o5 699001 5-26 300999 26 S5 44750 89428 650792 4-21 951476 1-05 699316 5-26 300684 25 36 44776 89415 65 I 044 4-20 951412 1-05 699632 5-26 3oo368 24 87 44802 89402 65i297 4-20 95 I 349 1.06 699947 5-26 3ooo53 23 38 44828 89389 661549 4-20 951286 1-06 700263 5-25 299737 22 89 44854 89376 65 1 800 4-19 951222 1-06 700578 5-25 299422 21 40 44880 89363 652052 4-19 951159 1-06 700893 5-25 299.01 20 19 ' 41 44906 89350 g. 652304 V.l 9-951096 1. 06 9.701208 5-24 10-298792 42 44932 89337 65^555 95io32 1-06! 7oi523 5-24 298477 18 48 44958 89324 652806 4-i8 950968 i-o6| 701837 5-24 298163 17 44 45 44984 45oio 893 1 1 89298 653o57 6533o8 4-i8 4-i8 950841 1-06 1-06 702152 702466 5-24 5.24 297848 297534 16 15 46 45o36 89285 653558 4-17 950778 I -06 702780 5-23 297220 14 47 45062 89272 6538o8 4-17 950714 1-06 703095 5-23 296905 13 48 45o88 89259 654059 4-17 95o65o I -06 Toin 5-23 296091 12 49 45ii4 89245 654309 4-i6 95o586 1-06 5-23 296277 11 50 45i4o 89232 654558 4-i6 95o522 i_-07 704036 5-22 _295964_ 10 61 45 1 66 89219 9.654808 4-i6 9-950458 1-07: 9.704350 5-22 .0-295650 ~9" 62 45192 89206 655058 4-i6 950394 1-07 704663 5-22 295337 8 53 45218 89193 655307 4-15 95o33o 1.07, 704977 5-22 295023 7 54 45243 89180 655556 4-15 950266 ..07! 703290 5.22 294710 6 55 45269 45295 89167 6558o5 4.15 g50202 1-07' 7o56o3 5-21 294397 5 56 89153 656o54 4-14 950 1 38 1.07' 705916 5-21 294084 4 57 45321 89140 656302 4-14 950074 11-07; 706228 5-21 293772 3 58 45347189127 656551 4-14 950010 1-07 706541 5.21 293459 2 59 45373189114 656799 4-13 949945 1-07 706854 5-21 293.46 1 60 45399 89101 657047 4-!3 949881 1.07 707166 5-20 292834 N. COS. N. sine. L. COS. D. 1" L.sine. 1 L. cot. D. 1" L. tang. 63° TRIGONOMETRICAL FUNCTI0NS.--27°. 57 Nat. Functions. Logarithmic Functions + VX ' N.sino.' N.C08. L. sine. D.l" L. COS. D.l"j L. tang. D.l" L.cot 45399 '89101 9-657047 4-i3 9.949881 1.07 9.707I66 5-20 10.292834 60 1 4342 3 B9087 657295 4-i3 949816 1-07 707478 5-20 292322 59 2 45431 t!9074 557042 4-12 949732 1.07 707790 5-20 292210 58 8 45477 89061 455o3 »9o48 657790 658037 4-12 949^«8 1.08 708102 5-20 291898 57 4 4-12 949623 1.08 708414 5-19 291606 56 5 45529 89035 658284 4-12 949558 1.08 706726 5-19 291274 65 6 45554 89021 658531 4-II 949494 1.08 709037 5-19 290963 54 7 45580 89008 658778 4-n 949429 1-08 709349 5-19 2900J1 53 8 456o6 86995 659025 4-11 949364 1.08 709660 5-19 290340 52 9 45632 HbgSi 659271 4-IO 949300 1. 08 709971 5-i8 290029 5i 10 11 45658 45684 88968 659517 4-10 949235 1.08 710262 5-18 289718 50 88955 9.659763 4-10 9-949179 1.08 9.710393 5.J8 10-289407 4'J 12 45710 88942 660009 4.09 949103 I -08 710904 5.18 289096 46 13 45736 88928 660253 4.09 949040 I -08 711215 5-i8 288785 47 14 45762 88915 66o5oi 4.09 948975 1.08 711025 5.17 288475 46 15 45787 88902 660746 4-09 948910 1.08 711836 5-17 288164 45 16 458i3 88688 660991 4-o8 948845 1.08 712146 5.17 287854 44 17 45830 88875 661236 4-o3 948780 1.09 712456 5.17 287544 43 18 45863 8S062 661481 4-o8 948715 1.09 712766 5.16 287234 42 19 45891 88848 661726 4-07 948650 1-09 713076 5.16 286924 41 20 Hi 45917 88835 661970 4-07 948584 1.09 713366 5.16 286614 40 'di- 43942 88822 9.662214 4-07 9.948019 1.09 9-713096 5-10 10-286304 iJ2 45968 88808 662459 662703 4-07 948454 1-09 714005 5-ID 285995 ss 23 43994 88795 4-o6 948388 1-09 714314 5-15 283086 87 24 46020 88782 662946 4-o6 948323 1-09 714624 5.13 285376 285067 284758 86 25 46046 88768 663190 663433 4-o6 948237 1.09 714933 5-15 35 26 46072 88755 4-o5 948192 1-09 715242 5-i5 31 27 46097 88741 663677 4-o5 948126 1.09 7i555i 5.14 284449 33 28 46123 88728 663920 4-o5 948050 1-09 7 1 5860 5.14 284140 32 29 46149 88715 664 I 63 4-o5 947993 I-IO 716168 5.14 283832 81 30 31 46175 88701 88688 664406 9.664648 4.04 4.04 ^947929 9.947863 I'lO 716477 5.14 283523 30 46201 I-IO 9.716783 5.14 10-263215 2ii 32 46226 88674 664891 4-04 947797 1. 10 717093 5-13 282907 28 33 46252 8S061 665 1 33 4-o3 947731 I. 10 717401 5.i3 282399 27 34 46278 88647 665375 4-o3 947663 I-IO 717709 5.13 2G2291 26 35 46304 88634 665617 4-o3 947600 I -10 7180.7 5.i3 2S1983 25 36 46330 88620 665859 4-02 947533 IIO 718325 5.13 2S1675 24 37 46355 88607 666 1 00 4' 02 947467 I . 10 718633 5-12 281367 23 38 4638i 88593 666342 4-02 947401 I .10 718940 5.12 281060 22 39 46407 88580 666583 4-02 947335 I. 10 719248 5.12 280732 21 40 41 46433 46458 86566 88553 666824 9.667065 4-01 4-01 947269 I. 10 I • 10 719555 5.12 280443 20 i'J 9-947203 9.719862 5-12 io-2«oi38 42 46484 88539 667305 4-01. 947136 I. II 720169 5-11 279831 18 43 46310 88526 667546 4-01 947070 I. II 720476 5-II 279524 17 44 46536 83512 667786 668027 4-00 947004 I-Il 720783 5-II 279217 16 45 46561 88499 4-00 946937 1. 11 721089 5-11 27891 1 15 4'5 46587 884o5 668267 4-00 946871 I.I. 721396 5-11 278604 14 47 4661 3 88472 6685o6 3.99 946804 i.u 721702 5-10 278298 13 48 46639 88458 668746 3.99 946738 I-Il 722009 5-10 277991 12 49 46664 88445 668986 3.99 946671 I- 11 7223i5 5-10 277683 U 50 46690^ 88431 669225 il99_ 946604 I-II I'll 722621 5-10 277379 10 zr 46716 88417 9.669464 1 3.98 9.946538 9.722927 5.10 10-277073 9 52 46742 88404 669703 3.98 946471 III 723232 5-09 276768 8 53 46767 88390 669942 3.98 946404 III 723538 5-09 276462. 7 54 46793 88377 883i3 670181 3.97 946337 III 723844 5.09 276156 6 55 46819 670419 3-97 946270 I. 12 724149 5.09 275851 5 56 46844 88349 670608 ■ 3-97 946203 1-12 724454 5.09 275546 4 57 46870 88336 670896 3-97 946 1 36 1-12 724759 5.08 275241 3 58 46896 83322 671134 3-96 946069 1-12 723063 5 -08 274935 2 59 46921 8S3o8 671372 3.96. 946002 I. 12 725369 5-08 274631 1 60^ 46947 88295 671609 3.96 945935 I.I2 725674 1 5.08 274326 N. 008. N.sine. L. COS. 1 D. 1" L. sine. L. oot 1 D. 1" L. tang. ' 62° 1 58 TRIGONOMETRICAL FUNCTIONS. — 28°. Nat. Functions. LcxiARiTHMic Functions -»- 10 1 ' N.sine.! N. cos. L. sine. D. 1" L. COS. D.l" 1 L. tang. D 1" L. cot , 46947 ; 88295 46973 88281 9-671609 3-96 9-945935 1-12. 9.725674 T 08 10.274326 60 1 671847 3- 95 945868 1-12 725979 5 08 274021 59 2 46909 88267 672084 3 95 945800 1.12 726284 5 07 273716 58 3 '47024 88254 672321 3 95 945733 1-12; 726588 5 07 27J412 57 A 47o5o 88240 672558 3 95 945666 I-I2I 726892 5 07 273108 56 5 47076 88226 672795 3 94 945598 1-12 727197 5 07 272803 55 6 47101 88213 673032 3 94 945531 I-I2 727501 5 07 272499 54 7 47127 88199 673268 3 94 945464 I-13; 727805 728109 5 06 272195 53 8 47153 :88i85 673505 3 94 945396 I-lS' 5 06 271891 52 9 147178188172 673741 3 93 945328 i.i3i 728412 5 06 271588 51 10 11 ; 47204,881 58 47229,8814! 9-674213 3 93 945261 1-13 728716 5 06 271284 50 3 93 9-945193 i-i3 9-729020 5 06 10-270980 49 12 47255 88i3o 674448 3 92 945i25 |.i3 729323 5 o5 270677 48 13 • 47281 : 88117 674684 3 92 945o58 i.i3 729626 5 o5 270374 47 14 : 47306! 88 io3 674919 3 92 944990 ...3 739929 5 o5 270071 46 15 47332 88089 675155 3 92 944922 i.i3 730233 5 o5 269465 45 16 : 47353 : 88075 675390 3 91 944854 i-i3 73o535 5 o5 44 17 ' 47383 i 88062 675624 3 91 944786 1-13 73o838 5 04 269162 43 18 ' 47409 ; 88048 675359 3 91 944718 i-i3 731141 5 04 268859 42 19 1 47434 1 88o34 676094 3 91 944650 i-i3 731444 5 04 268556 41 20 1 47460 1 88020 676328 3 90 944582 1-14 731746 5 04 268254 40 21 1 47486 1 88006 9-676562 T 90 9 -944514 1-14 9-732048 5 04 10-267952 o\l 22 1 4751 1 187993 676796 3 90 944446 1-14 732351 5 o3 267649 38 23 1 47537 1 87979 677030 3 00 89 944377 1-14 732653 5 o3 267347 37 24 j 47562 ] 87965 677264 3 944309 1-14 732955 5 o3 267045 36 25 1 47588 1 87951 677498 3 89 944241 I.. 4 733257 5 o3 266743 35 26 ! 47614 1 87937 677731 3 89 944172 I -14 733558 5 o3 266442 34 27 i 47639 1 87923 677964 3 88 944104 1-14 733860 5 02 266140 33 28 1 4766J 1 87909 678197 3 88 944o36 1-14' 734162 5 02 265838 32 29 : 47690 j 87896 678430 3 88 943967 1-14: 734463 5 02 265537 31 SO 477'6 87882 678663 3 3 88 "87 943899 1-14' i-i4| 734764 5 02 265236 30 31 1 47741 87^68" 9-678895 9-943830 9-735066 T 02 10-264934 29 32 47767 87854 679128 3 87 943761 i-u'i 735367 5 02 264633 28 S3 47793 47818 87840 679360 3 87 943693 1-15; 735668 5 01 264332 27 34 87826 679592 3 87 943624 i.i5l 735969 5 01 264031 26 35 47844 87812 679824 3 86 943555 1-15 736269 5 01 263731 25 36 47869 87798 68oo56 3 86 943486 i-i5 736570 5 01 263430 24 37 47895 87784 680288 3. 86 943417 '"i 736871 5 01 263129 23 33 1 47920 87770 68o5i9 3 85 943348 i-i5 737171 5 00 262829 22 39 1 47946 87756 680750 3 85 943279 1-15 737471 5 00 262529 21 40 41 1 47971 87743 680982 3 85 943210 i-i5 737771 5 00 262229 20 47997 48022 87729 9.6:^1213 3 85 9-943141 i-i5 9.738071 5 00 10-261929 19 42 87715 681443 3 84 943072 i-i5 733371 5 00 261629 18 43 48048 87701 681674 3 84 943oo3 i-i5 738671 4 99 261329 17 44 48073 87687 681905 3 84 942934 942864 i-i5 738971 4 99 261029 16 45 48099 \ 87673 682135 3 84 i-i5 739271 4 99 260729 15 46 48124 87659 632365 3 83 942795 1-16 739570 4 99 260430 14 47 481 5o • 87645 682595 3 83 942726 1-16 739870 4 99 26oi3o 13 48 48175 j 87631 682825 3 83 942656 I 16 740169 4 99 259831 12 49 48201 187617 683o55 3 83 942587 i-i6 740468 4 98 259532 11 50 51 148226 87603 683284 3 82 82 9425 n 9.942448 1.16 740767 4 98 259233 in 48202 §?i 9-683514 iT76 9.741066 I 98 10-258934 9 52 48277 683743 82 942378 1-16 741365 4 98 258635 8 53 i 483o3 87561 683972 82 942308 1-16 741664 4 98 258336 7 54 148328 87546 684201 81 942239 1-16 741962 4 97 258o38 6 55 ' 48354 87532 684430 81 942169 1-16 742261 4 97 257739 5 56 48379 87518 684658 81 942099 i-i6 742559 4 97 257441 4 57 ■ 48405 87304 684887 80 942029 1-16 742858 4 •97 257142 3 58 1 48430 87490 685ii5 3 -80 941959 1-16 743 1 56 4 •97 256844 2 59 ' 48456 87476 685343 3 .80 941889 1-17 743454 4 •97 256546 1 60 148481 87462 685571 3 -80 941819 1^7 i 74375a 4 .96_ 256248 N. COS. N.Bine. L. COS. D 1" L. sine. j L. cot. D . 1" L. tang. • 61° 1 TBIGONOMETRICAL FUNCTIONS. — 29*. 59 Nat. Functions. Logarithmic Functions + 10. i / jN.Biiie.|N. COB, L.Bine. D.l" L. COS. D.1" L. taiig. D.l" L.eot 48481 87462 9-685571 T 80 9.941819 1-17 9-743752 4 96 10-256248 60 1 485o6 87448 685799 3 79 941749 1-17 744o5o 4 96 255950 59 2 48532 87434 686027 3 79 941679 1.17 744348 4 96 253632 58 S 48557 87420 686254 3 79 941609 1.17 744645 4 96 255355 57 4 ! 48583 87406 686482 3 79 941539 1.17 744943 4 96 255o57 56 5 i 48608 87391 686709 3 78 941469 941398 1-17 745240 4 96 254760 55 6 48634 87377 686936 3 78 1-17 745538 4 95 254462 54 7 48659 87363 687163 3 78 941328 1-17 745835 4 95 254165 53 8 48684 l]lit 687389 687616 3 78 941258 1-17 746132 4 95 253868 52 9 48710 3 77 941187 1-17 746429 4 93 253571 51 10 48735 87321 687843 3 77 941117 1-17 746726 4 95 253274 50 11 48761 87306 0-688069 688295 T 77 9-941046 1-18 9.747023 4 94 10-252977 252681 49 12 48786 87292 3 ]l 940975 1-18 747319 4 94 48 13 4881 1 {87278 688521 3 940905 940834 1-18 747616 4 94 252384 47 U 48837 87264 688747 3 76 1-18 747913 4 94 252087 46 15 48862 87250 688972 3 76 940763 940693 1-18 748209 7485o5 4 94 251791 45 16 48888 87235 689198 3 76 i-i8 4 93 251495 44 17 48913 87221 689423 3 75 940622 1-18 748801 4 93 25II99 43 18 48938 87207 689648 3 75 94055 I 1-18 749393 4 93 250903 42 19 48964 87193 689873 3 7? 940480 1-18 4 93 250607 41 20 21 48989 87.78 690098 3 75 940409 i-i8 749689 4 93 25o3ii 40 49014 87164 9-690323 3 74 9-940338 1-18 9-749985 4 93 io-25ooi5 89 22 49040 87150 690548 3 74 940267 1-18 750281 4 92 249719 88 23 49065 87136 690772 3 74 940196 1-18 750576 4 92 249424 87 24 49090 87121 690996 3 74 940125 1-19 750872 4 92 249128 248833 86 25 49116 87107 691220 3 73 940054 1.19 751167 4 92 85 26 49141 87093 691444 3 73 939982 1.19 751462 4 92 248538 84 27 49166 87079 691668 3 73 939911 I- 19 751757 4 92 248243 83 28 49192 87064 691892 3 72 939840 1.19 752052 4 91 247948 82 29 49217 87050 692115 3 72 939768 1-19 7523-47 4 91 247653 81 30 31 49242 87036 692339 3 72 939697 1.19 1. 19 732642 4 91 247358 80 49268 87021 9-692562 3 72 9-939625 9-752937 4 91 10-247063 29 32 49293 87007 692785 3 7> 939554 1. 19 753231 4 91 246769 28 38 49318 86993 693008 3 71 939482 1.19 753526 91 246474 27 84 49344 86978 693231 3 71 939410 1-19 753820 90 246180 26 85 49369 86964 693453 3 71 939339 1.19 754115 90 245885 25 86 49394 86949 693676 3 70 939267 939195 1-20 754409 90 245591 24 37 49419 86935 693898 3 70 1-20 754703 90 245297 23 38 49445 86921 694120 3 70 939123 1-20 754997 90 245oo3 22 39 49470 86906 694342 3 70 939052 1-20 753291 755585 90 244709 244413 21 40 41 49495 86892 694564 3 69 938980 1-20 89 20 49521 86878 9-694786 3 69 't^t 1-20 9.755878 ^9 10-244122 19 42 49546 86863 695007 3 69 1-20 756172 89 243828 18 43 49571 86849 695229 3 69 938763 1-20 756465 ^9 243535 17 44 49596 86834 695450 3 68 938691 1-20 756759 ^ 243241 16 45 49622 86820 695671 3 68 938619 1-20 737052 89 242948 15 46 49647 868o5 695892 3 68 938547 1-20 757345 88 242655 14 47 49672 1 86791 696113 3 68 938475 1-20 757638 88 242362 13 48 49697 : 86777 696334 3 67 938402 1-21 737931 88 242069 12 49 49723 186762 696554 3 67 938330 1-21 758224 4 88 241776 24 J 483 11 50 51 49748 86748 696775 3 67 938258 1-21 758517 88 10 49773 86733 9-696995 3 67 9-938i85 1-21 9-758810 ' 4 "88 10-241190 9 52 49798 86719 697215 3 66 938u3 I-2I 759102 4 87 240898 8 53 49824 86704 697435 3 66 938040 I-21| 759395 4 87 240605 7 54 49849 86690 697654 3 66 $Xi 1-21 759687 4 ^7 24o3i3 6 55 ■49874 86675 697874 3 66 1-21 739979 4 87 240021 5 56 49899 86661 698094 3 65 937822 1-21 760272 4 87 239728 4 57 49924 86646 698313 3 65 937749 I-2I 760564 4 87 239436 3 58 499^0 86632 698532 3 65 937676 1-21 760856 4 86 239144 238852 2 59 1 4997^ 86617 698751 3 65 937604 I-2I 761148 4 86 1 60 5oooo 866o3 698970 3-64 937531 1-21 761439 4-86 238561 N.C08. N.sine. L. COS. 1 D. 1" L. sine. L.cot D.l" L.tang. » 60° 1 60 TKIGONOMETRICAL FUNCTIONS. — 30°. Nat. Functions. Logarithmic Functions + 10. f N.sine. N. COB. L. sine. D. 1" L.C08. D.l" 1 L. tang. D 1." L. cot 60 5oooo 866o3 9-698970 3 -64 9-937531 •21 9.761439 4-86 10. 238561 1 5oo25 86588 699189 3 .64 937458 1 •22 76173. 4^86 238269 59 2 5oo5o 86573 699407 3 • 64 937385 •22 762023 4 -86 237977 58 8 50076 86559 699626 3 64 937312 •22 762314 4^86 237686 57 4 5oioi 86544 699844 3 63 937238 -22 762606 4-85 237394 56 1 '^ 5oi26 86530 700062 3 63 937165 •22 762897 763 I 88 4^85 237103 55 ^ 5oi5i 865 1 5 700280 3 63 937092 -22 4^85 236812 54 ^ 50176 865oi 700498 3 63 937019 •22 763479 4^85 236521 53 / ^ 5o20I 86486 700716 3 63 936046 936872 -22 763770 4^85 236230 52 9 50227 86471 700933 3 62 •22 764061 4^85 235939 235648 51 10 50252 86457 7oii5i 3 62 936799 -22 764352 4.84 50 11 50277 86442 9-701368 3 62 0-936725 •22 9-764643 4-84 10-235357 49 12 5o3o2 86427 701585 3 62 936652 -23 764933 4-84 235067 43 13 50327 86413 701802 3 61 936578 -23 765224 4.84 234776 47 14 5o352 86398 86384 702019 3 61 9365o5 •23 765514 4-84 234486 4(5 15 5o4o3 702236 3 61 936431 1 -23 7658o5 4-84 234195 45 16 86369 702452 3 61 936357 -23 766385 4^84 233905 44 17 50428 86354 702669 702885 3 60 936284 1 •23 4^83 2336i5 43 18 50453 86340 3 60 936210 •23 766675 4-83 233325 42 19 50478 86325 7o3ioi 3 60 9361 36 1 •23 766965 4^83 233o35 41 20 5o5o3 86310 703317 3 60 936062 1 •23 767255 4^83 232745 40 3;» 21 5o528 86295 9-703533 3 59 9-935988 1 -23 9.767545 4-83 10-232455 22 5o553 86281 703749 3 59 935914 1 935840 1 -23 767834 4-83 232166 38 23 50578 86266 703964 3 59 -23 768124 4^82 231876 37 24 5o6o3 8625i 704179 704395 3 59 935766 1 •24 768413 4^82 23 1587 3(5 25 50628 86237 3 % 935692 1 •24 768703 4-82 231297 23 1 008 35 26 5o654 86222 704610 3 935618 I -24 769281 4^82 8i 27 50679 86207 704825 3 58 935543 1 •24 4^82 230719 83 28 50704 86192 7o5o4o 3 58 935469 1 935395 1 •24 769570 4^82 23o43o 32 29 50729 86178 705254 3 58 •24 769860 4^8i 23oi4o 31 30 31 50754 86 163 705469 3 T 57 57 935320 1 •24 770148 9.770437 770726 4-8i 229852 30 tA 86148 9-705683 9-935246 1 •24 4-8i 10-229563 29 32 86i33 705898 3 57 935171 I •24 4-8i 228985 28 83 50829 861 19 706112 3 57 935097 I •24 771015 4-8i 27 34 5o854 86104 706326 3 56 935022 I •24 77i3o3 4-8i 228697 228408 26 35 50879 86089 706539 3 56 934948 I •24 771592 4.81 25 36 50904 86074 706753 3 56 934873 I •24 77i8«o 4-80 228120 2i 37 50929 86059 86045 706967 3 56 934798 I -25 772168 4.80 227832 23 38 50954 707180 3 55 934723 I -25 772457 4^8o 227543 22 39 50979 86o3o 707393 3 55 934649 I -25 772745 4^8o 227255 21 40 5 1 004 8601 5 707606 3 55 934574 I •25 773033 4^8o 226967 20 41 51029 86000 9-707819 "3" "55" 9-934499 ' -25 9.773321 4-8o 10-226679 19 42 5io54 85985 708032 3- 54 934424 I -25 773608 4-79 226392 18 43 51079 85970 708245 3- 54 934349 I -25 773896 4-79 226104 17 44 5iio4 85956 708458 3 54 934274 I -25 774184 4-79 2258i6 16 45 51129 85941 708670 3. 54 934199 I 934123 I •25 774471 4^79 225529 15 46 5ii54 85926 708882 3 53 •25 774759 4-79 223241 14 47 51179 8591 1 85896 85881 709094 3 53 934048 I •25 775046 4-79 224954 13 48 5 1 204 709306 3. 53 933973 I •25 775333 v,% 224667 12 49 01229 709518 3. 53 933898 I • 26 775621 224379 11 50 5i254 85866 709730 3- 53 933822 I •26 775908 4-lS 224092 10 51" 51279 85851 9-709941 3. 52 9-933747 I -26 9.776105 776482 4-78 io-2238o5 ~9~ 52 5i3o4 85836 710153 3- 52 933671 I .26 4^78 2235i8 8 53 5i329 85821 710364 3- 52 933596 I -26 776769 777055 4^78 22323l 7 54 5i354 858o6 710575 3- 52 933520 I -26 4.78 222945 6 55 5i379 85792 710786 3. 5i 933445 I .26 777342 4-78 222658 5 56 5 1 404 85777 710997 3- 5i 93329? I -26 777628 4^77 222372 4 57 51429 85762 7 II 208 3- 5i -26 .26 771915 778201 4-77 222085 3 58 5 1454 85747 711419 3 5i 933217 I 4-77 221799 2 59 51479 85732 711629 3 5o 933141 I -26 778487 4-77 22l5l2 1 60 5i5o4 85717 71 1839 3-5o 933066 I .26 778774 4-77 221226 N. COS. N.sine. Ia cos. D.l" L. sine. L.cot. D.l" L. tang. ' 59° 1 TRIGOKOMETRICAL FUNCTIONS. — 31' 61 Nat. Functions. Logarithmic Functions + 10. ' N.fline. N. COS. L. sine. D. 1" L. COS. I ).l" L. tang. D. 1" L.cot. 5i5o4 85717 9-71 1839 3.5o 9.933066 I T^l 9-778774 4-77 10-221226 60 1 5i529 85702 7i2o5o 3-5o 932990 1 •27! 779060 4-77 220940 5y 2 5 1 554 85681 712260 3-50 932914 1 •27 779346 4-76 220054 58 8 5i579 85672 712469 3-49 932838 I •27 779632 4-76 220368 57 4 5 1 604 85657 712679 3-49 932762 I •27 780203 4-76 220082 ^56 6 51628 85642 712889 3-49 932685 1 •27 4-76 219797 1 55 1 6 5 1 653 85627 713098 3-49 932609 I •27 780489 4-76 2.9311 54 7 51678 85612 7i33o8 3-49 3-48 932533 I •27 780773 4 lb 2.9223 5.3 8 5 1 703 85597 713517 713726 932457 I •27 781060 4-76 218940 62 9 51728 85582 3-48 932380 I •27 781346 4-75 2.8654 61 10 51753 85567 713935 3-48 932304 I •27 •27 781631 4-73 218369 1 60 51778 8555i 9.714144 3-48 9-932228 I 9.781916 4-75 10.2.8084 ! 49 12 5i8o3 85536 714352 3-47 932i5i I •27 782201 4-75 2.7799 ! 48 IS 51828 85521 7.4561 3.47 932075 I -28 782486 4-73 217J14 1 47 14 5i852 855o6 714769 714978 3.47 931998 I .28 782771 1 4-75 2.7229 46 15 51877 85491 3-47 931921 1 • 28 783o56 4-73 216944 j 45 16 51902 85476 7i5i86 3-47 931845 I .28 783341 4-73 216609 44 17 51927 85461 715394 3.46 931768 I .28 783626 4-74 216374 1 4.; 18 51952 85446 7 1 56o2 3.46 931691 I .28 783910 4-74 216090 42 19 51977 85431 713809 ■ 3-46 931614 I .28 784195 4-74 2i58o5 1 41 20 52002 85416 __726o^ 3.46 931537 I ^1 784479 4-74 2l5521 40 21 32026 85401 9-716224 3.45 9-931460 I -28| 9.784764 4-74 10-215236 39 22 5205l 85385 716432 3-45 93i383 I -28, 785048 4-74 214932 08 23 52076 85370 716639 3-45 93i3o6 1 .28 7S?^f 4-73 214668 87 24 52IOI 85355 716846 3-45 931229 I -29 7856i6 4-73 214384 36 25 52126 85340 717053 3.45 93ii52 I •29 785900 4-73 214100 35 26 52i5i 85325 717259 3-44 931075 I .29 786184 4-73 2138.6 34 27 52175 853 10 717466 3-44 930998 1 •29 786468 4-73 213532 83 28 52200 85294 717673 3-44 930921 I •29 786752 4-73 213248 32 29 52225 85279 9.718291 3-44 930843 1 •29 787036 4-73 212964 31 80 81 52250 85264 3-43 3-43 930766 I -29 787319 4-72 212681 30 '29 52275 85249 9-930688 1 •29 9-787603 4-72 10.212397 82 52299 85234 718497 3-43 930611 I •29 787886 788170 4-72 212114 28 83 52324 852i8 718703 3.43 93o533 I •29 4-72 2ii83o 27 84 52349 85203 718909 3.43 930456 1 •29 788453 4-72 2 1 1 547 26 85 52374 85i88 719114 3-42 930378 1 -29 788736 4-72 2 1 1 264 25 36 52399 85173 719320 3-42 93o3oo 1 -30 789019 4-72 2.0981 24 37 52423 85i57 719525 3-42 930223 1 • 30 789302 4-71 210698 23 38 52448 85i42 719730 3.42 930145 •3o 789585 1 4-71 2 . 04 1 5 22 89 52473 85i?7 719935 3.41 930067 -30 789868 4-71 210l32 21 40 41 52498 85ii2 720140 3.41 929989 -3o -30 790i5i 4-71 209849 20 52522 85096 9-720345 3.41 9-929911 9.790433 4-71 10-209567 ly 42 52547 85o8i 720549 3-41 929833 -3o 790716 4-71 209284 18 43 52572 85o66 720754 3-40 929755 • 3o 790909 791281 4-71 209001 17 44 52597 85o5i 720958 3-40 929677 -3o 4-71 208719 16 45 52621 85o35 721162 3 40 929599 -30 791563 4-70 208437 15 46 52646 85o2o 721366 3.40 929521 -30 ! 791846 4-70 208154 14 47 52671 85oo5 721570 3-40 929442 •3o 792128 4-70 207872 13 48 52696 84989 721774 3-39 929364 -3i 792410 4-70 207590 12 49 52720 84974 721978 3.39 929286 -3i 792692 4-70 207308 11 50 52745 84959 722181 3.39 929207_ -31 ;___792974_ 4-70 207026 10 TT 52770 84943 "^22385" "3-39' 9-929129 -3i 9-793256 4-70 10-206744 9 52 52794 1 84928 722588 3-39 3.38 929050 -3i 793538 4-69 206462 8 53 52819 84913 52844 8489^ 722791 928972 -3i 793819 4.69 206181 7 64 722994 3-38 928893 -31 794101 4-69 205899 6 55 52869 84882 723197 3-38 928815 -3i 794383 469 2o56i7 5 56 52893 1 84866 723400 3-38 928736 • 31 794664 4-69 205336 4 57 52918 ;8485i 7236x)3 3-37 928657 -31 794945 4-69 2o5o55 8 58 52943 i 84836 7238o5 3-37 928578 • 3i mm 4-69 4-68 204773 2 59 52967 84820 724007 3-37 928499 -3i 204492 1 60 52992 84805 724210 3.37 928420 •3i 795789 4-68 20421 1 C N.co8.!N.8ine. L. COS. ■DA" L. sine. L.cot D.l" L. tang. ' 58° 1 1 (\2 TRIGONOMETRICAL FUNCTIOKS. — 32^ Nat. Function*. Logarithmic Functions + 10. / N.slne. N.cos. L. sine. D. 1" L. COS. D.I" L. tang. D.l" L. cot. 52992 84805 9-724210 3.37 9-928420 1-32 9-795789 4-68 10-204211 60 1 53017 84789 724412 3 •37 928342 1-32 i 796070 4-68 203930 59 2 53o4i 84774 724614 3 -36 928263 1-32 796351 4-68 203649 203368 58 3 53o66 84759 724816 3 -36 928183 1-32 796632 4-68 67 4 53Q9I 84743 725017 3 -36 928104 1-32 796913 4-68 203087 56 5 53ii5j 84728 725219 3 -36 928025 1-32 797194 4-68 202806 55 6 53140 84712 725420 3 -35 927946 1-32 797475 4-68 202325 54 7 53164 84697 725622 3 -35 927867 1-32 797755 4-68 202245 53 8 53189 84681 725823 3 -35 927787 1-32 798036 4-67 201964 i 52 1 9 53214 84666 726024 3 -35 927708 1-32 798316 4-67 2016^4 51 10 53238 84650 726225 3 -35 927629 1-32 798596 4-67 201404 50 11 53263 84635 9-726426 "3 34 9-927549 1-32 9-798877 4-67 10-201123 49 12 53288 84619 726626 3 34 927470 1-33 799137 4-67 200843 48 13 ! 53312 84604 726827 3 34 927390 1-33 799437 4-67 200563 47 14 i 53337 1 84588 727027 3 34 927310 1-33 799717 4-67 200283 46 15 !5336i 184373 727228 3 34 927231 1-33 799997 4-66 2oooo3 45 16 i 533S6 84557 727428 3 33 927151 1-33 800277 4-66 199723 44 17 5341 1 84542 727628 3 33 927071 1-33 800557 4-66 J 99443 43 18 53435 84526 727828 3 33 026991 1-33 8oo836 4-66 19Q164 42 19 53460 845 1 1 728027 3 33 IIX 1-33 801 1 16 4-66 198884 41 20 53484 84495 728227 3 33 1-33 801396 4-66 198604 40 39 21 53509 84480 9 -.728427 3 32 9-926751 1-33 '9"^T675~ 4-66 10-198325 22 53534 84464 728626 3 32 926671 1-33 801955 4-66 198045 38 23 53558 84448 728825 3 32 926591 1-33 802234 4-65 197766 37 24 53583 84433 729024 3 32 92651 1 1-34 8o25i3 4-65 197487 197208 36 25 53607 84417 729223 3 3i 926431 1-34 802792 4-65 35 26 53632 84402 729422 3 3i 926351 1-34 803072 4-65 196928 84 27 53656 84386 729621 3 3i 926270 1-34 8o335i 4-65 196649 33 28 5368i 84370 729820 3 3i 926190 1-34 8o363o 4-65 196370 32 29 53705 84355 730018 3 3o 9261 10 1-34 803908 4-65 196092 81 80 31 53730 53754 84339 730216 3 3o 926029 1-34 I -'34 804187 9-804466" 4-65 4-64 195813 10-195534 30 '^9 84324 9 -73041 5 3 3o 9-925949 32 53779 84308 - 73o6i3 3 3o 925868 1-34 804745 4-64 195255 28 83 53804 84292 73081 1 3 3o 925788 1-34 8o5o23 4.64 194977 194698 27 34 53828 84277 731009 3 29 925707 1-34 8o53o2 4-64 28 85 53853 84261 731206 3 29 923626 1-34 8o558o 4-64 194420 25 36 53877 84245 t3i4o4 7'3i6o2 3 29 925545 1-35 8o5859 4-64 194141 24 37 53902 84230 3 29 925465 1-35 806137 4-64 193863 23 88 53926 84214 731799 3 11 925384 1-35 806415 4-63 193585 22 39 53951 84198 84182 731996 3 9253o3 1-35 806693 4-63 193307 21 40 53975 732193 3 28 925222 1-35 1-35 806971 4-63 193029 20 41 54000 84167 9.732390 3" 28 9-925141 9.807249 4.'63 10-192751 19 42 54024 84i5i 732587 3 28 925060 1-35 807327 4-63 192473 18 43 54049 54073 84i35 732784 3 28 924079 924897 1-35 807805 4-63 192195 17 44 84120 732980 3 27 1-35 808083 4-63 191917 16 45 54097 84104 733177 3 27 924816 1-35 8oS36i 4-63 191639 15 46 54122 84088 733373 3 27 924735 1-36 8o8638 4-62 191362 14 i7 54146 84072 84057 733569. 3 27 924654 1-36, 808916 4-62 191084 18 48 54171 733765 3 27 924572 1-36: 809193 4-62 190807 12 49 54195 84041 733961 3 26 924491 1-36 809471 4-62 190529 11 60 54220 84025 734157 3 26 924409 9-924328 1-36 _A°9748_ 4-62 190252 10 9 51 54244 ! 84009 9-734353 3 26 Tr36 9-810025 4-62 10-189975 52 54260 ! 83994 734549 3 26 924246 1-36 8io3o2 4-62 189698 8 63 54293 83978 734744 3 25 924164 .-36 8io58o 4-62 189420 7 64 54317 83962 734939 3 25 924083 1-36 810857 4-62 189143 6 65 54342 83946 735i35 3 25 924001 1-36 811134 4-6i 188866 5 66 54366 83930 735330 3 25 923919 923^37 1-36 8ii4«o 4-6i 188590 4 67 54391 83915 735525 3 25 1-36 81 1687 4-6i i883i3 3 58 54415 1 83899 735719 3 24 923755 1.37 81 1964 4-6i i88o36 2 69 54440 83883 735914 3 24 923673 1-37 812241 4-6i 187759 1 60 54464 83867 736109 3-24 923591 1.37 8i25i7 4-6i 187483 N. COS. N. sine. L. COS. D. 1" • L. sine. Kcot D.l" L. tang. / 57° 1 TRIGONOMETRIC A 1. FUNCTIONS. — 33° 63 Nat. Functions. Logarithmic Functions + 10. 1 i ' iN.Bliie. N. COS. L. sine. D. 1" L. COS. I ).i" L. tang, j D. 1" L.cot. 54464 83867 9.736109 3.24 9.923591 I •37 9.812517 4-6i 10-187483 60 I 54488 83851 7363o3 3-24 923509 1 •37 812794 4-6i 187206 59 2 545 1 3 83835 786498 3-24 923427 I •37 813070 4-6i 186930 58 3 54537 838x9 736692 3-23 1 923345 1 •37 8x3347 8i3623 4 -60 186653 57 4 54561 838o4 736886 3-23 923263 1 •37 4-6o 186377 56- 5 54586 83788 737080 3-23 923181 I .37 813899 814175 4-6o X86101 55 6 54610 83772 737274 3-23 923098 I •37 4-6o 185825 54 7 ! 54635 83756 737467 3-23 923016 I •37 814452 4-6o 185548 53 8 I 54659 83740 737661 3.22 922933 1 .37 814728 4-6o 185272 52 9 1 54683 83724 ]IG 3-22 922b5l I •37 8i5oo4 4-60 X 84996 51 11 i 54708 ■ 83708 3-22 922768 1 -38 -38 8x5279 4-6o X84721 50 54732 83692 9.738241 "3 -2 2" 9-922686 1 9-8x5555 T-59~ 10.184445 49 12 j 54756 83676 738434 3-22 922603 1 -38 8i583i 4-59 184169 183893 48 13 54781 83660 738627 3.21 922520 I • 38 8x6107 4-59 47 14 54805 83645 738820 3-21 922438 1 -38 8x6382 4-59 x836i8 46 15 54829 83629 739013 3-21 922355 1 -38 8x6658 4-59 X 83342 45 16 54854 836i3 739206 3-21 922272 1 -38 8x6933 4-59 x83o67 44 17 54878 83597 739398 3-21 922189 1 -38 8x7209 4-59 182791 43 18 54902 8358i ?^???3 3-20 922106 1 -38 817484 4-59 182516 42 ly 54927 83565 3-20 922023 I -38 8i8o3d 4-58 182241 41 20 21 54951 54975 83549 "83533 739975 3-20 921940 -38 T39 18x965 40 9-740167 3-20 9-921857 1 9-8i83io 10-181690 89 22 54999 83517 740359 3-20 921774 1 .39 8x8585 4-58 181415 38 23 55o24 83501 74o55o 3-19 92I69I 1 .39 8x8860 4-58 181140 37 24 55048 83485 740742 3.19 921607 1 -39 819135 4-58 i8o865 36 25 55072 83469 740934 3.19 921524 1 •39 819410 4.58 180590 85 26 55097 83453 741125 3.19 921441 1 -39 819684 4. 58 i8o3i6 84 27 55i2i 83437 74i3i6 3.19 921357 1 -39 819959 4-58 180041 33 28 55145 83421 74i5o8 3-18 921274 1 -39 820234 4.58 X 79766 82 2y 55169 83405 741699 3-18 921190 1 •39 82o5o8 4-57 179492 81 30 31 55194 "55218 83389 83373 741889 3.18 921107 1 •39 820783 4-57 4-57 179217 80 9-742080 3- 18 9-921023 1 -39 9-821057 10.178943 29 32 55242 83356 742271 3-18 920939 1 920856 1 .40 82x332 4-57 178668 28 33 55266 83340 742462 3.17 .40 821606 4-57 178394 27 34 55291 83324 742652 3. .7 920772 1 -40 821880 4-57 178120 26 35 !553i5 833o8 742842 3.17 920688 1 .40 822x54 4-57 177846 25 36 55339 83292 743o33 3.17 920604 1 .40 822429 822703 4.57 X77571 24 87 55363 83276 743223 3.17 920520 1 .40 4-57 177297 23 38 ' 55388 83260 743413 3-i6 920436 1 .40 822977 4-56 X77023 22 39 : 55412 83244 a436o2 3-i6 920352 1 .40 823250 4-56 X767D0 21 40 41 \ 55436 ^5460" 83228 83212 743792 3-i6 920268 1 -40 823524 4-56 176476 20 9-743982 3.16 9-920184 ^0 9.82-3798 4-56 xo. 176202 19 42 55484 83195 744171 3.16 920099 1 • 40 824072 4-56 I75Q28 18 43 55509 55533 83179 744361 3-i5 920015 I -40 824345 4-56 175655 17 44 83i63 744550 3-i5 919031 1 •41 824619 4-56 X7538X 16 45 55557 83 1 47 744739 3-15 919846 1 •41 824893 4-56 175107 15 46 55581 83i3i 744928 3-i5 919762 1 -41 825x66 4-56 X74834 14 47 556o5 83ii5 745117 3-15 919677 1 •41 825439 825713 4-55 174561 13 48 55630 83098 745306 3.14 919593 1 •41 4-55 174287 12 49 55654 83o82 745494 3.14 919508 1 •41 825986 4-55 174014 11 50 51 55678 83o66 83^5o 745683 9^438^71 3.14 3.14 919424 1 •41 •41 826259 4-55 4-55^ 173741 10 9~ 55702 9-919339 9-826532 10.X73468 52 55726 83o34 746059 746248 3.14 919254 •41 826805 4-55 173195 8 53 1 55750 83oi7 3-13 919169 919085 •41 827078 4-55 172922 7 54 55775 83ooi 746436 3-13 •41 827351 4-55 172649 6 55 55799 55823 82985 746624 3-13 910000 918915 9i883o •41 827624 4-55 172376 5 56 82969 746812 3-13 •42 827897 4-54 172103 4 57 155847 82953 746999 747187 3-13 -42 828170 4-54 i7x83o 8 58 1 55871 82936 3.12 918145 [.42 828442 4-54 17x558 2 59 55895 82920 747374 3-12 918659 1-42 828715 4-54 17x285 1 60 559,9 82904 747562 3-12 918574 1-42 828987 4-54 171013 i| N. COS. N.sine L. COS. D.l" L.sine. 1 L.cot. . D.l" L. tang. ' 56° 1 64 TRIGONOMETRICAL FL'NCTIONS. — 34°. Nat. Functions. Logarithmic Functions + 10. 1 N.sine. N. COS. L. sine. D. 1" L. COS. D.l" L. tang. D.l" L. cot 55919 82904 9-747562 3 12 9-918574 1.42 9-828987 4 54 10.171013 60 1 55943 82887 747749 3 12 918489 1-42 829260 4 54 170740 i>\) 2 55968 82871 747936 3 12 918404 1.42 829532 4 54 170468 58 3 55992 82855 748123 3 11 9i83i8 1.42; 829805 4 54 170195 57 4 56oi6 82839 748310 3 II 918233 1.42 830077 4 54 169923 56 5 56040 82822 748497 748683 3 II 918147 1.42 83o349 4 53 169651 55 6 56o64 82806 3 II 918062 1.42 83o62i 4 53 169379 54 7 56o88 82790 748870 3 II 917076 917891 1.43 830893 4 53 169107 168835 53 8 56II2 82773 749036 3 10 1.43 83ii65 4 53 52 9 56i36 82757 749243 3 10 917805 I •43; 831437 4 53 168563 51 10 56 1 60 82741 749429 3 3 10 10 917719 1-43 831709 4 53 168291 50 11 56 1 84 82724 9-749615 9.917634 ?:i3 "9^83 1 98 r 4 53 10-168019 49 12 56208 82708 749B01 3 10 91754B 1.43 832253 4 53 167747 4S \13 56232 82692 749987 3 09 917462 1.43 832525 4 53 167475 47 14 56256 82675 750172 3 09 917376 1.43 832796 4 53 167204 46 15 56280 82659 82643 75o358 3 09 917290 1.43 833o68 4 52 166932 45 16 563o5 730343 3 09 917204 1.43 833339 4 52 166661 44 17 56329 82626 750729 3 ^ 917118 1-44 833611 4 52 166389 43 il8 56353 82610 750914 3 917032 1.44 833882 4 52 166118 42 19 20 56377 82593 751099 3 08 916946 1-44 8341 54 4 52 165846 41 56401 82577 731284 3 T 08 "08" 916859 1-44 834425 ± 52 165575 40 21 56425 82561 9 -75 1 469 9.916773 1-44 9.834696 4 "51" io-l653o4 8y 22 56449 56473 82544 75 1 654 3 08 916687 1-44 334967 4 52 l65o33 38 23 82528 751839 3 08 916600 1-44 835238 4 52 164762 37 24 56497 825ii 752023 3 07 9i65i4 1-44 835509 4 52 164491 86 25 56521 82495 752208 3 07 916427 1-44 835780 4 5i 164220 35 26 56545 82478 752392 3 07 916341 1-44 836o5i 4 5i 163949 163678 34 27 56560 56593 82462 752376 3 07 916254 1-44 836322 4 5i 33 28 82446 752760 3 07 916167 1.45 836593 4 5i 163407 32 29 56617 82429 752944 3 06 916081 1.45 836864 4 5i i63i36 31 30 31 56641 82413 753128 3 1 06 06 915994 9-915907 1.45 1.45 837134 4 5i 162866 30 56665 82396 82380 9-733312 9.837405 4 5i 10.162595 I'J 32 56689 56713 753495 3 06 9i582o 1.45 837675 4 5i 162325 2S 33 82363 753679 3 06 915733 1.45 837946 838216 4 5i 162054 27 34 56736 82347 753S62 3 o5 915646 1.45 4 5i 161784 26 85 56760 82330 754046 3 o5 915559 1-45 838487 4 5o i6i5i3 25 36 56784 82314 734-229 3 o5 915472 1-45 838757 4 5o 161243 24 87 568o8 lllV, 754412 3 o5 915385 1-45 839027 4 5o 160973 23 38 56832 754393 3 o5 915297 1-45 llf^l 4 5o 160703 2-2 89 56856 82264 754778 3 04 9i52io 1-45 4 5o 160432 21 40 56880 82248 754960 3 04 9i5i23 1.46 839838 4 5o 160162 20 41 56904 8T23r 9.755143 3 04 9.915035 1.46 9.840108 4 5o 10-159892 ! 19 1 42 56928 82214 755326 3 04 914948 914860 1.46 840378 4 5o 1 50622 18 43 56952 82108 82I8I 755508 3 04 1-46 840647 4 .50 159353 17 44 56976 755690 3 04 914773 1.46 840917 4 -49 159083 16 45 57000 82165 755872 3 o3 914685 1.46 841187 4 .49 13^813 15 46 57024 82148 756o54 3 o3 914598 1.46 841457 4 -49 158543 14 i7 57047 82132 756236 3 o3 914510 1.46 841726 4 -49 158274 13 48 57071 82115 756418 3 o3 914422 1.46 841996 4 •49 i58oo4 12 49 57095 82098 756600 3 o3 914334 1.46 842266 4 -49 157734 11 50 571 19 82082 _756782_ 9.736963 3 3 02 02 914246 1-47 1-47 842535 4 49 157465 10 51 57143 82065 9.914158 9.842805 4 49 10-157195 52 57167 82048 757144 3 02 914070 1-47 843074 4 49 156926 8 53 57191 82032 757326 3 02 913982 1-47 843343 4 .49 156657 7 54 57215 82015 751307 3 02 913894 1-47 843612 4 .49 156388 6 55 157238 81999 757688 3 01 9i38o6 1-47 843882 4 -48 i56ii8 5 56 157262 81982 757869 3 01 913718 1-47 844i5i 4 .48 155849 4 57 1 57286 81965 758o5o 3 01 9i363o 1-47 844420 4 48 155580 8 58 57310 81949 75823o 3 01 913541 1-47 844680 844958 4 48 i553ii 2 59 57334 81932 758411 3 01 913453 1-47 4 4S 1 55042 1 60 57358 81915 758591 3.01 913365 1-47 845227 4-48 154773 H.C08. N.sine. L. COS. D. 1" L, sine. L. cot. D. 1" L. tang. / 1 55° 1 TEIGOi^OMETlllCAL FUi^CTlONS. — 35^ 66 Nat. Functions. Logarithmic Functions + 10. N.sine. N.cos L. sine. D. 1" L. COS. D.l" L. tang. D.l" L.cot. 57358 81915 81899 9-758591 3-01 9-913365 1-47 9-845227 4-48 10.154773 1 54504 60 1 57381 ',11111 3.00 913276 1-47 845496 4-48 59 2 57405 81882 3.00 913187 I1.48 845764 4-48 154236 53 3 57429 8 1 865 759132 3-00 913099 1-48 846033 4-48 153967 15369^^ 1 53430 57 4 57453 81848 759312 3-00 9i3oio 1-48 846302 4-48 66 5 57477 8i832 759492 3-00 912922 1-48 846570 4-47 55 6 57301 8i8i5 759672 2-99 912833 ii-48 846839 4-47 i53i6i 54 7 57524 81708 81782 759852 2-99 912744 1-48 847 « 07 4-47 152893 53 8 57548 76003 I 2-99 912655 1-48 847376 4-47 152624 52 9 57572 81765 760211 2.99 912066 1-48 847644 4-47 152356 51 10 57596 81748 760390 2-99 912477 1-48 847913 4-47 4-47 152087 ioi5i8i9 50 4 'J 11 '& 81731 9-760560 760748 2.98' 9-912388 1-48 9.848181 12 81714 2.98 912299 1-49 848449 4-47 i5i5oi 4S 13 57667 81698 81681 760927 2.98 912210 I -40 848717 4-47 i5i283 47 14 57691 761106 2.98 912121 1-49 848986 4-47 i5ioi4 46 15 57715 81664 761285 2-98 9i2o3i 1-49 849254 4-47 I 50746 45 16 57738 81647 761464 2.98 91 1942 1-49 849522 4-47 1 50478 44 17 57762 8i63i 761642 2.97 911853 1-49 849790 4.46 l502I0 4-i 18 57786 57810 81614 761821 2-97 911763 1.49 85oo58 4.46 149942 1 42 I 19 81597 761999 2.97 91 1674 1-49 85o325 4.46 149675 41 20 57833 8i58o 762177 2.97 __9ii584_ 1-49 85o593 4.46 149407 4<» 21 57857 8 1 563 9.762356 l:V> 9-911495 r49 9.800861 4-46 10-149139 148871 3y 22 57881 81546 762534 911405 1-49 85II29 4-46 38 23 57904 8i53o 762712 2.96 9ii3i5 i-5o 85 I 396 4-46 148604 37 24 57928 8i5i3 762889 2.96 911226 i-5o 85 I 664 4.46 148336 36 25 57952 81496 763067 763245 2.96 9iii3t) i.5o 85i93i 4-46 148069 35 26 57976 81479 2.96 911046 i-5o 852199 4-46 147801 34 27 UIV3 81462 763422 2-96 910956 i-5o 852466 4-46 147534 33 28 81445 763600 2.95 910866 i-5o 852733 4-45 147267 146732 32 29 58047 81428 763777 2-95 910776 910686 1-50 85300I 4-45 31 30 31 58070 81412 763954 2.95 i-5o 853268 4-45 30 58094 81395 9-764131 2-95 9-910596 i-5o 9-853535 4-45 10-146465 2y 32 58ii8 81378 764308 2-95 9io5o6 1-50 8538o2 4-45 146198 28 33 58i4i 8i36i 764485 2-94 910415 i-5o 854069 4-45 14593 1 27 34 58i65 8x344 764662 2-94 910325 i-5i 854336 4-40 145664 26 35 58189 8i327 764838 2-94 910235 i-5i 8546o3 4-45 145307 i45i3o 2:. 36 58212 8i3io 765oi5 2-94 910144 i-5i 854870 4-45 24 37 58236 81293 765191 2.94 910004 i-5i 855137 4-45 144863 23 38 58260 81276 765367 2-94 909873 i-5i 855404 4-45 144596 22 39 58283 81259 765544 2.93 1-5. 855671 4-44 144329 21 40 41 58307 81242 765720 2.93 2.93 909782 9-909691 i.5i! i-5i| 855933 4-44 144062 20 58330 81225 9-765896 9.856204 4-4i 10.143796 143529 143263 ly 42 58354 81208 766072 2.93 909601 i-5i 856471 4.44 18 43 58378 81191 766247 2.93 909510 1-5, 856737 4-44 17 44 58401 81174 766423 2.93 909419 909328 1-51 807004 4-44 142996 16 45 58425 81I57 766598 2-92 1-52 857270 4-44 142730 15 46 58449 81140 766774 2-92 909237 1-52 857037 4.44 142463 14 47 58472 81123 766949 2-92 909146 1-52 857803 4.44 142197 13 48 58496 81106 767124 2.92 909055 1-52 858069 4.44 141931 12 49 58519 81089 767300 2-92 908964 1-52 858336 4.44 141664 11 50 51 58543 81072 767475 2.91 2-91 ^9o878T 1-52 iT5'2 858602 4-43 141398 10 9 58567" 8io55 9-767649 9.808868 4-43 io.i4ii32 52 58590 8io33 767824 2.91 908690 1-52 859134 4.43 140866 8 53 586 1 4 81021 768173 2.91 908599 1-52 859400 4-43 140600 7 54 58637 81004 2-91 908507 1-52 809666 4-43 140334 6 55 58661 80987 768348 2-90 908416 1-53 859932 4-43 140068 5 56 58684 80970 768522 2-90 908324 1-53 860198 4-43 139802 4 57 58708 80953 768697 2-90 908233 1-53 860464 4-43 139536 3 58 58731 80986 768871 2-90 908141 1-53 860730 4.43 139270 2 59 58755 80919 769045 2-90 908049 1.53 860995 4-43 139005 136739 1 60 58779 80002 769219 2-90 907958 1-53 861 261 4-43 N.C08. N.8ine. L. COS. D.l" L. sine. L. cot. D.l" L. tang. ' 54° 1 66 TRIGONOMETRICAL FUNCTIONS.— 36' Nat. Functions. LooARiTHMic Functions + 10. 1 :N.sine. N. COS. L. Bine. D. 1" L. COS. ] 0.1" L. fATng. Dl." L. cot 60 1 58779 80902 9.769219 769393 2 ?? 9.907058 1 907866 .53 9.861261 4-43 10. 188739 138473 1 588o2 80885 2 .53 861527 4 43 69 2 58826 80867 769566 2 89 907774 .53 861792 862058 42 138208 58 3 58849 8o85o 769740 2 89 907682 1 -53 42 13794a i7 4 58873 8o833 769913 2 89 907590 .53 862323 42 137677 56 5 58896 80816 770087 2 l^ 907498 •53 862589 42 i374u 55 6 i 58920 80799 770260 2 907406 -53 862854 42 137146 54 7 1 58943 80782 770433 2 88 907314 •54 863II9 42 i3688i 53 8 58967 80765 770606 2 88 907222 1 •54 863385 42 i365i5 52 9 5899^ 80748 770779 2 88 907129 •54 863650 42 i3635o 51 10 11 59014 80730 770952 2 88 907037 ^54 •54 863915 42 i36o85 50 49 59037 80713 9.771125 2 88 9.906945 906852 9.864180 42 io«i3582o 12 59061 80696 771298 2 87 •54 864445 42 135555 4S 13 59084 80679 771470 2 ?7 906760 •54 864710 42 135290 47 14 59108 80662 771643 2 87 & ! •54 864975 41 l35o25 46 15 591 3 1 80644 771815 2 87 •54 865240 41 134760 45 16 59154 80627 771987 2 87 906482 1 •54 8655o5 41 134495 44 17 59178 80610 772159 2 87 906389 1 • 55 865770 41 l3423o 43 18 59201 80593 772331 2 86 906296 .55 866035 41 133965 42 19 59225 80576 7725o3 2 86. 906204 1 .55 866300 41 133700 41 20 59248 8o558 772675 2 86 9061 U I .55 866564 41 133436 40 21 59272 8o54i 9.772847 2 86 9.906018 I .55 9.866829 T 41 10.133171 8 'J 22 59295 8o52/; 773018 2 86 905925 I • 55 867094 4 41 132906 83 23 59318 8o5o7 ]]lit: 2 86 9o5832 I -55 867358 4 41 132642 37 24 59342 80489 2 85 905739 I • 55 867623 4 4r 132377 86 25 5o365 80472 773533 2 85 905645 I .55 867887 4 41 i32ii3 85 26 59389 80455 773704 2 85 905552 I -55 868 1 52 4 40 i3i848 84 27 59412 80438 773875 2 85 905459 I .55 868416 4 40 i3i584 33 28 59436 80420 774046 2 85 905366 I • 56 868680 4 40 i3i32o 82 29 59459 80403 774217 2 85 905272 I • 56 868945 4 40 l3io55 31 80 59482 8o386 774388 2 2 84 84 9o5i79 I • 56 869209 4 4 40 40 :i3o79i 30 "sT 59506 8o368 9-774558 9 -905085 1 ~5"6 9.869473 io-i3o527 29 32 59529 8o35i 774729 2 84 904992 I • 56 869737 4 40 i3o263 23 33 59552 80334 774899 2 84 904898 I • 56 870001 4 40 129909 27 34 59576 8o3i6 775070 2 84 904804 I .56 870265 4 40 129735 26 35 59599 80299 80282 775240 2 84 9047 1 1 I .56 870529 870793 871057 4 40 129471 25 36 59622 775410 2 83 904617 1 • 56 4 40 129207 24 37 59646 80264 775580 2 83 904523 I .56! 4 40 128943 23 38 ^r4? 80247 775750 2 83 904429 I 904335 I •57 871321 4 40 128679 128415 22 39 8o23o 775920 2 83 •57 871585 4 40 21 40 59716 80212 776090 2 83 83" 904241 I •57 871849 4 39 39 I28i5i 20 19 41 i^it 80195 9.776259 2 9-904147 I 'i'^ 9-872112 4 10-127888 42 80178 776598 2 82 904053 I .57 872376 4 39 127624 13 43 59786 80160 2 82 903959 I •57 872640 4 39 127360 17 44 i 59809 80143 776768 2 82 903864 I •57 872903 4 39 127097 16 45 59832 80125 776937 2 S2 903770 1 •57 873167 4 39 126833 15 46 59856 80108 777106 2 82 903676 I 9o358i I •57 873430 4 39 126570 14 47 59879 80091 777275 2 81 •57 873694 873967 4 39 i263o6 13 48 59902 80073 777444 2 81 903487 I •57 4 39 i26o.<3 12 49 59926 8oo56 777613 2 81 903392 I •58 874220 4 39 1257uo 11 50 51" 1 59949 8oo38 777781 _2 81 903298 I ^58 874484 4 39 I255i6 10 1 59972 80021 9.7779D0 778119 2 81 9.903203 I •il 9-874747 4 39 10.125253 ~Y 52 1 59995 8ooo3 2 81 903 1 08 1 -58; 875010 4 39 124990 8 53 j 60019 79986 778287 2 80 9o3oi4 1 -58 875273 4 38 124727 7 54 j 60042 79968 778455 2 80 902919 I 902824 I -58 875536 4 38 124464 6 55 ' 6oo65 79951 778624 2 80 -58 875800 4 38 124200 5 56 1 60089 79934 778792 2 80 902729 I -?o 876063 4 38 123937 4 57 1 601 12 79916 778960 2 80 992634 • 1^ 876326 4 38 123674 3 58 1 60,35179899 l6oi58| 79881 779128 2 80 902539 1 19 876589 4 38 123411 2 59 779295 2 79 902444 19 876851 4 38 123149 1 60 60182 79864 779463 2-79 902349 1 • 59 877114 4.38 122886 N. COS. N. sine. L. COS. D. 1" L. sine. L.cot. dTi^ L. tang. ' 53° 1 TRIGONOMETKICAL FUNCTIONS. — 37". m Nat. Functions. Logarithmic Functions + 10. 1 ' N.sine. N. COS. L. sine. D. 1" L. COS. D.l" L. tang. D. 1" L.cot. 1 60182 79864 9-770463 2-79 9-902349 ^59 9-877114 4-38 10-122886 60 1 6oao5 79846 77903 1 2-79 902253 1.59 877377 4-38 122623 5y 2 60228 79820 779798 2-79 9021 58 1.59 877640 4-38 122360 58 3 602 5 1 798 n 779966 2-79 902063 1-59 877903 4-38 \l',tU 57 4 60274 79793 780133 2-79 901967 901872 1.59 87bi65 4-38 6t> 5 60298 79776 780300 2-78 1-59 878428 4-38 121572 55 6 6o32i 7v758 780467 2-78 901776 901681 1-59 878691 4-38 12.309 54 7 6o344 79741 780634 2-78 1-59 878953 4-37 12.047 53 8 60367 79723 780801 2-78 901585 1.59 8792.6 4-37 120784 52 9 60390 79706 780968 2.78 901490 1-59 879478 4-37 120522 51 10 60414 79688 781134 2-78 901394 1-60 879741 4-37 120259 50 11 60437 79671 9'78i3oi 2-77 9-901298 1.60 9-88uoo3 4-37 10.119997 49 12 60460 79653 781468 2-77 901202 1-60 880265 4-37 119735 48 13 60483 79t)35 781634 2-77 901106 1-60 880028 4-37 H9472 47 14 6o5o6 79618 781800 2-77 901010 1-60 880790 88.052 4-37 II92IO 118948 4G 15 6o52g 6o553 79600 781966 2-77 900914 1.60 4-37 45 16 79583 782132 2-77 900818 1-60 88.3.4 4-37 1 1 8686 44 17 60576 79565 782298 2-76 900722 1-60 88.576 4-37 118424 43 18 1 60599 79547 782464 2-76 900626 1 -60 881839 4-37 118161 42 19 160622 79530 782630 2.76 900529 160 882.01 4-37 117809 117637 41 20 1 60645 79512 782796 2.76 900433 9.900337 1.61 I 61 882363 4-36 40 21 i 60668 ; 79494 9-782961 2-76 9-882625 4-36 .0.117375 ~W 22 ! 6069 1 79477 783.27 2-76 900240 1-6. 882887 883148 4-36 I17113 116852 38 23 1 60714 79459 783292 2-75 900144 1.61 4-36 37 24 1 60738 79441 783458 2.75 1-61 8834 10 4-36 I16590 86 25 160761 79424 783623 2.75 ..6. 883672 4-36 116328 85 26 ! 60784 79406 783788 2-75 89985I 1.61 883934 4-36 116066 34 27 60807 79388 783953 2-75 899757 1-61 884196 4-36 ii58o4 83 28 6o83o 79371 784118 2-75 899660 1-61 884457 4-36 115543 32 29 6o853 79353 784282 2-74 899564 1-61 884719 4-36 115281 81 30 31 60876 1 79335 784447 2-74 899467 1-62 884980 4-36 ll5020 80 60899179318 9-7846.2 2-74 9-899370 1-62 9-885242 4-36 10-114758 29 3-2 160922 j 79300 784776 2-74 899273 1.62 8855o3 4-36 1 14497 28 83 60945 79282 784941 2-74 899176 1.62 885765 4-36 114235 27 34 ; 60968 79?'j4 785io5 2-74 ^t 1.62 886026 4-36 1 13974 26 3) ,60991 79''47 785269 2.73 1.62 886288 4-36 113712 25 36 !6ioi5 79229 785433 2-73 898884 1-62 886549 4-35 ii345i 24 37 |6io38 79211 785597 2-73 898787 1-62 886810 4-35 113.90 23 38 61061 79193 785761 2.73 898689 1-62 887072 4-35 112928 22 39 61084 79176 785925 2-73 898592 1-62 887333 4-35 112667 21 40 61107 1 79. 58J 786089 2-73 898494 ^63 887594 4-35 1 1 2406 20 41 6ii3o 79 '40 9-786252 2-72 9-898397 1.63 9.887855 4-35 10-U2.45 19 42 6ii53 79122 786416 2-72 898299 1.63 888116 4-35 1 1 1884 18 43 61 176 79105 786579 2-72 898202 1.63 888377 4-35 ...623 17 44 61 199 79087 786742 2-72 898104 1-63 888639 4-35 iii36i 16 45 61222 79069 786906 2-72 898006 1-63 888900 4-35 1 1 1 1 00 15 46 61245 7905 1 787069 2-72 897908 1-63 889.60 4-35 110840 14 47 61268 79Q33 787232 2-71 897810 1-63 889421 4-35 110579 iio3i8 13 48 61291 79016 787395 2.71 897712 1-63 889682 4-35 12 49 6i3i4 78998 787557 2.71 897614 1.63 889943 4-35 110057 11 50 51 61337 78980 787720 2-71 897516 1-63 ."-64 890204 4-34 109796 10 6i36o 78962 9-787883 2-71 9.897418 9-890465 4-34 10-109535 9 52 6i383 78944 788045 2-71 897320 1-64 890725 4-34 109275 8 53 61406 78926 788208 2.71 897222 1-64 890986 4-34 . 109014 108753 7 54 61429 X 788370 2-70 897123 I •641 891247 4-34 « 55 6i45i 788532 2-70 897025 1-641 891507 4-34 108493 6 56 61474 78873 788694 2-70 896926 ,.64| 891768 4-3', 108232 4 57 61497 78855 788856 2-70 896828 1-64! 892028 4-34 107972 3 58 6i52o 78837 7890.8 2-70 896729 1-64' 892289 4-34 1077 1 1 2 59 61543 788.9 789.80 2-70 89663i 1-64 892549 4-34 107451 1 60 6 1 566 78801 789342 2-69 896532 1-64 892810 4-34 107190 ~ N. COS. S^. sine. L. COS. D. 1" L. sine. 1 L. cot. D.l" L, tang. t 52° • 1 68 TRIGONOMETRICAL FUXCTIOXS. — 3§°. Nat. Functions. Logarithmic Functions + 10. ' N.sine. N. COS. L. sine. D. 1" L. COS. D.l" 1 L. tang. D.l" L.cot ! 61 566 78801 9.789342 2 69 9.896532 1-64 9.892810 4-34 10-107190 60 1 1 61589 78783 789504 2 .69 896433 1-65 1 893070 4-34 1 06930 69 2 61612 78765 789665 .69 896335 1-65 893331 4-34 106669 68 8 i 61635 78747 789827 69 896236 1-65 893591 4-34 106409 57 4 '6i658 78729 789988 69 896137 1.65 893851 4-34 106149 56 5 i6i68i 78711 790 '49 :^ 896038 1.65 8941 11 4-34 105889 55 6 161704 78694 7903 10 895939 895840 1.65 894371 4-34 105629 105368 54 7 j 61726 78676 790471 68 1.65 894632 4-33 53 8 61749 78658 790632 -68 895741 1-65 894892 4-33 io5io8 62 9 61772 78640 790793 -68 895641 1.65 895152 4-33 104848 51 10 61795 '61818 78622 78604 790954 •68 895542 1-65 1-66 895412 <-33 4-33' 104588 60 11 9-79II15 • 68 9-895443 9.895672 10-104328 12 61841 78586 791275 2 • 67 895343 1-66 895932 4-33 104068 4S |13 1 61864 78568 791436 2 -67 895244 1.66 896192 4-33 io38o8 47 U 1 61887 78550 791596 2 •67 895145 1.66 896452 4^33 103548 A<] 15 61909 78532 791757 2 •67 895045 1.66 896712 4-33 103288 45 -16 17 161932 78514 791917 2 67 894945 1-66 896971 4-33 io3o29 44 1 61955 78496 792077 2 67 894846 1-66 897231 4^33 102769 43 18 161978 78478 792237 66 894746 1-66 897491 4-33 102509 42 19 62001 78460 792397 66 894646 1-66 897751 4-33 102249 41 20 21 162024 ['62046 78442 78424 792557 66 894546 1-66 1.67 898010 4^33 101990 40 9-792716 66 9-894446 9-898270 4^33 io-ioi73o "3-7 22 1 62069 78405 7928-6 66 894346 1.67 89S530 4^33 101470 38 23 62092 78387 793o35 66 894246 1-67 898789 4-33 10121 1 37 24 62115 78369 793195 65 894146 1^67 899049 4-32 100951 36 25 1 62138 78351 793354 65 894046 1.67 899308 4^32 100692 35 26 62160 78333 793514 65 893946 1.67 899568 4^32 100432 34 27 ; 62183 7831 5 793673 65 893846 1^67 899827 4-32 100173 33 28 (62206 78297 793832 65 893745 1.67 900086 4-32 099914 32 29 62229 78279 79399. 794 1 DO 65 893645 1.67 900346 4-32 099634 31 30 31 6225l 78261 64 893544 1^67 900605 4^32 099395 30 62274 78243 9 -794308 2 64 9-893444 1.68 9.900864 4^32 io-099i36 098876 29 32 ! 62297 78225 794467 2 64 893343 1.68 901124 4-32 28 33 1 62320 78206 794626 2 64 893243 1.68 901383 4^32 098617 27 34 62342 78 1 88 794784 2 64 893142 1.68 901642 4-32 09S358 26 35 62365 78170 794942 2 64 893041 1.68 901901 4^32 098099 25 36 62388 78152 795101 2 64 892940 1.68 902160 4^32 097840 24 37 6241 1 78134 795259 2- 63 892839 1.68 902419 4-32 097581 23 38 62433 78116 795417 2 63 892739 1.68 902679 902938 4-32 097321 22 89 62456 7809S 795575 2 63 892638 1.68 4-32 097062 21 40 62479 78079 795733 1_ 63 892536 1.68 903197 4-3i 096803 20 41 62302 78061 9.795891 2 63 9.892435 .•69 9.903455 4-37" 10-096545 I'J 42 62524 78043 796049 2 63 892334 1.69 903714 4-3i 096286 18 43 62547 78025 796206 2 63 892233 1.69 903973 4-3i 096027 17 44 62570 78007 796364 2 62 892132 1.69 904232 4-3i 095768 16 45 62592 77988 796521 2 62 892030 1.69 904491 904750 4-3i 095509 15 46 62615 77970 796679 2 62 891929 1.69 4-3i 095250 14 47 62638 77952 796836 2 62 891827 1.69 9o5oo8 4^3i 094992 13 48 62660 77934 796993 2 62 891726 1.69 905267 4-3i 094733 12 49 62683 77916 77897 797 1 5o 2 61 891624 1.69 905526 4-3i 094474 11 50 62706 797307 2 61 891523 1^ 905784 4-3i 094216 10 51 62728" 77879 9-797464 2 61 9-891421 1-70 9.906043 43i 10-093957 9 52 62751 77861 797621 2 61 89.319 1.70 9o63o2 4-3. 093698 8 53 62774 77843 797777 2 61 891217 1.70 9o656o 4-3i 093440 7 54 62796 77824 797934 2 61 891115 1.70 906819 4-3i 093181 6 5» 62819 77806 798091 2 61 891013 1-70! 907077 4.31 092923 5 56 62842 77788 798247 2 61 890911 890809 1.70 907336 4-3i 092664 4 57 62864 77-769 798403 2 60 1-70 907594 4-31 092406 3 58 62887 77751 798560 2 60 890707 1.70 907852 4-3i 092148 2 59 62909 77733 798716 2 60 890605 1.70 908111 4-3o 091889 1 60 62932 77715 798872 2 60 89o5o3 1.70 908369 4-3o 091631 N. COS. N.alne. L. COS. 1d 1" L. sine. L.cot |D. 1" L. tang. ' 51° ' 1 TRIGONOMETRICAL FaKCTIOKS.— 3^^ 69 Nat. Functions. Logarithmic Functions + 10. 1 ' N.sine.N. COS. L. sine. D. ,1" L. COS. D 1" L. tang. Dl." L.cot 62932 77715 9.798S72 2 60 9 -890503 I 70 'ia 4-3o 10-091631 60 1 62955 77696 799028 2 60 890400 I • 7' 4-3o 091372 59 2 62977 77678 799 « 84 2 60 890298 1 7' 908886 4-3o 091 114 58 3 63ooo 77660 799339 2 59 890195 I 7' 909144 4-3o 090856 57 4 63022 77641 799495 2 59 890093 I 7' 909402 4 -30 090598 56 5 63o45 77623 799601 2 59 889900 I 889888 I 7> 909660 4-3o 090340 55 6 63o68 77605 799806 2 59 7' 909918 4-3o 090082 54 7 63090 77586 799962 2 59 889785 I 7' 910177 910435 4-3o 089823 53 8 63ii3 77568 Soon 7 2 59 889682 I 71 4-3o 089565 52 9 63i35 77550 800272 2 58 889579 1 7' 910693 910961 4-3o 089307 51 10 63 1 58 77531 800427 2 58 889477 I v_ 4-3o 089049 50 11 63 180 775i3 n.8oo582 2 58 9.889374 I 72 9-911209 4 -30 10-088791 ^y" 12 63203 77494 800737 2 58 889271 I 72 911467 4-3o 088533 43 13 63225 77476 800892 2 58 889168 I 72 911724 4-3o 088276 47 14 632^ 77458 801047 2 58 889064 I 72 91 1982 4-3o 088018 46 15 63271 77439 80I20I 2 58 888961 I 72 912240 4-3o 087760 45 16 63293 77421 8oi356 2 57 888858 I- 72 912498 4-3o 087502 44 17 633i6 77402 8oi5ii 2 57 888755 I- 72 912756 4-3o 087244 43 18 6333^ 77384 80 I 665 2 57 888651 I- 72 9i3oi4 4-29 086986 42 19 63361 77366 801819 2 57 888548 1- 72 913271 4-29 086729 41 20 63383 77347 801973 2 57 888414 I 73 913529 4-29 086471 40 3y "21 63406 77329 9.802128 2 57 9-888341 1- 73 9-913787 4-29 10-086213 22 63428 77310 802282 2 56 888237 I- 73 914044 4-29 085956 38 2:3 6345 1 77292 802436 2 56 888134 I- 73 914302 4-29 085698 37 24 63473 77273 802589 2 56 888o3o I 73 914560 4-29 085440 36 2.1 634q6 77255 802743 802897 2 56 887926 I- 73 914817 4-29 o85i83 35 26 635 1 8 77236 2 56 887822 I- 73 91 5075 4-29 084925 34 27 63540 77218 8o3o5o 2 56 887718 I- 73 915332 4-29 084068 33 28 63563 77 "99 803204 2 56 887614 I- 73 915590 4-29 084410 32 29 63585 77' -I 803357 2 55 887510 I- 73 915847 4-29 084153 31 80 636o8 77162 8o35ii 2 55 887406 I 74 916104 4-29 083896 30 29 31 63630 77^44 9-803664 2 55" 9-887302 I- 74 9-916362 4-29 10-083638 32 63653 77125 8o38i7 2 55 887198 I 74 916619 4-29 oS338i 28 33 63675 77107 803970 2 55 887093 I 74 916877 917134 4-29 o83i23 27 34 63698 77088 804123 2 55 886989 1 - 74 4-29 082866 26 35 63720 77070 804276 2 54 886885 I 74 917391 4-29 082609 25 36 63742 77o5i 804428 2 54 886780 1 74 917648 4-20 0S2352 24 37 63765 77033 8o458i 2 54 886676 I- 74 V,lTd 4-29 0S2095 23 38 63787 77014 804734 2 54 886571 I- 74 4-28 081837 22 39 63810 76996 804886 2 54 886466 1 74 918420 4.28 o8i58o 21 40 63832 76977 8o5o39 2 54 886362 1 75 918677 4-28 o8i323 10-081066 20 19 41 63854 76959 9-8o5i9i 2 54 9 -866257 1 75 9-918934 4-28 42 63877 76940 805343 2 53 8S6i52 I 75 9I9I9I 4-28 080809 18 43 63899 76921 805495 2 53 8S6047 I ii 919448 4-28 o8o552 17 44 63922 769.3 8o5647 2 53 880942 I "^i 919705 4-28 080295 16 45 63944 76884 805799 2 53 885837 I ^i 919962 4-28 o8oo38 15 46 63966 76866 805951 2 53 885732 I 7^ 920219 4-28 079781 14 47 I639B9 76847 76828 806 I o3 2 53 885627 I ^i 920476 4-28 079524 13 48 i64oii 806254 2 53 885522 I 75 920733 4-28 079267 12 49 1 64033 76810 806406 2 52 885416 I 7^ 920990 4-28 0790 1 11 50 i 64056 76791 806557 2 52 8853x1 I 76 921247 4.28 07B753 10 51 1 64078 76772 9 • 806709 T 52 9-885205 I "^5 9-92i5o3 4-28 10-07^497 9 52 ! 64100 76754 806860 2 52 885100 I '^i 921760 4-28 07S240 8 53 '64123 76735 80701 1 2 52 884994 I 884889 I 884783 I '^i 922017 4-28 o779-'3 7 54 64145 76717 807163 2 52 76 922274 4-28 077726 6 65 64167 76698 807314 2 52 "^5 922530 4-28 077470 5 56 64190 76679 807465 2 5i 884677 1 ^J 922787 4-28 077213 4 57 164213 76661 807615 2 •DI 884572 I 76 923044 4-28 076956 3 58 164234176642 807766 2 ■ 5i 884466 I 76 923300 4-28 076700 2 59 164256 76623 807917 2 .5i 884360 I 76 923557 4-27 076443 1 60 64279 76604 808067 2.5l 884254 I 77 9238i3 4-27 076187 N. cos.|N.8ine. 1 L. COS. D.l" L. sine. 1 L. cot. D.l" L. tang. ' 50° 1 70 IftlGONOMEtiliCAL FUNCTIONS. — 46°. Nat. Functions. LooABiTHMic Functions + 10. 1 In. sine 1 N. COS. L. sine. D. 1" L. COS. D.l" L. tang. D. 1" L. cot. 64279 76604 9-808067 2-5l 9-884254 "■v 9-9238i3 4 27 10.076187 GO 1 64301 76586 808218 2 ■5l 884148 •77 924070 4 27 073930 5'J 2 64323 76567 8o8368 2 •5l 884042 j •77 924327 4 27 075673 63 S 64346 76548 8o85i9 2 -5o 883936 •77 924583 4 27 075417 57 4 64368 76530 808669 2 -5o 883829 •77 924840 4 •27 075160 5() 64390 765ii 808819 2 -5o 883723 •77 925096 4 •27 074904 55 6 64412 76492 808969 2 -5o 883617 •77 925352 4 •27 074648 54 7 64435 76473 809 1 1 9 2 -5o 883510 •77 925609 4 •27 074391 074135 53 8 64457 76455 809269 2 -5o 883404 •77 925865 4 •27 62 9 64479 76436 809419 2 •49 883297 •78 926122 4 •27 073878 61 10 11 64501 76417 809569 2 2 -49 •49 883191 •78 926378 4 •27 073622 10-073366 50 "49" 64524 7^398 9-809718 9.883084 .78 9-926634 4 •27 12 64546 76380 809868 2 •49 882977 882871 -78 926890 4 •27 073110 48 13 64568 76361 810017 2 -49 .78 927147 4 27 072853 47 14 64590 76342 810167 2 •49 882764 .78 927403 4 •27 072597 46 15 64612 76323 8io3i6 2 48 882657 •78 927659 927915 4 27 072341 45 16 64635 76304 810465 2 48 882550 •78 4 27 072085 44 17 64657 76286 810614 2 48 882443 -78 928171 4 27 071829 071573 43 18 1 64679 76267 810763 2 48 8^2336 •79 928427 4 27 42 19 i ^'^■'oi 76248 810912 2 .48 882229 •79 928683 4 27 071317 41 20 64723 76229 811061 2 48 882121 _:79 928940 4 31. 07 1 060 40 21 1 64746 76210 9-811210 2 48 9-882014 1 •79 9.929196 929452 4 27 10-070804 89 22 1 64768 76192 8ii358 2 47 881907 1 •79 4 27 070548 38 23 64790 64812 76173 8ii5o7 2 47 881799 I •79 929708 4 27 070292 87 24 76154 8ii655 2 47 881692 I 881 584 I •79 929964 4 26 070036 86 25 64834 76135 81 1804 2 47 •79 930220 4 26 069780 35 26 64856 761 16 81 1952 2 47 881477 I •79 930475 4 26 069325 8t 27 64878 76097 812100 2 47 881369 • ■^ 930731 4 26 069269 83 28 64901 76078 812248 2 47 881261 I 930987 4 26 06001 3 82 29 64923 76059 812396 2 46 88ii53 !i -80! 931243 4 26 068757 81 80 64945 76041 812544 2 46 881046 I -80: 931499 4 26 o685oi 1 80 1 31 64967 76022 9-812692 2 46 9.880938 1 •80' 9-931755 4 26 10.068245 29 32 64989 76003 812840 2 46 88o83o I .80! 932010 4 26 067990 2S 33 65oii 75984 812988 2 46 880722 I -80 932266 4 26 067734 27 34 65o33 75965 8i3i35 2 46 880613 I -80 932522 4 26 067478 26 35 65o55 75946 8i3283 2 46 88o5o5 I -80 932778 4 26 067222 25 36 65o77 75927 8i343o 2 45 880397 ' .80 933o33 4 26 066967 24 87 65 1 00 75908 813578 2 45 880289 I -81 933289 4 26 066711 23 38 65122 75889 813725 2 45 880180 I -81 933543 4 26 066455 22 39 65i44 75870 813872 2 45 880072 I -81 933800 4 26 066200 21 40 65i66 7585 1 814019 2 45 879963 1 1 -81! 934056 4 26 065944 20 "liT 41 1 65i88 75832 9-814166 ^ 45 9-879855 ji -81 i 9-934311 4 26" 1 - 0656S9 065433 42 652IO 758i3 8i43i3 2 45 879746 I -81 934567 4 26 18 43 65232 75794 814460 2 44 879637 I -81 934823 4 26 065177 17 44 65254 75775 814607 2 44 879529 1 -8.1 935078 4 26 064922 16 45 65276 75756 814753 2 44 879420 I -8. 935333 4 26 064667 15 46 65298 7573s 814900 2 44 8793 1 1 I -81 9355S9 4 26 064411 14 47 65320 75719 816046 2 44 879202 I -82 935844 4 26 064 1 56 13 48 65342 75700 8i5i93 2 44 879093 I -82 936100 4 26 063900 1-2 49 65364 75680 815339 2 44 878084 I -82 936355 4 26 063645 11 50 51 65386 75661 8 1 5485 2 43 878S75 1 -82 936610 4 26 063390 10 654o8 75642 9-8i563i 2 43 9.878766 I .82 9-936866 4 25 io.o63i34 9 52 65430 75623 815778 2 43 878656 I -82 937121 4 25 062879 8 53 65452 75604 815924 2 43 878547 I -82 937376 4- 25 062624 7 54 65474 75585 816069 8i69i5 2 43 878438 I -82 937632 4- 25 062368 6 55 65496 75566 2- 43 878328 I -82 937887 4- 25 062113 5 56 655 18 75547 8i636i 2- 43 878219 'i -83 938142 4 25 o6i858 4 57 65540 75508 8i65o7 2- 42 878109 1 -83 938398 938653 4- 25 061602 3 58 65562 75509 8i6652 2- 42 877999 '• -83 4- 25 061347 2 59 65584 75490 816798 2- 42 877890 ,1 -83 938908 4- 25 061092 1 60 1 656o6 75471 816943 2-42 877780 :i -83 939163 4-25 060837 1 N.C08. N.sine. L. COS. D. 1" ! L.6ine. ! 1 L.cot. D.l" L.tang. 49° 1 n TRIGONOMETRICAL FUKCTIOKS. — 42' Nat. Functions. LOGAKITHMIC FUNCTIONS + 10. N. sine.j N. cos L. sine. D.l" L. COS. D.l" L. tang. D. 1" L. cot. 66qi3 1743. 4 9-825511 2-34 9-871073 1-90 1 9-954437 4-23 10.045563 60 1 66935 i 74295 825651 2-33 870960 1-90 j 954691 4-23 045309 o45o55 59 2 66906 74276 825791 2-33 870846 1-90 1 954945 4-23 58 3 66978 742 56 825931 2-33 870732 1-90 955200 4-23 044800 57 4 66999 74237 826071 2-33 870618 1-90 955454 4-23 044546 56 t 67021 74217 826211 2-33 870504 1-90 933707 4-23 044203 0440J9 55 6 67043 74198 826351 2-33 870390 1-90 955961 4-23 54 7 65064 74178 826491 2-33 870276 1-90 956215 1 4-23 043785 53 8 67086 74139 826631 2-33 870161 1-90 956469 4-23 043531 52 y 67107 74139 826770 2-32 870047 1-91 956723 4-23 043277 51 10 67129 74120 826910 2-32 869933 1^1 956977 9-957231 4-23 4-23 043023 50 11 67151 74100 9-827049 2-32 9-869818 1-91 10-042769 0425i5 4y 12 67172 74080 827189 827328 2-32 869704 1-91 957485 4-23 48 13 67194 74061 2-32 869589 1-91 957739 957993 4-23 042261 47 14 67215 74041 827467 2-32 869474 1-91 4-23 042007 46 15 67237 74022 827606 2-32 869360 1-91 958246 4-23 041754 o4i5oo 45 16 67253 74002 827745 827884 2-32 869245 1-91 958500 4-23 44 17 67280 73983 2-3l 869130 1-91 958754 4-23 041246 43 18 67301 73963 828023 2-3l 869015 1.92 959008 4-23 040992 42 19 67323 73944 828162 2-31 868900 1.92 959262 4-23 040738 41 20 21 6i344 i^n 8283oi 2-3l 868785 1-92 959516 4-23 040484 40 67356 73qo4 9-828439 8285i8 2-3l 9-868670 1.92 9-959769 960023 4-23 io-o4o23i ^9 oo 67387 738^5 2-3l 868555 1-92 4-23 039977 38 22 6:4C9 73865 828716 2-3l 868440 1-92 960277 4-23 039723 37 24 67430 73846 828855 2-3o 868324 1-92 960531 4-23 039469 039216 36 25 67452113826 828993 2-3o 868209 868093 1-92 960784 4-23 35 26 67473 i38o6 829131 2-3o 1-92 961038 4-23 038962 34 27 67495 73787 829269 2-3o 867078 867862 1-93 961 291 4-23 038709 038455 33 28 67516 73767 829407 2-3o 1-93 961545 4-23 32 29 67538 73747 829545 2-3o 867747 1.93 961799 962052 4-23 038201 81 30 31 67559 67580 73728 73708 829683 2 -30 867631 .-93 1-93 4-23 037948 30 9.829821 2-29 9-867515 9-962306 4-23 10-037694 29 32 67602 73688 829959 2-29 867390 867283 1.93 962560 4-23 037440 28 83 67t23 73669 830097 2-29 1-93 1-93 962813 4-23 037187 27 34 67645 73649 ( 830234 2-29 867167 963067 4-23 036933 26 85 67666 73629 83o372 2-29 867051 1-93 963320 4-23 o3668o 25 36 67688 73610 83o5o9 2-29 866935 1-94 963574 4-23 036426 24 37 67709 73390 83o646 2-29 866819 866703 1-94 963827 4-23 036173 23 33 67730 73570 830784 2-29 1-94 964081 4-23 i^ti 22 39 67752 73551 830921 2-28 866586 1-94 964335 4-23 21 40 67773 73531 83io58 2-28 866470 1-94 964588 4-22 035412 20 41 67795 ^5n 9-831195 2-28 9-866353 1-94 9-964842" 4-22 io-o35i58 19 42 67816 73491 83i332 2-28 866237 1-94 960095 j 4-22 o349o5 18 43 67831 73472 831469 2-28 866120 1-94 965349 4-22 03465 I 17 44 67859 73452 83 1606 2-28 866004 1-95 965602 4-22 034398 16 45 67880 73432 831742 2-28 865887 1-95 965855 4-22 o34i45 15 4H 67901 73413 831879 832015 2-28 865770 1.95 966109 4-22 033891 14 47 67923 733931 2-27 865653 1.95 966362 4-22 033638 13 48 67944 73373 832152 2-27 865536 1.95 966616 4-22 033384 12 49 67965 73353 832288 2-27 865419 1-95 966869 4-22 o33i3i 11 50 51 67987 73333 832425 2-27 865302 9 -865 1 85 1.95 1-95 _967.2^ 4-22 032877 _2o_ 68008 73314 9-832561 9-967376 4-22 10-032624 9 52 68029 73294 832697 2-27 86D068 1.95 967629 967883 4-22 032371 8 53 680 5 1 73274 832833 2-27 864950 1-95 4-22 o32ii7 1 7 54 68072 73254 832969 833 1 o5 2 26 864833 1-96 968136 4-22 o3i864 6 55 68093 13234 2 26 864716 1-96 968389 968643 4-22 o3i6!i 5 56 6811 5 73215 833241 2-26 864598 864481 1-96 4-22 o3i357 4 57 68i36 73195I 833377 2-26 1-96 968896 4-22 o3iio4 3 58 68157 73175 833512 2-26 864363 1-96 969149 969403 4-22 o3o85i 2 59 68179 73i55 833648 2-26 864245 1-96 4-22 o3o597 1 60 68200 73i35 833783 2-26 864127 ,-96 969656 4-22 o3o344 N. co8.;N.8ine. L. COS. D.l" L. sine. L. cot. D.l" L. tang. ~^ 47° 1 TBIGONOMETRICAL FUNCTION'S. — 43* 73 1 1 Nat. Functions. T,o<4AKiTHMic Functions + 10. 1 1 ' JN.wne. N.C08. L. sine. D.l" L.C08. I ).l" L. tang. D V L.cot. 68200 73i35 9-833783 2 26 9-864127 I • 96 9^969656 4 22 io^o3o344 60 1 .68221 73 1 16 8339,9 2 25 864010 I •96 969909 4 22 030091 029838 59 2 ; 68242 73096 834054 2 25 863892 1 •97 970162 4 22 58 3 1 68264 73076 834189 2 25 863774 I •97 970416 4 22 029584 57 4 6S285 73o56 834325 2 25 863656 1 •97 970669 4 22 029331 56- 5 ; 683o6 73o36 834460 2 25 863538 1 •97 970922 4 22 029078 55 6 168327 73016 834595 2 25 863410 1 •97 97, ,75 4 22 028825 54 7 1 68349 72996 834730 2 25 863301 , •97 97,429 4 22 028571 53 8 68370 72976 834865 2 25 863 1 83 •97 971682 4 22 0283 1 8 52 9 6839, 72957 834999 2 24 863o64 •97 971935 4 22 028065 51 10 11 168412 , 68434 72937 835,34 2 24 862946 •98 •98 972,88 4 22 027812 50 72917 9-835269 835403 T 24 9-862827 9-972441 4 22 ,0^027559 027306 49 12 68455 72897 2 •24 862709 862590 .98 972694 4 22 48 18 68476 72877 835538 2 24 •98 972948 4 22 027052 47 . 14 68497 72857 835672 -24 862471 .98 973201 4 22 026799 46 1 L 15 685 18 72837 835807 -24 862353 .98 973454 4 22 026546 45 I K 16 68539 728,7 835941 24 862234 • 98 973707 4 22 026293 44 I ■^ 17 68561 72797 836075 -23 862115 • 98 973960 4 •22 026040 43 I 18 68582 72777 836209 836343 •23 86,596 861877 86175s .98 9742,3 4 22 025787 42 ■ ^Hfid 686o3 72757 •23 .98 974466 4 •22 025534 41 ■ 20 21 68624 68645 72737 836477 -23 ^99 • 99 9747>9 9-974973 4 •22 02528, 40 72717 9-836611 "^ •23 9-86i638 4 •22 lO^O25027 3it '22 68666 72697 836745 •23 8615,9 •99 975226 4 •22 024774 88 23 68688 72677 836878 •23 861400 .99 975479 4 •22 024521 37 24 68709 72657 8370,2 -22 861280 •99 975732 4 22 024268 36 25 68730 72637 837,46 2 •22 861 161 • 99 975985 4 22 0240,5 35 26 68751 72617 837279 2 •22 861041 •99 976238 4 •22 023762 84 27 68772 72597 837412 2 •22 860922 860802 •99 976491 4 22 023509 83 28 68793 72577 837546 2 •22 •99 976744 4 22 023256 82 29 68814 72557 837679 •22 860682 ' {•00 976997 4 22 o23oo3 81 30 68835 72537 837812 •22 86o562 : J -00 9772D0 4 22 022750 80 31 68857 72517 9-837945 •22 9-860442 ' 2-00 9^9775o3 4 22 10-022497 29 32 68878 72497 838078 •21 86o322 ' 2-00 977756 4 22 022244 28 33 68899 72477 8382,1 •21 860202 : ^•00 978009 4 22 02,901 02,738 27 84 68920 72457 838344 •21 860082 : ^•00 978262 4 22 26 85 68941 72437 838477 ■21 859842 : ^•00 9785.5 4 22 02,485 25 86 68962 72417 838610 •21 J^OO 978768 4 22 021232 24 87 68983 72397 838742 •21 859721 : l-Ol 979021 4 22 020979 23 88 69004 72377 838875 •21 859601 : >oi 979274 4 22 020726 22 89 69025 72357 839007 •21 859480 : }0I 979527 4 22 020473 21 40 69046 72337 839,40 •20 859360 : >^0, 979780 4 22 020220 20 41 69067 72317 9-839272 20 9.859239 ; >-0, 9.980033 4 22 10-0,9967 19 42 69088 72297 839404 •20 850110 : 8588?7 i !0I 980286 4 22 oi97'4 18 43 69109 72277 839536 •20 •0, 980538 4 22 0,9462 17 44 69130 72257 839668 •20 >-0, 980791 4 21 0,9209 16 45 6915, 72236 839800 •20 858756 2 •02 981044 4 21 0,8956 15 46 69172 72216 839932 •20 858635 : •02 98,297 4 21 0,8703 14 47 69193 72196 840064 •19 8585,4 : •02 98,550 4 21 o,845o 13 48 69214 72176 840,96 .19 858393 2 •02 98,803 4 21 018,97 12 49 69235 72156 840328 •19 858272 5 858i5i : •02 982056 4 21 017944 11 50 69256 72136 840459 -19 ■02 982309 ^ 21 0,7691 10 9 51 69277 72116 9-840591 .19 9.858029 : •02 9^982562 4 21 ,o^o,7438 62 69298 72095 840722 .19 857908 c •02 982814 4 21 017186 8 58 69319 72075 840854 .19 857786 : >-02 983067 4 21 016933 7 54 69340 72055 840985 ■\l 857665 : •o3 983320 4 21 016680 6 55 69361 72035 841116 857543 '. • o3 983573 4 21 0,6427 5 56 69382 720,5 841247 •18 857422 : {•o3 983826 4 21 016,74 4 57 69403 7,995 841378 •18 857300 5 s.o3 984079 4 21 015921 8 58 69424 71974 841509 .18 857178 : ^o3 984331 4 21 015669 2 59 69445 71954 841640 •18 857056 : ^•o3 984584 4 21 oi54i6 1 60 69466 71934 84177' •18 856934 J ^o3 984837 J_ 31 oi5i63 N.cos. N.slne. L. COS. D .1" L. sine. L.cot D 1" L.tang: ' 46° 1 74 TRIGONOMETRICAL FUNCTIONS. — 44*. N>T. Functions. LOQAKITHMIC FUNCTIONS + 10. ' N.sine. N. COS. L. Sine. D 1" L. COS. D 1"! L. tang. Dl." L.cot 1 2 S 5 6 7 8 9 10 69466 69487 69:08 69529 69349 69570 69591 69612 69633 69654 69675 71934 71914 71894 7.873 71853 71833 71813 71792 71772 71752 71733 9-841771 841902 842033 842163 842294 842424 842555 842685 842815 842946 843076 2 2 2 2 2 2 2 2 2 2 2 .8 18 18 17 •7 •7 17 •17 17 •17 9.856934 8568.2 856690 856568 856446 856323 856201 856078 855956 855833 855711 2 2 2 2 2 I 2 2 2 2 o3 o3 04 04 04; 04' 04 04 04; 04 o5 9^984837 985090 985343 985596 985848 986101 986354 986607 986860 987112 987365 4 4 4 4 4 4 4 4 4 4 4 21 21 21 21 21 21 21 21 21 21 21 io.oi5i63 014910 014607 014404 oi4i52 0.3899 01 3646 0.3393 oj3i4o 012888 012635 60 5y 58 57 5t) 55 54 53 62 51 50 11 12 14 15 16 17 18 19 20 69696 69717 69737 69758 69779 69800 69821 69842 69862 69883 71711 7 1 691 71671 7i65o 7i63o 71610 71590 7.569 71549 7.529 9.843206 843336 843466 843595 843725 843855 843984 8441 14 844243 844372 2 2 2 2 2 2 2 2 2 2 .16 .16 •16 •16 16 .16 .16 •i5 • 15 .i5 9 •855588 855465 855342 855219 855096 854973 854850 854727 854603 804480 2 2 2 2 2 2 2 2 2 2 o5 o5 o5 o5 o5 06 06 06 9^987618 987871 988123 988376 988629 988882 989134 989387 989640 989893 4 4 4 4 4 4 4 4 4 4 21 21 21 21 21 21 21 21 21 21 10 012382 012129 01 1877 01 1624 011371 011118 010866 oio6i3 oio36o 010107 49 48 47 46 tl 43 42 41 40 21 22 23 24 25 26 27 28 29 30 69904 69925 69946 69966 69987 7000S 70029 70049 70070 70091 7.508 71488 71468 71447 71427 7.407 71386 71366 7.345 71325 9.844502 844631 844760 844880 845018 845147 845276 845405 845533 845662 2 2 2 2 2 2 2 2 2 2 • 15 •15 .i5 • i5 .15 .15 •14 •14 •14 •14 9^854356 854233 854109 853986 853862 853738 853614 853490 853366 853242 2 2 2 2 2 2 2 2 2 2 06 06 06 06 06 06 07 07 07 07 9.990145 990398 99o65i 990903 991 i56 991409 991662 99«9i4 992167 992420 4 4 4 4 4 4 4 4 4 4 21 21 21 21 21 21 21 21 21 21 10-009855 009602 009349 008844 oo85oi 008338 008086 007833 007580 39 38 37 36 85 34 38 82 31 30 29 28 27 26 25 24 23 22 21 20 31 82 33 34 85 36 37 38 89 40 70112 70132 70153 70174 70195 70215 70236 70207 70277 70298 7i3o3 71284 71264 71243 71223 71203 71182 71162 71141 71121 9-845790 845919 846047 846175 846304 846432 846560 846688 846816 846944 2 2 2 2 2 2 2 2 2 2 •14 •14 •14 •14 •14 .i3 .13 .i3 •13 .i3 9.853118 852994 852^69 852745 852620 852496 852371 852247 852122 851997 2 2 2 2 2 2 2 2 2 2 •07 07 07 07 08 08 08 08 9-992672 992925 993.78 993430 993683 993936 994189 994441 994694 994947 4 4 4 4 4 4 4 4 4 4 21 .21 .21 21 21 21 21 21 .21 21 10.007328 007075 006822 006570 oo63i7 006064 oo58ii 005559 oo53o6 oo5o53 41 42 43 44 45 46 47 48 49 50 70319 70339 7o36o 7o38i 70401 70422 70443 70463 70484 7o5o5 71100 71080 71059 71039 71019 70998 70978 70957 70937 70916 9.847071 847199 847327 847454 847582 847709 847836 847964 848091 848218 2 2 2 2 2 2 2 2 2 2 i3 • i3 • 13 12 12 12 12 • 12 .12 .12 9.851872 851747 85i622 85i497 85i372 851246 85II2I 850870 850745 2 2 2 2 2 2 2 2 2 2 08 08 08 09 09 09 09 09 09 09 9-995199 995452 995705 995957 996210 996463 996715 996968 997221 997473 4 4 4 4 4 4 4 4 4 4 21 21 21 21 21 21 21 21 21 21 10-004801 004548 004295 004043 003285 oo3o32 002779 002527 19 18 17 18 15 14 18 12 11 10 51 52' 53 54 55 56 57 58 59 60 70525 70546 70567 70587 70608 70628 70649 70670 70690 70711 70896 70875 70855 70834 70813 70793 70772 70752 70731 70711 9.848345 848472 848599 848726 848852 849232 849359 849485 2 2 2 2 2 2 2 2 2 2 .12 11 .11 • II • II • II 11 .11 .11 9.85o6i9 |2 850493 2 85o368 |2 85o242 2 85oii6 2 849990 2 849864 2 849738 2 849611 ,2 849485 2 09 10 10 10 10 10 10 10 10 ■0 9.997726 997979 998231 998484 998737 998989 999242 999495 999747 10 • 000000 4 4 4 4 4 4 4 4 4 4 21 21 21 21 21 21 21 21 •21 -21 10-002274 002021 001769 ooi5i6 00 1 263 OOIOII 000758 ooo5o5 000253 lO-OOOOOO 9 8 7 6 5 4 3 2 1 N. COS. N.sine. L. COS. D.l" L.slne. 1 II L.cot. D.l" L. tang. 1 45° 1 TABI.E III PKEOISE CALCULATION FUNCTIONS NEAR THEIR LIMITS. *6 SINES OF SMALL AKGLE^. log. sin. a;"= 4.685575 + log. x - diff. FOR THE SINES OF SMALL ANGLES. Angles. 1 5' 5o" 20' 20" 23' 5o" 29' 5o" 32' 3o" 35' 37' 2C" 89' 3o ' 4i' 3o' 43' 20" 45' 10" ^t .. 48' 40" 5o' 20" 52' 53' 3o' 55' 56' 3o ' 58' 59' 20" I*' 00' 40 " 2' 3' 20 " 4' 40" 5| 5o" 8' 10" 9 20" 10' 3o" 1 1 ' 40" 12' 5c" 14' 1 5' 16' 10 " 17' 10' 18' 10' 19' 20" 20' 2C' 21' 20" 22' iO" 23' 20" 24' 20" 25' 10" 26' 10' 27' 10" 28' 10" 29' 29' 5o" Seconds. Dia*. Angles. Seconds o 540 950 1220 i43o 1620 1790 1930 2100 2240 2370 2490 2600 2710 J820 2920 3020 3l20 32IO 33oo 3390 3480 356o 3640 3720 3800 388o 3930 4020 4090 4160 423o 43co 4370 4440 45oo 4570 463o 4690 4760 4820 4880 494c 5ooo 5o6o 5iio 5170 523o 5200 5340 5390 21 22 23 24 25 26 27 28 3o 3i 32 33 34 35 36 II 39 40 41 42 43 44 45 46 % 49 \° iq 5o" 3o' 5o" 3i'4o" 32' 3o" 33' 3o" 34' 20" 35' 10" 36' 36' 5o" 37' 40" 38' 3o" 39' 3o" 40' 20" 41' 10" 41' 5o" 42' 40 ' 43' 3o" 44' 10" 45' 45' 5o" 46' 3o" %'"" 48' 5o" 49' 3o" 5o' 20" 5r 5r5o" 52' 3o" 53' 10" 54' 54' 40" 55' 20" 56' 10" 56' 5o" 57' 3o" 58' 10" 58' 5o" 59' 3o' 20 00' 10' 5o' I ' 40" 2' 20' 3' 3' 35" 4' 10" 4' 5o' 5' 3o' 6' 10' 6' 5o' 7' 3o' 5390 545o 55oo 555o 56io 566o 5710 5760 58io 586o 5910 5970 6020 6070 6110 6160 6210 6260 63oo 635o 6890 6440 6480 653o 6570 6620 6660 6710 6750 6790 6840 6880 6920 6970 7010 7o5o 7000 7i3o 7170 7210 725o 73oo 7340 7380 741 5 745o 7490 753o 7570 7610 7650 Diff. 5o 5i 52 53 54 55 56 % 61 62 63 64 65 66 u 69 70 71 72 73 74 73 76 t 81 82 83 84 85 86 89 90 91 92 93 94 95 96 99 Angles. 7' 3o" 8' 10" 8'45" 9' 20' 10' 10' 40' n'lS" n' 5o" 12' 3o" i3' 5" i3'4o' 14' 20" 1 5' i5'35" 16' 10" 16' 45" 17' 20" 17' 55" 18' 3o" 19' 5" 19' 40" 20' i5 " 20' 5o" 21' j5" 22' 22' 35" 23' 10" 23' 45" 24' 20" 24' 55" 25' 3o" 26' 26' 35" 27' 5" 27' 40" 28' 10" 28' 45' 29' 1 5' 29' 5o" 3o' 20" 3o' 55 ' 3i' 25" 32' 32' 3y' 33' 5" 33' 35" 34' 5" 34' 40" 35' 10" 35' 40" 36' 1 5" I Seconds. I 765c I 7690 1725 7760 7800 ■'840 7875 7910 7950 7985 8020 8060 8100 81 35 8170 8203 8240 8275 83io 8345 838o 841 5 8450 8485 8520 8555 S590 8625 8660 8695 8730 8760 8795 8825 8H60 8890 8925 8955 8990 9020 9055 oo85 9120 91 5o 9185 9215 9245 9280 9810 9340 9375 Diff. TAN^GENTS AN^D COTANGENTS OF SMALL ANGLES. 77 I log tan. x"= 4.685575 + log. X + diff. log cot. x"= 15.314425 - -log. X - diff. FOR TANGENTS AND COTANGENTS OF SMALL ANGLES. Angles, o" Seconds. Diff. Angles. Seconds. Diff. Angles. Seconds. Diff. lo 3'3o" 38io 5o 5i 52 53 54 55 i«3o' 10" 5410 7' 'o" 43o 4' 10" 385o 3o' 3o" 543o 100 11' 10 " 670 8do ' 4' 5o" 3890 3r 5460 lOI 14' »o" 2 3 4 5 5' 3o" 3930 3i'3o" 5490 102 loS 104 io5 17' »9 1020 1440 6' 6' 40" 3960 4000 32' 32' 20" 5520 5540 21' 1260 6 7' 20" 4040 56 ll 32' 5o" 5570 106 23' i38o V5o" 8' 3o" 4070 33' lo- 5590 24' 5o" 1490 I 4110 ss' 40" 5620 \ll 26' 3o" 1590 9' 4140 34' 10" 565o 27' 5o" ,6?o 9 10 9' 40" 4180 34' 3o" 5670 109 110 29' 20" 1640 10' 20" 4220 61 62 63 64 65 35' 5700 III 3o' 40" ** 10' 5o" 425o 35' 20" 5720 32' 1920 12 i3 14 i5 11' 3o" 4290 35' 5o" 5750 1 12 iiS ii4 ii5 33' 10" 34' 20" 1990 2060 12' 12' 3o" 4320 435o 36' 10" 36' 40" 5770 58oo 35' 3o" 2i3o 16 i3' 10" 4390 66 u 69 70 37' 10" 58So 116 36 40" 2200 i3' 40" 4420 37' 3o" 585o 37' 5o" 38' 5o" 2270 \l 14' 10" 445o 38' 588o \\l 233o 14' 5o" 4490 38' 20" 5900 39' 5o" 2390 19 20 i5' 20" 4520 38' 5o" 5930 119 120 40' 5o" 245o 1 5' 5o" 455o So' 10" 5950 41' 5o" 25l0 21 16' 20" 4580 71 39' So" 5970 121 122 1 23 124 125 42' 5o" 2570 263o 22 23 n' 4620 ?3 40' 6000 43' 5o" \i'°" 465o 40' 20" 6020 44' 40" 2680 24 25 4680 40' 5o" 6o5o 45' 40" 2740 26 18' 3o" 4710 76 4i' 10" 6070 126 \ll 46' 3o" 2790 19' 4740 41' 40" 6100 47' 20" 48' 10' 2840 2890 11 19' 3o" 20' 4770 4800 42' 42' So" 6120 6i5o 49' 2940 11 20' 3o" 483o ll 42' 5o" 6170 \ll 49' 5o 2990 3 1 32 33 34 35 21' 4860 8i 82 83 84 85 43' 10" 6190 iSi l32 i33 1 34 .35 5o' 40 ■ 3040 21' 3o" 4890 43' 40" 6220 5i'3o" 3090 22' 4920 44' 6240 52' 20" 3i4o 22' 3o" 4950 44' So" 6270 53' 3i8o 23' 4980 44' 5o" 6290 53' 5o" 323o 36 ll 39 40 23' 3o" 5oio 86 ll 89 90 45' 20" 6320 i36 \ll .39 140 54' 40" 3280 24' 5o4o 45' 40" 6340 55' 20" 3320 24' 3o" 5070 46' 6360 56' 3360 25' 5ioo 46' 20" 638o 56' 5o" 3410 25' So" 5i3o 46' 40" 6400 57' 3o" 58' 10" 3430 26' 5i6o 47' 10" 6430 141 142 143 144 145 3490 353o 41 26' 3o" 5190 91 iP"" 5450 58' 5o" 42 43 44 45 26' 5o" 5210 92 93 95 6480 59 3o" 3570 27' 20" 5240 48' 20" 6500 1" 0'20" 3620 27' 5o" 5270 48' 40" 6520 1' 366o 46 8 28' 20" 53oo 96 49' 6540 146 r4a" 3700 28' 40" 5320 49' 20" 6560 2' 10" 2'5o" 3J3o 3770 29' 10" 29' 40" 535o 538o ll 49' 40" 5o' 10" 9580 6610 J' 3o" 38io 49 36' 10" 5410 99 5o' So" 663o 78 TAJS^GENTS AND COTAXGENTS OF SMALL ANGLES. log. tan. A"= 4.685575 + log. A+ diff. log. cot. A"= 15.314425 - log. A- diff. FOR TANGENTS AND COTANGENTS OF SMALL ANGLES. Angles. 4'4o' St-conde. 663o 665o 6670 669c 672c 674c 6760 5780 6800 683o 685o 6870 6890 6glO 6930 6960 6970 6990 7010 7o3o 7060 7080 7100 7120 7140 7160 7180 7200 7220 7240 7260 7i8o 7300 7320 7340 7360 7380 7400 7420 7440 7<6o 7480 7600 7520 7540 7560 7580 7600 7620 7640 7660 Diff. i5o i5i l52 i53 1 54 1 55 1 56 1 58 159 160 161 162 i63 164 i65 166 167 168 169 170 171 172 173 174 175 176 \n 181 182 1 83 184 i85 186 188 189 190 191 192 193 194 195 196 198 199 Angles. 8' 1 5" 8' 3o" 8 5o" 9' 10" 9' 3o" 9 5o" 10' 10" 10' 20" 10' 40" 10' 55" 11' i5" ir35" ir55" 12' i5" 12' 35" 12' 55" i3' i5" i3'35" i3' 5o" 14' 10" 14' 3o" 14' 45' i5' 5' i5' 20' 1 5' 40' i5'55' i6' i5' 16' 3o' 16' 5o' '''' % 17 25' 18' i5' 18' 35' 18' 55' 19' i5' 19' 3o' 19' 45' 20' 5' 20 20' 20' 40' 20' 55' 21' i5' 21' 3o' 21' 45' 22' 5' 22' 20' 22' 35' Seconds. 7660 7680 7695 7710 7730 7750 7770 7790 7810 7820 7840 7855 7875 7895 7935 7955 7975 801 5 8o3o 8o5o 8070 8o85 8io5 8120 8140 8i55 8175 8190 8210 8225 8245 8260 8280 8295 83iD 8335 8355 8370 8385 8405 8420 8440 8455 8475 8490 85o5 8525 8540 8555 Diff. 200 201 202 2o3 204 205 206 207 208 209 210 211 212 2l3 214 2l5 216 219 220 221 222 223 224 225 226 227 228 229 23o 23 1 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 248 249 Angles. 22' 35" 22 55" 23' 10" 23' 3l" 23' 45 24' 24' 20" 24' 35" 24' 55" 25' 10" 25' 25" 25' 45" 26' 26' 20" 26' 35" 26' 5o" 27' 10" 27' 25" 28' i5" 28' 35" 28' 5o" 29' 10" 29' 25" 29' 40" 3o' 3o' 1 5' 3o' 3o' 3o' 5o' 3r 5' 3r2o' 3r3i)' 3r55' 32' 10' 32' 25' 32* 40' 32' 55' 33' i5' 33' 3o' 33' 45' 34' 34" i5' 34' 3o' 34' 45' 35' 35' 20' 35' 35' 35' 5o' 36' 5' 36' 20' Seconds. 8555 8575 8590 8610 8625 8640 8660 8675 8b9> 8710 8725 8745 8760 8780 8795 883o 8845 8865 8880 8895 8915 8930 8950 8965 898c 9000 901 5 9o3o 9o5o 9065 9080 9095 9ii5 9i3o 9145 9160 9175 9195 9210 9225 924c 9255 9270 9285 9300 9330 9335 9350 9365 9380 y^^^k^k^^^^^^^t ^ ^^* m ^wtfi *d I TABLE IV OOKTAININe THE NATURAL TANGENTS AND COTANGENTS EVERY DEGREE AND MINUTE OF THE QUADRANT. 80 NATURAL TANGENTS. s 0° V 2° 3° 4° 6° 6° 7° 8° 9° 000000 017455! 03492 1 o524o8 069927 087489 io5io4 122785 i4o54i 158334 60 I Z o'^i^ 5212 2699 070219 iir^ 5398 3q8o 0837 8683; 59 2 018037! 328; 55o3 3^5 o5ii 5692 5987 3375 ii34 8981! 58 3 873 5795 0804 8368 3670 I43i 9279 ^7 4 001 164 619' 6086 3575 \t 8661 6281 3966 1728 95771 56 5 454 910 6377 3866 8954 6575 4261 2024 9876 55 6 745 019201 6668 4i58 1681 9248 6869 7i63 4557 2321 i6oi74i 54 I oo2o36 % 6960 445o 1973 9541 4852 2618 047 2 1 53 327 618 725i 4742 2266 9832 7458 5i47 5443 2915 0771 52 9 020074 7542 5o33 2558 090127 1752 8046 3212 1069 5i 10 909 365 7834 5325 285i 0421 5738 35o8 13681 5o II oo32oo 656 8125 56i7 3143 0714 8340 7o34 38o5 1666' 49 1965' 48 12 % 947 8416 .5909 3435 1007 8635 6329 6625 4102 i3 021238 8707 6200 3728 i3oo 8920 9223 4399 2263 47 U 004072 529 8999 6492 . 6784 4020 1 887 6920 4696 2562: 46 i5 363 820 TX 43 13 9518 7216 4993 2860! 45 i6 465 022111 7076 46o5 2180 9812 75i2 5290 5587 3159144 3458 43 n 945 402 9873 7368 4898 2474 110107 Cl ih 005236 984 04.0164 7660 54?3 2767 0401 5884 3756; 42 '9 527 0456 7951 3o6i 06,5 8399 6181 4055; 41 20 818 023275 io38 8243 5775 3354 86^4 6478 4354 40 21 006109 566 8535 6068 3647 8900 9286 6776 4652 39 495 1 138 22 400 857 i33o 8827 636 1 394. 2168 7073 23 690 981 024148 1621 9119 6653 4234 9582 7370 525o' 37 24 439 1912 941 1 6946 4528 9877 7667 5549' 36 4848 35 25 007272 730 2204 9703 7238 4821 2463 130173 0469 0765 7964 26 563 025022 2495 2787 060287 7531 5ii5 2757 8262 6147 34 27 854 3i3 7824 5408 3o52 8559 6446 33 28 008145 604 3078 0579 8116 5702 3346 1061 8856 67451 32 29 436 026186 lii: 087. 8409 6% 6583 3641 1357 9154 7044 3 I 3o 727 ii63 8702 3936 i652 945 1 7343; 3o 3i 009018 477 3952 1455 8904 9287 423o 1948 9748 7622 29 32 309 768 4244 1747 6876 4525 2244 1 50046 7941 28 8240 27 33 600 027059 4535 2039 9580 7170 7464 4820 2540 o343 34 010181 35o 4827 5ii8 233i 9873 5ii4 2836 0641 8539 26 35 641 2623 080 I 65 7757 5409 3i32 0938 8838 25 36 472 933 5410 2915 0458 8o5i 5704 3428 1236 9137 24 11 763 028224 5701 3207 0751 8345 5999 3725 i533 9437 23 oiio54 5i5 6284 3499 1044 8638 6294 6588 4021 i83i 9736 22 39 345 806 ?l i336 8932 4317 2426 170035 21 40 636 °''% 6576 1629 9226 6883 46i3 o334 20 41 927 6867 4667 1922 9519 7178 iu 2724 o634 ig 0933 18 42 012218 679 7159 22l5 98.3 "7^1^ 3022 43 509 970 745o 4959 25o8 1 001 07 11^ 55o2 3319 i233ii7 44 800 030262 8o33 525i 2801 0401 5798 3617 i532; 16 45 013091 553 382I 844 5543 33?6 0693 0989 8358 6094 3915 i83iii5 46 8325 5836 8653 6390 6687 42i3 2i3i i4 % 6731 o3ii35 964I 426 8617 6128 3679 1282 8948 45io 243o' ,3 8908 6420 3972 1576 92£ 6983 4808 2730 1 2 49 (JI4254 717 9200 6712 4265 1870 9538 ]l]l 5io6 3o3o 1 1 5o 545 o32009 'S 7004 4558 2164 9833 5404 3329 10 5i 836 300 7296 485 1 2458 120128 7872 5702 3629 9 52 •i5i27 418 882 000075 7589 5i44 2752 0423 l',^ 6000 53 0366 7881 8173 5437 3o46 0718 6298 54 709 033173 0658 5730 3340 ioi3 8761 6596 4528; 6 55 016000 465 0949 8465 6023 3634 i3o8 9058 6894 4828, 5 56 291 756 1241 8738 63i6 3928 i6o4 9354 7192 5l27 4 57 582 034047 1 533 9o5o 6609 4222 1899 965 1 ]% 5427 3 58 873 338 1824 9342 6902 45i6 2194 2489 9948 5427 2 ii 017164 63o 2Il6 9635 7196 4810 140244 8086 6027 1 89' 88° 87° 86° 85° 84° 83° 82° 81° 80^ Na tural C 0-tange nts. £?>>-83 4-85 4.86 4.87 4-88 4.89 4-91 4.93 4-96 4-98 NATURAL TANGENTS. 81 J.- 10° 176327 11° i 12° 1 13° 14° 15°" 10° 17° 18° 10° "1 ig438o', 212557 23o868' 249328 ^'^6? 286745 3o573i 324920 344328 60 I 6657; 4682! 2861 1175I 9637 7060 6049I 5241 4653J 5o 4978 58 a 6927 4984 3i65 14^1 9946 8573 8885 7375 6367 5563 3 7227 5286 l^?1 1788 250255 ^, 6685 5885 53o4 57 4 7527 5588 2094 o564 9197 7003 6207 6528 5630 56 5 7827 &127 5890 $1] 2401 ^%l 9509 8320 7322 5955 55 6 6192 2707 9821 8635 7640 6850 6281 54 I 8427 6494 4686 3oi4 1492 270133 8950 7959 7172 6607 53 8727 9028 6796 4990 3321 1801 0445 9266 7494 69331 52 1 9 7099 5294 3627 2111 0757 9581 7817 328139 7259 347583 5i lO 179328 197401 '"^ 233934 252420 271069 289896 308914 5o 11 9628 E5 4241 2729 i382 290211 9233 8461 79" 8237 8563 '^ 12 9928 6208 4548 3039 3348 1694 0527 9552 8783 i3 180229 83o8 65ia 4855 2006 0842 9871 310189 o5o8 9106 47 14 o529 8610 6817 4162 3658 23i9 1158 9428 8889! 46 9216 45 i5 0829 8912 7121 5469 3968 263 1 1473 n8^ 9731 i6 n3o 92i5 7426 5776 4277 2944 0827 330073 9542.1 44 \l- i43o 9517 nil 4587 3256 1146 0396 9868! 43 1731 9820 6390 4897 3569 2420 1465 0718 350195 42 •9 203l 200122 8340 6697 5207 3882 2736 1784 1041 0522! 41 20 182332 200425 218643 237004 2555i6 274194 293o52 3i2io4 331364 35o848| 40 21 2632 0727 895c 7312 5826 4507 3368 2423 1687 1175 39 22 2933 io3o 9254 til 6i36 4820 3684 2742 2010 i5o2 38 23 3234 1 333 9559 6446 5i33 4000 3062 2333 1829 37 24 3534 1635 9864 8234 6756 5446 43 16 338i 2656 2156 36 25 3835 1938 220169 8541 7066 5759 4632 3700 2679 2483 35 26 4i36 2241 0474 8S48 'lU] 6072 4948 4020 33oa 2810 34 27 4437 2544 0779 9156 6385 5265 4340 3620 3i37 33 28 4737 2847 1084 9464 ???? 6698 558i 4659 3949 3464 32 29 5o38 3149 1380 221695 9771 7011 296213 4979 4272 3791 3i 3o 185339 2o3452 240079 2586 18 277325 3 15299 33459a 354119 3o 3i 5640 3755 2000 o386 8928 7638 653o 5619 4919 4446 ll 32 5941 4o58 23o5 0694 9238 7951 6846 5939 5242 4773 33 6242 436 1 2610 1002 9549 8265 7163 6258 5566 5ioi 27 34 6543 4664 2916 i3io 9859 8578 7480 6578 589c 5429 26 35 6844 4967 3221 1618 260170 8891 7796 8ii3 6899 6213 57361 23 36 7145 5271 3526 1925 0480 9205 7219 6537 608^' 24 39 7446 5574 3832 2233 0791 9519 8430 7539 6861 6412, 23 8048 6180 4i37 4443 2341 2849 1102 i4i3 9832 280146 8747 9063 7859 8179 718a 33783' 6740 22 7068 21 40 188349 206483 224748 i43iD7 261723 280460 299380 3i85oo 357396 20 41 8651 6787 5o54 3466 2o34 0773 1087 9697 8820 8157 42 8952 7090 5360 3774 2345 300014 9141 8481 43 9253 7393 5665 4082 2656 1401 o33i 9461 8S06 838o! 17 44 9555 7697 5971 4390 Itl 3589 1715 0640 0966 9782 9i3o 8708: .6 45 9856 8000 6277 4698 2029 2343 320103 9454 9037 1 1 5 46 190157 83o4 6583 5007 1283 0423 340 1 o>; 9365) M ^2 0459 8607 6889 53 1 5 3900 2657 1600 0744 9694' i3 48 0760 8911 7194 5624 4211 2971 1918 1065 0428 360022! 12 49 1062 9214 7500 5932 4523 3286 2235 i386 0752 o35i! 11 5o 191363 209518 227806 246241 264834 283600 302553 321707 2028 341077 360679 10 5i 1665 9822 8112 6549 5i45 3914 2870 1402 1008 9 52 1966 210126 8418 6858 5457 5768 4229 4543 3i88 2349 1727 1337: 8 53 2268 0429 0733 8724 7166 35o6 2670 2032 1666 7 19951 6 2324! 5 54 2570 9o3i 7475 7784 8092 6079 4857 3823 IZ 2377 55 2871 1037 9337 9643 6391 5172 5487 4141 2702 56 3173 i34i 6702 4459 3634 3027 2653 j 4 ll 3475 1645 9949 8401 7014 58oi 4777 3955 3332 2982' 3 33i2| 2 tl^l iin 23025D 8710 7326 6116 509D S3? 3677 5, 0562 9019 7637 643 1 541 3 4002 3641 1 7r 78° 77° 76° 75° 74° 73° 72° 71° 70° .S Natural Co-tangents. P.] tol i.5.o.| 5.o5 5.09 5.i3 1 5.17 1 5-22 i 5.27 5-33 1 5-39 5.46 82 NATURAL TANGEJ^TS. 20° 21° 22° 23° 24° 25° 26° 27° 28° 29° I 2 363070 43oo 4629 383864 404026 4365 4703 424475 4818 5i62 445229 4663o8 6662 7016 8453 509525 510258 2456 554309 4689 5070 5450 60 3 iiU 4866 5042 55o5 6275 7371 88i3 0625 2829 32oJ u 4 5200 5380 5849 6624 7725 8080 9534 5831 5 56i8 5534 5719 6o58 6192 6536 X 1359 1726 m 6212 55 6 5948 5368 8434 4902?6 6593 54 I 6278 6202 63o7 6736 6880 8020 8789 2093 4324 ^1 53 6608 6536 7224 9144 0617 491339 2460 4698 52 9 6938 6871 7075 7568 8369 ,6lZ 2828 5072 558ii8 5i 10 367268 387205 407414 427912 8256 i^? "I'il 535446 5o II 7598 7540 7753 470209 1700 5821 l^, ^ 12 7928 7874 8209 8601 9418 o564 2061 3930 6195 i3 0259 8945 9768 0920 2422 4298 4666 6570 9263 8 14 8589 8544 877^ 9289 450117 1275 i63i 2784 6045 9645 i5 8919 8879 9111 9634 0467 3i45 5o34 7319 560027 45 i6 9200 9214 9450 997? 0817 1986 3507 5402 1694 8o6g 8445 0409 44 \l 9581 9549 9790 4ioi3o 43o323 1167 2342 3869 ^'^2?, 0791 43 991 1 9884 0668 i5i7 1868 IX 423 1 6i38 1174 42 '9 370242 390219 0470 ioi3 4593 65o7 516875 8820 1 556 41 20 370573 390554 410810 43 1 358 452218 473410 539195 561939 2322 40 21 0804 1^ ii5o 1703 2568 3766 7244 9571 39 22 1235 1490 i83o 2048 2919 4122 5679 7613 540J22 2705 38 23 1 566 i56o ^i 3269 % 6042 It, 3o88 37 24 l^l 223l 2170 3620 6404 0698 3471 36 25 25ll 3o84 3971 4322 5l;l 6767 8720 I 0741 3854 35 26 2559 2567 2903 285 1 3430 5548 7i3o TM i45o 4238 34 27 2890 3192 3532 3775 4673 5905 Ull 8218 1826 4621 33 28 3222 3239 4121 5o24 6262 9828 2203 5oo5 32 20 3553 3574 3873 4467 5375 6619 520197 520567 542QD6 5389 565773 6157 3i 3o 373885 393910 414214 434812 455726 ''% 498582 3o 3i 4216 iiu 4554 5i58 6078 8945 93o8 r^i] It 32 4548 4895 5236 55o4 6429 7690 \i 6541 33 4880 nil 5850 6781 8047 8403 9672 1677 6025 73io U 34 521 1 5577 tl'B 7i32 5ooo35 2047 4463 35 5543 5592 5919 7484 8762 l^n 2417 4840 7694 6079 25 36 5875 5928 6260 ^^ 7836 8188 9120 i]ii 5218 24 U 6207 6265 6601 m 1127 5595 8464 23 6539 6601 6943 7582 8540 l8?5 3528 Ito 8849 22 39 6872 6938 7284 7929 8892 480193 48o55i j«99 9234 21 40 377204 397273 417626 438276 459244 502219 2583 524270 546728 569619 20 41 7536 7611 K 8622 9596 0909 4641 7106 570004 \t 42 1869 8201 7948 6285 ^ 46??^? 1267 r. 50I2 7484 0390 43 865 1 1626 5383 7862 8240 0776 1161 \l 44 8534 8622 X 9663 o654 ti 3677 5754 45 8866 8960 4400 II 1006 4041 6125 8619 l93' 23lo 2705 i5 46 llf. ^?I 9677 420019 o358 1359 1712 vz 4406 t^ '1 9755 14 i3 9864 9971 o36i 2o65 3419 3778 484137 7240 12 49 380197 38o53o 4oo3o9 400646 0704 1 400 2418 55o2 7612 55oi34 3092 II 5o 421046 441748 462771 5o5867 ''ltd 55o5i3 'nUl 10 5i o863 0984 l322 1389 2095 3i24 4496 4855 6232 0892 t 52 \lt 173T 2443 3478 383 1 6598 8728 i65o 4252 53 1660 2074 2791 3i39 5214 k 9100 4638 I 54 1 863 'M 2417 4i85 5514 5933 9473 2o3o 5o26 55 2196 253o 2759 3487 3835 4538 9845 2409 5413 5 56 2673 3l02 4892 6293 6653 530218 2789 5800 4 U 2863 3oii 3445 4i83 5246 8427 87oi 9169 0591 0963 i336 3169 6187 6575 6962 3 lltl 3350 3788 4532 5600 7013 3549 2 59 3688 4i32 4880 .5954 7373 3929 » m^" 68° G7° 6G° 05° 64° 63° 62° 61° 60° Natural Co-tangents. P. P. to 1". 3-53 560 568 5.76 585 5-95 6 05 6 16 6.28 640 1 NATURAL TANGENTS. 83 a i 30° 31° 32° 33° 34° 35° 36° 37° 38° arl 577350 600861 624869 649408 674509 700208 726543 753554 781286 809784 60 I 7738 i65- 5274 9821 4o32 0641 6987 4010 1754 810266 59 2 8126 6488 650235 5355 1075 7432 4467 4023 538o 2223 0748 58 3 85i4 4 8qo3 2049 2445 foS ^IS .509 ^11 2692 3i6i 1230 1712 57 56 5 V^Vo 2842 6894' 1477 6627 8767 92i3 5837 363 1 2195 55 6 323q 7299 1892 7o5i 2812 6204! 4100 2678 54- 7 580068} 3635 7704! 23o6 8110 2721 7475 3246 9658 6751I 4570' 3i6i 53 8 04571 4o32 ?n 368 1 730104 72091 5o4o 3644; 52 9 08461 4429 85i6; 3i36 4116 o55o 7667 55ioi 4128151 10 581235; 604827 62802 1 j 653551 678749 704551 730996 758125 785981 8 4612 5o II 1625 5224 9327 3966 4382 9«74 4987 1443 8583 645 1 55L tl la 2014 5622 9734 680025 5422 •i^ 9041 6022 9500 7394 i3 24o3 6019 630140 52i3 5858 6o65 ii 14 ml 6417 681 5 0546 04 5o 6204 6730 2783 9950 7865 6549 i5 ^^^j ^11 0876 323o 760418 8336 7034 45 16 3573 72K l302 7166 3678 ?!S 8808 tt 44 \l 30^3 8010 1767! 6461 1728 7603 8o39 4125 9280 43 4353 21741 6877 258 1 1 7294 2i54 4573 1706 2256 9752 8491 42 •9 4743 8408 258o| 8476 5o2i 790225 8976 41 20 21 585i34 5524 608807 9205 6329881 657710 33961 8127 ^1S ''^ll 735469 5017 762716 3176 3636 790697 1170 819463 40 22 5915 63o6 9604 38o4 8544 3860 9788 6366 1643 23 6iooo3 421 1 8961 9379 9796 4287 710225 68i5 a 2117 0922 37 24 ^l o4o3 4619 4714 0663 7264 2590 1409 36 23 0802 5o27 5142 IIOI 7713 8162 5oio 3o64 'X 35 20 7479 1 201 5436 660214 5569 1 539 5486 3538 34 27 7870 8262 1601 5844 o63i t^,ti ^ 8611 5941 4012 2872 33 s8 2001 6253 1049 9061 6403 4486 3360 32 29 8653 2401 6661 66l886 6853 2854 951 1 6865 4961 3848 3i 3o 589045 612801 637070 687281 713293 739961 767327 795436 824336 3o 3i 9437 3201 8298 23o4 m '^'.i 740411 7789 bVsl 4825 li 32 9829 36oi 2723 0862 8252 53i4 33 590221 4002 3i4i 8567 8995 461 1 l3l2 8714 6862 58o3 ll 34 06 1 3 4402 8707 3560 5o5o 1763 9177 7337 6292 6782 35 1006 48o3 9117 3970 9425 5930 6370 2214' 9640 7813 8290 8766 25 36 i3q8 5204 9527 4398 9854 2666 770104 7272 7762 8252 24 37 1791 56o5 9937 640347 4818 690283 3U7 o567 23 38 2184 6006 5237 0713 6810 3569 io3i 9242 22 39 2577 64o8 0757! 5657 1143 725o 4020 1495 9719 8743 21 40 592970 3363 616809 641 167 1578 666077 691572 "Itll 744472 771959 2423 800196 829234 20 41 7211 6497 2003 4925 0674 ii5i 9725 \t 42 3757 7613 8oi5 i 2433 8573 %ll 2888 83021 6 43 4i5o 2863 9014 3353 1629 0707 n 44 4544 8417 2810 3294 9455 6282 38i8 2107 2585 1199 16 45 if^] 8819' 3222 8179' 3725 720339 6735 4283 1691 2183 i5 46 922 ij 3633 8599 41 56 7189 4748 3o63 14 % 5725 9624' 4044 9020 4587 5oi8 0781 7642 6096 5214 3542 2676 3169 i3 6120 620026 4456 9442 1223 568o 4021 12 49 65i4 0429 4868 9863 5450 1 665 8540 749003 6146 45oo 3662 II 5o V^ 620832 645280 670284 695881 722108 776612 'nill 834x55 10 5i 1235 5692 0706 63i3 255o 9458 7078 4648 I 52 7698 i638 6104 1128 6745 1^6 75?l66 -545 8012 5938 5i42 53 8093 8488 2042! 65i6 i55o 7177 6418 5636 I 54 2445 6929 2849: 7342 2394 7610 8042 'il'i 0821 8479 6898 6i3o 55 8883 1276 1731 8946 S 6624 5 56 9278 3253 27D5 8168 2817 8475 4766 9414 7119 4 u 9674 3657 3240 X 5210 2187 9881 7614 8109 3 600069 0465 4061 858 1 3662 5654 2642; 780349 8821 2 59 4465 8994 4085 9774 6098 3o98| 0817 93o3 8604 I 59° 58° 57° 66° 55° 54° 53° 52° 51° 50° ^ Natural Co-tangeuts. 1 fok^'' 6.67 6.83 6.97 7-U 7.31 7.50 7.70 7.9a 8.24 1 84 NATURAL TANGENTS. i 40° 41- 42° 48° 44° 45° 46° 47° 48' 49° 839100 86928J 870309 900404 9325 1 5 965689 I-OOOOO I.03553 1-07237 1-11061 i.i554i 29 33 5087 5763 7402 950071 3842 .879 ao57 55oi 3.63 72231 28 33 559, 6383 7938 0624 44i4 5563 9322 3228 7292! 27 34 6095 680a 8474 ..78 4987 5d6o 5624 9386 3295 736. 1 26 35 6599 733, 9010 .733 5685 945o 336. 7430' 25 36 7104 7843 8363 9547 2287 6i34 3ai6 5747 5809 95.4 3428 75oo 24 11 7608 8m3 930084 2842 6708 9578 3494 7569 2J 8883 06a I lit] 954508 7282 5870 9642 356. 763H 22 39 8619 9403 1.59 7857 988432 3295 5932 9706 3627 7708 2, 40 859124' 889924 93.697 3335 i-023d5 .•05994 '■•^m 1-13694 376. 3828 1-17777 20 7846, .0 79«6 18 41 963oi 890446 5o64 9007 34.4 6od6 43 860 1 36 0967 2773 5621 9582 3474 3533 6. .7 & 43 0643 1489 33i2 61-77 6734 990158 6.79 3894 7986 .7 8o55 .6 44 1148 aoia 385i 0735 3593 a653 624. 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Q 52 2003 9718 '■|H nil 845o 50.5485 429-7181 8 53 3i6o 7-01174 2637 4io5 8.00948 2848 1681 8040 9007 27-0566 5i-3o32 491-106, 7I 54 4321 8307 2048 4645 2715 4899 52-0807 572-957 6 55 5486 4756 2417 14-6079 5645 8821 687.349 5 56 6655 ^o5^ 8546 6674 '■'T,i 2789 o655 3i63 1235 6656 7117 53.7086 54-56i3 859.436 4 ll 7839 8600 7678 8711 9372 1I45-C2 1718-87 3437-75 3 9007 8.10536 6141 3540 1821 28.1664 55.4415 2 59 6.30189 7.10038 2481 8781 3919 2411 9755 3994 56.35o6 • 9- 8° 7° 6° 5° 4° 8° 2° 1° 0° 1 Natural Co-tangents. ^•f>'7-8o 22*19 28.46 37-83 5-28 7-88 iS.oi .... .... .... THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS ^h'^J-^^o ^^^^^^^° ^°« FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOUR^ oCe^RD^^^. ^° ^^-^^ - -- SEVENT^H^^D^I!; C7 n adaj. I. tb II. ai E T rap" and rap Scl SER 17 1933 SEP 18 1933 OCT 23 1933 MAR 18 1935 MAR 27 Wm SEP 16 1936 NOV o 1S3« K (•< NOV 19 1936 DEC 22 1936 NOV 24 1938 MAR 16 1941 H SfP 14 ,944 ^^ ]6Ju!'57GB REC'D LD JUL 2 1957 and have procured their introduction inio iuum u LD 21-50m-l,' are »re- ical the I its eos:- this hem. ntv. Jaxbs W. Thompson, A.M., Prin. of CentreviUe Academy, Maryland. Any of the above sent by mail, post-paid, on receipt of price. F^b YC 22302 Sheldon d: Compan/y's TextSooks, HlWa Elements of Rhetoric and Composition By D. J. Hill, A.M., President Lewisburg University, authctTj of the Science of Rhetoric. Beginning with the selection of'^a | theme, this book conducts the learner through every process j of composition, including the accumulation gf material, its | arrangement, the choice of words, the construction of sentences, the variation of^ expression, the use of figures, the formation of ^ of (pf}j:3l 800561 06-f^^- UNIVERSITY OF CALIFORNIA LIBRARY ' Sliy, auu r iKJi'ctyoKjt. \jt j.'M.^x^^ ^ ^. ^^^ , . . 12mo, cloth. Elements of Political Economy By Francis Watland, D.D., late President of Brown Uni- versity. 12rao, cloth, 403 pages. Recast by Aaron L. Chapin, D.D., President of Beloit College. No text-book on the subject has gained such general accept- ance, and been so extensively and continuously used, as Dr. Wayland's, Dr. Chapin has had chiefly in mind the icants of the class-room, as suggested by an experience of many years. His aim has been to give in full and proportioned, yet clear and compact statement, the elements of this important branch of science, in their latest aspects and applications. J