INMEMORIAM FLORIAN CAJORl i^l^ti ^. ^^' >2/^.^6 ?^^..„- 1 -y AN ELEMENTARFrr;:- GE OMETR Y AND TRIGONOMETRY WILLIAM F. BRADBURY, A. M., HOPKINS MASTER IN THE CAMBRIDGE HIGH SCHOOL; AUTHOR OF A TREATISE ON TMOOItOBTRrBT AND SUaVETING, AND OF AN ELEMENTARY ALGEBRA. BOSTON: PUBLISHED BY THOMPSON, BROWN, fH8 pewits is wholly in the surface. ' ; J0.{ ^ Soiid ha"^ .length, breadth, and thickness. 13* Scholium. The boundaries of solids are surfaces; of surfaces, lines; the ends of lines are points. 14. A Theorem is something to be proved. 15. A Problem is something to be done. 16. A Proposition is either a theorem or a problem. 17. A Corollary is an inference from a proposition or state- ment. 18i A Scholium is a remark appended to a proposition. 19. An Hypothesis is a supposition in the statement of a proposition, or in the course of a demonstration. 20. An Axiom is a self-evident truth. AXIOMS. 1. If equals are added to equals, the sums are equal. 2. If equals are subtracted from equals, the remainders are equal. 3. If equals are multiplied by equals, the products are equal. 4. If equals are divided by equals, the quotients are equal. 5. Like powers and like roots of equals are equal. G. The whole of a magnitude is greater than any of its parts. 7. The whole of a magnitude is equal to the sum of all its parts. 8. Magnitudes respectively equal to the same magnitude are equal to each other. 9. A straight line is the shortest distance between two points. BOOK I. ANGLES, LINES, POLYGONS. ANGLES. DEFINITIONS. 1« An Angle is the difference in direction of two lines. If the lines meet, the point of meeting, B, ^-^-^"^ is called the vertex ; and the lines A Bj B C, ^ .^^r^^___/7 the sides of the angle. If there is but one angle, it can be designated by the letter at its vertex, as the angle B ; but when a number of angles have the same vertex, each angle is designated by three letters, the middle letter showing the vertex, and the other two with the middle letter the sides ; as the angle ABC, 2. If a straight line meets another so as to make the adjacent angles equal, each of these angles is a right angle ; and the two lines are perpendicular to each other. Thus, AC D and D C B, being equal, are right an- gles, and A B and B G are perpendicular to each other, 3. An Acute Angle is less than a right angle; && U C B. 4f An Obtuse Angle is greater than a right A angle ; as A C JS. Acute and obtuse angles are called oblique angles. 4 PLANE GEOMETRV^. 5t The Complement of an angle is a right angle minus the given angle. Thus (Fig. in Art. 7), the complement oi AG D mACF— ACD = DGF. 6t The Supplement of an angle is two right angles minus the given angle. Thus (Fig. Art. 7), the supplement oi AC D is {ACF-\-FCB) — AGI)z=DCB. THEOKEM I. 7» The sum of all the angles formed at a point on one side of a straight line^ in the same 2'dane, is equal to two right angles. Let D G and E G meet the straight line A B 2ii the point G ; then AGD + J)GF + FGB = two right angles. At G erect the perpendicular, GF; then it is evident that AGI)-\-DGF+FGB = AGD + I)GF+FGF + EGB = AGF-i-FGB = two right angles. 8. Corollary 1. If only two angles are ^ formed, each is the supplement of the other. For by the theorem, ACD -{-D C B=itwQ right angles ; A ^ B therefore AG Dz=. two right angles — D GB^ or D C B=z two right angles — A CD. 9. Corollary 2. The sum of all the angles formed in a plane about a point is equal to four right angles. Let the angles ^jS/), DBE, EBF, F BGj GB A, be formed in the same plane about the point B. Produce A B -y then the sum of the angles above the line A C is equal to two ' right angles ; and also, the sum of the angles below the line AC ia equal C BOOK I. to two right angles (7) * ; therefore the sum of all the angles at the point B is equal to four right angles. THEOREM II. 10» If at a point in a straight line two other straight lines upon opposite sides of it make the sum of the adjacent angles equal to two right angles, these two lines form a straight line. Let the straight line D B meet the two lines, AB, BG, so as to make ABD-\-DBC=^i^o right angles : then A B and B C form a straight line. ^ For if ^ ^ and B C diO not form a straight line, draw B U 80 that A B and B E shall form a straight line ; then ABI)-\-DBB= two right angles (7) ; but by hypothesis, ABI)-\-I)BC= two right angles ; therefore DBE = DBC the part equal to the whole, which is absurd (Axiom 6) ; there- fore AB and B C form a straight line. THEOREM III. 11. If two straight lines cut each other ^ the opposite, 'or vortical, angles are equal. Let the two lines, AB, CD, cut each other at E ; then AEC = DEB. For A E D m the supplement of both A AECaadBEB (8)', therefore AEC=DEB In the same way it may be proved that AED — CEB * The figures alone refer to an article in the same Book ; in referring to an article in another Book the number of the Book is prefixed. PLANE GEOMETRY. THEOREM IV. 12. Two angles whose sides have the same or opposite directions are equal. 1st. Let BA and B C^ including the angle B, have respectively the same direc- tion as ED and EF^ including the angle E ; then angle B = angle E. For since B A has the same direction as ED, and BC the same as EF, the differ- ence of direction of B A and B C must be the same as the difference of direction of E D and E F; that is, angle B ==: angle E. 2d. Let B A and B 0, including the angle B, have respectively opposite di- rections to E F and E D, including the angle E ; then angle B = angle E. Produce I) E and FE so as to form the angle G^^^; then (11) GEH—BEF and GEHz=ABC by the first part of this proposition ; therefore angle B z=l angle E. PARALLEL LINES. 13. Definition. Parallel Lines are such as A B have the same direction \ Oi&AB and CD. ^ ^ 14. Corollary. Parallel lines can never meet. For, since parallel lines havG^ the \ame dir^ctk)n,-4f\ they coincide^ at one point, they would coincide throughout and form one and the same straipjht line. Conversely, straight lines in the same plane that never meet, however far produced, are parallel. For if they never meet they cannot be approaching in either direction, that is, they must have the same direction. BOOK I. 7 15, Axiom. Two lines parallel to a third are parallel to each other. ]6t Definition. When parallel lines are cut by a third, the angles without the parallels are called external; those within, internal ; thus, AGE, EGB, CHF, FEB are ex- ternal angles; A G H, BGH, G H C, G H D are internal angles. Two in- ternal angles on the same side of the secant, or cutting line, are called internal angles on the same side ; 2i?> AG H smdi G H C, or B G H and GHD. Two internal angles on opposite sides of the secant, and not adjacent, are called alternate internal angles ; as A G H ^n^ GHD, ox BGH 2,ndiGHC. Two angles, one external, one internal, on the same side of the secant, and not adjacent, are called opposite external and in- ternal angles ; 2,^ E G A and G H C, or E G B and GHD. THEOREM V. 17. If a straight line cut two parallel lines, 1st. The opposite external and internal angles are equal. 2d. The alternate internal angles are equal. 3d. Tlie internal angles on the same side are supplemembs of each other. Let E F cut the two parallels A B and CD ; then ^\^g 1st. The opposite external and ~ I internal andes, EGA and GHC, ^ ^„ G — ■ ^^^v:; — D or EGB and GHD, are equal, \.^ since their sides have respectively the same directions (12). 2d. The alternate internal angles, AGH and GHD, or BGH and GHC, are equal, since their sides have opposite directions (12). 8 PLANE GEOMETRY. 3d. The internal angles on the same side, ^6^^ and G H G, OT BG li and G H D, are supplements of each other ; for AGE is the supplement of A G E (8), which has just been proved equal to GIIG. In the same way it may be proved that BGU and GHD are supplements of each other. THEOREM VI. CONVERSE OP THEOREM V. 11 8 1 If d Straight line cut two other straight lines in the same plane, these two lines are parallel, 1st. If the opposite external and internal angles are equal. 2d. If the alternate internal angles are equal. 3d. If the internal angles on the same side are supplements of each other. Let E F cut the two lines A B and GD so astomake^(?i?= G H B, or AG H = GHD, or BGII and GUI) supplements of each other ; then A B m parallel to CD. For, if through the point G a line is drawn parallel to CD, it will make the opposite external and internal angles equal, and the alternate internal angles equal, and the internal angles on the same side supplements of each other (17); therefore it must coincide with A B -, that is, AB is parallel to CD. -x PLANE FIGURES. DEFINITIONS. 19". A Plane Figure is a portion of a plane bounded by lines either straight or curved. When the bounding lines are straight, the figure is a polygon, and the sum of the bounding lines is the perimeter. BOOK I. 9 20o An Equilateral Polyg^on is one whose sides are equal each to each. 21. An EquiaLgular Polygon is one whose angles are equal each to each. 22. Polygons whose sides are respectively equal are mutually equilateral. 23. Polygons whose angles are respectively equal are mutu- ally equiangular. Two equal sides, or two equal angles, one in each polygon, similarly situated, are called homologous sides, or angles. 24. Equal Poljgons are those which, being applied to each other, exactly coincide. 25. Of Polygons, the' simplest has three sides, and is called a triangle ; one of four sides is called a quadrilateral ; one of five, a pentagon ; one of six, a hexagon ; one of eight, an octagon ; one of ten, a decagon. TRIAKGLES. 26. A Scalene Triangle is one which has no two of its sides equal ; as J -5 (7. ^ ^ 27. An Isosceles Triangle is one which has two of its sides equal ; SiS I) JE F. DL \F G 28. An Equilateral Triangle is one whose L-ides are all equal ; a.s I G IT, 10 PLANE GEOMETRY. 29t A Bight Triangle is one which has a right angle ; SiS JK L. The side opposite the right angle is called the hypothenuse. 30. An Obtuse-angled Triangle is one which has an obtuse angle; as MNO. j^-. 31* An Acute-angled Triangle is one whose angles are all acute; as D £J F. Acute and obtuse-angled triangles are called oblique-angled triangles. 32 » The side upon which any polygon is supposed to stand is generally called its base; but in an isosceles triangle, as DjEF, in which D E = E F, the third side D F i^ considered the base. THEOREM yil. The sum of the angles of a triangle is equal to two right angles. Let A BChe 2i triangle ; the sum of its three angles, A, B, C, is equal to two right angles. Produce A G, and draw CD par- allel to ^ ^ ; then J)CE=A, be- ing external internal angles (17); BG D ■=. B, being alternate internal angles (17) ; hence DGE-\-BGD-\-BGA=A-\-B-{-BGA but DGE -\- BG D -\- BGA — two right angles (7) ; therefore A-\-B-\-BGA^= two right angles. 34. Cm. I. If two angles of a triangle are known, the thin! can be found by subtracting their sum from two right angles. BOOK I. 11 35 • Cor. 2. If two triangles have two angles of the one respectively equal to two angles of the other, the remaining angles are equal. 36* Cor. 3. In a triangle there can be but one right angle, or one obtuse angle. . 37. Cor. 4. In a right triangle the sum of the two acute angles is equal to a right angle. 38. Cor. 5. Each angle of an equiangular triangle is equal to one third of two right angles, or two thirds of one right angle. 39. Cor. 6. If any side of a triangle is produced, the exte- p rior angle is equal to the sum of the two interior and opposite. THEOREM VIII. 40. If two triangles have two sides and the included angle of the one respectively equal to two sides and the included angle of the other y the two triangles are equal in all respects. In the triangles ABC, DEF, let the side AB equal DE, AC equal DF, and the angle A equal the angle D ; then the triangle ABC i% equal in all re- spects to the triangle D E F. Place the side A B on its equal D E^ with the point A on the point D, the point B will be on the point E^ SiS A B is equal to D E ; then, as the angle A is equal to the angle D, A C will take the direction D F, and as ^ (7 is equal to D F, the point C will be on the point F ; and BC will coincide with E F, Therefore the two triangles coincide, and are equal in all re- spects. 12 PLANE GEOMETRY. THEOREM IX. 41. If two triangles have two angles and the included side of the one respectively equal to two angles and the included side of the other, the two triangles are equal in all respects. In the triangles ABC and D E F, let the angle A eqnal the angle D, the angle C equal the angle F, and the side A C equal D F ; then the triangle AB C m equal in all respects to the triangle DBF. Place the side A C on its equal D F, with the point A on the point D, the point C will be on the point F^ ?i% AC is equal to D F ; then, as the angle A is equal to the angle D, A B will take the direction D E ; and as the angle C is equal to the angle F, CB will take the direction FE ; and the point B fall- ing at once in each of the lines D E and F E must be at their point of intersection E. Therefore the two triangles coincide, and are equal in all respects. THEOREM X. 42. In an isosceles triangle the angles opposite the equal sides are equal. In the isosceles triangle A B C let B A B and B C he the equal sides ; then the angle A is equal to the angle C. Bisect the angle A B C hj the line B D ; then the triangles ABB and BCD are equal, since they have the two sides AB, B D, and the included angle ABB equal respectively to BC, B Z), and the included angle BBC (40) ; therefore the angle A=:^ C. 43. Cor. 1. From the equality of the triangles ABB and BCD, AD = DC, and the angle ADB = BDC; that is, the BOOK I. 13 line bisecting the angle opposite the base of an isosceles triangle bisects the base at right angles and also bisects the triangle ; also the line drawn from the vertex perpendicular to the base of an isosceles triangle bisects the base, the vertical angle, and the triangle. And, conversely, the perpendicular bisecting the base of an isosceles triangle bisects the angle opposite, and also the triangle. 44* Cor. 2. An equilateral triangle is equiangular. THEOREM XI. 45. If two angles of a triangle are equal, the sides opposite are also equal. In the triangle ABC let the angle A equal the angle C ; then ^^ is equal to^a Bisect the angle A B C hy the line B D. Now by hypothesis the angle ~" ' D A is equal to the angle C, and by construction the angle A B D is equal to the angle D BC ] therefore (35) the angle A D B m equal to the angle B D C ; and the two triangles AB D, D BC, having the side B D common and the angles including B D respectively equal, are equal (41) in all respects ; therefore ABz=iBC. 46. Corollary. An equiangular triangle is equilateral. THEOREM XII. 47« The greater side of a triangle is opposite the greater angle ; and, conversely, the greater angle is opposite the greater side. In the triangle ^ i5 C let ^ be greater than C ; then the side AC is greater than A B. At the point B make the angle CBD equal to the angle C ; 14 PLANE GEOMETKY. then (45) DB=iDC and A C = AD + D C=AD + DB But (Axiom 9) j^ AD+DByAB therefore AC^AB Conversely. Let A C '^ A B ; then the angle A B C ^ C. Cut oE AD^=AB and join B D ; then as A D = A B, the angle A B D = A B B (42) ; and ADByC (39) ; therefore ABD-yC] hut ABCyABB; therefore A B y G. y ■ /y THEOREM XIII. 48. Two triangles mutually equilateral are equal in all respects. Let the triangle A BG have A B, BG, G A respec- tively equal to AD, DG, G A of the triangle ABG \ then ABG \^ equal in all respects to ADG. Place the triangle ADG so that the base A G will co- incide with its equal A G, but so that the vertex D will be on the side of A (7, opposite to B. Join B D. Since by hy- pothesis AB z= A D, AB D is an isosceles triangle ; and the angle ABD = ADB (42) ; also, since BG = G D, BG D is an isosceles triangle ; and the angle D BG = G D B ; there- fore the whole angle ABG=^ADG', therefore the triangles ABG and ADG, having two sides and the included angle of the one equal to two sides and the included angle of the other, are equal (40). • BOOK I. 15 49* Scholium. In equal triangles the equal angles are oppo- site the equal sides. THEOREM XIV. 50« Tivo right triangles having the hypothenuse and a side of the one respectively equal to the hypothenuse and a side of tie other are equal in all respects. Let A B G have the hypothenuse A B and the side B C equal to the hypothe- nuse BD and the side BC oi BDC', then are the two triangles equal in all respects. Place the triangle BDC %q> that the side B G will coincide with its equal BG^ then G D will be in the same straight line with A G (10). An isosceles triangle A B I) is thus formed, and B G being perpendicular to the base divides the triangle into the two equal triangles ABG and B D G (43). THEOREM XV. 51. If from a point without a straight line a perpendicular and oblique lines he drawn to this line, 1st. The perpendicular is shorter than any oblique line. 2d. Any two oblique lines equally distant from the perpendicu- lar are equal. 3d. Of two oblique lines the more remote is the greater. Let A be the given point, BG the given line, A D the perpendicular, and AE, AB, AG oblique lines. B^^ Ist. In the triangle A D E, the an- ^ gle A D E being a right angle is'greater than the angle A E D; therefore A D ^ and D B. Since CA = C B, ^/ D A =zDB (51). 2d. Let E be any point without the perpen- dicular ; draw EA and EB, and from the point D, where E A cuts D C, draw D B. The an- gle ABE^ABD = BAD; hence, in the triangle AEB since the angle A B E :> B AE, E A y E B (47). BOOK I. 17 QUADRILATERALS. DEFmmONS. 54. A TrapeziTim is a quadrilateral which has no two of its sides' parallel ; as A B C D. 55. A Trapezoid is a quadrilateral ^ ^ ^ .;f which has only two of its sides parallel ; mJSFGff. H' \ 56* A Farallelogpram is a quadrilateral whose opposite sides are parallel ; as IJ K L, or MN P, or Q R S T, ov U V W X. I J 57. A Bectangle is a rij^ht-angled parallel- ogram ; 3iS UK L. 58. A Sqnare is an equilateral rectangle ; asMJVOF. 59. A Rhomboid is an oblique-angled par- allelogram ; as Q E S T. 60. A Rhombus is an equilateral rhomboid ; aaUVWX. Q T u r J X w 61. A Diagonal is a line joining the vertices of two angles not adjacent ; as> D B. 18 PLANE GEOMETRY. THEOREM XVII. 62. In a parallelogram the opposite sides are equal, and the opposite angles are equal. Let ABC D he a parallelogram ; then B C will AB = DC,BC — AD, the angle /^T 7 A=zCraxidiB = D. I ^"'^^^... / Draw the diagonal B D. As B C and A D A D are parallel, the alternate angles C B D and B D A are equal (17); and as ^ ^ and DC are parallel, the alternate angles AB D and B D C Sive equal ; therefore the two triangles A B D and B D C, having the two angles equal, and the in- cluded side ^i> common, are equal (41) ; and the sides opposite the equal angles are equal, viz. : A B =z D C and BC =^ A D ] also the angle A =z C, and the angle ABC=ABD + DBC = BDC-\-BDA=ADC 63* Cor. 1. The diagonal divides a parallelogram into two equal triangles. 64* Cor. 2. Parallels included between parallels are equal. THEOREM XVIII. 65. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. Let A B CD be a quadrilateral having B O B C equal and parallel to A D ; then / ''"^>.,^ / AB CD is a parallelogram. / ^'^ -., / Draw the diagonal B D. As B C m A D parallel to AD, the alternate angles CBD and BDA are equal (17); therefore the two triangles CBD and BDA, having the two sides C B, B D, and the included angle CBD respec- tively equal to the two sides AD, D B, and the included angle A D B, are equal (40), and the alternate angles A B D and B D C are equal ; therefore A B is parallel to Z> C (18), and A B C D is a parallelogram. BOOK I. 1-9 THEOREM XIX. 66* The line joining the middle points of the two sides of a trapezoid which are not parallel is parallel to the two parallel sides, and equal to half their sum. Let ^i^ join the middle points of the sides AB and CD, which are not parallel, of the trapezoid A B C D ; then 1st. EF is parallel to BC and AD. j^ G C Through F draw G H parallel to ^ ^, meeting AD produced in H. The an- E^ gles GFC and DFR are equal (11); also the angles G C F and FD II (17) ; D H and the side G F \s, equal to FD ; therefore the triangles GFC and D F H are equal (41), and GF=FII=z\GH But ^ ABG H\^a parallelogram, GIIz=BA (62) ; therefore FII==iBAz=AE therefore AE F H is a parallelogram (65), and E F is parallel to A D, and therefore also to B C. 2d. EF=i(AD + BC) For as AEFH and EBGF are parallelograms EF=AH=AD+Dff and also EF—BG=BC—G.C Now, as the two triangles GFC and D F H are equal, GC = D H ; therefore, if we add the two equations, we shall have 2 EF=AD-\- BC or EF=i(AD + BC) E 20 PLANE GEOMETRY. THEOREM XX. 67 » Th£ sum of the interior angles of a polygon is equal to tvrlce as many right angles as it has sides minus two. Let ABC D E F be the given polygon ; q the sum of all the interior angles A, B, C^ /^'^^^^^^^'^\ D, E, F, is equal to twice as many right j^:''l -^D angles as the figure has sides minus two. y "~^-,., / For if from any vertex A, diagonals A (7, V AD, AE, are drawn, the polygon will be divided into as many triangles as it has sides minus two ; and the sum of the angles of each triangle is equal to two right angles (33) ; therefore the sum of the angles of all the triangles, that is, the sum of the interior angles of the polygon, is equal to twice as many right angles as the polygon has sides minus two. PRACTICAL QUESTIONS. 1. Do two lines that do not meet form an angle with each other ? Two lines not in the same plane ? 2. Does the magnitude of an angle depend upon the length of its sides ? 3. If a right angle is 90°, what is the complement of an angle of 27° ? of 51° ? of 91° ? of 153° ? What is the supplement of an angle of 13° ? of 83° ? of 97° ? of 217° ? 4. If three of four angles formed at a point on the same side of a straight line, in the same plane, contain respectively 15°, 27°, and 99°, how many degrees does the fourth angle contain ? 5. If five of six angles formed in a plane about a point are respectively 11°, 53°, 74°, 19°, and 117°, how many degrees are there in the sixth angle ? 6. On opposite sides of a line A B are two lines making with A B, at the point A, the first an angle of 29°, and the second an angle of 61° ; how are these two lines related ? BOOK I. 21 7. Can two polygons, each not equilateral, be mutually equilateral ? 8. Can two polygons, each not equiangular, be mutually equiangular 9. If two angles of a triangle are respectively 32° and 43°, how many degrees are there in the remaining angle ? 10. If one acute angle of a right triangle is 24°, how many degrees rre there in the other acute angle ? 11. How many degrees in each angle of an equiangular triangle ? 12. How many degrees in each angle at the base of an isosceles triangle whose vertical angle is 14° ? 13. How many degrees in each acute angle of a right-angled isosceles triangle ? 14. If one of the angles at the base of an isosceles triangle is double the angle at the vertex, how many degrees in each ? 15. If the angle at the vertex of an isosceles triangle is double one of the angles at the base, how many degrees in each i 16. Two triangles mutually equilateral are mutually equiangular (48). Are two triangles mutually equiangular also mutually equilateral ? 17. Is a square a parallelogram ? Is a parallelogram a square ? 18. Is a rectangle a parallelogram ? Is a parallelogi-am a rectangle ? 19. How many sides equal to one another can there be in a trapezoid ? How many in a trapezium ? 20. How many degrees in each angle of an equiangular pentagon ? an equiangular hexagon ? octagon ? decagon ? dodecagon ? 21. If the parallel sides of a trapezoid are respectively 8 feet and 13 feet in length, how long is the line.joinmg the middle points of the other two sides ? 22. If one of the angles of a parallelogram is 120°, how many are there in each of the other angles ? 22 PLANE GEOMETRY. EXERCISES. The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the prop- ositions in the following books, and can be omitted if thought best. 68i Two angles whose sides have, one pair the same, the other opposite directions, are supplements of each other. (12.) (8.) 69« Any side of a triangle is less than the sum, but greater than the difference, of the other two. (Axiom 9.) 70. The sum of the lines drawn from a point within a triangle to the extremities of one of the sides is less than the sum of the other two sides. Produce one of the lines to the side of the triangle. (Axiom 9.) 71 • The angle included by the lines drawn from a point within a triangle to the extremities of one of the sides is greater than the angle included by the other two sides. Produce as in (70). (39.) 72. The angle at the base of an isosceles triangle being one fourth of the angle at the vertex, if a perpendicular is drawn to the base at its extreme point meeting the opposite side produced, the triangle formed by the perpendicular, the side produced, and the remaining side of the triangle, is equilateral. 73. If an isosceles and an equilateral triangle have the same base, and if the vertex of the inner triangle is equally distant from the ver- tex of the outer and the extremities of the base, then the angle at the base of the isosceles triangle is J or j of its vertical angle, accord- ing as it is the inner or the outer triangle. 74. Prove Theorem VII. by first drawing a line through B par- allel to A C. 75. Prove Theorem VII. by drawing a triangle upon the floor, walking over its perimeter, and turning at each vertex through an angle equal to the angle at that vertex. BOOK I. 23 76. Only one perpendicular to a straight line can be drawn from a point. (Two cases. 1st. When the point is without the line. 2d. When the point is within the line.) 77« Two straight lines perpendicular to a third are parallel. (13.) 78. If a line joining two parallels is bisected, any other line drawn through the point of bisection and joining the parallels is bisected. 79. If two triangles have two sides of one respectively equal to two sides of the other, but the included angles unequal, the third side of the one having the included angle greater is greater than the third side of the other. Place the triangles as in the figure; draw BE bisecting the angle C BD, and join C and E. 80. (Converse of 79.) If two triangles have two sides of one respectively equal to two sides of the other, but the third sides un- equal, the included angle of the one having the third side greater is greater than the included angle of the other. (Prove it by proving any other supposition absurd.) 81. Prove in Theorem XIII. the angles of the two triangles equal by reference to (80) then that the triangles are equal by (40) or (41). 82. (Converse of part of 62.) If the opposite sides of a quad- rilateral are equal, the figure is a parallelogram. 83. (Converse of part of 62.) If the opposite angles of a quadri- lateral are equal, the figure is a parallelogram. 84. (Converse of 63.) If a diagonal divides a quadrilateral into two equal triangles, is the figure necessarily a parallelogram ? 85. The diagonals of a parallelogram bisect each other. 86. (Converse of 85.) If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 87. The diagonals of a rhombus bisect each other at right angles. 24 PLANE GEOMETRY. 88. (Converse of 87.) If the diagonals of a quadrilateral bisect each other at right angles, the figure is a rhombus or a square. 89. The diagonals of a rectangle are equal. 90. The diagonals of a rhombus bisect the angles of the rhombus. 91 . Straight lines bisecting the adjacent angles of a parallelogram are perpendicular to each other. 92. From the vertices of a parallelogram measure equal distances upon the sides in order. The lines joining these points on the sides form a parallelogram. 93. Prove Theorem XX. by joining any point within to the ver- tices of the polygon. 94. If the' sides of a polygon, as A B OD EF, are produced, the sum of the angles a, &, c, c?, e, /, is equal to four right angles. 95. If a pavement is to be laid with blocks of the same regular form, that is, blocks whose faces are equiangular and equilateral, prove that their upper faces must be equilateral triangles, S(|uares, or hexagons. (67 ; 9.) 96. If two kinds of regular figures, with sides of the same length, are to be used at each angular point, show that the pavement can be laid only with blocks whose upper faces are, 1st. Triangles and squares. 2d. Triangles and hexagons. 3d. Triangles and dodecagons. 4th. Squares and octagons. IIow many of each must there be at each angular point? 97. If three kinds of regular figures, with sides of the same length, are to be used at each angular point, show that the pavement can be laid only with blocks whose upper faces are, 1st. Triangles, squares, and hexagons. 2d. Squares, hexagons, and dodecagons. How many of each must there be at each angular point ? RATIO AND PEOPOETIOK DEFINITIONS. (It is necessary to understand the elementaiy principles of ratio and pro- portion before entering upon the Books that are to follow. It, is therefore introduced here, but not numbered as one of the Books of Geometry, as it belongs properly to Algebra. Reference to the propositions in ratio and proportion will be made by the abbreviation Pn., with the number of the article annexed.) 1* Ratio is the relation of one quantity k> another of the same kind ; or it is the quotient which arises from dividing one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one an- other with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to 6 is written, a : 6, or - ; read, a is to b, or a divided by b. 2i The Terms of a ratio are the quantities compared, whether simple or compound. The first term of a ratio is called the antecedent, the other the consequent ; the two terms together are called a couplet. 3t An Inverse or Reciprocal Ratio of any two quantities is the ratio of their reciprocals. Thus, the direct ratio of a to 6 is a : ft, that is, - ; the inverse ratio of a to 6 is - : -, that is, 1 1 & , - -i- T = - or 6 : a. aba 4i Proportion is an equality of ratios. Four quantities are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth. 2 26 PLANE GEOMETRY. The equality of two ratios is indicated by the sign of equality (=), or by four dots (: :). Thus, a : 5 = c : c?, or a : 6 : : c : o?, or - = - ; read a to 6 equals c to c?, or a is to 6 as c is to d, or a divided by h equals c divided by d. 5. In a proportion the antecedents and consequents of the two ratios are respectively the antecedents and consequents of the proportion. The first and fourth terms are called the extremeSy and the second and third the means. 6i When three quantities are in proportion, e. g. a : 6 = ft : c, the second is called a mean proportional between the other two ; and the third, a third proportional to the first and second. 7i A proportion is transformed by Alternation when antece- dent is compared with antecedent, and consequent with conse- quent. 8. A proportion is transformed by Inversion when the ante- cedents are made consequents, and the consequents antece- dents. 9t A proportion is transformed by Composition when in each couplet the sum of the antecedent and consequent is compared with the antecedent or with the consequent. lOi A proportion is transformed by Division when in each couplet the difference of the antecedent and consequent is com- pared with the antecedent or with the consequent. 11, Axiom. Two ratios respectively equal to a third are equal to each other. RATIO AND PROPORTION. 27 THEOREM I. 12f In a proportion the product of the extremes is equal tc the product of the means. Let a : h = c : d that is - = - b ^ d Cleariug of fractions ad=z be 13* Scholium. A proportion is an equation; and making the product of the extremes equal to the product of the means is merely clearing the equation of fractions. THEOREM II. 14. If the product of two quantities is equal to the product of two others, the factors of either product may he made the extremes, and the factors of the other the means of a proportion. Let a = he Dividing hy hd h^^ 1 that is a : b = c : d THEOREM in. 15. If four quantities are in proportion, they loill he in pro- portion hy alternation. Let a '.h =^ c : d By (12) ad = bc By (14) a:c = h:d 28 PLANE GEOMETRY. THEOREM IV. 16i If four quantities are in proportion^ they will he in pro- portion by inversion. Let a : h = c : d By (12) ad = bc By (14) . b -.a^d'.c THEOREM V. 17« If four quantities are in proportion^ they will he in pro- portion by composition. Let a : b == c : d that is Adding 1 to each member or that is THEOREM VI. 18i If four quantities are in proportion, they will he in pro- portion by division. Let a -.h ■=^ c \ d that is b ' d 1 + ^- ■l^' a-\-b b ~ c-^d ■ d a + b :b-=: c + d\ :d Subtracting 1 from each member or that is a c b~ d 1-^-2-' a — b c — d b ~ d a — h : h =2 c — d \d RATIO AND PROPORTION. 29 19. Corollary. From (17) and (18), by means of (15) and (H), If a :h =z c : d then a-^-b : a — b=:c -{-d :c — d THEOREM VII. 20» Equimidtiples of two quantities have tlie same ratio as the quantities themselves. T, a ma b mb that is a :b =. ma : mb 21 • Corollary. It follows that either couplet of a proportion may be multiplied or divided by any quantity, and the result- ing quantities will be in proportion. And since by (15), if a :b =z ma : mb, a : m a=b : mb or ma : a = mb : b, it follows that both consequents, or both antecedents, may be multiplied or divided by any quantity, and the resulting quan- tities will be in proportion. THEOREM VIII. 22 • If four quantities are in proportion, like powers or like roots of these quantities will be in proportion. Let a '.b -~ c '. d au X • a c that is T = -; d Hence -j- = -- 5« d^ that is cr*» : 6" = c** : c?" Since n may be either integral or fractional, the theorem is proved. 30 PLANE GEOMETKY. THEOREM IX. 23. If any number of quantities are proportional^ any antece- dent is to its consequent as the sum of all the antecedents is to the stem of all the consequents. Let a :b^=c : d=^e :f Now ab = ab (A) and by (12) ad^bc (B) and also , a/=6e (C) Adding (A), (B), (C) a{b + ^+/)=6(a+c + e) Hence, by (14) a:6=a + c + 6:6 + 6?+/ ' THEOREM X. 24 • //' there are two sets of quantities in proportion, their pro- ducts, or quotients, term by term, will be in proportion. Let a : b =z c : d and , e\f=^g:h By (12) ad = bc (A) and ehz=fg (B) Multiplying (A) by (B) adeh = b cfg (C) Dividing (A) by (B) ^=)^ (D) From (C) by (U) ae\bf=cg\dh 1 « /T^x abed and from (D) - : - = - : 7 ^ ' t f g h BOOK II EELATIONS OF POLYGONS. DEFINITIONS. 1, The Area of a polygon is the measure of its surface. It is expressed in units, which represent the number of times the polygon contains the square unit that is taken as a standard. 2. Equivalent Polygons are those which have the same area. B 3*. The Altitude of a triangle is the perpendic- ular distance from the opposite vertex to the base, or to the base produced ; as B D. it The Altitude of a parallelogram is the perpendicular distance from the opposite side to the base ; as IK. 5( The Altitude of a trapezoid is the perpendicular distance between its paral- lel sides ; as P /?. ' E K H M P N THEOREM I. 6* Two polygons mutually equiangular and equilateral are equal. Let ABGDE F and GHIKLM be two poly- gons having the sides A AB,BC, CD,DE,EF, FA and the angles A, B, D a PLANE GEOMETEY. (7, D, E, F of the one re- spectively equal to the sides G II, R I, I K, K L, L M, M G, and the angles G, H, /, K, L, M of the other j then is the poly- gon ABGD EF equal to the polygon Gil I XL M. For if the polygon ABG D E F is applied to the polygon GHIKLM so that A B shall he on G H with the point A on G, B will fall on H, as A B and G H are equal ; and as the angle B is equal to the angle //, B G will take the direction HI; and as BG is equal to ///, the point G will fall on I\ and so also the points D, E, F will fall on the points K, L, M\ and the polygon ABG D E F will coincide with the polygon GHIKLM, and therefore be equal to it. THEOREM II. 7i The area of a rectangle is equal to the product of its basA and altitude. A H I J K D Let ABGD be a rectangle ; its area B P Q K C =zAD X AB. Suppose A B and A D to he divided into any number of equal parts, A E, EF, AH, HI, &c., and through the points of division, lines EL, F M, HO, IP, &c. be drawn parallel to the sides of the rectangle ; then the rectangle will be divided into squares ; these squares will be equal to each other (6). If one of the equal parts, A E, represents the linear unit, then one of the squares, A E S H, represents the square unit ; and there will be as many square units in the rectangle A E L B as there are linear units in A I) -, and as many square units in the rectangle ABGD as there are square, units in AE LD multiplied by the number representing the number of linear units in A B i that BOOK II. 33 is, the area of the rectangle is equal to the product of its base and altitude, that is = A B X AB. 8. Scholium. If A I) and A B have no common measure, the linear unit may be taken as small as we please, that is, so small that the remainders will be infinitesimal, and can be neg- lected. 9. Corollary. The area of a square is the square of one of its sides. THEOREM III. 10. The area of a parallelogram is equal to the product of its base and altitude. Let D F hQ the altitude of the paral- E B F lelogram A BG D \ then the area of ABGDz=AD X DF. At A draw the perpendicular A E meet- A D ing C B produced in E ] AEFD is a rectangle equivalent to the parallelogram ABC D. For the two triangles AE B and DFC, having the sides A E, A B equal respectively to the sides D Fj D C (I. 64), and the included angle E A B equal to the included angle FD C (I. 12), are equal. Adding D F C to the common part ABFD gives the parallelogram ABC D) and adding its equal AEB to the common part ABFD, gives the rectangle AEFD; therefore the parallelogram A B C D is equivalent to the rectangle AEFD; but the area of the rec- tangle z= A D X D F (J); therefore the area of the parallelo- gram =zAD X D F. THEOREM IV. 11, The area of a triangle is equal to half the product of its base and altitude. , Let BDhe the altitude of the triangle ABC; then the area of ABC=:iACX BD. 34 PLANE GEOMETRY. Draw CE parallel to AB, and BE parallel to A C, forming the parallelogram ABEC. The triangle A B C i^ one half the parallelogram ABEC (I. 63) ; the area of the parallelogram z=z AC y^ B D (10) ; therefore the area of the triangle = J ^ (7 X B D. 12. Cor. 1. Triangles are to each other as the products of their bases and altitudes. For if A and a represent the alti- tudes of two triangles T and t^ and B and h their bases, their areas are J ^ X B and \a y^.h; therefore T -.t^z^Ay B \\ayb or (Pn. 21) T '.t=zAX B '.aXh 13* Cor. 2. Triangles having equal bases are as their alti- tudes ; those having equal altitudes as their bases. For in the proportion above, if ^ = 6, or ^ = a, the equals can be can- celled from the second ratio (Pn. 21). THEOREM V. 14. The area of a trapezoid is equal to half the product of its altitude and the sum of its parallel sides. Let EE be the altitude of the trape- zoid ABCD; then the area oi ABCD ^^EEX (BC + AD). Draw the diagonal B D ; it will di- vide the trapezoid into two triangles, A B D, BCD, having the same alti- tude E E a^ the trapezoid. By (11) the area of BCD=:\ EF X BC and the area of A B D z= ^ E E X A D Therefore the area of the trapezoid = \ E E X {B C -\- AD). 15. Corollary. As (I. 66) the line joining the middle points of the sides AB and CD of the trapezoid = ^{BC -{- AD), BOOK II. 35 therefore the area of a trapezoid is equal to the product of its altitude and the line joining the middle points of the sides which are not parallel. THEOREM VI. 16* A line drawn parallel to one side of a triangle divides the other sides proportionally. In the triangle A B C let D JS he drawn parallel to B C ; then Ai: :£C=AI) :DB Draw I) C and B £ ; the triangles A DB and £J D C, having the same vertex D and their bases in the same straight line A G, have the same altitude; therefore (13) ADE\ EDG = AE'.EC And the triangles A D E and D E B, having the same vertex E and their bases in the same straight line A B, have the same altitude; therefore (13) ABE: DEB = AD: DB But the triangles EDO and DEB are equivalent (11), since they have the same base D E and the same altitude, viz., the perpendicular distance between the two parallels D E and B C. Therefore (Pn. 11) A E '. EG — AD : DB 17. Corollary, k^ A E : E G ^=A D : D B by (Pn. 17) A E -{- E G : A Ez=A D -\- D B : AD that is, AG: AEz=AB: AD 7 THEOREM VII. CONVERSE OF THEOREM VI. 18. A line dividing two sides of a triangle proportionally is parallel to the third side of the triangle. In the triangle ABG if D E divides A B and A G so that AB : AD = A G : A E, then DEm parallel to B G. 36 PLANE GEOMETRY. For if D E is not parallel to B C, through D draw D F parallel to BC; then (17) AB'.ADz=2AC:AF A^ But by hypothesis AB.AD = AG :AE Now as the first three terms of these two proportions are the same, their fourth terms must be equal ; that is, A F = A E, the part equal to the whole, which is ab- surd (Axiom 6) ; therefore £> E is parallel to B C. 19. Definition. Similar Polygons are those which are mutu- ally equiangular, and have their homologous sides, that is, the sides including the corresponding angles, proportional. THEOREM VIII. 20. Two triangles mutually equiangular are similar. In the two triangles ABC, DEF, let the angle A = D, B = E,SiiidC = F; then the triangles are similar. As the triangles are mutually equiangular, we have only to prove the homologous sides proportional. Cut off A G and A II equal respectively to BE and BF, and join Gil; the triangle AG II is equal to B E F (I. 40), and the angle A G II = E ; but E=B; therefore A G II = B, and Gil is parallel to ^C (I. 18); and (17) AB:AG==AC:Aff or AB:BE=AC:BF In like manner it may be proved that AB:BE = BC:EF=:AC:BF 21. Cor. Two triangles whose homologous sides are equally inclined to each other are similar. For if one of the triangles is BOOK II. 37 turned through an angle equal to the angle of inclination of the sides, the sides of the triangles become respectively parallel ; they are therefore mutually equiangular (I. 12) and similar (20). THEOREM IX. 22. The altitudes of two similar triangles are proportional to the homologous sides. Let BG and E H he the alti- tudes of the similar triangles ABC imdi DEF) then BG'.EH — AB'.DE — AC :DFz=BC :EF For the two right triangles ^ G ^ ^ ABG, DEH are mutually equiangular (I. 35), and similar (20) ; therefore BG'.EH=:AB'.DE=AC'.DF=BC:EF THEOREM X. 23 • Two triangles having an angle of the one equal to an angle of the other, and the sides including these angles proportional, are similar. In the triangles ABC.DEF let the angle A = D and AB '.DE—AC'.DF then the triangles ABC and D E F are similar. Cut off ^ 6^ and A II re- spectively equal to D E and D F, and join G H; the AGH = DEF, and the angle A G II = E (I. 40). By hypothesis AB ': D E = A C : BF or AB '.AG — AC .AH that is, the sides AB, AC are divided proportionally by the triangle 38 PLANE GEOMETRY. line G H -, therefore GH is parallel to B C (18), and the angle AGH = £ (1.17); hut the angle AGH = E; therefore B =z E, and the two triangles are mutually equiangular and therefore similar (20). THEOREM XI. 24 • In a right triangle the perpendicular drawn from the ver- tex of the right angle to the hypothenuse divides the triangle into two triangles similar to the whole triangle and to each other. In the right triangle A B C if B I) is B drawn from the vertex B of the right /^\^ angle perpendicular to the hypothenuse / j ^\^^ A C, the two triangles A B D, B G D 2irQ ^ /_i — ^C similar to A B G and to each other. The two right triangles A B D and A B G have the acute an- gle A common ; they are therefore mutually equiangular (I. 35), and similar (20). The two right triangles A B G and B G D have the acute angle G common ; therefore they are mutually equiangular and similar. The two triangles A B D and B G D^ being each similar to A B G, are similar to each other. 25. Gor. 1. Since A B G and A B I) sere similar triangles AG :AB = AB:AJ) And since A BG and BG D are similar AG \GB—GB'.GD that is, if in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypothenuse, either side about the right angle is a mea7i proportional between the whole hypothenuse and the adjacent segment. 26. Gor. 2. ks AB D and B G D are similar triangles AD ■.DB = DB :DG BOOK II. 39 that is, in a right triangle the perpeiidicidar from the vertex of the right angle to the hypothenuse is a mean proportional between the segments of the hypothenuse. THEOREM XII. 27. The square described on the hypothenuse of a right triangle is equivalent to the sum of the squares described upon the other two sides. Let ABCheoi triangle right- angled at B ; then Tu' = Jb'-{- bc^ On the three sides construct squares, draw B D perpendicu- lar to A (7, and produce it to FE\ DC EL is a rectangle whose area is (7) C^X CD=ACy^ CD The area of the square (9) BIKC — BU'' But (25) AG'.BC- or ACXCDzz that is, the square BIKG is equivalent to the rectangle DOEL. In the same way the square AG H B can be proved equivalent to the rectangle AD L F \ therefore the sum of the two rec- tangles, that is, the square AG E F m equivalent to the sum of the squares BIKG and A G H B ; or AG' = AB^ + bV- 28. Gorollary. Since Tg'' = Tb'-\- BC'' bg^=^Tg^ — Tb' BG BG and S/AG'' AB^ PLANE GEOMETRY. THEOREM XIII. 29i Similar triangles are to each other as the squares of their homologous sides. Let ABC SLiid DBF be two ^ similar triangles ; then ABC :DBF=AC^ : ITF^ Draw B G and FII perpendic- ular respectively to A C and I) F ; then (22) ^ BG.EH this multiplied by the proportion I AC '.IDF=AC :DF gives \ACXBG:IDFXFH=AC'':DF^ hwt I AC X BG m the area of ^ ^ C', and ^ D F X E H i% the area of i> ^ i^ (11) ; therefore ABC :J)FFz=zTC^ -.WF.^ THEOREM XIY. 30* Similar polygons can he divided into the same number of similar triangles. Let A B C J) E F and GHIKLM be similar poly- gons ; they can be divided into the same number of sim- ilar triangles. F E From the homologous vertices A and G draw the diagonals ACj A D, A E, G Ij G K, and G L ; these diagonals divide the polygons as required. For, as the polygons are similar, the an- gle B = H, and^^ : GH=.BC .HI-, therefore the trian- gles ABC and G H I ixre similar (23). As the triangles ABC and GHI are similar, the angle BC A = H I G ', but the whole angle BCD=: HIK; therefore the angle ACE== GIK; and as the trian^^les ABC and GHI are similar BOOK II. 41 BC '.HI=AC '.GI But BC :HI=zCD :IK Therefore AC \GI=CD .IK and AC B and G I K 2xq similar (23). In like manner it can be proved that the other triangles are similar each to each. THEOREM XV. 31* The perimeters of similar polygons are to each other as the homologous sides ; and the polygons as the squares of the howMo- gous sides. Let ABC D E F and ^^^-tt^x ^ I GHIKLM hQ two similar /C^^^^^^^^'^X /^^\ polygons. A^".:i[ 7 ^ ^v---- - ^ 1st. Their perimeters are \ ""--,. / \- — -1:1;^ to each other vl^ AB : GH F E * For as the polygons are similar AB : GH=BC '.HI= CD .IK, (fee. Therefore (Pn. 23) AB^ BC-\-CD,kQ.'. GH-^HI-\-IK,kc. = AB :GH that is, the perimeters oi ABC D EF and GHIKLM are as AB : GH. 2d. A B C D E F : G H I K L M =zAB^ : gTi'' From the homologous vertices A and G draw the diagonals AC, AD, AE,GI,GK,2indiGL', the polygons will be divided into the same number of similar triangles (30) ; therefore (29) ABC : GUI=TC^ :~I^ and ACD '.GIK=irZ''.G~f Therefore AB C \ G H I = AC D -. GIK In like manner AC D : G I K = AD E - G K L and ADE'.GKL = AEF'.GLM Hence (Pn. 23) ABC-\-ACD-\- ADE + AEF : GHI + GIK + GKL + GLM=ABC.GHI But ABC :GHIz=TB^:GH'' 42 PLANE GEOMETRY. Therefore the sums of the triangles, that is, the polygons themselves, are to each other as the squares of the homologous sides. 32. Definition. A Regular Polygon is one that is both equi- angular and equilateral. THEOREM XVI. 33* Regular polygons of the same number of sides are similar. Let A BCD IJF ^nd y f ^ GHIKLM be two reg- ular polygons of the same number of sides ; they are similar. They are mutually p e equiangular; for the sum of their angles is the same (I. 67); and each angle is equal to this sum divided by the number of angles which is the same. The homologous sides are proportional ; for as the polygons are regular, AB = £C=CD, &c., and GH=HI = IK, &c., therefore ^^ : GH = BC : HI=CD : IK, &c. THEOREM XVII. 34. There is a point in a regular polygon equidistant from its vertices, and also equidistant from its sides. Let ABCDlEF be a regular polygon. Bisect the angles A and B hy A and £ 0. As the whole angles A and B are each less than two right angles, the sum of A B and ABO is less than two rig^t angles ; therefore A and B can- not be parallel (I. 17), but will meet. 9f7: BOOK II. ' 43 Suppose them to n]@>an the point \ then is equidistant from' the vertices A, B, C, D, E, F, and also from the sides A B, B C, CD, &i Draw OC, OB, OE, OF. OA = OB (I. 45). As B bisects the whole angle B, the angle B A = B C ; therefore the triangle A B = B C (I. 40), and OC =OAz= OB. In like manner it can be proved that OD=OE=OF=zOA equidistant from the vertices of the polygon. As the triangles A B, B C, C D, &c. are equal, their altitudes are equal, that is, the bases are equidistant from the vertex 0. 35* Scholium. is called the centre, and the perpendicular G the apothem of the polygon. 36* Corollary. In regular polygons of the same number of sides, the apothems are as the homologous sides ; therefore the perimeters of regular polygons of the same number of sides are as their apothems ; and the polygons as the squares of their apothems. THEOREM XVIII. 37. The area of a regular polygon is equal to half the product of its perimeter and apothem. For, if a regular polygon is divided into triangles by lines drawn from the centre to the several vertices, the area of each triangle is equal to half the product of its base and the apo- them of the polygon (11); therefore the area of the polygon is equal to half the product of the sum of the bases, that is, to half the product of its perimeter and its apothem. 44 PLANE GEOMETEY. PRACTICAL QUESTIONS. 1. What is the perimeter and the area of a rectangle 25 by 35 inches ? 2. What is the area of a parallelogram whose base is 20 feet and altitude 12 feet ? 3. What is the area of a triangle whose base is 14 feet and altitude 8 feet? 4. What is the square surface of a boq,rd 15 feet long, and 16 inches wide at one end and 9 inches at the other ? What kind of a figure is it ? 6. What integral numbers will express the sides and hypothenuse of a right triangle ? 6. How far from a tower 40 feet high must the foot of a ladder 50 feet long be placed that it may exactly reach the top of the tower ? 7. The foot of a ladder 67 feet long stands 40 feet from a wall ; how much nearer the wall must the foot be placed that the ladder may reach 10 feet higher ? 8. If a ladder 108 feet long, with its foot in the street, will reach on one side to a window 75 feet high, and on the other to a window 45 feet high, how wide is the street ? 9. A has an acre of land one of whose sides is 20 rods in length ; B has a piece of land of exactly similar form containing 9 acres. What is the length of the corresponding side of B's ? 10. What is the distance on the floor from one corner to the opposite comer of a rectangular room 16 by 24 feet ? 11. If the height of the above room is 10 feet, what is the distance from the lower corner to the opposite upper comer ? 12. Find the length of the longest straight rod that can be put into a box whose inner dimensions are 12, 4, and 3. 13. What is the altitude of an equilateral triangle whose side is 12 feet ? 14. If the bases of two similar triangles are respectively 100 and 10 feet, how many triangles equal to the second are equivalent to the first ? 15. How many times as much paint will it take to cover a church whose steeple is 120 feet in height as to cover an exact model of the church whose steeple is 10 feet in height ? 16. What is the area of a right-angled triangle whose hypothenuse is 125 feet and one of the sides 75 feet ? BOOK II. 45 EXERCISES. The following Theorems, depending for their, demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the prop- ositions in the following books, and can be omitted if thought best. D C E 38. The square on the sum J. C of two straight fines A B, B C is equivalent to the squares on A B (tnd B C, together with twice the rectangle AB.BC. Or, algebraically, ii^ a = AB, and b = B C, {a-{-bf = a?-\-2ab-\-b^ A B~ C 39* Corollary. The square on a line is four times the square on naif of the line. ab !^ F / H a? ab 40» The square on the diJBference J. (7 of two straight lines AB^ B C is equivalent to the squares on AB and BC. diminished by twice the rectangle AB.BC. Or, algebraically, if a= AB, and b =. BC, {a — by = a* — 2 ab -\- h" 41 * The rectangle contained by the sum and difference of two lines A B, B C is equivalent to the difference of their squares. Or, algebraically, if a ^ ^ ^ and b = BC (a -I- 6) (a -- 6) = a« — b' Produce ABso that BD = BC. K D G E L F 1 H A E G B B T L K I H C B D 42* Parallelograms are to each other as the products of their bases and altitudes. (10.) 43 » Parallelograms having equal bases are to each other as their altitudes; those having equal altitudes are as their bases. 44. Where must a line from the vertex be drawn to bisect a tri- angle? (13.) 45. Two or more lines parallel to the base of a triangle divide the other sides, or the other sides produced, proportionally. 46 PLANE GEOMETRY. 46* Lines joining the middle points of the adjacent sides of a quadrilateral form a parallelogram ; and the perimeter of this* paral- lelogram is equal to the sum of the diagonals of the quadrilateral. Draw the diagonals. (18.) 47« Lines drawn from the vertex of a triangle divide the opposite side and a parallel to it proportionally. 48« State and prove the converse of 47. 49t A B CD is a parallelogram ; E and F the middle points of A B and CD. BF and Z>^ trisect the diagonal A G. 50* If two triangles have two sides of the one equal respectively to two sides of the other, and the included angles supplementary, the triangles are equivalent. 51* The diagonals divide a parallelogram into four equivalent tri- angles. Two triangles standing on opposite sides are equal. 52* If the middle points of the sides of a triangle are joined, the area of the triangle thus formed is one fourth the area of the original triangle. 53* Every line passing through the intersection of the diagonals of a parallelogram bisects the parallelogram. 54* If a point within a parallelogram is joined to the vertices, the two triangles formed by the joining lines and two opposite sides are together equivalent to half the parallelogram. Through the point draw lines parallel to the sides of the parallelo- gram. ^5t State and prove the proposition if the point named in 54 is without the parallelogram. 56* The area of a trapezoid is equal to twice the area of the tri- angle formed by joining the extremities of one non-parallel side to the middle point of the other. 57* Two triangles are similar if two angles of the one are equal respectively to two angles of the other. 58* Two triangles are similar if their homologous sides are pro- portional. BOOK IL 47 59« Definition. When a point is taken on a given line, or a given line produced, the distances of the point from the extremities of the hne are called the segments. If the point is within the given line, the sum of the segments, if in the line produced, the difference of the segments, is equal to the line, 60i The line bisecting any angle, interior or exterior, of a triangle, divides the opposite side into segments; which are proportional to the adjacent sides. Let B be the bisected angle of a triangle ABC. Through G draw a line parallel to the bisecting line and meeting A B. If the interior angle at B is bisected, A B must be produced ; if the exterior angle, A C. In the latter case, if E is the point where the bisecting line meets A C produced, the segments of the base (59) are ^^ ami CE. (1.17.) (1.45.) (16.) 61. Two triangles having an angle of the one equal to an angle in the other are to each other as the rectangles of the sides containing the equal angles ; or ABC:ADE=ABXAC:ADX AE Draw BE. (13.) (Pn. 24.) (Pn. 21.) 62. Prove Theorem XII., first drawing G Cand BF; then prov- ing the triangles AGO and A BF equal. Turn the triangle A BF on the point A in its own plane till A B coincides with A G ; where will 7^ be? (7,11.) 63. Prove that if G H, KI, and L B, in the figure above, are produced, they will meet in the same point. 64. Prove Theorem XII., first producing FA to GIT, and pro- ducing Gff, KI, and i^^ till they meet. 65. Prove Theorem XII., first constructing the squares on oppo- site sides of ^ i? and B C from that on which they are drawn in the figure in Art. 62; moving the square A GHB on AB, i\. distance 48 PLANE GEOMETRY. equal to i? C in the direction BA ; then proving that these squares are divided into parts that can be made to coincide with the parts of the square on A C. B 66. If A is an acute angle of the triangle ABC^ and BD is the perpendicular from i? to J. (7, then BC^=zAB^-\-AC^ — 2ACX AD 67« If A is an obtuse angle of the triangle B ABC, and BD is the perpendicular from B BC' = AB^-{-A0' + 2A0X AD ^ 68. Show that if the angle A becomes a right angle, both 66 and 67 reduce to the same as 27 ; and if becomes a right angle, both reduce to the same as the second equation in 28. 69» If a line is drawn from the vertex of any angle of a triangle to the middle of the opposite side, the sum of the squares of the other two sides is equivalent to twice the square of the bisecting line to- gether with twice the square of a segment of the bisected side. Draw a perpendicular from the same vertex to the opposite side. (66, 67.) 70» The sum of the squares of the four sides of a parallelogram is equivalent to the sum of the squares of thef diagonals. (69.) (39.) 71. In the figure in Art. 62 draw HI, KB, FG. The triangle HIB is equal, and the triangles CKF, GAF are equivalent to ABC. 72. The squares of the sides of a right triangle are as the seg- ments of the hypothenuse made by a perpendicular from the vertex of the right angle. 73. The square of the hypothenuse is to the square of either side as the hypothenuse is to the segment adjacent to this side made by a perpendicular from the vertex of the right angle. 74. The side of a square is to its diagonal as 1 : V^2; or the square described on the diagonal of a square is double the square itself. 75. (Converse of 30.) Two polygons composed of the same num- ber of similar triangles similarly situated are similar. BOOK III. THE CIECLE. DEFINITIONS. 1. A Circle is a plane figure bounded by a curved line called the circumference, every point of which is equally distant from a point within called the centre ; ^^ A B D E, a 2. The Radius of a circle is a line drawn from the centre to the circum- ference ; as C J). 3t The Diameter of a circle is a line drawn through the centre and termi- nating at both ends in the circumfer- ence ; as ^ i>. 4« Corollary. The radii of a cir- cle, or of equal circles, are equal ; also the diameters are equal, and each is equal to double the radius. 5* An Arc is any part of the circumference ; as A F B. 6* A Chord is the straight line joining the ends of an arc ; as A B. 7.. A Segment of a circle is the part of the circle cut off by a chord ; as the space included by the krc AF B and the chord AB. 8. A Sector is the part of a circle included by two radii and the intercepted arc ; as the space BCD. 9. A Tangent (in geometry) is a line which touches, but does not, though produced, cut the circumference ; as G B. 50 PLANE GEOMETRY. A tangent is often considered as terminating at one end at the point of contact, at the other where it meets another tan- gent or a secant. 10. A Secant (in geometry) is a line lying partly within and partly without a circle ; ^^ G E. A secant is generally considered as terminating at one end where it meets the concave circumference, and at the other where it meets another secant or a tangent. THEOEEM I. 11, In the same circle, or equal circles, equal angles at the cen- tre are subtended by equal arcs ; and, conversely, equal arcs sub- tend equal angles at the centre. Let B and E be equal angles at the centres of the two equal circles ACG and DFH] then the arcs AC and DF are equal. Place the angle ^ on q the angle E ; as they are equal they will coincide ; and as B A and B C are equal to E D and E F, the point A will coincide with D, and the point C with F ; and the arc A G will coincide with D F, otherwise there would be points in the one or the other arc imequally distant from the centre. Conversely. If the arcs A C and D F are equal, the angles B and E are equal. For, if the radius ^ ^ is placed on the radius D E with the point B on E, the point A will fall on i>, ^% A B =^ D E -, and the arc A C will coincide with D F, otherwise there would be points in the one or the other arc unequally distant from the centre ; and as the wcq A C =. D F, the point C will fall on F ; therefore B G will coincide with E F, and the angle B be equal toi;. BOOK m. 51 THEOREM II. 12o in the same or equal circles, equal chords subtend equal arcs ; and, conversely, equal arcs are subtended by equal chords. Let ABC and D E F he two equal circles ; if the arcs AB and D E are equal, the chords A B and D E are equal ; and conversely, if the chords A B and D E are equal, the arcs A B and D E are equal. For, if the centre of the circle A BC h placed on the centre o{ D E F with the point A of the circumference on the point I), if the arcs or the chords are equal, B will fall on E ; and in either case the chords and arcs will coincide, otherwise there would be points in the one or the other circumference unequally distant from the centre. THEOREM III. 13i Angles at the centre vary as their corresponding arcs, IjQtACD, DCE, ECFhe equal an- gles at the centre C ; then the arcs A £>, D E, E F &re equal (11); then the an- gle ACE is double the angle A C D, and the arc A E double the arc A D ; and the angle A C F is three times the angle AC B, and the arc A F three times the arc AB\ and if the angle A C G \9, m times the angle A CD, the arc A G \% m times the arc AD', that is, the angle varies as the arc, or the arc as the angle. 52 PLANE GEOMETRY. 14« Cor. 1. As angles at the centre vary as their arcs, or arcs as their corresponding angles, either of these quantities may be assumed as the measure of the other. The measure of an angle is, then, the arc included between its sides arid described from its vertex as a centre. \5% Cor. 2. As the sum of all the angles about the point C is equal to four right angles (I. 9), one right angle, H C A, is measured by one quarter of the circumference, or by a quadrant. THEOREM IV. 16. The radiics perpendicular to a chord bisects the chord and the arc subtended by the chord. Let C E hQ a* radius perpendicular to the chord ^ ^ ; it bisects the chord A B, and also the arc A E B. Draw the radii C A and C B and the chords A E and E B. As equal oblique lines are equally distant from the perpen- dicular, AD — DB{1.^2); and as E is E a point in the perpendicular to the middle of A B, it is equally distant from A and B (I. 53) ; therefore the chords and hence (12) the arcs A E, E B are equal. 17. Corollary. The perpendicular to the middle of a chord passes through the centre of the circle, and of the arc ; and the radius drawn to the centre of an arc bisects its chord perpendic- ularly. DEFINITIONS. 18. An Inscribed Angle is one whose vertex is in the circum- ference and whose sides are chords ; ^^ABC'm the outer circle 19. An Inscribed Polygon is one whose sides are chords. BOOK III. 53 Thus ABGDEF is inscribed in the outer circle, the circle is said to be circumscribed about the polygon. 20. A Circumscribed Polygon is one whose sides are tangents. Thus ABC D E F \% circumscribed about the inner circle. In this case the circle is said to be inscribed in the polygon. THEOREM V. 21 1 An inscribed angle is measured by half the arc included by its sides. 1st. When one of the sides B D is a. diameter ; then the angle B is measured by half the arc A D. Draw the radius C A, and the triangle AC B is isosceles, C A and C B being radii ; therefore the angle A = B (I. 42). But the exterior angle AC D is equal to the sum of the two angles A and B (I. 39) ; therefore the angle B is equal to half the angle AC D -, the angle AC D is measured by the arc J i> (14) ; therefore the angle B is meas- ured by half the arc A D. 2d. When the centre is within the angle, draw the diameter B C. By the preceding part of the proposition the an- gle ^ 5 (7 is measured by half the arc ^(7, and (7^ Z) by half CD; therefore ABC -\-CBD,ox ABD.is measured by half AC -\- CD, ox half the arc A D. 54 PLANE GEOMETRY. 3d. When the centre is without the angle, draw the diameter B C. By the first part of the proposition the an- gle ABC is measured by half the arc AC, wcidi BBC by half DC; therefore ABC — DBC, or A B D, is measured XjhtliAC — DC, or half the arc A D. 22. Cor, 1. All the angles ABC, ADC, inscribed in the same segment are equal; for each is measured by half the arc A EC 23 • Cor. 2. Every angle inscribed in a semicircle is a right angle; for it is measured by half a semi-circumference, or by a quadrant (15). THEOREM yi. 24. Every eqvMateral polygon inscribed in a circle is regular. Let ABC DEE be an equilateral B ^ .0 polygon inscribed in a circle ; it is also equiangular and therefore regular. For the chords ^ ^, BC, CD, &c. being equal, the arcs A B, BC, CD, &c. are ^qual (12); therefore the arc AB -\- the arc B C will be equal to the arc BC -{- the arc C D, &c, ; that is, the angles B, C, &c. aTe in equal segments ; therefore they are equal (22), and the poly- gon is equiangular and regular. THEOREM VII. 25* An infinitely small chord coincides with its arc. Let ^ ^ be an infinitely small chord ; it coincides with the arc A D B. BOOK III. "Draw the diameter CD perpendicu- lar to the chord A B ; and draw A C and AD ; CAD is a right-angled tri- angle (23) ; therefore (II. 26) CE '.AEz=AE :ED that is, E D i^ the same part of AE that AE h of C E. But ^ ^ is half the infinitely small chord A B (16), and ^ ^ is infinitely small in comparison with C E -, therefore EDh infinitely small in comparison with A E^ that is, the point E is on D, and the chord A B coincides with the axcADB. THEOREM VIII. 26* A circle is a regular ^polygon of an infinite number of sides. If the circumference of a circle is divided into equal arcs, each infinitely small, the infinitely small chords of these arcs would form a regular polygon (24) of an infinite number of sides ; and as each chord would coincide with its arc (25), the polygon would be the circle itself 27. Scholium. It might be supposed that although the dif- ference between each chord and its arc is infinitesimal, yet as there is an infinite number of these differences their sum would not be infinitesimal and ought not to be neglected ; that is, that the perimeter of the polygon and the circumference of the circle differed by a finite quantity. But each chord is in- finitely small compared with the diameter of the circle, or is equal to 7-7; and the difference between each chord and its arc is infinitely smaller than the chord itself, or is equal to — - ; and an infinite number of these differences is equal Inf. X Inf to -p-^ , X Dif. = T-7 ; that is, the difference between the Inf. X Inf. *^ Inf. perimeter of the polygon and the circumference of the circle is infinitesimal. 56 PLANE GEOMETRY. THEOREM IX. 28 • Circumferences of circles are to each other as their radii, or as their diameters ; and the circles themselves as the squares of their radii, or the squares of their diameters. For circles are regular polygons of an infinite number of sides (26) ; and if the circumferences of circles are divided into the same infinite number of arcs, the polygons formed by their chords, that is, the circles themselves, are regular polygons of the same number of sides, and are therefore similar (II. 33) ; and the apothems of the polygons are the radii of the circles ; therefore the circumferences of the circles are as their radii (II. 36), or as twice their radii, that is, as their diameters ; and the circles themselves as the squares of their radii, or the squares of their diameters. 29. Cor. 1. If (7 and c denote the circumferences, i? and r the corresponding radii, and D and d the corresponding diame- ters, we have C : c —R :rz=D : d or C '. R = c : r and C '. D ^= c \ d That is, the ratio of the circumference of every circle to its ra- dius or to its diameter is the same, that is, is constant. The constant ratio of the circumference to its diameter is denoted by TT (the Greek letter p). C Cor. 2. ^ C = 7rJD = 27rR THEOREM X. 31 1 The area of a circle is equal to half the product of its cir- c^imference and its radius. The area of a regular polygon is half the product of its perim- eter and its apothem (II. 37) ; a circle is a regular polygon BOOK III. 57 of an infinite number of sides (26) ; the circumference of the circle is the perimeter of the polygon, and its radius is the apothem ; therefore the area of a circle is half the product of its circumference and its radius. 32. Corollary. If (7 = the circumference, D = the diame- ter, B = the radius, and A =. the area of a circle, we have A = \G X R But (30) (7=:27r^ = 7rZ> Therefore A = lX^'ivRxK = nR'' or A^z^^I) X^ = lirD^ M^ THEOREM XI. C 'TT^f^^ 33* The side of a regular hexagon inscribed in a circle is equal to the radius of the circle. In the circle whose centre is C draw the chord A B equal to the radius ; ^ ^ is the side of a regular hexagon inscribed in a circle. Draw the radii CA and GB; CAB'm an equilateral, and therefore an equiangu- lar triangle ; hence the angle C is equal to one third of two right angles, or one sixth of four right angles : that is, the arc ^ ^ is one sixth of the whole circumference, or the chord A B the side of a regular hexagon inscribed in the circle (12 and 24). 34. Corollary. The chord of half the arc A B would be the side of a regular dodecagon ; the chord of one quarter of the arc A B, the side of a regular polygon of twenty-four sides j and so on. 3* 68 PLANE GEOMETRY. PROPOSITION XII. PROBLEM. 35* The chord of an arc given to find the chord of half the arc. Let AB he the given chord, A D the chord of half the arc ADB, and R denote the radius. Draw the diameter DF, the radius AC, and the chord A F. The triangle ADF is right angled at A (23) ; then (II. 25) DF:AI) = AD :DF or AD'' = DFXI>E='1I{XDE and AD = sJYYy^WE Now DE = DG —GE = R — GE and (11. 28) GE = \l A G^ — A E' z=z sjE^ — A FA therefore DE = R — sjR^ — A E^ Substituting this value of D E in AD = sI'iYyODE we have AD = Sj2 R^ — 2R s/R' — A E^ 36* Gor. 1. If (7 denote the given chord, c the chord of half the arc, the equation becomes = y 2 i?2 _ 2 i? Jr^ 4 = V2 R^ —Rsj4. R^ G' 37. Gor. 2. If the diameter D, that is, 2 R, is unity, the equation in (36) becomes c = Vi - i sjr^^G'^ BOOK III. 59 PROPOSITION XIII. • PROBLEM. 38t To find the arithmetical value of the constant n. From (30) C = 7rD; ifZ) = l, this equation becomes C = tt. If then we can find the circumference of a circle whose diame- ter is unity, we shall have the value of tt. If the diameter is unity, radius is one half, and the side of a regular hexagon inscribed in the circle is one half (33), and the perimeter of the hexagon is 6 X 2 ^^^ ^• As the diameter is unity, and the side of the inscribed hexa- gon one half, we can find the side of the regular inscribed dodec- agon from the equation in (37) : = V/.5 — .433 = ^."067 = .2588+ The perimeter of the inscribed dodecagon is therefore 12 X .2588+ =3.105+. By using the side of the dodecagon = .2588+, as C, or .067 = C% from the same equation we can find the side of a regular inscribed polygon of twenty-four sides : -.483 = V^.0T7 = . 13038 The perimeter of the inscribed polygon of twenty-four sides is therefore 24 X .13038 = 3.129. By continuing this process we approximate to the circumfer- ence, that is, to the value of tt. 60 PLANE GEOMETRY. 39. Scholium. By other more expeditious methods the value of TT has been found accurately to two hundred and fifty places of decimals. For practical purposes it is sufi&ciently accurate to call 7r = 3.14159. PRACTICAL QUESTIONS. 1. What is the circumference of a circle whose radius is 10 feet ? '^« 2. What is the diameter of a circle whose circumference is 57 rods ? 3. What is the area of a circle whose radius is 40 feet ? 4. What is the area of a circle whose circumference is 18 inches ? 5. What is the circumference of a circle whose area is 116 square feet ? 6. The radii of two concentric circles are 40 and 54 feet ; what is the area of the space bounded by their circumferences ? 7. A has a circular lot of land whose diameter is 95 rods, and B a simi- lar lot whose area is 750 square rods ; compare these lots. 8. What is the difference between the perimeters of two lots of land each containing an acre, if one is a square and the other a circle ? 9. What is the area of a square inscribed in a circle whose area is a square metre ? 10. What is the area of a regular hexagon inscribed in a circle whose area is 567 square feet. 11. If a rope an inch in diameter will support 1,000 ^inds, what mjpist be the diameter of a rope of like material to support 4, (M^ pounds ? . j*' 12. If a pipe an inch in diameter will fill a cistern in 25 minutes, how long will it take a pipe 5 inches in diameter ? 13. If a pipe an inch in diameter will empty a cistern in an hour, how long will it take this pipe to empty the cistern if there is another pipe one third of an inch in diameter through which the fluid runs in ? Ans. 67^ minutes. 14. If a pipe 3 inches in diameter will empty a cistern in 3 hours, how long will it take the pipe to empty the cistern if there are 3 other pipes each an inch in diameter through which the fluid runs in. Ans. 4^ hours. BOOK m. 61 EXERCISES. The following Theorejd!s, depending for their demon§tpatton upon those already demonstrated, ace introd«ced as exercises for the pupil. In some o^them reiferenc.es"are made to tlie propositions upon which the demonstraS^srfTJepends. They are not connected with the prop- ositiojaeritt tlje 'following books, and can»i^e omitted if thonirht best. . - "' ^ «•-<■* circumference. . , .,,.J XiPr^EVsery. diameter bisects the c^e a^ the circi 41. ' III l iikl ii limM 1 1 11 ^ l lf llllli r ( i f two p^n^. (4.) (I. 51.) 42. The diametei^'lir^reektei^jJl^Ii, anx.^fe 43. In the same or equal circles, when the sum of the arcs is less than a circumfer- ence, the greater arc is subtended by the greater chord; and, conversely, the greater At chord is subtended by the greater arc. Draw ^ a (21.) (1.47.) What is the case when the sum of the arcs is greater than a circumference ? 44. In the same circle equal chords are equally distant from the centre ; and of two unequal chords the greater is nearer the centre. 45. The shortest and the longest line that can be drawn from any point to a given circumference lies on the line that passes from the point to the centre of the circle. 46. Two parallels cutting the circumference of a circle intercept equal arcs. 47. A straight line perpendicular to a diameter at its extremity is a tangent to the circumference. Draw GB. (1.51.) ^! 48. The lines joining the extremities of two diameters are parallel. 49. If the extremities of two chords are joined, the vertical, or opposite, triangles thus formed are similar. 62 PLANE GEOMETRY. 50« If twQ circumferences cut each other, the chord which joins their points of intersection is bisected at right angles by the Hne join- ing their centres. (17.) 51* If two circumferences touch each other, their centres and point of contact are in the same straight hne, perpendicular to the tangent at the point of contact. (47.) 52» The distance between the centres of two circles whose cir- cumferences cut one another, is less than the sum, but greater than the difference, of their radii. 53» Every angle inscribed in a segment greater than a semicircle is acute ; and every angle inscribed in a segment less than a semicir- cle is obtuse. (21.) 54« The angle made by a tangent and a chord is measured by half the included arc. Draw the diameter A B. (47.) (21.) 55 1 The angle formed by two chords cut- ting each other within the circle is measured by half the sum of the arcs intercepted by its sides and by the sides of its vertical angle. Join B C (in lower figure). (21.) 56. By moving the point of intersection of the two chords, show that (14) and (21) can be deduced from (55). 57 • The segments of two chords inter- secting within a circle are reciprocally pro- portional ; that is,AE: BE = ED: EG. Join AD,BG. (21.) (II. 20 ) 58. The opposite angles of a quadrilateral inscribed in a circle are supplementary. (21.) 59. A quadrilateral whose opposite angles are supplementary, and no other, can be mscribed in a circle. 60. Lines through the point of contact »f two circumferences that are tangent to each other are cut proportionally by these circumfer- ences. (22.) (II. 20.) 61 . The area of a sector is equal to half the product of its arc by the radius of the circle. (31.) BOOK III. 63 62* Show how to find the area of a segment of a circle. 63* The area of a circumscribed polygon is equal to half the pro- duct of its perimeter by the radius of the circle. 64* A tangent is a mean proportional between a secant drawn from the same point and the part of the secant without the circle. Join A D, DC. (54 ; 21.) (II. 57.) 65* The angle formed by two secants, two tangents, or a secant and a tangent cutting each other without the circle, is measured by half the difference of the in- tercepted arcs. Join CF. (I. 39.) (21.) 66i By moving the point of intersec- tion, show that (21) can be deduced from (65). Show also that (46) can be deduced from (65). 67» Two secants drawn from the same point are to each other inversely as the parts of the secants without the circle. Join GF, D G. (21.) (II. 57.) 68. Two tangents drawn to a circumference from the same point are equal. Join B E. Figure in (66.) (54.) 69« A perpendicular from a circumference to the diameter is a mean proportional be- tween the segments of the diameter. Join AB,BG. (23.) (II. 26.) 70. If from one end of a chord a diame- ter is drawn, and from the other end a per- pendicular to this diameter, the chord is a mean proportional be- tween the diameter and the adjacent segment of the diameter. Join A B (23.) (11.25.) Tl» The sum of the opposite sides of a circumscribed quadrilat- eral is equal to the sum of the other two sides. (68.) BOOK IV. GEOMETRY OF SPACE. PLANES AND THEIR ANGLES. DEFINITIONS. 1 , A straight line is perpendicular to a plane when it is per- pendicular to every straight line of the plane which it meets. Conversely, the plane, in this case, is perpendicular to the line. The /oa^ of the perpendicular is the point in which it meets the plane. 2. A line and a plane are parallel when they cannot meet though produced indefinitely. 3« Two planes are parallel when they cannot meet though produced indefinitely. THEOREM I. 4( A plane is determined, 1 st. By a straight line and a point without that lin^ ; 2d. By three points not in the same straight line ; 3d. By two intersecting straight lines. 1st. Let the plane MN, pass- ing through the line A B, turn upon this line as an axis until it contains the point C ; the posi- tion of the plane is evidently de- termined; for if it is turned in either direction it will no longer contain the point (7. BOOK IV. 65 2d. If three points, A, B, C, not in the same straight line are given, any two of them, as A and B, may be joined by a straight line ; then this is the same as the 1st case. 3d. If two intersecting lines A B, AC are given, any point, C, out of the line A B can be taken in the line A C ; then the plane passing through the line A B and the point C contains the two lines A B and A C, and is determined by them. 5t Corollary. The intersection of two planes is a straight line ; for the intersection cannot contain three points not in the same straight line, since only one plane can contain three such points. THEOREM li. 6* Ohlique lines from a point to a plane equally distant from the perpendicular are equal ; and of two ohlique lines unequally distant from the perpendicular^ the more remote is the greater. Let AC, A D be oblique lines drawn to the plane if iV at equal distances from the perpendicular AB: 1st. A C= AB; {or the trian- gles ^^(7, ABB are equal (I. 40). 2d. Let ^ i^ be more remote. From BF cut off BE = BB and draw A E ; then A F "^ AE (L 51); and AE = AB = AC; therefore AF> ABor AC. 7. Cor. 1. Conversely, equal oblique lines from a point to a plane are equally distant, from the perpendicular; therefore they meet the plane in the circumference of a circle whose cen- tre is the foot of the perpendicular. Of two unequal lines the greater is more remote from the perpendicular. 8. Cor. 2. The perpendicular is the shortest distance from a point to a plane. d ¥ \ /' 1 -• '• / B \\e\^ / / \ A/ ^ N 66 GEOMETKY OF SPACE. THEOREM III. 9. The intersections of two parallel planes vrith a third plane are parallel. Let A B and C D he the intersec- B tions of the plane A D with the par- \ /\ \ allel planes M N and P Q; then A B \ / \ \ and CD are parallel. \ \ For the lines A B and G D cannot \ \^ meet though produced indefinitely, \ \ / \ since the planes M N and P Q in which \ \/ \ they are cannot meet ; and they are in C ^ the same plane A D ; therefore they are parallel. 10. Corollary. Parallels intercepted between parallel planes are equal. For the opposite sides of the quadrilateral A D be- ing parallel, the figure is a parallelogram ; therefore AC =.B D. THEOREM IV. lit If two angles not in the same plane have their sides paral- lel and similarly situated, the angles are equal and their planes parallel. I Let ABC and DE F he two angles A/" in the planes M iV and P Q, having their sides A B, B C respectively paral- lel to JDUf E F, and similarly situated ; then 1st. Since BA has the same direc- p tion as E D, and B C the same as \ ^ E F, the difference of direction of ^ ^ \ D^ ^^ and B C must be the same as the differ- ence of direction of E D and E F; that is, angle B = angle E. 2d. The planes of these angles are parallel. For, since two intersecting lines determine a plane (4), the plane of the lines A B and B C must be parallel to the plane of the lines D E and E F, Sis AB and B C are respectively parallel to D E and E F. BOOK IV. 67 THEOREM V. 12. // two straight lines are ciU hy parallel planes, they are divided proportionally. Vc^\ \ , V _ Vjr\ M i-4 N Let A B and C D he cut by the parallel M planes M N, PQ, and R S, in the points A, E, B, and C, F, D-, then AE:EB = GF:FD ^ For, drawing A D meeting the plane P Q in 6r, the plane of the lines A B and AD ^ cuts the parallel planes PQ and ES in E G and B D ; therefore E G and B D are parallel (9), and we have (XL 16) AE :EB=:AG : GD The plane of the lines A D and G D cuts the parallel planes MN and PQ'mAG and GF; therefore AG'm parallel to GF', and we have AG :GD=zGF :FD Hence we have (Pn. 11) AE :EB=:GF '.FD bo GEOMETRY OF SPACE. EXERCISES. The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the prop- ositions in the following books, and can be omitted if thought best. 13* An infinite number of planes can pass through a given line. (4.) 14« There can be but one perpendicular from a point to a plane. 15« A line perpendicular to each of two lines at their point of intersection is perpendicular to the plane of these lines. (4.) (I. 76.) 16( Parallel lines are equally inclined to the same plane. 17« State the converse of (16). Is it true? 18* Lines parallel to a line in a given plane are parallel to the plane. 19. State the converse of (18). Is it true ? 20* Parallel planes are equally inclined to the same straight line. 21 1 State the converse of (20). Is it true? BOOK V. POLYEDKONS. DEFINITIONS. 1. A Polyedron is a solid bounded by planes. The bounding planes are called faces ; their intersections, edges ; the intersections of the edges, vertices. 2t The Volume of a solid is the measure of its magnitude. It is expressed in units which represent the number of times it contains the cubical unit taken as a standard. 3* Equivalent Solids are those which are equal in volume. 4* Similar Solids are those whose homologous lines have a constant ratio. {Corollary.) It follows that similar solids are bounded by the same number of similar polygons similarly situated. PRISMS AND CYLINDERS. 5* A Prism is a polyedron two of whose faces are equal polygons having their homologous sides parallel, and the other faces parallelograms. (Corollary.) The lateral edges are equal. The equal parallel polygons are called bases; as ^^ and CD. 6. The Altitude of a prism is the perpendic- ular distance between its bases ; SiS B F. 70 SOLID GEOMETRY. 7. A Right Prism is one whose other faces are perpendicular to its bases, faces are rectangles. {Corollary ) Its lateral 8. A prism is called triangular, quadrangular, or pentagonal, according as its base is a triangle, a quadrangle, or a pentagon; and so on. 9. A Parallelopiped is a prism whose bases are parallelograms. {Corollary.) It follows that all its faces are parallelograms. A Right Parallelopiped is a right prism whose bases are parallelograms. 10. A Rectangular Parallelopiped is a right parallelepiped whose bases are rectangles. {Corollary.) All its faces are rectangles. 11. A Cube is a parallelopiped whose faces are all squares. {Corollary.) It follows that its faces are all equal, and the parallelopiped rectangular. . 12. A Cylinder is a right prism whose bases are regular polygons of an infinite number of sides, that is, whose bases are circles. A cylinder can be described by the revolution of a rectangle about one of its sides which remains fixed. The side oppo- site the fixed side describes the convex surface, and the other two sides the two circular bases. Thus the rectangle AB C D revolving about B C would describe the cylinder, the side A D the convex sur- face, and AB, D C the circular bases. 13. The Axis of a cylinder is the straight line joining the centres of the two bases ; or it is the fixed side of the rectangle whose revolution describes the cylinder ; as 5 (7. BOOK V. 71 THEOREM I. 14, The convex surface of a right prism is equal to the perime- ter of its 6ase multiplied hy its altitude. Let ^ ^ be a right prism ; its convex surface is equal to FG-\-GH-\-HI^IK-\-KF multiplied by its altitude A F. For the convex surface is equal to the sum of the rectangles AG, BH, CI, &c. The area of the rectangle AG = FG X ^F; the area of BH=zGHXBG] of CI=HIx C H -, and so on. But the edges AF, B G, C H, &c. are equal to each other and to the altitude of the prism ; and the bases of these rectangles together form the perimeter of the prism. Therefore the sum of these rectangles, that is, the con- vex surface of the right prism, is equal to the perimeter of its base multiplied by its altitude. 15* Corollary. As a cylinder is a right prism (12), this demonstration includes the cylinder. If, then, R = the radius of the base, and A = the alti- tude of a cylinder, the convex surface = 2 tt BA. THEOREM II. 16t The sections of a prism made hy parallel planes are equal polygons. Let the prism ^ JjTbe intersected by the par- allel planes LN and Q S ; then LN and ^*S' are equal polygons. For LM, M N, NO, &c. are respectively parallel to Q R, R S, ST, &c. (IV. 9), and similarly situated ; therefore the angles L, M, N, 0, P are respectively equal to the angles Q, R, S, T, U (IV. 11); and the polygons LN and QS are mutually equiangu- lar. Also the sides LM, M N^ NO, &c. are 72 SOLID GEOMETRY. respectively equal to Q R, RS, ST, &c. (I. 62). Therefore the polygons, being mutually equiangular and equilateral, are equal (11. 6). 17t Cor. 1. A section made by a plane parallel to the base is equal to the base. 18. Cor. 2. A section of a cylinder made by a plane paral- lel to the base is a circle equal to the base. THEOREM III. 19. Prisms equivalent. having equivalent bases and equal altitudes are Let A C and FH be two prisms having equal altitudes and their bases B O, G H equivalent ; the prisms are equivalent. Let DU and IK he sections made by planes respectively par- allel to the bases BG and GH; these sections are respectively equal to the bases (17); there- fore the section DE m equivalent to IK, at whatever distance from the base either may be. If, therefore, the planes of these sec- tions move, remaining always parallel to the bases, as the sec- tions will always be equivalent, it is evident that in moving over an equal length of altitude the sections will move over equal volumes ; therefore, as the altitudes are equal, the prisms are equivalent. 20. Corollary. Any prism is therefore equivalent to a rec- tangular prism having an equivalent base and an equal altitude. BOOK V. 73 THEOREM IV. ^ 21 . The volume of a rectangular parallelopiped is equal to the product of its three dimensions. Let -4 2> be a rectangular parallelopiped ; then its volume is equal toBCxBExBA. Suppose B Fj the linear unit, is contained in B C four times, in B E five times, and in B A seven times ; then dividing B C, B E, B A respectively into four, five, and seven equal parts, and passing planes through the several points of division parallel to the sides of the parallelopiped, there will be formed a number of cubes equal to each other (19), and each equal to the cube whose edge is the linear unit. It is evident also that the whole number of cubes is equal to the product of the three dimensions, or 4 X S X 7 = 140. This demonstration is applicable, what- ever the number of units in the linear dimensions may be. Therefore the volume of a rectangular parallelopiped is equal to the product of its three dimensions. 22t Scholium. If the three dimensions are incommensur- able, the linear unit can be taken infinitely small, that is, so small that the remainder will be infinitesimal and can be neg- lected. 23* Cor. 1. As the base is equal to B C X B E, the volume of a rectangular parallelopiped is equal to the product of its base by its altitude. 24* Cor. 2. The volume of a cube is equal to the cube of its edge. SOLID GEOMETRY. THEOREM V. 25* The volume of any prism is equal to the product of its base by its altitude. For any prism is equivalent to a rectangular parallelopiped, having an equivalent base and the same altitude (20) ; and the volume of the equivalent rectangular parallelopiped is equal to the product of its base by its altitude ; therefore the volume of any prism is equal to the product of its base by its altitude. 26. Corollary. As a cylinder is a right prism, this demon- stration includes the cylinder. If, therefore, R = the radius of base, A = the altitude, and V ==. the volume of a cylinder, THEOREM VI. 27. Similar prisms are as the cubes of their homologous lines. Let A D and B R he similar prisms whose altitudes are IK and MN. Let V represent the vol- ume of A JD, and v the volume of Uff; then V:v = IK^ : MN^ = AC^: UG^ = CO'' : GF'' For (25) V=CD X IK and v=GII X MN, therefore V'.v=CDXlK'. GHXMN But(n. 31) CD :GHz=CO'' and (4) IK:MN=CO\ Multiplying the last two proportions together we have CDXIK.GHX MN=CO^:GI^ therefore (Pn. 11) V -. v = C O"" -. G P"" But in similar solids homologous lines have a constant ratio (4) ; therefore F : v as the cubes of any homologous lines. BOOK V. 75 PYRAMIDS AND CONES. DEFINITIONS. 28. A Pyramid is a polyedron bounded by a polygon called the base, and by triangular planes meeting at a common point called the vertex. 29. A pyramid is called triangular, quad- ^ rangular, 'pentagonal^ according as its base is a triangle, a quadrangle, or a pentagon ; and so on. 30. The Altitude of a pyramid is the perpendicular distance from its vertex to its base ; as ^ ^. 31 . A Bight Pyramid is one whose base is a regular polygon and in which the per- pendicular from the vertex passes through the centre of the base. 32. The Slant Height of a right pyramid is the perpendicu- lar distance from the vertex to the base of any one of its lateral faces ; as ^ C. 33. A Cone is a right pyramid whose base is a regular polygon of an- infinite number of sides, that is, whose base is a circle. A cone can be described by the rev- olution of a right triangle about one of its sides which remains fixed. The other side describes the circular base, and the hypoth- enuse the convex surface. Thus the right triangle ABC revolving about A B would describe. the cone, BC the base,, and the h^^pothenuse A C the convex surface. 34. The Axis of a cone is the line from the vertex to the centre of the base ; or it is the fixed side of the right triangle whose revolution describes the cone ; as A B, » 76 SOLID GEOMETRY. 3.5* Corollary. The axis of a cone is perpendicular to the base, and is therefore the altitude of the cone. 36* A Frustum of a pyramid is a part of the pyramid included between the base and a plane cutting the pyramid parallel to the base ; aa D £!. 37 • The Altitude of a frustum is the per- pendicular distance between the two parallel planes or bases ; 38t The Slant Height of a frustum of a right pyramid is the perpendicular distance between the parallel edges of the as 6^ a f THEOREM YII. 39* If « pyramid is cut by a plane parallel to its hose, 1st. The edges and altitude are divided proportionally ; 2d. The section is a polygon similar to the base. Let A-BCDEF be a pyramid whose al- ^ titude is A N, cut by a plane G I parallel to the base ; then 1 St. The edges and the altitude are di- vided proportionally. For suppose a plane passed through the vertex A parallel to the base; then the edges and altitude, being cut by three parallel planes, are divided proportion- ally (IV. 12), and we have AB'.AG = A C -.AH^AD '.AI=AN'.AM 2d. The section GI '\% similar to the base BD. For the sides of G I are respectively parallel to the sides of B D (IV. 9), and similarly situated ; therefore the polygons GI, BD are mutually equiangular. Also, as (rX is parallel to BF, BOOK V. 77 and LK to F E, the triangles ^ ^i^and AG L qxq similar, and the triangles A F E and ALK ) therefore GL'.BF=AL'.AF,2.ndiLK:FE=AL'.AF Therefore GL : BF=zLK: FE In the same manner we should find LK:FE=zKI'.ED = IH'.DC,kG, Therefore the polygons G I and B D are similar (II. 19). 40t Corollary. A section of a cone made by a plane parallel to the base is a circle. y^ THEOREM VIII. 41* The convex surface of a right pyramid is equal to the perimeter of its base multiplied by half its slant height. A Let ^-5 GB EF be a right pyramid whose slant height \^ AH\ its convex surface is equal to B G ■\- G B -\- B E -\- E F -^^ F B multiplied by half of A H. The edges AB, AG, AD, AE, A F, be- ing equally distant from the perpendicular A J^ (11. 34), are equal (IV. 6) ; and the bases BG, G D, BE, &c. are equal; there- fore the isosceles triangles ABG, AGB, ABE, (fee. are all equal (I. 48) ; and their altitudes are equal. The area oi ABG \^ BG X J ^ // (II. 11); oi AG B'mG By, J ^ -^ ; and so on. Therefore the sum of the areas of these triangles, that is, the convex surface of the right pyramid, m{BG-\-GD-{-DE-^EF+FB)\AH. 42* Gorollary. As a cone is a right pyramid (33), this dem- onstration includes the cone. If, therefore, R = the radius of the base, and aS' = the slant height of a cone, its convex surface = 2 n R\ S =:. n R S If a plane parallel to the base and bisecting the altitude be 78 SOLID GEOMETRY. drawn, as the section will be a circle (40) with a radius and cir- cumference one half the radius and circumference of the base, therefore, if r' = the radius of this section, the convex surface = 2irr' S , ' THEOREM IX. 43t The convex surface of a frustum of a right pyramid is equal to the sum of the perimeter of its two bases multiplied hy half its slant height. Let G D he the frustum of a right pyra- B mid ; its convex surface is equal to G H -\- /^ i "-- / HI-\-IK+KL -\-LG+ BC+CD fYpf\ + Di:-\-BF+ FB multiplied hy half bU'T'^A^A MN. \ V'^D N\i \ X The lateral faces of a frustum of a right ^ ^ Y pyramid are equal trapezoids (39 ; 11. 6) ; and their altitudes are all equal. The area oi G G (II. 14) is (GH+ BG) X IMN; of HD is {HI + G D) X \MN', and so on. Therefore the sum of the areas of these trapezoids, that is, the convex surface of the frustum of the right pyra- mid, is GH-\-HI^IK -\- KL^ LG -\-BG + GD + DE-^- EF -\- FB multiplied by half M N. 44* Gor. 1. If the frustum is cut by a plane parallel to its two bases, and at equal distances from each base, this plane will bisect the edges G B^ HG, ID, &c. (39) ; and the area of each trapezoid is equal to its altitude multiplied by the line joining the middle points of the sides which are not parallel (II. 15). Therefore the convex surface of a frustum of a right pyramid is equal to the perimeter of a section midway between the bases multiplied by its slant height. 45* Cor. 2. As a cone is a right pyramid (33), this demon- stration includes the frustum of a cone. If, therefore, R and BOOK V. 79 r = the radii of the two bases of the frustum of a cone, and AS'=its slant height, its convex surface =z(2irE-{-2irr)^S={irIi-\-irr)S If r' = the radius of a section midway between and parallel to the bases, the convex surface = 2irr' S ^ THEOREM X. 40* If two pyramids having equal altitudes are cut by planes parallel to their bases and at equal distances from their vertices^ the sections are to each other as their bases. Let A-BC DBF Sind G-UIK be two pyra- mids of equal altitudes AT, GW, cut by the planes LM N P and ^i?AS' parallel respectively to the bases and at equal distances from the vertices A and G, then LMNOP:QRS=:BCDEF.HIK For as the polygons LMNOP and BCDEF are similar (39) LMNOP : BCDEF = LP' : BF^ = aV : AB^= AV^ : If* In like manner QMS: HIK=z G'y'' :G~W^ But SLS AV = G Y ?ind A T = GW til GrGiOTP LMNOP \BCDEF=QRS',HIK or (Pn. 16) LMNOP '.QRS=BCDEF '.HIK 47i Corollary. If two pyramids have equal altitudes and equivalent bases, sections made by planes parallel to their bases and at equal distances from their vertices are equivalent. /, 80 ^ SOLID GEOMETRY. THEOREM XI. 48« Pyramids having equivalent bases and the same altitude are equivalent. Let A-BC DBF sind G-H I K be pyramids having equivalent bases and equal altitudes ; then the two pyramids are equivalent. For, if at equal dis- tances from the vertex sections are formed by planes parallel respective- K ly to their bases, these sections are equivalent (47). If now the planes forming these sections be supposed to move, remaining always parallel to the bases, and each keeping the same distance from^ the vertex as the other, these sections, always being equiv- alent to each other, will move over equal volumes ; therefore, as the altitudes are equal, the pyramids must be equivalent. THEOREM Xn. 49. A triangular •pyramid is one third of a triangular prism of the same base and altitude. Let C-D E Fhe a triangular pyramid and A B C-D E F be a triangular prism on the same base D E F -, then C-D E F is one ihirdio^ ABC-DEF, Taking away the pyramid C-D E F i\iQVQ remains the quadrangular pyramid whose ver- tex is C and base the parallelogram ABED. Through the points A, C, E pass a plane ; it will divide the pyramid C-A BED into two triangular pyra- mids, which are equivalent to each other (48), since their bases are halves of the parallelogram ABED, and they have the BOOK V. / 81 same altitude, the- perpendicular from their vertex C to the base ABED. But the pyramid C-A B E, that is, E-A B C, is equivalent to the pyramid C-DEF, as they have equal bases ABC and D E F, and the same altitude (48). Therefore the three pyramids are equivalent and the given pyramid is one third of the prism. 50. Corollary. The volume of a triangular pyramid is equal to one third the product of its base by its altitude. THEOREM XIII. 51* The volume of any pyramid is equal to one third of the product of its base hy its altitude. Let A-BC D E F be any pyramid; its volume is equal to one third the product of its base BCBEFhy its altitude A K Planes passing through the vertex A and the diagonals of the base B £>, BE, will divide the pyramid into triangular pyramids whose bases together compose the base of the given pyramid and which have as their common altitude A N, the altitude of the given pyramid. The volume of the given pyramid is equal to the sum of the volumes of the several tri- angular pyramids, which is equal to one third of the sum of their bases multiplied by their common altitude; that is, is equal to one third of the product of the base BCD FE by the altitude A N. 52% Cor. 1. As a cone is a right pyramid (33), this demon- stration includes the cone. A cone, therefore, is one third of a cylinder, or of any prism, of equivalent base and the same altitude. If 7? = radius of the base, A = the altitude, and V = the volume of a cone, V = ^n B^A. 53« Cor. 2. The ratio of similar pyramids to one another is the same as that of similar prisms ; that is, as the cubes of homologous lines. 82 SOLID GEOMETRY. THE SPHERE. DEFINITIONS. 54 • A Sphere is a solid bounded by a curved surface, of which every point is equally distant from a point within called the centre, A sphere can be described by the revolution of a semicircle about its diameter which remains fixed. 55i The Radius of a sphere is the straight line from the cen- tre to any point of the surface. 56t The Diameter of a sphere is a straight line passing through the centre and terminating at either end at the surface. 57. Corollary. All the radii of a sphere are equal ; all the diameters are equal, and each is double the radius. / I THEOREM XIV. 58. Every section of a sphere made hy a plane is a circle. Let ABB be a section made by a plane cutting the sphere whose centre is G \ then is J ^ i) a circle. Draw C E perpendicular to the plane, and to the points A, D, F, where the plane cuts the surface of the sphere, draw CA, CD, C F. As C A, CD, C F are radii of the sphere they are equal, and are therefore equally distant from the foot of the perpendicular C E (IV. 7). Therefore E A, ED, EF are equal, and the section A B D is a circle whose centre is E. 59. Corollary. If the section passes through the centre of the sphere, its radius will be the radius of the sphere. 60. Definition. A section made by a plane passing through the centre of a sphere is called a great circle. A section made by a plane not passing through the centre is called a small circle. BOOK V. 83 THEOREM XV. 61 • The surface of a sphere is equal to the product of its diam- eter hy the circumference of a great circle. Let ABCDEF be the semicircle by whose revolution about the diameter A F, the sphere may be described; then the surface of the sphere is equal to the diameter A F multi- plied by the circumference of the circle whose radius is 6^ ^4, or = ^ ^ X circ. G A. Let A B C D E F\)Q a regular semi-decagon inscribed in the semicircle. Draw G per- pendicular to one of its sides, as B C. Draw B K^ OP, C L, D M, EN perpendicular to the diame- ter A F, and B H perpendicular to G L. The surface described by ^ C is the convex surface of the frustum of a cone, and is equal to BG X circ. PO (45). But the triangles ^(7 i7 and P G are similar (IL 21) ; therefore BG :BHoyKLz=GO . P or (IIL 28) BG : KL = circ. G : circ. P .'. BGX circ. PG — KLX circ. G G That is, the surface described hy B G is equal to the altitude KL multiplied by circ. G 0, or by the circumference of the cir- cle inscribed in the polygon. In like manner it can be proved that the surfaces described by ^ ^, G B, D E, and E F are respectively equal to their altitudes A K, LM, M N, and N F multiplied by circ. G 0. Therefore the entire surface described by the semi-polygon will be equal to {AK^KL + LM + MN-\-NF)c\rc. GO = AFX circ. GO This demonstration is true, whatever the number of sides of the semi-polygon ; it is true, therefore, if the number of sides is infinite, in which case the semi-polygon would coincide with the semicircle ; and the surface described by the semi-polygon would be the surface of the sphere, and the radius of the in- 84 SOLID GEOMETRY. scribed polygon would be the radius of the sphere. Therefore we have the surface of the sphere equal to ^1 i^ X circ. G A 62* Corollary. Let S = the surface of the sphere, C = the circumference, Ji = the radius, J) = the diameter, then we have (III. 30) (7 = 2 tt 7?, or tt /> Therefore S=2'itR X 2 i? = i'lrR^ or tt D^ That is, the surface of a sphere is equal to the square of its diame- ter multiplied hy 3.14159. THEOREM XVI. 63* The volume of a sphere is the product of its surface hy one third of its radium. A sphere may be conceived to be composed of an infinite num- ber of pyramids whose vertices are at the centre of the sphere, and whose bases, being infinitely small planes, coincide with the surface of the sphere. The altitude of each of these pyramids is the radius of the sphere, and the sum of the surfaces of their bases is the surface of the sphere. The volume of each pyra- mid is the product of the area of its base by one third of its altitude, that is, of the radius of the sphere (51) ; and the vol- ume of all the pyramids, that is, of the sphere, is, therefore, the product of the surface of the sphere by one third of its radius. 64. Cor. 1. Let V =z the volume of the sphere, and R, D, and aS^ the same as in (62). Then, as (62) F=47ri?2 X \R=z^itR\ or Jtt/)'' That is, the volum.e of a sphere is the cube of the diam,eter multi- plied hy .5236. 65. Cor. 2. As in these equations f it and \ it are constant, the volumes of spheres vary as the cubes of their radii, or as the cubes of their diameters. -rf • ^ BOOK V. - 85 \^)ji PRACTICAL QUESTIONS. ^ 1. How many square feet in the convex -8urf3,c«"of a rfght prism whose altitude is 2 feet, and whose base is o. regular hj^agon of which each side is 8 inches long ? How many square feet in the whol^surface ? ^^ 2. The radius of the base of a cylinder is 6 inches, and its altitude 3 feet ; how many square feet in the whole surface ? /^~ / y / 3. What is the number of feet in the bounding planes of a cube whose edge is 5 feet ? The number of solid feet in the cube ? - /^fiT 4. What is the number of feet in the bounding planes of a right par- allelopiped whose three dimensions are 4, 7, and 9 feet ? The number of cubic feet in the parallelopiped ?,-~-5 A- \_ 5. What is the number of cubic feet in the right prism whose dimen- sions are given in the first example ? 6. What is the number of cubic feet in the cylinder whose dimensions are given in the second example ? 7. The altitude of a prism is 9 feet and the perimeter of the base 6 feet. What is the altitude and perimeter of the base of a similar prism one third as great ? 8. What is the ratio of the volumes of two cylinders whose altitudes arc as 3 : 6, if the cylinders are similar ? What, if the bases are equal ? What, if the bases are as 3 : 6 and the altitudes equal ? r"^ How many square feet in the convex surface of a right pyramid whose slant height is 3 feet, and whose base is a regular octagon of which each side is 2 feet long ? 10. How many square feet in the convex surface of a cone whose slant height is 5 feet and whose base 4ias a radius of 2 feel. } How many square feet in the whole surface ? 11. How many cubic feet in a right quadrangular pyramid whose alti- tude is 10 feet, and whose base is 3 feet square ? 12. How many cubic feet in the cone whose dimensions are given in the tenth example ? 13. The slant height of a frustum of a right pyramid is 6 feet, and the perimeters of the two bases are 18 feet and 12 feet respectively ; what is the convex surface of the frustum ? 14. What would be the slant height of the pyramid whose frustum is given in the preceding example ? 15. What is the whole surface of a frustum of a cone whose altitude i.s 8 feet, and of whose bases the radii are 11 feet and 5 feet respectively ? 86 SOLID GEOMETKY. 10. The altitude of a pyramid is 25 feet, and its base is a rectangle 8 feet by 6 ; how many cubic feet in the pyramid ? 17. The altitude of a cone is 20 feet, and the radius of its base 5 feet ; how many cubic feet in the cone ? 18. How many cubic feet in a frustum of the cone given in the preced- ing example, cut ofi' by a plane 5 feet from the base ? 19. How far from the base must a cone whose altitude is 12 feet be cut off so that the frustum shall be equivalent to one half of the cone ? 20. How many square feet in the surface of a sphere whose radius is 6 feet ? 21. How many cubic feet in a sphere whose radius is 8 feet ? 22. What is the ratio of the volumes of two spheres whose radii are as 4 : 8 y 23. Are spheres always similar solids ? Are cones ? 24. What is the least number of planes that can enclose a space ? EXERCISES. 66. The convex surfaces of right prisms of equal altitudes are as the perimeters of their bases. (14.) 67 • The opposite faces of a parallelepiped are equal and parallel. 68. The four diagonals of a parallelepiped bisect each other. 69. A plane passing through the opposite edges of a parallelepiped bisects the parallelepiped. • 70. In a rectangular parallelepiped the diagonals are equal ; and the square of each is equal to the sum of the squares of the three dimensions. 71. In a cube the square of a diagonal is three times the squa^re of an edge. 72. Prisms are to each other as the products of their bases by their altitudes. (25.) 73. Prisms with equivalent bases are as their altitudes; with equal altitudes, as their bases. (72.) BOOK V. 87 74. Polygons formed by parallel planes cutting a pyramid are as the squares of their distances from the vertex, (39 ; II. 31.) 75. Pyramids are to each other as the products of their bases by their altitudes. (51.) 76. Pyramids with equivalent bases are as their altitudes ; with equal altitudes, as their bases. (75.) 77« How can Theorem VIII. be proved from Theorem IX. ? 78. If a pyramid is cut by a plane parallel to its base, the pyra* mid cut off will be similar to the whole pyramid. (39 ; 4). 79« In a sphere great circles bisect each other. 80« A great circle bisects a sphere. (54.) 81. The centre of a small circle is in the perpendicular from the centre of the sphere to the small circle. 82. Small circles equally distant from the centre of a sphere are equal. 83. The intersection of the surfaces of two spheres is the circum- ference of a circle. 84. The arc of a great circle can be made to pass through any two points on the surface of a sphere. (IV. 4.) 85. Definition. A plane is tangent to a sphere when it touches but does not cut the sphere. 86. Prove that the radius of a sphere to the point of tangency of a plane is perpendicular to the plane. (IV. 8.) 87. As the serai-decagon revolves about A F^ «- what kind of a solid is described by the triangle p A BKl What by the trapezoid KC? By LD'i ^ 88. The surface described by the line A B = ^ AKX circ. GO. Draw from G a perpendicular to A B, and from the point where it meets A B a. perpendicular to ^ A F. (42.) F 89. The surface described by the line C D = L M y, circ. GO. (15.) 00 SOLID GEOMETRY. 90» Definition. The surfaces described by the arcs AB, BC, CD, &c. are called zones. 91. The area of a zone Is equal to the product of its altitude by the circumference of a great circle. 92. Zones on the same or equal spheres are as their altitudes. 93. The surface of a sphere is four times the surface of one of its great circles. (62; III. 32.) 94. Definition. A polyedron is circumscribed about a sphere when its faces are each tangents to the sphere. In this case the sphere is inscribed in the polj^'edron. 95. The surface of a sphere is equal to the convex surface of the circumscribed cylinder. (62 ; 15.) 96. Definition. A Spherical Sector is the solid described by any sector of a semicircle as the semicircle revolves about its diameter. 97. The volume of a spherical sector is equal to the product of the surface of the zone forming its base by one third of the radius of the sphere of which it is a part. 98. A Spherical Segment is a part of a sphere included by two parallel planes cutting or touching the sphere. When one plane touches and one cuts the sphere, the spherical segment is called a spherical segment of one base ; when both cut, a spherical segment of two bases. 99. How can the volume of a spherical segment of one base be found? A spherical segment of two bases? 100. A sphere is two thirds of the circumscribed cylinder. 101. A cone, hemisphere, and cylinder having equal bases and the same altitude are as the numbers 1, 2, 3. .^ BOOK VI PROBLEMS OF CONSTRUCTION. In the preceding demonstrations we have assumed that our figures were already constructed. The Problems of Construc- tion given in this Book depend for their solution upon the prin- ciples of the preceding Books. In some of the problems the construction and demonstration are given in full ; in others the construction is given and the propositions necessary to prove the construction referred to in the order in which they are to be used, and the pupil must complete the demonstration. In a few instances references are made to the Exercises appended to the previous Books. In such cases either the propositions to which reference is made can be demonstrated or the problem omitted. PROBLEM 1. 1, To bisect a given straight line. Let ^ ^ be the given straight line. From A and B as centres with a radius greater than half of A B, describe arcs cutting one another at C and D ; join C and D cutting AB hi U, and the line ^ -ff is bisected at U. For C and D being each equally distant from A and B, the line CD must be perpendicu- lar to ^ ^ at its middle point (converse of I. 50). C D B 90 PLANE GEOMETKV. PROBLEM 11. 2. From a given point without a straight line to draw a per- pendicular to that line. Let G be the point and A B the line. From (7 as a centre describe an arc cutting ^ ^ in two points E and F ; with ^ E and F as centres, with a radius greater A >>..^ ■■'p ' ^ than half E F, describe arcs intersecting at jD. Draw CD, and it is the perpen- dicular required (converse of I. 53). PROBLEM in. 3* From a given point in a straight line to erect a perpendicu- lar to that line. Let C be the given point and A B the F given line. ^ With (7 as a centre describe an arc cutting ^ ^ in i> and E \ with B and E as centres, with a radius greater than A-\ q )-B D C, describe arcs intersecting at F. Draw C Fy and it is the perpendicular required (converse of L 53). Second Method. With C as a centre de- scribe an arc D E F ; take the distances BE and EF equal to CD, and from E and F as centres, with a radius greater than half the distance from E to F, de- scribe arcs intersecting at G. Draw G G, / / z> c and it is the perpendicular required (III. 33 ; III. 16 -, III. 15). BOOK VI. 91 Third Method. With any point, D, without the line A B, with a radius equal to the distance from D to C, describe an arc cutting AB at E ; draw the diameter ED F. Draw C F, and it is the perpen- dicular required (III. 23). PROBLEM IV. 4t To bisect a given arc, or angle. 1st. Let AB he the given arc. Draw the chord A B and bisect it with a perpendicu- lar (1; III. 16). 2d. Let C be the given angle. With (7 as a centre describe an arc cutting the sides of the angle in A and B ; bisect the arc A B with the line CD, and it will also bi- sect the angle C{IU. 11). PROBLEM V. 5t At a given point in a straight line to make an angle equal to a given angle. Let A be the given point in the line A B, and C the given angle. With C as a centre describe an arc DF cut- ting the sides of the angle C; with ^ as a centre, with the same radius, describe an arc; with ii^ as a centre, with a radius equal to the distance from D to E, describe an arc cutting the arc FG. Draw AG. The angle J = C (III. 12; m. 11). 92 PLANE GEOMETRY. PROBLEM VL 6* Through a given point to draw a line parallel to a given straight line. Let C be the given point, and A B the given line. From G draw a line (7i> to ^^; at (7 in the line DC make an angle D C E equal to C D A (5) ; CE is parallel to ^ ^ (I. 18). PROBLEM Vn. 7» Two angles of a triangle given, to find the third. Draw an indefinite line AB-, at any point C make an angle AG D equal to one of the given angles, and DG E equal to the other (5). Then ECB'm the third angle (I. 7 ; I. 33). PROBLEM VIII. 8. The three sides of a triangle giveriy to construct the triangle. Take A B equal to one of the given sides ; with ^ as a centre, with a radius equal to another of the given sides, describe an arc, and with ^ as a centre, with a radius equal to the remaining side, describe an arc inter- secting the first arc at G. Draw A G and G B^ and AGB ia evidently the triangle required. BOOK VI. PROBLEM IX. 93 3 /- / 1/ 9e Two sides and the inchided angle of a^trian^e ^iven, to construct tke triangle. , DrS^ A B equal to one of the given sides ; C at B make the angle ABC equal to the given angle (5), and take BG equal to the other given side ; join A and (7, and ABC m evi- dently the triangle required. PROBLEM X. 10. Two angles and a side of a triangle given, to construct tlie triangle. If the angles given are not both adja- cent to the given side, find the third angle by (7). Then draw A B equal to the given side, and at B make an angle ABC equal to one of the angles adjacent to A B, and at A make an angle B AC equal to the other angle adjacent to A B, and A BC i^ evidently the triangle required. PROBLEM XL 1 1 , Two sides of a triangle and the angle opposite one of them given, to construct the triangle. Draw an indefinite line AC ; at A make the angle CAB equal to the given angle, and take A B equal to the . j side adjacent to the given angle ; with B as a centre, with a radius equal to the other given side, de- scribe an arc cutting A C. If the given angle A is acute, D- ^G 94 PLANE GEOMETRY. 1st. The given side B C, opposite the given angle, may be less than the other given side ; then -^ the arc described from B as a centre will cut AC in two points, C and D, on the same side of A, and, drawing ^^-^A,---' BC and BD, the triangles ABC and ABD (whose angle BDA is the supplement of the angle BCA), both satisfy the given conditions. 2d. The given side opposite the given angle may be equal to the perpendicular B E ; then the arc described from i? as a centre will tonch A (7, and the right triangle AB E h the only one that can satisfy the given conditions. 3d. The side opposite the given angle may be greater than the other given side ; then the arc described from ^ as a centre will cut A C in C, and in another point on the other side of A. In this case there can be but one triangle ABC satisfying the given conditions, the triangle formed on the opposite side of A B containing not the given angle but its supplement. 4th. If the given angle is not acute, the given side opposite the given angle must be greater than the other given side, and, as in the last case above, there can be but one solution. 12* Scholium, If the side opposite the given angle A is less than the perpendicular, or if the given angle is not acute, and at the same time the side opposite the given angle is less than the other given side, the solution is impossible. 13. Corollary. From this and the preceding Problem and Theorems VIII., IX., and XIV. of Book I., it follows that witli the exception of the ambiguity pointed out in the first part of this Problem, two triangles are equal if any three parts, of which one is a side, of the one are equal to the corresponding parts of the other. BOOK VI. 95 PROBLEM XII. 14* To find the centre of a given circumference or of a given arc. Let ABB he the given circumference, or arc. Draw any two chords not parallel to each other, a,s A B, B D, and bisect these chords by the perpendiculars C B and CF. These perpendiculars will intersect at the centre of the circumference or arc (III. 17). 15 1 Scholium. By the same construction a circumference may be made to pass through any three given points ; or a cir- cle circumscribed about a given triangle ; or about a given regular polygon (II. 34). PROBLEM XIIL 16t To inscribe a circle in a given triangle. Let ABChe the given triangle. Bisect any two of its angles, as A and G, D^ where the two bisecting lines meet, as a centre, with a radius equal to the distance of D from any one of the sides, describe a circle, and it will be the circle required. Draw the perpendicular DE, DF, DG. The angles at A are equal by construction, and the angles AED and AFD are each right angles ; therefore the triangles AD E and AFD are mutually equiangular (I. 35), and the hypothenuse A D m common; therefore the triangles are equal (I. 41), and D E = D F. In like manner D E = D G. Therefore the circle described from /> as a centre with the radius D E will pass through the points 7^ and G ; and since the angles at Ef F, G are right angles, the sides of the triangle A BG are With the point A 96 PLANE GEOMETRY. tangents; therefore the circle EFG m inscribed in the tri- angle ABC (III. 20). 17. Scholium. The lines bisecting the angles of a triangle all meet in the same point. PROBLEM XIV. 18. Through a given point to draw a tangent to a given cir- cumference. 1st. If the given point is in the circumference. Erect a perpendicular to the radius at the given point (III. 47). 2d. If the given point is without the circumference. Join the given point A with the centre C of the given circle B D E ; on ^ C as a diameter describe a cir- cle cutting the given circle in B and D. Draw A B and A B, and each will be tangent to the given circle through the given point. For drawing the radii CB, CD, the angles .5, D are each right angles (III. 23) ; therefore A B, A B are tangents to the given circle. 19i Corollary. The tangents A B, AD are equal (I. 50). PROBLEM XV. 20* Upon a given straight line to describe a segment of a circle which shall contain a given angle. Let ^ ^ be the given straight line. At B make the angle ABB equal to the given angle (5). Draw B C perpen- dicular to D B ; bisect A B in B, and from B draw EC perpendicular to A B. From C, the point of intersection of BC and EC, with a radius equal to G By describe a circle AG B F ^ B FA is the segment required. BOOK VI. 97 A B is ?i chord (I. 53). And as ^ Z) is perpendicular to the radius C B at B, B D is a tangent to the circle, and hence the angle A B I) m measured by half the arc AG B (III. 54) ; and any angle B F A inscribed in the segment B F A is ako meas- ured by half the arc AG B (III. 21), and is therefore equal to the angle A B D or the given angle. 21* Corollary. If the given angle is a right angle, the re- quired segment would be a semicircle described on the given line as a diameter. PROBLEM XVI. 22t To divide a given line into parts proportional to given lines. Let it be required to divide A A B into parts proportional to Jf, N, 0. Draw at any angle with ^ ^ an indefinite line A C. From A cut o^ A D, D E, E F equal respectively to i¥, N, 0. Join B to F, and through D and E draw lines parallel to B F. These parallels divide the line as required (II. 16). 23. Corollary. By taking J/, N, equal, the given line can be divided into equal parts. PROBLEM XVII. 24. To find a fourth proportional to three given lines. Let it be required to find a fourth proportional to M, N,0.^ A Draw at any angle with each other the indefinite lines AF, AG. From AF cut off AB=zM, BC = y, and 98 PLANE GEOMETRY. from A G cut off A I) =z 0. Join B D and through G draw G E parallel to B D \ then D E i& the required fourth pro- portional (II. 16). 25% Gorollary. By taking A B equal to Jf, and A D and B G each equal to N^ a third proportional can be found to M and N. PROBLEM XVIII. 26 • To find a mean proportional between two given lines. Let it be required to find a mean M N proportional between M and N. From an indefinite line cut off" AB = M, BG = J^; on AG as a diameter describe a semicircle, and at B draw B D perpendicular to A G. tional required. Join AD, D G. BB m the mean propor- (III. 23 ; 11. 26.) 27. Definition. When a line is divided so that one segment is a mean proportional between the whole line and the other segment, it is said to be divided in extreme and mean ratio. PROBLEM XIX. 28. To divide a given line in extreme and mean ratio. Let it be required to divide A B m extreme and mean ratio. At B draw the perpendicular BG = \ AB; join AG', cut off G D = G B, A E =z A D, QXidi A B m divided at E in extreme and mean ratio. For, describe a circle with the cen- tre G and radius G B and produce A G to meet the circumfer- ence in Ft then ^1 i^ is a secant and AB ^ tangent of the circle DFB, and therefore (III. 64) BOOK VI. AF'.AB = AB:AD 99 and (Pn. 18) AF—AB'.ABz=AB — AD:AD But AB=2CB=:DF therefore AF— AB = AF— DF=AD = AE and the proportion becomes AF :AB = FB :AF or (Pn. 16) AB:AF = AF:FB PROBLEM XX. 29. Through a given point ivithin the sides of a given angle to draw a line so that the segments included between the j^oint and the sides of the angle may he in a given ratio. MN Let it be required to draw through the point D within the angle ^ a line so that AD '.DC =^M :N. Draw D E parallel to A B. Find EC 2, fourth proportional to M, N, and BE (24); join C to D, and produce CD to J, and AC \^ the line required (XL 16). PROBLEM XXI. 30* The base, an adjacent angle, and the altitude of a triaiufle given, to construct the triangle. At A of the base A B draw an indefi- nite line A C making the angle A equal to the given angle ; at any point in A B, as D, draw the perpendicular DE equal ^ ^ to the given altitude ; through E draw EF parallel to ^ J5 cut- ting AC in G', join G j5, and A G B 'i& the triangle required. 100 PLANE GEOMETRY. PROBLEM XXIL 31 « To construct a parallelogram, having the sum of its base and altitude given, tohick shall be equivalent to a given square. On A B, the given sum, as a diame- ter, describe a semicircumference. At any point, as B, \n A B draw the perpen- dicular B C equal to a side of the given square ; through C draw C D parallel to A B, cutting the circumference in D ; draw D E perpendicular io AB. A E, E B are one the base and the other the altitude of the parallelogram required (26). 32. Scholium. If the side of the square is greater than half the sum of the base and altitude, the construction is im- possible. PROBLEM XXIII. 33* To construct a parallelogram having the difference between its base and altitude given, which shall be equivalent to a given square. On A B the given difference, as a diameter, describe a ch'cumference. At A draw the perpendicular A Z)' equal to a side of the given square ; join D with the centre C, and produce D C to E. D F, D E are one the base and the other the altitude of the paral- lelogram required (III. 64). PROBLEM XXIV. 34* To construct a square equivalent to a given parallelogram. Find a mean proportional between tjie altitude and base of the given parallelogram (26), and it will be a side of the re- quired square. Q^^y^- BOOK Nl. 101 PROBLEM XXV. 35* To construct a square equivalent to a given triangle. Find a mean proportional between the base and half the altitude (26), and it will be a side of the required square. PROBLEM XXVI. 36.^ To construct a square equivalent to a given circle. Find a mean proportional between the radius and the semi- circimiference, and it will be a side of the required square. PROBLEM XXVII. 37. To construct a square equivalent to the sum of two given squares. Construct a right triangle (9) with the sides adjacent to the right angle equal respectively to the sides of the given squares ; the hypothenuse will be a side of the required square (II. 27). 38. Scliolium. By continuing the same process we can find a square equivalent to the sum of any number of given squares. PROBLEM XXVIII. 39. To construct a square equivalent to the difference of two given squares. Construct a right .triangle (11), taking as the hypothenuse a side of the greater square, and for one of the sides adjacent to the right angle a side of the other square ; the third side of the triangle will be a side of the required square (II. 28). 102 PLANE GEOMETRY. PROBLEM XXIX. E D F For the triangles BCD 40i To construct a triangle equivalent to a given polygon. Let A Dhe the polygon. ^ Draw B D cutting off the triangle ^C D\^xk^\\^ Q draw CF parallel to ^ Z> meeting £ 1) produced in F; JQia ^ F, .afn.d a. polygon A B F E is formed with one side less than the given polygon and equivalent to it. and B F D, having the same base B D, and the same altitude, are equivalent ; adding to each the common part A B D E, we have ABODE equivalent to A B F E. In like manner a polygon with one side less can be found equivalent to A B F E, and by continuing the process the sides may be reduced to three, and a triangle obtained equivalent to the given polygon. 41. Scholium. Since by (35) a square can be found equiva- lent to a given triangle, by (40) and (35) a square can be found equivalent to any polygon. PROBLEM XXX. 42« On a given line to construct a polygon similar to a given polygon. Let AD \)Q the given poly- gon and JfZ the given line. Draw the diagonals A E^ AD, AO. At M and L make the angles G ML and F E G LM equal respectively to A FE and A E F, and a triangle G L M will be formed similar to A E F. In like manner on G L construct a triangle similar to AD E) on G K one similar to AO D ; on GI one similar to ABO ; and the polygons A D, BOOK VI. 103 KG, being composed of the same number of similar triangles similarly situated, are similar (11. 75), PROBLEM XXXI. 43* Tim similar polygons being given, to construct a similar polygon equivalent to their sum, or to their difference. Find a line whose square shall be equivalent to the sum (37), or to the difterence (39), of the squares of any two homologous sides of the given polygons, and this will be the homologous side of the required polygon (11. 31). On this line construct (42) a polygon similar to the given polygons. PROBLEM XXXIL 44 « To construct a square which shall he to a given squa^'e in a given ratio. On any line ^ (7, as a diameter, describe a semicircumference ABC; divide the line A C at the point D so that A D : D C in the given ra- tio. Perpendicular to AC draw D B meeting the circumference at B ; join B A, B C, and on BC, produced if necessary, take BF= a, side of the given square. Through F draw JE F parallel to A C, meeting B A in E, and B E is & side of the required square. For as -5 is a right angle (TIL 23), we have (IL .72) BF^ .BF^ — EG : GF But as ^i^ is parallel to A C, we have (II. 47) EG .GF=AD '.DC therefore (Pn. 11) BE^'.BF^=zAD .DC 104 PLANE GEOMETRY. PROBLEM XXXIII. 45* To inscribe a square in a given circle. Draw two diameters AC, B D 2! right angles to each other, and join A B, B C, CD, DA] ABCn is the required square (III. 23 ; III. 12). 46* Corollary. By bisecting the arcs AB, BC, CD, DA, and drawing the chords of these smaller arcs, a regular octagon will be inscribed in the circle. By continuing this bisection regular polygons can be inscribed having the number of their sides 16, 32, 64, and so on. PROBLEM XXXIY. 47t To inscribe a regular hexagon in a given circle. TaKe A B equal to the radius of the given circle, and it will be a side of the hexagon required (III. 33). 48. Corollary. By drawing A C, CD, D A 2in equilateral triangle will be in- scribed in the circle. By bisecting the arcs AB, BC, &c., and continuing this bisection as in (46), and drawing the chords of these smaller arcs, regular polygons can be inscribed having the number of their sides 12, 24, 48, 96, and so on. PROBLEM XXXV. 49. To inscribe a regular decagon in a given circle. Divide the radius ^ ^ in extreme and mean ratio at the point D (28), and take BC = A D, the greater segment, and it will be the side of the required decagon. BOOK VI. 105 Draw AC, CJD. The triangles ACB, DCB are similar (II. 23); for they have the angle B common, and by construction AB :AD=iAD : D B but AD = BG therefore AB:BC = BC:BD B C Therefore, as A C B is isosceles, I) C B is also isosceles, and CD = C B ; therefore also C J) = I) A, and A C D is an isos^ celes triangle, and the angle A = A C B. But the exterior- angle BI)C = A-\-ACI) = twice the angle A. Therefore, as ^ = BB C, B = twice the angle A. But B = AC B; therefore the sum of the three angles A, B, and A C B is equal to five times the angle A ; or the angle A is one fifth of two right angles, or one tenth of four right angles; therefore the arc B C is one tenth of the circumference, and the chord B C a side of a regular decagon inscribed in the circle. 50« Corollary. By drawing chords joining the alternate vertices a regular pentagon will be inscribed. By proceeding as in (46) regular polygons can be inscribed having the number of their sides 20, 40, 80, and so on. PROBLEM XXXVI. 51. To inscribe a regular polygon of fifteen sides in a given circle. Find by (47) the arc A C equal to a sixth of the circumference, and by (49) the arc A B equal to a tenth of the circumference, and the chord B C will be a side of the poly- gon required. For i - T^ = -iij 52. Corollary. Proceeding as in (46) regular polygons can be inscribed having the number of their sides 30, 60, and so on. 106 PLANE GEOMETRY. PROBLEM XXXVIL 53* To circumscribe about a given circle a polygon similar to a given inscribed regular polygon. Let A D hQ the given inscribed poly- gon. Through the points A, B, C, B, E, F draw tangents to the circumference. M^ These tangents intersecting will form the polygon required. For the triangles AGB, B H 0, &c. are isosceles (19); and as the arcs AB, B (7, &c. are equal, the angles GAB, GBA, HBC, RGB, &c. are equal (III. 54) ; therefore, as the bases AB, BG, &c. are equal, these isosceles triangles are equal. Hence the angles G^ H, I, Ky L, M are equal, and the polygon MI is equiangular; and as GB=zBH=HC = GI, &c., GH=HI,ko.', therefore the polygon MI is equilateral and regular (11. 32). It is also sim- ilar to ^ i> (II. 33) ; and as its sides are tangents it is circum- scribed about the circle. 54 • Corollary. As (45-52) regular polygons can be in- scribed having the number of their sides 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, and so on, regu- lar polygons having the number of their sides represented by these numbers can also be circumscribed about a given circle. EXERCISES. 5^» From two given points to draw two equal lines meeting in a given straight line. (I. 53.) 56. Through a given point to draw a line at equal distances from two other given points. 57. From a given point out of a straight line to draw a line mak- ing a given angle with that Hne. (I. 17.) BOOK VI. 107 58. From two given points on the same side of a given line to draw two lines meeting in the first line and making equal angles with it. 59. From a given point to draw a line making equal angles with the sides of a given angle. 60. Through a given point to draw a line so that the parts of the line intercepted between this point and perpendiculars from two other given points shall be equal. If the three points are in a straight line, the parts equal what ? 61 . From a point without two given lines to draw a line such that the part between the two lines shall be equal to the part between the given point and the nearer line. When is the Problem impossible ? 62. To trisect a right angle. 63. On a given base to construct an isosceles triangle having each of the angles at the base double the third angle. -4-64. To construct an isosceles triangle when there are given 1st. The base and opposite angle. 2d. The base and an adjacent angle. 3d. A side and an opposite angle. 4th. A side and the angle opposite the base. 65. The base, opposite angle, and the altitude given, to construct the triangle. (III. 22.) (20.) When is the Problem impossible ? 66. The base, an angle at the base, and the sum of the sides given, to construct the triangle. When is the Problem impossible ? 67. The base, an angle at the base, and the diflference of the sides given, to construct the triangle. 1st. When the given angle is adjacent to the shorter side. 2d. When the given angle is adjacent to the longer side. When is the Problem impossible ? 68. The base, the difference of the sides, and the difference of the angles at the base given, to construct the triangle. 108 PLANE GEOMETRY. 69« The base, the angle at the vertex, and the sum of the sides given, to construct the triangle. When is the Problem impossible ? 70» The base, the angle at the vertex, and the difference of the sides given, to construct the triangle. 71 • On a given base to construct a triangle equivalent to a given triangle. 72. With a given altitude to construct a triangle equivalent to a given triangle. 73. Two sides of a triangle and the perpendicular to one of them from the opposite vertex given, to construct the triangle. 74. Two of the perpendiculars from the vertices to the opposite sides and a side given, to construct the triangle. 1st. When one of the perpendiculars falls on the given side. 2d. When neither of the perpendiculars falls on the given side. 75. An angle and two of the perpendiculars from the vertices to the opposite sides given, to construct the triangle. 1st. When one of the perpendiculars falls from the vertex of the given angle. 2d. When neither of the perpendiculars falls from the vertex of the given angle. 76. An angle and the segments of the opposite side made by a I perpendicular from the vertex given, to construct the triangle. It* G-iven an angle, the opposite side, and the line from the given -wertex to the middle of the given side, to construct the triangle. When is the Problem impossible ? 78. An angle, a perpendicular from another angle to the opposite side, and the radius of the circumscribed circle given, to construct the triangle. When is the Problem impossible ? 79. To divide a triangle into two parts in a given ratio, 1st. By a line drawn from a given point in one of its sides. 2d. By a line parallel to the base. ,v^^' yyyvryvt/^^ BOOK VI. 109 80«/To'*t!iksect a triangle by straight lines drawn from a point witbm to the vertices. ^ / 81« Parallel to the base of a triangle to draw a line equal to the sum of the lower segmentg oLtbe two sides. 82. Parallel to the base of a triangle to draw a line equal to the diflference of the lower segments of the two sides. 83. To inscribe in a given triangle a quadrilateral similar to a given quadrilateral. 84. To divide a given line so that the sum of the squares of the parts shall be equivalent to a given square. V85 t To construct a parallelogram when there are given, 1st. Two adjacent sides and a diagonal. 2d. A side and two diagonals. 3d. The two diagonals and the angle between them. 4th. The perimeter, a side, and an angle. 86i To construct a square when the diagonal is given. 87 • To construct a parallelogram equivalent to a given triangle and having a given angle. 88. To draw a quadrilateral, the order and magnitude of all the ddes and one angle given. Show that sometimes there may be two different polygons satisfy- ing the conditions. 89. To draw a quadrilateral, the order and magnitude of three sides and two angles given. 1st. The given angles included by the given sides. 2d. The two angles adjacent, and one adjacent to the unknown side. 3d. The two angles being opposite each other. 4th. The two angles being both adjacent to the unknown side. In any of these cases can more than one quadrilateral be drawn ? 90. To draw a quadrilateral, the order and magnitude of two sides and three angles given. 1st. The given sides being adjacent. 2d. The given sides not being adjacent. 110 PLANE GEOMETRY. 91 • In a given circle to inscribe a triangle similar to a given triangle. 92. Through a given point to draw to a given circle a secant such that the part within the circle may be equal to a given line. 93 • With a given radius to draw a circumference, • 1st. Through two given points. 2d. Through a given point and tangent to a given line. 3d. Through a given point and tangent to a given circumference. 4th. Tangent to two given straight lines. 5th. Tangent to a given straight line and to a given circumference. 6th. Tangent to two given circumferences. State in each of these cases how many circles can be drawn, and when the construction is impossible. 94 • To draw a circumference, 1st. Through two given points and with its centre in a given line. 2d. Through a given point and tangent to a given line at a given point. 3d. Tangent to a given line at a given point, a,nd also tangent to a second given line. 4th. Tangent to three given lines. 5th. Through two given points and tangent to a given line. 6th. Through a given point and tangent to two given lines. 95* To draw a tangent to two circumferences. There can be drawn, 1st. When the circles are external to each other, four tangents. 2d. When the circles touch externally, three. 3d. When the circles cut, two. 4th. When the circles touch internally, one. 6th. When one circle is within the other, none. PLANE TRIGONOMETKY. Af 4:jhapter i. ^PKELIMINAEY. C ^ LOGARITHMS. 1( Logarithms are exponents of the powers of some number which is taken as a base. In the tables of Logarithms in common use, the number 10 is taken as the base, and all numbers are con- sidered as powers of 10. And, since 10*^ = 1, that is, since the Logarithm of 1 is 0, 10^ = 10, " " " 10 " 1, 102=: 100, " " " 100 " 2, 10»=1000, " « " 1000 " 3, &c., 11 » Rule. From the Logarithm of the dividend subtract the Logarithm of the divisor, and the remaiiider will he the Logarithm of the quotient. E. g. 1. Divide 78.46 by 0.00147. Log. of 78.46 0.00147 1.894648 3.167317 Quotient, 53374.1. 2, Divide 0.0014 by 756. Log. 4.727331 ,5>4^'' Log. of 0.0014 Va^^ " 756 Quotient, 0.000001852. 3.146128 2.878522 Log. 6.267606 Negative numbers can be divided in the same manner as positive, taking care to prefix to the quotient the proper sign, according to the rules of Algebra. 3. Divide .7478 by 0.00456. 4. Divide 5000 by 0.00149. 6. Divide 0.00997 by 64.16. 6. Divide —14.55 by 543. 7. Divide —465 by —19.45. Ans. 163.99+. Ans. 0.00015539+. Ans. — 0.0267955-I-. Ans. 23.9074-I-. INVOLUTION BY LOGARITHMS. 12. Rule. Multiply the Logarithm of the number by the exponent of the power required. 1. Find the 15th power of 1.17. Log. of L17 Ans. 10.638. 0.068186 15 Log. 1.022790 LOGARITHMS. 2. Find the 5th power of 0.00941. Log. of 0.00941 3.973590 5 Ans. 0.000000000073782. Log. 11.867950 3. Fipd the 4th power of 0.0176. Ads. 0.000000095951+. 4. F'md the 9th power of 1.179. Ans. 4.401765+. Negative numbers are involved in the same manner, taking care to prefix to the power the proper sign, according to the rules of Algebra. 5. Find the 3d power of —0.017. Ans. —0.000004913. 6. Find the 6th power of —14. Ans. 7529536. EVOLUTION BY LOGARITHMS. 13« Rule. Divide the Logarithm of the number by the expo- nent of the root required. Negative numbers are evolved in the same manner, taking care to prefix to the root the proper sign, according to the rules of Algebra. For the sake of convenience, where the character- istic of a Logarithm is negative, and not divisible by the index of the root, we can increase the negative characteristic so as to make it divisible, providing we prefix an equal positive number to the decimal part of the Logarithm. E. g. 1. Find the 5th root of 0.0173. Log. of 0.01 73 is 2.238046, which is equal to 5 + 3.238046, and dividing this by 5 gives 1.647609, which is the Log. of 0.4442. 2. Find the 3d root of 80.07. Ans. 4.31013+ 3. Find the 8th root of 0.0764. Ans. .72508+ 4. Find the 7th root of —17. Ans. —1.49891+ 5. Find the 5th root of —0.00496. Ans. —0.34601+ lit Instead of subtracting one Logarithm from another, it is sometimes more convenient to add what it lacks of 10, — which difference is called the complement, — and from .11 10 PLANE TRIGONOMETRY. the sum reject 10. The result is evidently the same. For X — y = a?-|-(10 — y) — 10. The complement is easiest found by beginning at the left of the Logarithm of the number, and subtracting each figure from 9, except the last significant fig- ure, which must be subtracted from 10. . In proportion, therefore, we have the following rule : Add the complement of the Logarithm of the first term to the Logarithms of the second and third terms, and from the sum reject 10. E. g. 1. Find a fourth proportional to 14, 175, and 7486. Complement of Log. of 14, 8.853872 175, 2.243038 7486, 3.874250 Ans. 93575. Log. 4.971160 2. Given the first three terms of a proportion, 416, 584, and 256, to find the fourth. Ans. 359.38+. 3. Find the value of 179 X 4968 -r- 489. Ans. 1818.552+. 4. Find the value of ^1-^^^}1 . Ans. 4.7776+ d04 X 513 6. Find the value of V(0.1739) _^ 331.9 (v/2.04 + V^l.203/ Ans. 0.000197055. 23.3X6.764 X^ 6. Find the value of "^''^-^ - X9.97 Ans. 838.965+ 7. In a system whose base is 4, what is the Logarithm of 41 of 161 of 641 of 21 of 81 of 1 1 ofjl of J1 of J1 ofOI 8. Solve the equation 125* = 25. X X Log. 125 = Log. 25 _ Locr. 25 __ 1.39794 _ 2 . ^ — Log. 125 ~ 2:09691 ~ 3' ^^^* 9. Solve the equation 2048* = 16. TRIGONOMETRIC FUNCTIONS. 11 CHAPTER II. TRIGONOMETRIC FUNCTIONS. DEFINITIONS. 15. Trigonometry is that branch of mathematics which treats of methods of computing angles and triangles. 16. Plane Trigonometry treats of methods of computing plane angles and triangles. 17. The circumference of every circle is divided into 360 equal parts, called degrees ( ° )/ea:clr degree into 60 equal parts, called minutes ( ' ), and each minute into 60 equal parts, called seconds ( '' ). 18. As angles at the centre vary as their arcs, or arcs as their corresponding angles, the measure of an angle is the arc included between its sides and described from its vertex as a centre (Geom., III. 14). 19. As the sum of all the angles about the point C is equal to four right angles, one right angle, A C B, would be measured by one quarter of the circumference, or 90° (Geom., III. 15). 20. The Complement of an arc or angle is 90° minus this arc or angle. Thus, the arc AD is the complement of D B, and the angle A C D of D C B. When an arc or angle is greater than 90°, its complement is negative. 21. The Supplement of an arc or angle is 180° minus this arc or angle. Thus, the arc EAD is the supplement of D B, and the angle E C D of D C B. When an arc or angle is greater than 180,° its supplement is negative. 12 PLANE TKIGONOMETRY. A 22» * The Sine of an arc or angle is the line drawn from one end of the arc, perpendicular to the diameter pass- ing through the other end; or it is half the chord of double the arc. Thus, D F is the Sine of the arc D B, or of the angle D C B. 23. The Versed Sine of an arc or angle is that part of the diameter which is between the foot of the sine and the arc. Thus, B F is the Versed Sine of the arc B D, or of the angle BCD. 24* The Cosine of an arc or angle is the sine of the com- plement of the arc or angle, or the radius minus the versed sine of the arc or angle. Thus, H D = C F is the Cosine of the arc B D, or of the angle BCD. (Co in Cosine, &c., stands for complement.) 25. The Tangent of an arc or angle (in Trigonometry) is tbe line touching one extremity of the arc, and terminated by a line drawn from the centre through the other extremity. Thus, B I is the Tangent of the arc B D, or of the angle BCD. 26. The Cotangent of an arc or angle is the tangent of the complement of the arc or angle. Thus, A K is the Cotangent of B D, or of the angle BCD. 27. The Secant of an arc or angle (in Trigonometry) is the line drawn from the centre through one end of the arc, and terminated by the tangent to the other end. Thus, C I is the Secant of B D, or of the angle BCD. 28. The Cosecant of an arc or angle is the secant of the complement of the arc or angle. Thus, C K is the Cosecant of B D, or of the angle BCD. * Those who prefer the Analytical Method will turn from this point to Chapter IV. /J TRIGONOMETRIC FUNCTIONS. 13 29. The Sine, Tangent, and Secant of the supplement of an arc are (irrespective of the signs) the same as for the arc itself. The Sine and Cosine of an arc form the two sides of a right- angled triangle whose hypothenuse is the Radius of the arc. The Radius and Tangent of an arc form the two sides of a right-angled triangle whose hypothenuse is the Secant of the arc. 30i Suppose the Radius C D to move in the plane of the circle about the centre C : let it move so that the arc 'B D and the angle BCD become ; then the Sine, Tangent, and Versed Sine of the arc or angle become 0; the Secant and Cosine equal to Radius ; the Cosecant and Cotangent infinite. If C D moves toward A until the arc B D or the angle BCD becomes 30°, then, if Radius is unity, D F, or Sine 30°= J For the Sine D F is half the chord of double the arc, that is, is half of D G, which, as it subtends sixty degrees, or one sixth of the circumference, is equal to Radius (Geom., III. 33). If C D moves until the arc B D or the angle BCD becomes 45°, then the triangle CDF becomes isosceles ; and if Radius is unity, we have FD^ + FC^ = 2 DF'' == C D^ Hence D F = C D y^J or Sine 45° = v'^ = ^V'2 The Tangent, in this case, equals the Radius. If C D moves until B D, or B C D, becomes 60°, then since DF = VCD2 — CF2 and C F = Sine of A D = Sine 30° = ^ D F, or Sine 60° = V^f^ = y'f = ^ y/S If C D moves until the arc B D, or the angle BCD, becomes 90°, then the Sine, Versed Sine, and Cosecant become equal to Radius ; the Cosine and Cotangent ; the Secant and Tangent infinite. 14 PLANE TRIGONOMETRY. If we suppose C D to move until the point D passes entirely round the circumference, it will be easy to trace the changes in the length of the Sine, Cosine, &c. 31 1 The centre C is the absolute zero point. If we consider a line extending in one direction plus, a line extending in the opposite direction should be con- sidered minus. It has been agreed to consider the trigonometric lines which extend from EB upward, or from A L to the right, plus ; therefore all those extending from B E down- ward, or from A L to the left, must be considered minus. It will on inspection be found that these lines change their direction at the point where they become 0, or infinite. Therefore, the algebraic signs of the Sines, Co- sines, &c., change from plus to minus, or minus to plus, as each passes the point where it becomes 0, or infinite. These changes in the signs will be found to be as follows : 1st quadr. Sine and Cosecant -f- Cosine and Secant -f- Tangent and Cotangent -\- 2d quadr. + 3d quadr. 4th quadr. + + 32. By -various methods, the Sines, Cosines, Tangents, and Cotangents have been calculated for every minute of the Quadrant, with Radius as unity ; and the logarithms of these numbers have been taken from the table of logarithms, and, with 10 added to the characteristic, to avoid negative charac- teristics (that is, the radius assumed is 10000000000), have been arranged in the table entitled Logarithmic Sines and Tangents. 33* To find the Logarithmic Sine, Cosine, &c., of any arc or angle. In the tables the degrees up to 45° are at the top of the page, TKIGONOMETRIC FUNCTIONS. 15 and the minutes on the left ; above 45° (since the Sine, or Tan- gent, of any arc is the Cosine, or Cotangent, of its complement), the degrees are at the bottom of the page, and the minutes on the right. In the first and second columns, marked D., is the rate of variation per second for the columns at their left, and in the third, marked D., the rate of variation for the columns on both sides of it. In the columns marked D. the last two figures are to he considered as decimals. It must be remembered that, as the arc or angle increases, the Sines and Tangents increase, while tlie Cosines and Co- tangents decrease. E. g. The Log. Sine of 37° 10' is 9.781134 74° 50' '* 9.984603 1. Requires, the Log. Sine of 41° 14' 25''. Log. Sine of 41° 14' is 9.818969 Number to be added for 2^" is 25 X 2.4 = 60 Ans. 9.819029 2. Required, the Log. Cosine of 65° 24' b". Log. Cosine of 65° 24' is 9.619386 Number to be subtracted for 5" is 5 X 4:. 6 = 23 ; IL n r ' Ans. 9.619363 31* To find the degrees, minutes, and seconds corresponding to any Logarithmic Sine, Cosine, &c. . Find in the column with the given title (that is. Sine, Cos., Tan., or Cot.) the given logarithm ; if the title is at the top, take the degrees at the top and the minutes on the left ; but if the title is at the bottom, take the degrees at the bottom and the minutes on the right. If the given logarithm is not found exactly, take the degrees and minutes corresponding to the next less logarithm for Sines and Tangents, next greater for Cosines and Cotangents ; divide the difference of these two logarithms by the corresponding tabular difference D., and the quotient will be the additional number of seconds. 16 PLANE TRIGONOMETRY. E. g. 1. Eequited, the degrees, minutes, and seconds corre- sponding to the Log. Sine 9.874321. 9.874321 Log. Sine 48° 28'', 9.874232 1.87) 89.00 48 Ans. 48° 28' 48". 2. Eequired, the degrees, &c. corresponding to Log. Cotan- gent 9.911302. Log. Cotangent 50° 48', 9.911467 9.911302 4.3) 165.0 38 Ans. 50** 48' 38". 3. Find the Log. Sine of 13° 10' 31". 4. Find the Log. Sine of 76° 10' 49". 6. Find the Log. Cosine of 87° 51' 42". 6r Find the Log. Cosine of 175° 43' 44". 7. Find the Log. Cotangent of 17° 16' 14". 8. Find the Log. Cotangent of 49° 15' 27". 9. Find the Log. Tangent of 43° 5^ 44". 10. Find the Log. Tangent of 113° 21' 5". 11. Given Log. Sine 8.898611, to find the degrees, &c. cor- responding. 12. Given the Log. Tangent 9.47864, to find the degrees, &c. corresponding. 13. Given the Log. Sine 9.90543, to find the degrees, &c. cor- responding. 14. Given the Log. Cosine 9.996087, to find the degrees, &c. corresponding. 15. Given the Log. Cosine 9.846321, to find the degrees, &c. corresponding. 16. Given the Log. Cotangent 10.5673 to find the degrees, &c. corresponding. 17. Given the Log. Cotangent 9, to find the degrees, &c. cor- responding. SOLUTION OF PLANE TRLAJ^GLES. 17 CHAPTER III. SOLUTION OF PLANE TRIANGLES. 35* In every plane triangle there are six parts, three sides and three angles. Of these, any three being given, provided one is a side, the others can be found. RIGHT-ANGLED TRIANGLES. 36* In a right-angled triangle, one of the six parts, viz. the right angle, is always given; and if one of the acute angles is given, the other is known; therefore, in a right-angled triangle, the number of parts to be considered is four, any two of which being given, the others can be found. We may have four cases, according as there are given, 1. The hypothenuse and an acute angle ; 2. A side about the right angle and an acute angle ; 3. A side about the right angle and the hypothenuse ; 4. The sides about the right angle. All these cases can be solved by the following Theorem : THEOREM I. 37 • In any right-angled plane triangle, Ist. Radius is to the hypothenuse as the sine of either acute angle is to the opposite side; 2d. Radius is to either side as the tangent of the adjacent acute angle is to the opposite side. Let ABC be a triangle, right- angled at C. Let h represent the hypothenuse, and a and h the sides opposite the angles A and B respec- tively. With either angle, as A, as a centre, and any radius, (which 18 PLANE TKIGONOMETEY. radius will represent the radius of the tables,) describe the arc DE; from D draw D F perpendicular to A e, and draw E G parallel to D F. Then D F will be the tabular sine, A and G E the tabular tangent of the angle A. From the similar triangles ADF, AGE, and ABC, we have, 1st. AD : AB = DF : BC b ^ that is, R : h=z sin. A : a a. 2d. AE : ACr=GE : BC -x that is, R : 6 = tan. A : a • 38. Corollary 1. As sin. A = cos. B R : A = COS. B : a 39. Corollary 2. If radius is unity these proportions will give, a -==.11 sin. A a = b tan. A a=.h cos. B Case I. 40. Given the hypothenuse and an acute angle. / From Theorem I. R : A = sin. A : a ' R : A = sin. B, or cos. A : h Ex. 1. Given h 255, A 57° 14', to find a a and h. By Logarithms, Radius 10. :A255 2.406540 = sin. A 57° W 9.924735 : a 214.42 2.331275 SOLUTION OF PLANE TRIANGLES. 19 Radius 10. : h 255 2.406540 Oj^ COS. A 57° 14' 9.733373 "0 : 6 138.01 2.139913 Ud n Ex52. Given h 1676, A 67° 13', to find a and h. { a 1545.23. Ans. h 649.03. Ex. 3. Given h 78.4, B 15° 51', to find a and h Ans. f a 75.42. /7-X./"""' 16 21.41. Case II. ' "- *7 41* 6^m?i a sic?e about the right angle and an a^ute angle. From Theorem I. sin. A : a = R : A R : A = sin. B : 6 U ^ ^ Ex. 1. Given a 195, B 64° 43', to find h and h, sin. A = cos. 64° 43' comp. 0.369476 2.290035 2.659511 10. 2.659511 9.956268 2.615779 Ex. 2. Given h 1075, B 75° 49', to find a and h. ^^g ia 271.68. \ h 1108.79. Ex. 3. Given 6 17.45, A 47° 31', to find a and h. ^g I a 19.05. ^' I A 25.84. '.a 195 R : h 456.57 R :A = sin. B 64° 43' : 6 412.84 20 PLANE TRIGONOMETRY. Case III. 42. Given a side about the right angle and the hypothenuse. From Theorem I. A : R = a : sin. A R : A = sin. B : h Ex. 1. Given h 24.5, a 17.4, to find the other parts. h 24.5 :R = a 17.4 comp. 8.610834 10. 1.240549 : sin A 45° 15' 5'' 9.851383 B = 90° — 45° 15' 5" = 44° 44' 55" R : h 24.5 = sin. B 44° 44' 55'' 10. 1.389166 9.847571 . : 6 17.248 1.236737 Otherwise h can be found from the formula b^ = h^ — a*. .-. b = Ex.2. Given A 172.8, b 14.17, a) (A - a) to find the other parts. /-A 85° 17' 47". Ans. ^ B 4° 42' 13". 1 a 172.218. Case IV. 43 • Given the sides about the right angle. From Theorem I. b : R = a : tan. A sin. A : a = R : A Ex. 1. Given a 195, 6 147, to find the other parts. b 147 comp. 7.832683 :R 10. = al95 2.290035 •. tan. A 52° 69' 22" 10.122718 SOLUTION OF PLANE TRIANGLES. 21 B = 90° — 52° 59' 22'' = 37° 0' 38^ sin. A 52° 59' 22'' comp. 0.097712 : a 195 2.290035 = R . lO : h 244.2 2.387747 Otherwise h can be found from the formula h^ = a^ -\- h^ ; then the angles by using the first proportion of Theorem I. Ex. 2. Given a 189, h 14, to find the other parts. A 85° 45' 49^. Ans. i B 4° 14' 11". 189.518. 44* In the following Examples two parts of a right-angled triangle are given, and the others required. a 888.896. 1. Given 6 217, A 915. Ans. ^ A 76° 16' 52". 3. 'If Given a 17.94, A 15° 39'. Ans. | B 13° 43' 8". A 5° 9' 34". 2. Given a 174, h 1927. Ans. \ B 84° 50' 26". 1934.89. h 64.038. h 66.503. 4. Given ^ 47.9, A 59° 17'. Ans. ( « ^1'1'^^8. I h 24.467. A 23° 36' 42'; 5. Given a 298, h 744. Ans. { B 66° 23' 18". If 681.712. A 35° 57' 32". 6. Given a 9.75, 6 13.44. Ans. {B W" 2' 28". h 16.60. ( 03042. 7. Given 6 0.02518, A 34° 7' 10". Ans. /^^-^^^^^ I h 0.030 ^ 22Ns^/^ PLANE TRIGONOMETRY. OBLIQUE-ANGLED TRIANGLES. 45* In solving oblique-angled triangles, there are four cases There may be given, 1. Two angles and a side ; 2. Two sides and an angle opposite one of them ; 3. Two sides and the included angle • 4. The three sides. For solving these we demonstrate the three following Theorems. THEOEEM II. 46. In dny filane triangle^ the sides have the same ratio as the sines of the opposite angles. Let a, 5, c represent the sides oppo- site the angles A, B, C, respectively. Then a :h '. €•=. sin. A : sin. B : sin. C From B draw B D perpendicular to 6. Then A B D and B D C being right-angled triangles, from Theo- rem I. we have E : a = sin. C : B D .-.Ex BD = asin. C R :c = sin. A :BD .'.Ex BD = csin. A Therefore a sin. C = c sin. A or a \ c-=z sin. A : sin. C In like manner it can be shown that a : b = sin. A : sin. B h : c =i sin. B : sin. C Therefore a : 6 : c = sin. A : sin.^ B : sin. C 47. Scholium. If one of the angles, as C, should become ;i SOLUTION OF PLANE TRIANGLES. 23 right angle, then c will become the hypothenuse, and sin. C radius, and the proportion will become a : hz=i sin. A : R or R : A r= sin. A : a which is the same as the first proportion in Theorem I. THEOREM III. 48« In any plane triangle, the sum of any two sides is to their difference, as the tangent of half the sum of the opposite angles is to the tangent of half their difference. Let ABC be a plane triangle ; then BC + BA : BC — B A = tan. ^ {k. + Q) : tan. J (A — C) Produce A B to D, making B D equal to B C, and join D C. Take B F equal to B A, draw A F and produce it to E. AD = Bc4-BA FC = BC — BA The sum of the two angles BAF and B F A is equal to the sum of B A C and B C A, as each sum is the supplement of ABC; therefore, as A B is equal to B F, BAF = HA + C) If from the greater of two quantities we subtract half their sum, the remainder will be half their difference ; therefore, EAC = J(A — C) As B D is equal to B C, the angles B D C and BCD are equal ; and D A F is equal to B F A, and B F A to C F E ; therefore the triangles A D E and F E C are mutually equiangular ; hence the two angles at E are equal, and A E is perpendicular to D C ; and if with A E as radius and A as a centre, an arc is de- scribed, D E becomes the tangent of D A E, and EC of E A C. By similar triangles we have (Geom. II. 19) AD:FC = DE:EC or BC + BA :BC — BA = tan. J (A + C) : tan. J (A — C) 24 PLANE TRIGONOMETRY. THEOREM IV. 49i If from any angle of a plane triangle a perpendicular he drawn to the opposite side or base, then the sum of the segments of the base will be to the sum of the other two sides as the difference of these sides is to the difference of the segments of the base. From B, in the triangle ABC, .^^ draw BD perpendicular to AC. ^^ \ \ Then ^.^ \ \ AD + DC:AB + BC = AB — BC:AD — DC For BC2 — D C2 = B D2 = AB2 — AD2 or AD2 — D C2 __ ^ £2 _ 5(^2 As the product of the sum and difference of two quantities is equal to the difference of their squares, we have (A D + D C) (A D -- D C) = (A B + B C) (A B — B C) or AD + DC:AB + BC = AB— BC:AD — DC 50* Scholium. When the perpendicular falls within the triangle, the sum of the segments of the base is equal to the whole base ; when without, the difference. ' Case I. 51 • Given two angles and a side. From Theorem II. sin. A : sin. B : sin. Q z= a :h : c Ex. 1. Given A 48°, C 55° 17', a 417, to find the other parts. ^ & B = 180° — (55° 17' + 48°) = 76° 43' sin. A 48° comp. 0.128927 : sin. B 76° 43' 9.988223 = a417 2.620136 : b 546.12 2.737286 SOLUTION OF PLANE TRIANGLES. 26 sin. A 48° comp. 0.128927 : sin. C 55** 17' 9.914860 = a417 2.620136 : c 461^24 2.663923 Ex. 2. Given A 95° 4VB 25° 14', c 49.17, to find the other parts. J C 59° 42'. Ans. \ a 56.727. i h 24.278. Case II. 52* Given two sides and an angle opposite to one of them. From Theorem II. a -.h : c ■=■ sin. A : sin. B : sin. C Ex. 1. Given a 55, c 49.87, A 25° 44', to find the other parts. a 55 comp. 8.259637 : c 49.87 1.697839 r= sin. A 25° 44' 9.637673 :sin. C23° 11' 2" 9.595149 B = 180° — (23° 11' 2^ + 25° 44') = 131° 4' 58" sin. C 23° 11' r comp. 0.404851 : sin. B 131° 4' 58^ 9.877234 = c 49.87 1.697839 : h 95.483 1.979924 53* If B C, the side opposite the given angle, is less than the other given side AB, and the given angle a- is acute, there are two triangles which satisfy the conditions, viz. ABC and ABD, in which the angles B C A and B D A are supplements of each other. The Log. sine obtained in working such an example represents 2 D^--- 26 PLANE TRIGONOMETRY. either the angle BC A, or its supplement BD A (Art. 29). If the given angle A is obtuse, or ths side opposite the given angle is greater than the other given side, there is but one solution (Geom., VI. 11). Whenever the solution is impossible (Geom., VI. 12), the Log. sine obtained in working the example will be greater than radius, which is absurd! Ex. 2. Given a 95.5, c 173.2, A 27° 4', to find the other parts. /C 55° 36' 47"', rC 124° 23' 13"^ Ans. } B 97° 19' 13'^, or -| B 28° 32' 47^ (6 208.17, ib 100.29. Case III. 54* Given two sides and the included angle. From Theorem III. a-^c'.a — c^ tan. J (A + C) : tan. | (A — C) From Theorem II. sin. A : sin. B . sin. Q ■= a \h : c Ex. 1. Given a 976, c 89, B 51° 17', to find the other parts. J (A + C) = J (180° — 51° IT) r= 64° 21' 30'' a-\-c= 1065 comp. 6.972650 : a — ^ = 887 2.947924 =:tan. J (A + C) = tan. 64° 21'30" 10.318746 : tan. J (A — C) = tan. 60° 2' 36" 10.239320 Half the sum plus half the difierence gives the greater angle A 124° 24' 6"; half the sum minus half the difference, the less C 4° 18' 54". sin. C 4° 18' 54" comp. 1.123553 : sin. B 61° 17' 9.892233 = c89 1.949390 : 6 922.94 2.965176 SOLUTION OF PLANE TRIANGLES. 27 Ex. 2. Given a 91, b 104, C 14° 30', to find the other parts. rA 55° 5' 37''. Ana. ^B 110° 24' 23^ ic 27.783. Case IV. 55 1 Given the three sides. B From B let fall a perpendicular upon b. /i ^\. a From Theorem IV. / j ^^^^^ 6 :a + c = a — c :DC — DA ^ D b ^ The angles A and C can then be found as in Art. 42. Ex. 1. Given a 125, 6 135, c 75, to find the angles. b 135 comp. 7.869666 :a + c :DC- 200 50 - D A 74.0741 2.301030 1.698970 1.869666 c D C, therefore, is 104.53, can now be found. and DA a A 30.46 ; the angles /-A 66° 2' A and 7"+ \th '. -^ - f i^~-<5 ' )^-fH> • ^C 33° 14' 56"+ 56. The sum of any two sides must be greater than the remaining side, otherwise the triangle is impossible. If the perpendicular is drawn to the longest side, it will fall within the triangle. The shorter segment of the base is adjacent to the shorter side. Ex. 2. Given a 347, b 642, c 476, to find the angles. A 39° 11' 14^5. B 80° 43' 43".5. , V C 60° 5' r. a- 28 PLANE TRIGONOMETRY. MISCELLANEOUS EXAMPLES. 1. Given A 45° 4', B 75° 35', c 457, to find the other parts. ( C 59° 21'. Ans. \ a 376.06. " [h 514.48. 2. Given a 454, c 753, A 45° 25', to find the other parts. 3. Given a 57, 6 89, C 75° 4', to find the other parts. / A 36° 32' 37^. Ans. \ B 68° 23' 23^ ^. ( c 92.495. f 4^ Given a 41, h 74, c 63, to find the other parts. (A 33° 37' 26^. Ans. ^B88° 4' 12^ ( C 58° 18' 22^ 6. Given h 75, a 35, to find the other parts. /A 27° 49' b\ Ans. ■] B 62° 10' 55". ( h 66.332. 6. Given h 919, A 37** 37', to find the other parts. Ans [^ 560.94. I h 727.95. 7. Given h 45.3, A 34° 23', to find a and h. Ans / ^ 30.998. ^^' \h 54.890. 8. Given a 40, 6 57, c 97, to find the other parts. 9. Given a 0.(55377, h 0.06607, A 45°, to find the other parts. /B60° 19' 34", /B 119° 40' 26". Ans. \ C 74° 40' 26", or J C 15° 19' 34". (c 0.07334, I c 0.0201. 10. Given a 54, 6 35, B 97° 15', to find the other parts. NOMETMC FUNCTIONS. 29 CHAPTER IV. GONOMETRIC FUNCTIONS. ANALYTICAL METHOD * DEFINITIONS. 57» Instead of considering the Sine, Tangent, &c. as lines, having a certain position in a circle, and varying not only as the arc, but also as the radius, we consider them, in this system, as ratios, varying only as the angle, and capable of being repre- sented by certain lines in a circle only when the radius is unity. 58. The Sine of an angle is the ratio of the side opposite it in a right-angled triangle to the hypothenuse. That is, if in any right-angled triangle A B C we represent the hypothenuse by A, and the sides opposite the angles A and B by a and h respectively, sm. A = T sin. B = (1) 59. The Tangent of an angle is the ratio of the side oppo- site it in a right-angled triangle to the side adjacent. That is tan. A = -r b tan. B = - a (2) 60. The Secant of an angle is the ratio of the hypothenuse to the side adjacent to the angle. That is sec. A = 7 sec. B = (3) 61. The Cosine, Cotangent, Cosecant of an angle are respec- tively the sine, tangent, and secant of its complement. * Those who have taken the Geometrical Method can omit Chapters IV. andV. • 30 PLANE TRIGONOMETRY. Therefore, as the acute angles of a right-augled triangle are complements of each other, we shall have A = sin. B = - h, B r= sin. A = T COS. COS. cot. A = tan. B = cot. B =: tan. A = cosec. A = sec. B = - a cosec. B = sec. A = t (i) 62. By inspecting these equations it will be seen that the sine and cosecant of an angle are reciprocals of each other ; so also the cosine and secant, and the tangent and cotangent. That is sin. A — cosec. A cos. sec. A tan A- \ cot. A or i5osec. A.=: - or sec. A = or cot. A = sin. A l__ COS. A 1 tan. A (5) 63* The sine, cosine, dhc, vary only as the angle ; that is, for a given angle they are constant. Let A D E and A B C be any two ^ right-angled triangles, having a com- mon angle A ; they are equiangular and similar. Hence DE:DA = BC:BA, or ^ = t^-; = E C DE_BC DA~"BA that is, the sine of the angle A is constant, whatever the length of the sides. In the same way it can be proved that the cosine, tangent, (fee. of a given angle are constant. TRIGONOMETRIC FUNCTIONS. 31 64'« Tlie sine, cosine, ^ C F 1 sin. C = DC cos. p CF HD CD~~ CD tan. DF BI FC~~BC cot. p CF HD ^ — FD~-HC" AK ~ AC sec. CD_CI ^""CF — CB cosec. CD CD ^~~DF~CH~ CK "CA If CA, CD, CB that is, radius, becomes unity, we shall have sin. C = D F COS. C = D H tan. C = B I cot. C = A K ' sec. C = C I cosec. C = CK In the Geometrical Method, ivithout limiting the radius to unity, these lines are defined as the sine, cosine, From B draw BD perpendicular to h. Then ABD and BDC being right-angled triangles, by (37) we have c sin. A = B D = a sin C hence a : c =z sin. A : sin. C In like manner it can be shown that a : b = sin. A : sin. B b : c = sin. B : sin. C Therefore a : b : c = sin. A : sin. B : sin. C 101. If the perpendicular falls without the triangle, the angles B C A and BCD, be- ing supplements of each other, have the same sine, and B D = a sin. B C D = a sin. C SOLUTION OF PLA'NE TRIAJ^GLES. 47 THEOREM.ni. 102. In any plane triangle, the sum of any two sides is to their differ ence, as the tangent of half the sum of the opposite angles is to the tangent of half their difference. Let ABC (Art. 103) be a plane triangle ; then a -\- c : a — c = tan. J (A + C) : tan. J (A — C) By (41) we have a \ c •=: sin. A : sin. C By composition and division (Geom. Pn. 19) a -\- c : a — c=z sin. A -\- sin. C a 4- c sin. A 4- sin. C or — ' — = ' a — c sin. A — sin. C But by (25) sin. A -f- sin. C tan. ^ (A -}- C) sin. A — sin. C tan. ^ (A — C) a -\- c tan. ^ (A -f- C) C) or a -\- c : a sin. A — sin. C Therefore a — c tan. ^ (A b = tan. 1 (A 4- C) : tan. J (A — C) (42) THEOREM IV. 103* In any plane triangle, the cosine of any angle is equal to the sum of the squares of the two adjacent sides minus the square of the opposite side, divided by twice tJie product of the adjacent sides. In the triangle ABC CD = 5 — AD CD2 = 52 — 25AD + AD2 Adding BD^ to both members, we have (Geom., II. 27) But by (40) A D = c cos, A .-. a^ = 6^* -|- (T* — 2 6c cog. A — 6' -f c* — q* 2 6c Therefore A = (43) 48 PLANE TRiaONOMETRY. 104. For greater convenience infusing logarithms (43) can be changed by subtracting both' members from unity and reducing, as follows : 1 — cos. A 2hc—y'—(?^a^__o}—{h — cf 2bc ~~ 2bc (a — h -{- c) (a -{- b — c) But by (35) ~~ 2 6c 1 — cos. A = 2 sin.^ J A (44) Substituting in (44) this value of 1 — cos. A, and also put- tmg 8 -, and reducing, we have In like manner sin. J A = 4/ sm. sm. h V = iC = y/. c^- -b){s- -c)] bc '(.9- -a)(s- -c) ac c^- -a)(s- -b) ab (45) 105» Adding both members of (43) to unity, we have 1 -|- COS. A __ 2bc-\-b^-\-(^ — a^ __ (b -}- cy — a* 2b 2 be __ (6 -f c + g) (6 -f c — g) 26c (46) But by (35) 1 + cos. A = 2 cos.^ J A Substituting in (46) this value of 1 -|- cos. A, and the value of s as in Art. 104, and reducing, we have In like manner cos. cos. cos. s (s — g) b~c ,9 (s — b) c) (47) SOLUTION «JF PLANE TtllANGLES. 49 106. Dividing equations (45) \)j (47) in order, by means of (9) we have ^ Y s (s — a) 2 y s (s — b) tan.iC^. A^-;^^^-^> ^ y s (s — c) (48) Case I. 107* (riven two angles and a side. By (41) sin. A : sin. B : sin. C = a : b : c Ex. 1. Given A 48°, C 55° 17', a 417, to find the other parts. ° B = 180° — (55° 17' + 48°) = 76° 43' sin. A 48° : sin. B 76° 43' = a417 : b 546.12 sin. A 48° : sin. C 55° 17' = a417 : c 461.24 comp. 0.128927 9.988223 2.620136 2.737286 comp. 0.128927 9.914860 2.620136 2.663923 Ex. 2. Given A 95° 4', B 25° 14', c 49.17, to find the other parts. C 59° 42'. Ans. ^ a 56.727. b 24.278. 50 PLAlifE TRIGONOMETRY. Case II. 108i Given two sides and an angle opposite to one of them. By (41) a :h : €■= sin. A : sin. B : sin. C Ex. 1. Given a 55, c 49.87, A 25° 44', to find the other parta a 55 comp. 8.259637 : c 49.87 1.697839 = sin. A 25° 44' 9.637673 : sin. C 23° ir 2"' 9.595149 B = 180° — (23° 11' r 4- 25° 44') = 131° 4' 58" sin. C 23° 11' r comp. 0.404851 : sin. B 131° 4' 58^' 9.877234 = c 49.87 1.697839 : h 95.483 1.979924 109. If B C, the side opposite the given angle, is less than the other given side AB, and the given angle A- is acute, there are two triangles which satisfy the conditions, viz. ABC and A B D, in which the angles B C A and B D A are supplements of each other. The Log. sine obtained in working such an example represents either the angle BCA, or its supplement BDA (Art. 80). If the given angle A is obtuse, or the side opposite the given angle is greater than the other given side, there is but one solution (Geom., VI. 11). Whenever the solution is impossi- ble (Geom., VI. 12), the Log. sine obtained in working the ex- ample will be greater than unity (10. in the tables), which is impossible (Art. 58). ,-''C SOLUTION *0F PLANE TRIANGLES. 51 Ex. 2. Given a 95.5, c 173.2, A 27° 4', to find the other parts. / C 55° 36' 47'', ( C 124° 23' 13''. Ans. \ B 97° 19' 13", or <| B 28° 32' 47". (6 208.17, 16 100.29. Case III. llOt Given two sides and the included angle. By (42) . a -{- c : a — c =: tan. J (A -|- C) : tan. J (A — C) < By (41) sin. A : sin. B : sin. C =: a : h : c Ex. 1. Given a 976, c 89, B 51° 17', to find the other parts. J (A + C) = J (180° — 51° 17') = 64° 21' 30"" a-\-c= 1065 comp. 6.972650 : a _ c = 887 • 2.947924 = tan. |(A + C) — tan. 64° 21' 30" 10.318746 : tan. i (A — C) i= tan. 60° 2' 36" 10.239320 Half the sum plus half the difference gives the greater angle A 124° 24' 6"; half the sum minus half the difference, the less C 4° 18' 54". sin. C 4° 18' 54" comp. 1.123553 : sin. B 51° 17' 9.892233 = c 89 1.949390 : 6 922.94 2.965176 Ex. 2. Given a 91, 6 104, C 14° 30', to find the other parts. /A 55° 5' 37". Ans. ^B 110° 24' 23". I c 27.783. 52 PLANE TRlGONOIikTKY. Case IV. lilt Given the three sides. By (48) tan. J A = 4 /- tan, _ l{s-h){s-c) (s — a) -a) {s — c) s{s — b) Ex. 1. Given a 125, b 135, c 75, to find the angles. s 167.5 comp. 7.775985 comp. 7.775985 comp. 7.775985 s — a 42.5 comp. 8.371611 1.628389 1.628389 s — b 32.5 1.511883 comp. 8.488117 1.511883 s—c 92.5 1.966142 1.966142 comp. 8.033858 2)19.6256^1 2)19.858633 2)18.950115 Log. tangents 9.8128105 9.9293165 9.4750575 ^ A 33° 1-3.6" i B 40° 21' 28.3" ^ C 16° 37' 28.1" A 66° 2' 7.2" B 80° 42' 56.6" C 33° 14' 56.2" 112. The angles can also be obtained from formula (45) or (47). As the sines differ from each other more for angles be- tween 0° and 45°, the cosines for angles between 45° and 90°, (45) is preferable when the half-angle is less than 45°, and (47) when the half-angle is more than 45°. But (48) is more accu- rate for aU angles, and requires but four logarithms. 113* The sum of any two sides must be greater- than the remaining side, otherwise the triangle is impossible. Ex. 2. Given a 347, b 542, c 476, to find the angles. (A39° ir 14''.5. Ans. \ B 80° 43' 43'^5. ( C 60° 5' r. For Miscellaneous Examples see page 28. PEACTieAL APPLICATIONS. 53 CHAPTER VI. PRACTICAL APPLICATIONS. DEFINITIONS. Ill* A Horizontal Plane is a plane which is tangent to the earth's surface, and every line in this plane is a horiz(mtal line. 115* A Vertical Line is a line which is perpendicular to a horizontal plane, and every plane including in its surface such a line is a vertical plane. 116. A Horizontal Angle is one that has the plane of its sides horizontal. 117. A Vertical Angle is one that has the plane of its sides vertical. 118. An Angle of Elevation is a vertical angle having one side horizontal, and the inclined side above it ; as C AB (Art. 120). 119. An Angle of Depression is a vertical angle having one side horizontal, and the inclined side below it ; as F B A. HEIGHTS AND DISTANCES. PROBLEM I. 120. To determine the height of a vertical object standing on a horizontal plane. . Suppose it is required to find the height of BC. From the foot of B C measure any con- venient distance CA, and at A take the angle of elevation CAB. Then, in the right-angled triangle ABC, all the angles and the side AC are known, and BC can be found D C 54 PLANE TK|Gt)NOIl!jETRY. 121 • Secoiid Method. WitLout ineasur- ing the angle of elevation C A B, -the 'height of B C may be found, as follows : Cut a stake equal in length to the dis- tance of your eye from the ground ; move away from B C, until, by taking the position A D, with your head at A and the stake D E standing perpen- dicular at your feet, you can just see, in a line with the top of the stake, the top of the object. Then, since AD is equal to DE, AC is equal to CB (Geom. II. 20) ; and if A C is measured, C B becomes known. This method is used in finding the heights of trees, or the length of that part of the tree which is fit for timber. 122 1 Third Method. When shadows are cast by objects, still another method can be used. Measure the height of any convenient vertical object and the length of its shadow, and also the length of the shadow of the object whose height is sought. Then (Geom. II. 20) the length of the shadow of the one is to its height as the length of the shadow of the other to its height. 123i If the height B C is known, by taking any position, as A, and measuring the angle BAC, the distance AC can be found. 124. If from the top of the tower the angle of depressio-i F B A, which is equal to BAC, is taken, then, if B C is known, A C can be found ; if A C is known, B C can be found. Ex. 1. The distance A C is 100 feet, and the angle BAC 41° 15' ; what is the height of the tower 1 Aus. 87.6976 ft. PROBLEM II. 125. To find the height of a vertical object standing on an in- clined plane. PRACflCAI APPLICATIONS. 55 Measure any convenient distance from the object, as C D, and the angles of elevation of both D C and D B. Then all the angles, and the side D C, of the triangle D B C, are known, and A C can be found. Ex. 1. If DC is 55 feet, and the angle of elevation of DC is 10°, and of D B 36°, whal is the height of the tree 1 Angle B D C = 36° — 10° = 26° Angle D B C = 90° — 36° == 54° Ans. 29.8 ft. 126« Second Method. If two stations be taken in the line AC, as A and D, and the distance A D and the angles of eleva- tion of A B, D B, and D C are measured, then in the triangle A B D aU the angles and the side A D are known, and B D can be found ; then in the triangle B D C all the angles and the side B D are known, and B C and C D can be found. Ex. 2. If A D is 11.5 feet, and the angle of elevation of A B 38°, of D B 41°, and of D C 9°, what is the height of the tree 1 Angle BAD = 29°, BDC = 32°, ABD=:3°, BCD=f:99°. \ ^^ /'^j .■?■• Ans. 57.J^6 ft. y> / ij ^ PROBLEM III. 127. to find the height of an inaccessible object above a hori- zontal plane. The first method is pre- cisely the same as the last method in Prob. II., the angle of elevation of A D being 0°. 56 PLANE TRIGONOME^Y. 128. Second Method. Rule. Divide the distance between the stations by the difference of the natural cotangents of the angles of elevation. Demonstration.* With B as the centre, and B A as radius, describe the arc AE; produce BC till it meets the arc at E; at the point E draw E G tangent to the arc, and produce B D and BA toH andG. AD :GH = BD AD :GH=:BC BC = BE X AD GH But B E is radius or unity, and G H is the difference between the tangents of the angles G B E and H B E, that is, between the, cotangents of B A C and B D C : AD .-. BC cot. BAG — cot. BDC Ex. 1. If the distance AD is 97 feet, and the angles of ele- vation of AB and DB are respectively 37° 22' and 56° 10', what is the height B P ] t cot. 37° 22' 1.30952 cot. 56° 10' 0.67028 129. Third Method. 0.63924)97.00000(151.7 ft. Ans. Measure any base line AD, and the 'i^ Analytical. AC = BCtan. ABC DC = BC tan. DBC .-. AC— DC = AD = BC(tan. ABC — tan. DBC) BC = AD cot. BAG — cot. BDC + To find the Nat. Cot. from the Log. Tables aubtract 10 from the char- acteristic of the Log. Cot., and then find its corresponding natural number from the Table of Logarithms. PRACTICAL APPLICATIONS. 57 horizontal angles CAD and CD A, and the vertical an- gle CAB. Then, in the triangle A CD, we have one side and all the angles, and A C can be found ; then, in the right-angled triangle ABC, we have one side and all the angles, and the height B C can be found. Ex. 2. At the point A, I took the angle of elevation of the top of the tower B C 34° 45' ; then, turning at a right angle, I measured off A D 40 feet, and measured the angle ADC 58°. What is the height of the tower 1 Ans. 44.4 ft. PROBLEM lY. ISOi To find the distance of an inaccessible object. Measure a horizontal base line A C, and the angles BAC and ACB. Then, in the triangle ABC, we have one side and all the angles, and A B or C B can be found. Ex. 1. If A C is 20 chains, the angle A 25°, AB1 and C 92°, what is the distance Ans. 22.43 chs. PROBLEM V. 131 • To find the distance between two objects separated by an impassable barrier. Take any station C, from which both A and B are visible and accessible. Measure the angle ACB, and the sides AC and CB. Then, in the triangle ABC, we have two sides and the included angle, to find the third side A B. 58 PLANE trigonometry: 132t If the point A can be seen at B, the angle ABC can also be measured, and then only one side, A C or C B, whichever is most convenient, need be measured. Then, in the triangle ABC, we have one side and all the angles,, and A B can be found. Ex. 1. If AC is 44.4 chains, C B 50 chains, and the angle C 39° 25', what is the distance AB1 Ans. 32.268 chs. 133i If there is an elevated object, whose height is known, in the line A B produced, from which the points A and B can both be seen, by taking the angles of depression of A and B, the distance A B can be found by a rule the reverse of that given in Art. 128, viz. : Multiply the difference of the natural cotangents of the angles of depression hy the height of tlie object. Ex. 2. Wishing to know the width of a river, from the top of a tower 197 feet above the level of the river, I found the angle of depression of the nearer edge 54" 10', of the farther 48° 37' j what was the width of the river ] Ans. 31.3 ft. PROBLEM VI. 131* To find the distance between two inaccessible objects. Measure any convenient base line AD, and the angles BAD, BDA, CDA, and CAD. Then, in the triangle ABD, we have all the angles and the side A D, and B D can be found. In the triangle ACD we have all the angles and the side A D, and C D can be found. Then, in the triangle BCD, the two sides B D and D C and the included angle are known, and B C, the distance required, can be ^-^"^^ Ex. 1. If Ai>-ir40"roHriong, the angle BAD 100°, ADB 51°, CDA 120°, and CAD 55°, what is the distance between BandC] Ans. 355.05 rds. (q^ (H^U C lATIONS. 59 PROBLEM VII. 135* To find the distances, from a given pointy of three objects whose distances from each other are known. Let D be the given point, and A, B, C three points whose distances from each other are known; it is required to find the distance from D to the several points. The angles ADB and BDC must be measured. Then describe a circumfer- ence through the three points A, D, C; draw AB, BC, AC, AD, BD,CD; from A and C draw lines to E, the point where B D cuts the circumference. In the triangle A E C the side A C is given, and all the angles are known ; for E C A = E D A, and CAE= CDE (Geom., IIL 22); therefore AE can be found. In the triangle ABC, the three sides being given, the three angles can be found. Then, in the triangle ABE, we know the sides AB, AE, and the included angle B AE (= BAC — EAC), and the angle ABE can be found. Then, in the triangle ABD, all the angles become known, and the side AB is given ; there- fore AD and BD can be found ; then C D can also be found. 13S. If the point B is between D and the line A C, the angle B A E = B A C 4- E A C. But in this case the distances A B and B C cannot be the same as when B is beyond the line A C, unless B D cuts A C at right angles. If, however, B D cuts A C at right angles, and the position of B is not known, though the distances of A and C from D can be found, the distance of B will be ambiguous ; B may be in either of two points in the line B D. If the angle B is the supplement of ADC, the point B will fall on E, and BCA =BDA and CAB = CDB. In this case D may be anywhere in the arc ADC, and the distances 60 PLANE TRIGONOMETRY. A D, B D, C D cannot be determined from the data given. Also, if A, B, C, D are in the same straight hne, the distances cannot be determined. ^ ,^ x •Ex. 1. If AB (Fig. Art.l35) is 50, B C 65, AC 38.62 > chains in length, the angle ADB 10°, BD C 12° U', what are ' / the distances A D, B D, C D 1 ^J ' '^^'- -d:!: - ^^ Ans. AD 100, BD 145.37, CD 84.828 chs. DETERMINATION OF AREAS. 137» The Areas of triangles, parallelograms, and trapezoids, when their altitudes are given, can be found bj application of the principles already demonstrated in Geometry. But by Trigonometry the areas of these polygons can be found when in the triangle and parallelogram, in place of the base and alti- tude, two adjacent sides and an angle, and in the trapezoid the sides and two opposite angles, are given. PROBLEM I. 138. To fifid the area of a parallelogram. "--^ Rule I. Multiply the base hy the altitude. (Geom., II. 10.) Ex. 1. How many square yards are there in the sides, floor, and ceiling of a rectangular room, 20 feet long, 16 feet wide, and 10 feet high] 139. If two adjacent sides and an angle are given, the area can be found by . Rule II. Multiply together the two adjacent sides and the sine of the included angle. For, by Theorem I., ED = ADXsin.A A_ Area = AB X ED = AB X AD X sin. A If the work is done by logarithms, ten must be taken from the index of the log. sine. (Art. 32.) PKAOTICAL APPLICATIONS. Gl Ex. 2. What is the area of a parallelogram whose adjacent sides are 475 and 355 feet, and the included angle 49° ? '^\ 7 7/^ PROBLEM II. HO. To find the area of a triangle. Rule I. Multiply one half the base hy the altitude, (Geom., 11. 11.) 141. As a triangle is half a parallelogram of the same base and altitude, when two sides and the included angle are given, the area can be found (139) by Rule II. Multiply together tJie two sides and half the sine of the included angle. Ex. 1. What is the area of a triangle whose two sides are 76 and 14 rods, and4:he included angle 71° ir f, f/^f^tc /^ 142* When the three sides are gi^en, an angle can be found, and then the area by the last rule ; or, without finding an angle, the area can be found by the following rule : Rule III. From half the sum of the three sides subtract succes- sively the three sides ; mmdtiply together these three remainders and the half-sum, and extract the square root of the product. Let a, b, c denote respectively the sides opposite the angles A, B, C. DC =:AC — AD DC2 = AC2— 2 ACX AD + AD^ A^-^ ^ ^C Adding B D^ to both members, by Geom., II. 27, we have BC2 = AC2 + AB2 — 2 ACX AD or ^2 _ 52 _j_ ^2 _ 2 5 X AD b^±^-^ 62 PLANE TRIGONOMETRY. But (Geom., II. 28) B D^ = A B^ — A D^ AC X BD h lWc^~{c^-\-a^ — hy 2 ~~2y 4&^ ^4 62 c« — (& 2 -I- c« — g^)^ 16 As the product of the sum and difference of two quantities is equal to the difference of their squares, we have 4 h^^— {V^J^c^-a'f=(^ &c — [6^+c^— a2J)X(2 &c+[6^+c^— a^j) But 2&C — (S'^ + c* — a«) = a« — (&"— 2 6c + c«)=:a2— (& — c)« and «8_(6_c)2=(a4-[& — c]) X (a — [& — c]) = (a+&— c) X (a+c— &) and so also s/ a -\- h -\- c (a-f-J-f c) X (6+c — a) X (g-j-c — & ) X («+ ^ — ^ .-. Area = i / ^q Putting i we have Area =z^s (s — a) {s — b) {s — c) Ex. 1. What is the area of a triangle whose sides are 45, 55, and 60 feet 1 5 = ^ = 80 Log. 1.903090 80 — 45 = 35 " 1.544068 80 — 55 = 25 " 1.397940 80 — 60 = 20 " 1.301030 2 )6.146128 3.073064 Ans. 1183.2 sq.ft. PRACTICAL APPLICATIONS. 63 PROBLEM III. 143« To find the area of a trapezoid. Rule I. Multiply half the sum of the parallel sides by the per- pendicular distance between them. 144* If the angles are known, we can use Rule II. Divide the trapezoid by a diagonal^ and find the area of each triangle (141) ; their sum mill be the area required. 145. If the length of the diagonal is known, the area of these triangles may be found by (142). This rule applies equally well to a trapezium. Ex. 1. Find the area of a trapezoid whose parallel sides are 97 and 84, and the perpendicular distance between them 47 feet. :^ U / 6 ^ ^2-- Ex. 2. Find the area of a figure whose four sides are succes sively 27, 77, 28, and 85 rods in length, the angle between the first and second side 93°, and between the third and fourth 76° 15'. Ans. 13 acres, 2 roods, 33.97 sq. rds. Ex. 3. Find the area of a trapezium whose sides are succes- sively 35.8, 13.32, 35.84, and 17.8 rods, and the line from the beginning of the first to the end of the second side 38.9 rods. Ans. 3 acres, 1 rood, 36.25-J- sq. rds. PROBLEM IV. 1 46. To find the area of any polygon. If the diagonals necessary to divide the figure into triangles have been measured, the area can be found by finding the sum of the areas of the several triangles. When these diagonals are not known, the method generally used is called the rectangidar method. The exposition of this method belongs more properly under the head of surveying. / 64 PLANE TKIGONOMETRY. MISCELLANEOUS EXAMPLES. 1. The distance up the inclmed surface of a hill whose angle of elevation is 7°, is 25 rods ; the hill descends on the other side to the same level, with an inclination of 15°. How many pickets three inches wide, placed three inches apart, will it take to build a fence over the hill 1 Ans. 1194. 2. Having measured a horizontal line from the base of a ver- tical tower to the distance of 160 feet, I find that the angle of elevation of the top of the tower is 21°. What is the height of the tower 1 Ans. 61.418 ft. 3. Having measured from the base of a hill, whose inclina- tion is 65°, 25 rods on a horizontal plane, I find that the angle of elevation of the top is 16° 25^ What is the altitude of the hill above the horizontal plane 1 Ans. 8.538 rds. 4. Two observers, A and B, at sea, a mile apart, take at the same time the angles of elevation of a meteor which appears due west of each. A finds the angle 21° 50', B 19° 30'. ' What is its altitude 1 Ans. 3.049 ms. 5. From a steeple 65 feet above the level of an adjacent pond, the angle of depression of one edge is 40° 10', of the other 21° 30'. What is the width of the pond, and its distance from the church*'? Ans. Width 88 ft., distance 77 ft. 6. When a tree 25 feet high casts a shadow 100 feet long, what is the sun's altitude 1 Ans. 14° 2' 10". 7. From the base of a tower, the angle of elevation of the vj^ top of a second tower is 30°, and from the top of the first, \, which is 175 feet high, the angle of depression of the top of the second is 10°. If both stand on the same horizontal plane, what is the height of the second] Ans. 134.05 ft. 8. In the ruins of Persepolis there stand two upright col- / / ^ PRACTICAL APPLICATIONS. umns, one G4, the other 50 feet flV>r>Arf> |Ka -pln^; m ^ nm dq. tweeii these, on the same plane, stands a statue, whose head is 86 feet from the summit of the lower, and 97 feet from that of the higher column, and the distance from the foot of the lower column to the centre of the base of the statue is 76 feet. What is the distance between the tops of the columns 1 Ans. 157.03 ft. y^ 9. If the horizontal parallax of the sun, that is, the angle at the centre of the sun, subtended by the radius of the earth (3962 miles) is 8".5776 (log. sine 5.6189407) what is the dis- tance of the sun from the earth 1 Ans. 95273760.9 ms. 10. If the angle at the earth, subtended by the sun's diam- eter, is 32', and the distance as above, what is the diameter of the sun] Ans. 886845.5 ms. 11. If the moon's horizontal parallax is 57' 9'', what is its distance from the earth"? Ans. 238341.8 ms. 12. If the moon subtends an angle at the earth of 31' 7''', and its distance is as above, what is its diameter ] Ans. 2157+ nas. 13. If the annual parallax, that is, the angle at the object, subtended by the radius of the earth's orbit, of the nearest fixed star (a Centauri) is nearly 1" (log. sine 4.685575), what is its minimum distance ] Ans. 19650000000000. 147. The angle which a line makes with the meridian is called the bearing of the line. Thus, if N S is the meridian and the angle NAB 40°, NAC 75°, SAD 45°, the bearing is of A B, N. 40° W, or of B A, S. 40° E. A C, N. 75° E., or of C A, S. 75° W. AD, S. 45° E., or of D A, N. 45° W. 66 PLANE TRIGONOMETRY. When the bearing of two lines which form an angle is given it is easy to find by inspection the number of degrees in the angle. Thus, the angle BAG =40° + 75°= 115° BAD = 40° + (180 — 45°) = 175° C A D = 180° — 75° — 45° = 60° 14. Having run a line 85 rods S. 45° E., I came to an impas- sable marsh, and, seizing a man round to the other side of the marsh, I placed him exactly in the line ; then, running N. 20° E. 20 rods, I found that the bearing of the man was S. 24° E. ; then, going to the man, I ran the line (S. 45° E.) to the corner 44 rods farther. What is the whole length of the line? Ans. 167.76 rds. 15. Wishing to find the distance between two objects just visible in the distance, I measured a base line 100 rods due east, and, at the west end of the line, found the bearing of both ob- jects ; one N. 17° W., the other N. 45° E. ; at the other end, one N. 23° 30' W., the other N. 40° E. What is the distance] Ans. 871.92 rds. 16. Coming into a harbor I observed a tower, eastward of it a steeple, and still farther eastward a cliff. The bearing of the tower was found to be N. 17° E., of the steeple N. 20° E, and of the cliff N. 26'' 30' E. From a chart, the three objects were found to be in one straight line ; from the tower to the steeple the distance was set down as 25 rods, and from the steeple to the cliff 54/^ rods. What is my distance from each object 1 Ans. Tower, 477.68 rods. ; steeple, 477.06 rods. ; cliff, 480.2 rods. A TABLE OF LOGARITHMS OF NUMBERS. a.7 N. Log. i N. Log. IN. Log. |N. 1 Log. 1 0.000000 26 1.414973 51 1.707570 76 1.880814 2 0.301030 27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158 53 1.724276 78 1.892095 4 0.602060 29 1.462398 54 1.732394 79 1.897627 5 0.698970 30 1.477121 55 1 . 740363 80 1.903090 6 0.778151 31 1.491362 56 1.748188 81 I 908485 7 0.845098 32 1.505150 57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.954243 34 1.531479 59 1.770852 84 1.924279 1ft 1.000000 35 1.544068 60 1.778151 85 1.929419 li 1.041393 36 1.556303 61 1.785330 86 1.934498 12 1.079181 37 1.568202 62 1.792392 87 1.939519 lU 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 39 1.591065 64 1.806180. 89 1.949390 15 1 176091 40 1.602060 65 1.812913 90 1.954243 16 1 204120 41 1.612784 66 1.819544 91 1.959041 17 1.230449 42 1.623249 67 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.64.3453 69 1.838849 94 1.973128 S<0 1.301030 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 1.991226 24 1.3802J1 1.397540 49 1.690196 74 1.8692.32 99 1.995635 25 50 1.698970 75 1.875061 100 2.000000 In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the annexed first two figures of the Logarithm in the second column stand in the next lower line. LOGARITHMS OF NUMBERS. N. 1 |l|2|3i4i5i.6i7|8|9|D. ! 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 000000 4321 8600 012837 7033 021189 5306 9384 033424 7426 0434 4751 9026 3259 7451 1603 .5715 9789 3826 7825 1787 .5714 9606 3463 7286 1075 4832 8557 2250 5912 0868 5181 9451 3680 7868 2016 6125 .195 4227 8223 2182 6105 9993 3846 7666 1452 .5206 8928 2617 6276 9904 3503 7071 .611 4122 7604 1059 4487 7888 1263 4611 7934 1231 4504 7753 0977 4177 7354 .508 3639 6748 9835 2900 5943 8965 1967 4947 7908 0848 3769 6670 9552 2415 5259 80S4 0892 3681 6453 9206 1943 1301 5609 9876 1100 8284 2428 6533 .600 4628 8620 2576 6495 .380 4230 8046 1829 5580 9298 2985 6640 .266 3861 7426 .963 4471 7951 1403 4828 8227 1599 4944 8265 1560 4830 8076 1298 4496 7671 .822 3951 7058 .142 3205 6246 9266 2266 5244 8203 1141 4060 6959 9839 2700 5542 8366 1171 3959 6729 9481 2216 1734 6038 .300 4521 8700 2841 6942 1004 5029 9017 2969 6885 .766 4613 8426 2206 5953 9668 3352 7004 .626 4219 7781 1315 4820 8298 1747 5169 8565 1934 5278 8595 1888 5156 8399 1619 4814 7987 1136 4263 7367 .449 3510 6549 9567 2564 5541 8497 1434 4351 7248 .126 2985 5825 8647 1451 4237 7005 9755 2488 2166 6466 .724 4910 9116 3252 7350 1408 5430 9414 3362 7275 11.53 4996 8805 2582 6326 ..38 3718 7368 .987 4576 8136 1667 5169 8644 2091 5510 8903 2270 .5611 8926 2216 .5481 8722 1939 5133 8303 1450 4574 7676 .756 3815 6852 9868 2863 5838 8792 1726 4641 7536 .413 3270 6108 8928 1730 4514 7281 ..29 2761 2598 6894 1147 5360 9532 3664 7757 1812 5830 9811 3755 7664 1538 5378 9185 2958 6699 .407 4085 7731 1347 4934 8490 2018 .5518 8990 2434 5851 9241 2605 5943 9256 2544 5806 9045 2260 .5451 8618 1763 4885 7985 1063 4120 7154 .168 3161 6134 9086 2019 4932 7825 .699 3555 6391 9209 2010 4792 7556 .303 3033 3029 7321 1570 5779 9947 4075 8164 2216 6230 .207 3461 7748 1993 6197 .361 4486 8571 2619 6629 .602 4540 8442 2309 6142 9942 3709 7443 1145 4816 8457 2067 5647 9198 2721 6215 9681 3119 6531 9916 3275 660i 9915 3198 6456 9690 2900 6086 9249 2389 5507 8603 1676 4728 7759 .769 3758 6726 9674 2603 .5512 8401 1272 4123 6956 9771 2567 5346 8107 .8.50 3577 3891 8174 2415 6616 .775 4896 8978 3021 7028 .998 4932 8830 2694 6524 .320 4083 7815 1514 5182 8819 2426 6004 95.52 3071 6562 ..26 3462 6871 .253 3600 6940 .245 3525 6781 ..12 3219 6403 9564 2702 .5818 8911 1982 .5032 8061 1068 4055 7022 9968 2895 .5802 8689 1.5.58 4107 '239 ..51 2846 5623 8382 1124 3848 432 428 424 419 416 412 408 404 400 396 393 389 386 382 379 376 372 369 366 363 360 357 355 351 349 346 343 340 338 335 333 330 328 325 323 321 318 315 314 311 309 307 305 303 301 299 297 295 293 291 289 287 285 283 281 279 278 276 274 272 041393 5323 9218 053078 6905 000698 4458 8186 071882 5547 4148 8053 1924 5760 9563 3333 7071 .776 4451 8094 1707 5291 8845 2370 .5866 9335 2777 6191 9579 2940 6276 9586 2871 6131 9368 2580 5709 8934 2076 5196 8294 1370 4424 7457 .469 3460 6430 9380 2311 5222 8113 .985 3839 6674 9490 2289 5069 7832 .577 3305 079181 082785 6360 9905 093422 6910 100371 3804 7210 110590 9543 3144 6716 .258 3772 7257 0715 4140 7549 0926 4277 7603 0903 4178 7429 06,55 3858 7037 .194 3327 6438 9527 2594 5640 8664 1667 4650 7613 0555 3478 6381 9264 2129 4975 7803 0612 3403 6176 8932 1670 11.3943 7271 120574 3852 7105 130334 3539 6721 9879 143015 146128 9219 152288 5336 8362 161368 4353 7317 170262 3186 176091 8977 181844 4691 7.521 190332 3125 .5899 8657 201397 N. 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 D. 2^ LOGARITHMS OF NUMBERS. 3 Y jV. ^ A /I y^ > — ~.~_.^ [nTI 1 1 |cilV7 4fl|^ 176 Pl \ S \ 9ND.1 160 204120 4391 >1*63 493^1 52iR 5475 5746 6016 6286, 6556 271 161 6826 7096 7365 7634 7904 8173 8441 8710 8970 9247 269 162 9515 9783 ..51 .319 .586 .853 1121 1388 1654 1921 267 163 212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 266 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 165 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 262 166 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 261 167 2716 2976 3236 3496 3755 4015 4g74 4533 4792 5051 259 168 6309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 169 170 7887 8144 0704 8400 0960 8657 1215 8913 1470 9170 1724 9426 1979 9682 2234 9938 2488 .193 2742 256 254 23U449 171 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 172 5528 5781 6033 6285 6537 6789 7041 7292 7544 7/95 252 173 8046 8297 8548 8799 9049 9299 9550 9800 ...50 .300 250 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249' 175 3038 3286 3534 3782 4030 4277 4525 4772 .5019 5266 2433 176 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 177 7973 8219 8464 8709 8954 9198 9443 9687 9932 .176 245 178 250420 0664 0908 ri5i 1395 1638 1881 2125 2368 2610 243 179 180 2853 3096 5514 3338 5755 3580 5996 3822 6237 4064 6477 4306 6718 4548 6958 4790 7198 5031 7439 242 241 255273 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 238 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 184 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 185 7172 7406 7641 •;'875 8110 8344 8578 8812 9046 9279 234 186 9513 9746 9980 .213 .446 .679 .912 1144 1377 1609 233 187 271842 2074 2306 2538 2770 3001 3233 34M 5772 3696 3927 232 188 4158 4389 4620 4850 5081 5311 5542 6002 ■6232 230 189 190 6462 6692 6921 9211 7151 9439 7380 9667 7609 9895 7838 . 123 8067 .351 8296 .578 8525 .806 229 228 278754 8982 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 225 194 7802 802b 8249 8473 8696 8920 9143 9366 9589 9812 223 195 200035 0257 0480 0702 0925 1147 1369 1591 1813 2034 222 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 221 197 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 198 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 199 8853 9071 9289 9507 9725 9943 .161 .378 .595 .813 218 200 30lOoO 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 201 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 202 5351 5566 5781 5996 6211 6425 6639 6854 7068 7282 215 203 7496 7710 7924 8137 8.351 8564 8778 8991 9204 9417 213 204 9630 9843 ..56 .268 .481 .693 .906 1118 1330 1.542 212 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 211 206 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 208 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 209 210" 320146 0354 2426 0562 2633 0769 2839 0977 3046 1184 3252 1391 3458 1598 3665 1805 3871 2012 4077 207 206 322219 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 213 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 213 8380 8583 8787 8991 9194 9398 9601 9805 ...8 .211 203 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 218 84561 8656 8855 9054 9253 9451 9650 9349 ..47 .246 199 219 340W4I 0642 0841 1039' 1237 1435 1632' 1830' 2028' 2225' 198 j ! N. 1 1 1 1 2 ' 3 i 4 1 5 1 6 !i 7 1 8 i 9 1 n. 1 'tip LOGARITHMS OF NUMBERS. ' N. 1 1 1 [ 2 1 .3 1 4 1 5 1 6 1 7 1 8 1 9 1 D. ^20" 342423. 2620| 2817| 3014 3212, 3409, 3606 3802, 3999i 4l96i 197 \f A . 221 4392 4589)4785 4981 5178 5374 6570 5766 5962 6157 196 ^ \ ^ 222 6353 6549 6744 6939 7135 7330 7526 i 7720 7915 8110 195 L T r 223 8305 8500| 8694 8889 9083, 9278 9472, 9666 9860 ...54 194 IS5 \ 224 350248 0442 0636 0829 1023 1216 1410 1603 1796 1989 193 f K V 225 2183; 2375 2568 2761 2954 3147 3339^3532 3724 3916 193 ^< 226 4i08| 4301 4493 4685' 4876 6068 6260, 6452 5643 6834 192 227 6026! 6217 6408 6599, 6790 6981 7172 7363 7654 7744 191 ^28 7935! 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 i 229 23C 9835 ..25 1917 .215 2105 .404 2294 .693 2482 .783 2671 .972 1161 2859 i .3048 1350 3236 1.539! 189 3424 1 188 \ 1 361728 r J i 23! 3612 3800 3988 4176 4363 4551 4739 4926 5113 53011 188 \ \ i 23t 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169i 1871 J- \ ^ 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 [I? 234 9216 9401 9587 9772 9958 .143 .328 .513 .698 .883 185 235 371068! 1253 1437 1622 1806 1991 2176 2360 2544 2728 184 236 2912 30U6 ^280 3464 3047 3831 4015 4198 4382 4565! 1841 V » jo 237 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 i \ 238 6577 6759 6942 7124 7306 7488 7670! 78.52 8034 8216 182 239 240 8398 8580 039;i 8761 8943 0754 9124 0934 9306 1115 9487 9668 1296 1476 9849 1656 ..30 1837 181 181 i\^ v^ 380211 0573 1 \\ V 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 V \\ ^ 242 3815 3995 4174 4353 4533 4712 4891 6070 5249 5428 179 i V ' 243 5606 5785 5964 6142 6321 6499 6677 6866 7034 7212 178 v^ 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 89891 178 245 9166 9343 9520 9698 9875 ..51 .228 .405 .582 .7591 177 ^ V "hf 1 246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2521 176 IV 1 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 ^\^ 'r 248 4452 4627 4802 4977 5152 5326 5501 6676 5850 6026 176 <.x:> "\ 249 250 6199 6374 6548 8287 6722 8461 6896 8634 7071 8808 7245 8981 7419 9154 7592 9328 7766 9501 174 173 397940 8114 iW 251 9674 9847 ..20 .192 .365 .638 .711 .883 1066 12281 173 1 252 401401 1573 1745 1917 2089 2261 2433 26051 2777 2949 172 \^ I 253 3121 3292 3464 3635 3807 3978 4149 4320 '4492 4663 171 } Vr I 254 4834 5005 5176 5346 5517 6688 58581 6029 6199 6370 171 S V\' ^ 255 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 \ 256 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 h\> I 257 9933 .102 .271 .440 .609 .777 .946 1114 1283 1451 169 258 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 \ 259 260 3300 3467 5140 3635 5307 3803 5474 3970 5641 4.37 5808 4305 5974 4472 6141 4639 6308 4806 6474 167 167 "if v 414973 ^f r 261 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 i ^ A 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 V t V 263 9956 .121 .286 .451 .616 .781 .946 1110 1275 14,39 166 lY ^vV 264 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164 ^ iX 265 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 In, V i 1 \ 266 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 163 267 6511 6674 6S36 6999 7161 7324 7486 7648 7811 7973 162 4 iv i 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 162 t vVi 269 9752 9914 ..75 .236 .398 .559 .720 .881 1042 1203 161 c\ v\ "STiJk 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 161 H.^^ 27r 2969 3130 3290 3450 3610 3770 3930 4090! 4249 4409 160 HV 272 4569! 4729' 4888! 50481 6207 .5367 5526! 6685' 6844 6004 1.59 273 6163 6322i 6481 6640 6798! 6957 7116172751 7433 7592 159 274 7751 790'^ 8067 8226' 8384 8542 870118859 9017 9175 158 \\ ^ \\ 275 9333] 9491 9648! 9806' 99641 . 1221 .279 .437 .594 .762 158 \\ ^ V 276 440909 1066 1224' 13811 1538! 1695! 18521 2009' 2106! 2323 157 \M \ 277 ' 248<» '?637 2793 2950! 3106 3263 3419! 3^76 3732 3889 157 hi 278 4045 4201 4357, 4513 4669 4825 49Sl! 5137 5293' 5449 156 279 5604 5760 5915160711622616382 6537 6692 6848 7003 155] Z!^ |ll2l3(4|5|6|7!8|9lD. 1 n^n^-n^Jc N. I 1) I I I 2 i 3 280 447158 7313 281 8706 8861 282 450249 0403 283 1786 1940 284 3318 3471 285 4845 4997 286 6366 6518 287 7882 8033 288 9392 9543 289 460898 1048 290 462398 25481 291 3893 4042' 292 5383 5532 293 6868 7016 294 8347 8495 295 9822 9969 296 297 471292 27)56 1^38 2903 298 4216 4362 299 5671 5816 7266 300 477121 301 8566 8711 302 480007 0151 303 1443 1586 304 2874 3016 305 4300 4442 306 5721 5863 307 7138 7280 308 8551 8692 309 9958 ..99 310 491362 1502 311 2760 2900 312 4155 4294 313 5544 5683 314 6930 7068 315 8311 8448 316 9687 9824 317 501059 1196 318 2427 2564 319 3791 3927 320 505150 5286 321 6505 6640 322 7856 7991 323 9203 9337 324 510545 0679 325 1883 2017 326 1218 3351 327 4548 4681 323 5874 6006 329 7196 7328 330 618514 8646 331 9828 99.59 332 521138 1269 333 2444 2575 334 3746 3876 335 6045 5I74| 336 633Q 6469 33/ 7630 7759 338 8917 9045 339 530200 0328' 7468 9015 0557 2093 3624 5150 6670 8184 9694 1198 2697 4191 5680 7164 8643 .116 1585 3049 4508 6962 7411 8855 0294 1729 3159 4585 6005 7421 8833 .239 1642 3040 4433 5822 7206 8586 9962 1333 2700 4063 5421 6776 8126 9471 0813 2151 3484 4813 6139 7460 8777 ..90 1400 270 4006 5304 6598 7888 9174 0456 LiLi I 1 .5557 691 8260 9606 0947 2284 3617 4946 6271 7592 ! 8909 .221 1530 2835 4136 5434 6727 8016 9302 0584 LOGARITHMS OF NUMBERS. ~N. 1 |l|2|3|4|5|6|7|8!9|D. ! 340' 531479 1607 i:34 1862 1990 211712245 2372 2500 2627 128 341 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 127 342 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 127 343 5294 5421 5547 5674 5800 5927 6053 6180 6306 6432 126 344 6558 6685 6811 6937 7063 7189 7315 7441 7567 7693 126 345 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 126 346 9076 9202 9327 9452 9578 9703 9829 9954 ..79 .204 125 347 540329 0455 0580 0705 0830 0955 1080 1205 1330 1454 123 348 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 349 350 2825 2950 4192 3074 4316 3199 4440 3323 3447 4688 3571 3696 4936 3820 5060 3944 5183 124 124 544068 4564 4812 351 5307 5431 5555 5678 5802 5925 6049 6172 6296 6419 124 352 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 353 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 123 354 9003 9126 9249 9371 9494 9616 9739 9861 9984 .106 123 355 550228 0351 0473 0595 0717 0840 0902 1084 1206 1328 122 356 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 357 2068 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 358 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 359 360 5094 5215 5336 5457 6664 5578 6785 5699 6905 5820 7026 5940 6061 7267 6182 7387 121 120 550303 6423 6544 7146 361 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 120 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 363 9907 ..26 .146 .265 .385 .504 .624 .743 .863 .982 119 364 561101 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 365 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 119 366 5481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 367 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 US 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 369 370 7026 568202 7144 8319 7262 8436 7379 8554 7497 8671 7614 8788 7732 8905 7849 7967 8084 9257 118 117 9023 9140 371 9374 9491 9608 9725 9842 9959 ..76 .193 .309 .426 117 372 570543 0660 0776 0893 1010 1126 1243 1359 1476 1592 117 373 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 M6 374 2872 2988 3104 3220 3336 3452 3568 3684 3800 3915 116 375 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 376 5188 5303 5419 5534 5650 5765 5880 5996 6111 6226 115 377 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 115 378 7492 7607 7722 7836 7951 8006 8181 8295 8410 8525 115) 379 380 8639 579784 8754 9898 8868 ..12 8983 .126 9097 .241 9212 .355 9326 .469 9441 9555 .697 9609 .811 114 114 .583 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 114 382 2063 2177 2291 2404 2518 2631 2745 2858 2972 3085 114 383 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 113 384 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 385 6461 5574 5686 5799 5912 6024 6137 6250 6362 6475 113 386 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 112 387 7711 7823 7935 8047 8160 8272 8384 8496 S608 8720 112 388 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 389 390 9950 ..61 .173 1287 .284 1399 .396 1510 .507 .619 1732 .730 .842 1955 .953 2066 112 111 691065 1176 1021 1843 391 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 392 3286 3397 3508 3618 3729 3S40 3950 4061 4171 4282 111 393 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 110 394 5496 5606 5717 5827 5937 6047 6157 626V 6377 6487 110 395 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 110 396 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 110 397 8791 8900 9009 9119 9228 9337 9446 9556 9665 9774 109 398 9883 9992 .101 .210 .3191 .428 .537 .646 .755 .864 109 399 600973 10S2 1191 1299 14081 1517 1625 1734 1843 1951 109 N. 1 Il|2|3i4l5i6|7l8|9in. J LOGARITHMS OF NUMBERS. N.~ 1 1 1 1 2 1 3 ! 4 1 . 5 1 6 1 7 ! 8 1 9 I D ! 400 602060 1 2169 2277 2386,2494,26031 2711 2819 2928 3036' JOS' 401 3144 J253 3361 3469 3.577 3686 3794 3902 4010 4118 108 102 4226 4334 4442 4550 4658 4766 4874 4982 6089 5197 108 403 5305 5413 5521 5628 5736 5844 .5951 6059 6166 6274 108 404 635r 6489 6596 6704 6811 6919 7026 7133 7241 7.3481 107| 405 7455 7562 7669 7777 7884 7991 8098 8205 8313 8419 107 406 8526 8633 8740 8847 8954 1 9061 9167 9274 9381 9488 107 407 95941 9701 9808 9914 ..2li .128 .234 .341 .447 ..5.54 107 408 610660 1723 0767 0873 0979 1086 1192 1298 1405 1511 1617 106 409 410 1829 2890 1936 2996 2042 3102 2148 3207 22h4 3311 2360 .3419 2466 3525 2572 36.30 2678 106 612784 3736 106 4'.: 3842 3947 4053 41.59 4264 4370 4475 4.581 4686 4792 106 412 4897 5003 5108 5213 5319 5424 5529 5634 5740 .58451 105 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 68951 105 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 79431 105 415 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 116 9093 9198 9302 9406 951 ll 9615 9719 9824 9928 ..32 104 417 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 101 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 104 419 420 2214 2318 3353 2421 3456 2525 3559 2628 3663 2732 3766 2835 2939 3042 3146 104 623249 3809 3973 407614179! 103 421 4282 4385 4488 4591 4695 4798 4901 5004 61071.52101 103 422 5312 5415 5518 5021 5724 58271 5929 6032 6 1 35 1 6338 103 423 6340 6443 6546 6648:6751 6853! 6956 7058 7161 72631 103 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 82871 102 425 8389 8491 8593 8695 8797 8900 9002 9104 9206 93081 102 426 9410 9512 9613 9715 9817 9919 ..21 .123 .2241 .326 102 427 630428 0530 0631 0733 0835 0936 1038 1139 12411 1342 102 428 1444 1545 1647 1748 1849 1951 2052 21.53 2255! 2356 101 429 2457 2559 2660 2761 2862 2963 .3064 3105 3266 i 3367 101 430 633468 3569 3670 3771 3872 3973 4074 4175 4276 4376 100 431 4-177 4578 4679 4779 4880 4981 5081 5182 5283 5383 100 432 5484 5584 5685 5785 .5886 5986 6087 6187 6287 6388 100 43'3 6488 6588 6688 6789 6889 6989 7089 7189 7290] 7390 100 434 7490 7590 7690 7790 7890 7990 8090 8190 82901 8389 99 435 8489 8589 8689 8789 8888 8988 9088 9188 9287^ 9387 99 436 9486 9586 9686 9785 9885 9984 ..84 .183 .283 .382 99 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 99 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 439 440 2465 2563 3551 2662 3650 2761 3749 2860 3847 2959 3946 3058 4044 3156 4143 3255 3354 4242 4340 99 98 643453 441 ^ 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 442 5422 5521 6619 5717 5815 .5913 6011 6110 6208 6306 93 443 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 444 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 98 445 smo 8458 85.55 8653 8750 8848 8945 9043 9140 9237 97 lie 9335 9432 9530 9627 9724 9821 9919 ..16 .113 .210 97 117 650308 0405 0502 0.599 0696 0793 0890 0987 1084 1181 97 HS 1278 1375 1472 1569 1666 1762 1859 19.56 2053 2150 97 149 2246 2343 2440 2536 2633 2730 2826 2923 3019|3I16 97 450 C5?i\3 3309 3405 3502 .35981 3695 3791 3888 3984! 4080 96 451 4H7 4273 4369 4465 4562 4658 4754 4850 494 6 .50 42 96 452 5138 5235 .5331 5427 .5523 5619 5715 .5810 .5906 60021 961 453 5098 6194 6290 6386 6482 6577 6673 6769 6864 69601 96 1 454 7056 7152' 7247 7343 74381 7534 7629 7725 7820 7916 96 455 8011 8107 8202 8298 8393 8488 8.584 6679 8774 8870 95 456 8965 9060 9155 9250 9346 944 1 9536 9631 9726 9821 95 457 9916 ..11 .106 .201 .296 .391 .486 ..581 .676 .771 95 458 660895 0960 1055 11.50 1245 1.339 14,34 1529 1623 1718 95 459 1813 1907 2002 2096 2191 228C 2380 2475 2569 2663 95 1 ^ 1 1 1 2 i 3 1 4 1 5 1 6 1 7 i 8 1 9 1 D. i U LOGARITHMS OF NUMBERS. -^\ ;i|2|3|4!5|6|7|8|9!D. 1 460 662758 2852| 2947 1 3041, 3135 3230, 3324, 34l8i 3512| 3607 94 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 94 462 4642 4736 4830 4924 5018 5112 5200 5299 5393 .6487 94 463 »581 5675 5769 5862 5956 6050 6143 6237 6331 6424 94 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 94 165 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 93 466 8386 8479 9572 8665 8759 8852 8945 9038 9131 9224 93 467 9317 9410 9503 9596 9689 9782 9875 9967 ..60 .153 93 468 670246 0339 0431 0524 0617 0710 0802 0895 0988 1080 93 469 170 1173 1265 2190 1358 2283 1451 1543 1636 2560 1728 2652 1821 2744 1913 2005 93 92 072098 2375 2467 2836 2929 471 472 b02l 3942 3113 4034 3205 4126 3297 4218 3390 4310 3482 4402 3574 3666 4586 37581 3850 4677, 4769 92 92 Vm 473 486 1 4953 6045 5137 5228 5320 5412 5503 5595 5687 92 174 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 175 6694 6785 6870 6968 7059 7151 7242 7333 7424 7616 91 476 7607 7698 7789 7881 7972 8063 8154 8246 8336 8427 91 477 8518 8609 8700 8791 8882 8973 9064 9165 9246 9337 91 178 9428 9519 9610 9700 9791 9882 9973? ..63 .1.64 .245 91 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 91 480 681241 1332 1422 1513 1603 1693 1784 1874 1964 2066 90 481 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4766 90 484 4845 4935 5025 5114 5204 5294 5383 5473 5563 5662 90 485 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 89 486 6036 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 487 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 89 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 489 490 9309 690196 9398 0285 9486 0373 9575 9664 0550 9753 0639 9841 9930 ..19 .107 89 89 0462 0728 0816 0905 0993 491 1081 1170 1258 1347 1435 1624 1612 1700 1789 1877 88 492 1965 2053 2142 2230 2318 2406 2494 2583 2671 2769 98 493 2847 2935 3023 3111 3199 3287 3375 3463 3651 3639 98 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 495 4605 4093 4781 4868 4956 5044 5131 5219 5307 5394 88 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 87 497 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 87 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 499 500 8101 698970 8188 8275 9144 8362 9231 8449 9317 8535 9404 8622 9491 8709 9578 8796 9664 8883 9751 87 87 9057 501 9838 9924 ..11 ..98 .184 .271 .368 .444 .531 .617 87 502 700704 0790 0877 0963 1050 1130 1222 1309 1395 1482 86 503 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 86 504 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 86 50£ 3291 3377 3463 3549 3635 3721 3807 3895 3979 4066 86 596 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 86 507 6008 5Qtf4 5179 5265 5350 5436 5522 5607 5693 6778 86 503 5864 5949 C035 6120 6206 6291 6376 6462 6547 6632 85 509 510 6718 707570 6803 7656 6888 6974 7826 7059 7144 7996 7229 8081 7316 8166 7400 8251 7485 86 85 7740 7911 8336 511 8421 8506 8591 8676 8761 8846 8931 9015 9863 9100 9185 85 512 9270 9355 9440 9524 9609 9694 9779 9948 ..33 85 513 710117 0202 0287 0371 0456 0540 0625 0710 0794 0879 86 514 0963 1048 1132 1217 1301 1385 1470 1.554 1639 1723 84 515 1807 1892 1976 2060 2144 2229 2313 2397 2481 26C6 84 516 2650 2734 2818 2902 2986 30701 3154 3238 3323 3407 84 517 3491 3575 3650 3742 3826 3910 3994 4078 4162 4246 84 518 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 519 5167 5251 53351 5418 5502 5586 5669 5753' 5836 5920 841 'nT 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 D.l LOGARITHMS OF NUMBERS. '^nT 1 {1|2|3|4|5|6|7|8|9|D^ 520 716003 6087 6170 62.54 6337 6421 6.504, 65«8| 6671 67.54 ^3 521 6838 6921 7004 7088 7171 72.54 73381 7421 7504 7587 83 522 7671 7754 7837 7920 8003 8086 8169| 8253; 8336 8419 83 523 8502 8585 8668 8751 8834 ;8917 9000, 9083! 9165 9218 83 524 9331 9414 9497 9580 9663 i 9745 982819911 9994 ..77 83 525 720159 0242, 0325 0407 0490 ; 0573i 0655| 0738 0821 0903 83 526 0986 1068) 1151 12.33 1310 1398 1481 1.563 1646 1728 82 527 1811 1893 1975 2058 2140 2222 2305i 2387 2469 2552 82 528 2634 2716! 2798 2881 2963 3045 3127 3209 3291 ,337 1 82 529 3456 3538, 3620 3702 3784 3866 3948 4030 4112 4194 82 530 724276 4358' 4440 4522 4604 4685 4767 4849 4931 .5013 82 531 5095 5 1 76 1 .5258 5340 5422 5503 5585 5667 6748 6830 82 532 5912 59931 6075 61.56 6238 6320 6401 6483 6564 6646 82 533 6727 6809; 6890 6972 7053 7134 7216 7297 7379 7460 81 534 7541 76231 7704 7785 7866 7948 8029 8110 8191 8273 81 535 8354 843518516 8597 8678 8759 8841 8922 9003 9084 81 536 9165 9246,9327 9408 9489 9570 9651 9732 9813 9893 81 537 9974 ..55 .136 .217 .298 .378 .459 •540 .621 .702 81 538 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 81 539 540 1589 1C69 1750 1830 2635 1911 1991 2796 2072 2152 2956 2233 3037 2313 3117 81 '80 732394 2474 2555 2715 2876 541 3197 3278 3358 34.38 3518 3598' 3679 3759 3839 3919 80 542 3999 4079 4160 4240 4320 4400' 4480 4560 4640 4720 80 543 4800 4880 4960 5040 5120 52001 52 ro 5359 5439 5519 80 544 5599 5679 5759 5838 5918 5998! 6078 6157 6237 6317 80 545 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 546 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 547 7987 8067 8146 8225 8305 8384 8403 8543 8022 8701 79 548 87S1 8860 8939 9018 9097 9177 9266 9335 9414 9493 79 549 550 95/2 9651 0442 9731 0521 9810 9889 0678 996^ ..47 .126 0915 .206 .284 79 79 740363 0600 0757 0836 0994 1073 551 1152 1230 1.309 1388 1467 1546 1624 1703 1782 1860 79 552 1939 2018 2096 2175 2254 2332 2411 2489 2568 2646 79 553 2725 2804 2882 2961 3039 3118 3196 3275 .3353 3431 78 554 3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5153 5231 5309 .5387 5465 5543 5621 5699 6777 78 557 6855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 558 6G34 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 559 560 7412 7489 7567 7645 8421 7722 8498 7800 8576 7878 8653 7955 8731 8033 8808 8110 8885 78 77 748188 82G6 8.343 561 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 502 9736 9814 9891 9968 ..45 .123 .200 .277 .3.54 .431 77 563 750508 0586 0663 0740 0817 0894 0971 1048 1125 1202 77 564 1279 1356 1433 1510 1.587 1664 1741 1818 1895' 1972 77 565 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 7? 566 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 567 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 568 4348 4425 4.501 4578 46,54 4730 4807 4883 4900 .5036 76 I 569 570 5112 5189 5951 5265 6027 5341 6103 .5417 5494 6256 5570 6332 .5646 6408 .5722 5799 6484 6.560 76 76 755875 6180 671 6636 6712 6788 6864 6940 7016 7092 7168 7244' 7.320 76 572 7396 7472 7548 7624 7700 7776 7851 7927 8003 8079 76 573 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 574 8912 8988 9063 9139 9214 9290 9366 9441 9517, 9.592 76 575 9668 9743 9819 9894 9970 ..45 .121 .196 .272! .347 75 576 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 76 677 1176 1251 1326 140t4 1477 15.52 1627 1702 1778, 1853 75 578 1928 2003 2078 2153 2228 2303 2378 2463 2529 2604 76 579 2679 27.5412829 2904- 2978 30.53 3128 3203 3278 3353' 75 1 _N^_ U 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 i 9 ■ i>. 6 6. J 10 LOGARITHMS OF NUMBERS. N. J |l|2|3i4|5|6|7|8|9|D. 1 580 763428. 3503 3078, 3653 3727 3802: 3877, 395:^ 4027 4101 75 581 4176 4251 4326 4400 4475 455C 462^^ 4699 4774 4848 75 682 4923 4998 5072 5147 5221 5296 537( 5445 5520 5594 75 583 5669 5743 5818 5892 5966 0041 6il£ 6190 6264 6338 74 584 6413 6487 6562 6636 6710 6785 685JJ 6933 7007 7082 74 J85 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 74 586 7898 7972 8046 8120 8194 8268 8342 841618490 8564 74 587 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 588 937'. 9451 9525 9599 9673 9746 9820 9894 9968 .42 74 589 590 770115 0189 0263 0999 0336 0410 1146 0484 1220 0557 1293 0631 0705 1440 0778 1514 74 74 770852 0926 1073 1367 591 1587 1661 1734 1808 1881 1955 2023 2102 21 r5 2248 73 592 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 594 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 4517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 596 5246 5319 5392 5465 5538 5610 1 5683 5756 5829 5902 73 597 5974 6047 6120 6193 6265 6338 1 6411 6483 6556 6629 73 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 600 7427 7499 8224 7572 8296 7644 7717 7789 8513 7862 8585 79.34 8006 8730 8079 8802 72 "72 770151 8368 8441 86.58 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 602 9596 9669 9741 9813 9885 9957 ..29 .101 .173 .245 72 603 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 72 604 1037 1109 1181 1253 1324 1396 1468 1.540 1612 1684 72 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 606 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 607 3189 3260 3332! 3403 3475 3546 3618 3689 3761 3832 71 608 3904 3975 4046,4118 4189 4261 4332 4403 4475 4.546 71 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 5^59 Jl 610 785330 6401 5472 5543 5615 5686 5757 5828 5899 .5970 71 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 612 6751 6822 6893 6964 7035! 7106 7177 7248 7319 7390 71 613 7460 7531 7602 7673 7714 7815 7885 7956 80271 8098 71 614 8168 8239 8310 8381 8451 8522 8593 8663! 8734 8804 71 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 616 9581 9651 9722 9792 9863 9933 ...4 ..74 .144 -.215 70 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 0988 1059 1129 1199 1269 1340 1410 1480 i:50 1620 70 619 620 1691 1761 1831 2532 1901 2602 1971 2672 2041 2111 2812 2181 2882 2252 29?.2 2322 3022 70 "70 792392 2462 2742 621 3092 3162 3231 3301 3371 3441 3511 3581 3651 3721 70 622 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 .5045 5115 70 624 5185 5254 5324 5393 5463' 5532 5602 5672 5741 5811 70 625 5880 5949 6019 6088 6158! 6227 6297 6366 6436 6505 69 626 6574 6644 67131 67821 6852 6921 6990 7060 7129 7198 69 627 7268 7337 7406 7475! 7545^ 7614 7683 7752 7821 7890 69 628 7960 8029 8098 8167 8238 1 8305 8374 8443 8513 8582 69 629 630 8651 8720 9409 8789 9478 8858 9547 8927 8996 96 10 9685 9065 97.54 9134 9823 9203 9892 9272 9961 69 69 799341 631 800029 0098 0167 0236 0305 0373 0442 0511 0580 0648 69 632 0717 0786 0854 0923 0992, 1061 1129 1198 1266 1335 69 633 1404 1472 1541 1609| 1678 1747 1815 1884 1952 2021 €9 634 2089 2158 2226 2295] 2363 2432 2500 2568 2637 2705 69 635 2774 2842 2910 2979 3047|3116 3184 3252 3321 3389 68 636 3457 3525 3594 3G62 3730 3798 38G7 3935 4003 4071 68 637 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 68 638 4821 4889! 4957 5025; 5093 5lBl 5229 52U7I 5365 .5433 68 039 5501155691 5637 5705 57731 5841' 5908i 5976: 6044' 61 12' 68 | Jld |l|2i3|4l5|6|7|8|9|D. 1 ^^/), 5-6'^^ LOGARITHMS OF NUMBERS. -"• 1 \ ^ 1 2 1 3 4|5 |6|7|8|9|U. 1 (540 806180 6248 6316| 6384 0451 6519 6587 6055, 0723 6790 68 041 6858 6926 699417061 7129 7197 7204 7332 7400 7407 68 642 7535 7603 7670 7738 7800 7873 7941 8008 8076 8143 68 043 8211 8279 8346:8414 8481 8549 80101 8084 8751 8818 67 644 8886 80 "iS 9021 9088 9150 9223 92901 9358 9425 9492 67 645 9560 9627 9094 9702 9829 9896 9904 ..31 ..98 .105 67 646 810233 0300 0367 0434 0501 0569 0036 0703 0770 0837 67 647 0904 0971 1039 1100 1173 1240 1307 1374 1441 1508 67 648 1575 1642 1709 1770 1843 1910 1977 2044 2111 2178 67 649 6-50 2245 2312 2379 2445 2512 2579 3247 2646 3314 2713 3381 2780 3448 2847 3514 67 "07 812913; 2980 3047 3114 3181 651 3581 3648 3714 3781 3948 3914 3981 4048 4114 4181 67 652 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 653 4913 4980 5046 5113 5179 5246 5312 5378 5445 .5511 66 654 6578 5644 5711 5777 5843 ,5910 5976 0042 6109 6175 66 655 6241 6308 6374 0440 0500 m 6639 0705 6771 6838 66 656 6904 6970 7036 7102 7109 T3'01 7307 7433 7499 66 657 7565 7631 7698 7704 7830 7896 7962 8028' 8094 8160 66 658 8226 8292 8358 8424 8490 8550 8622 8088 8754 8820 66 659 660 8885 819544 8951 9017 9676 9083 9149 9807 9215 9281 9939 9340 ...4 9412 ..70 9478 66 66 9610 9741 9873 .1.36 661 820201 0267 0333 0399 0404 0530 0595 0001 0727 0792 66 662 0858 0924 0989 1055 1120 1180 1251 1317 1382 1448 66 663 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 65 664 2168 2233 2299 2304 2430 2495 2560 2020 2091 2756 65 665 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 666 3474 3539 3605 3070 3735 3800 3865 3930 3996 4061 65 667 4126 4191 4256 4321 4380 4451 4516 4581 4646 4711 65 668 4776 4841 4906 4971 5030 5101 5166 5231 5296 5361 65 669 670 5426 826075 5491 5556 6204 5021 5080 6334 5751 0399 5815 6464 5880 0528 5945 6593 6010 6658 65 65 6140 6269 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 05 672 7369 7434 7499 7563 7628 7092 7757 7821 7886 7951 65 673 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 675 9304 9368 9432 9497 9561 9025 9090 9754 9818 9882 64 676 9947 ..11 ..75 .139 .204 .208 .332 •396 .460 .525 64 677 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 678 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 64 679 680 1870 832509 1934 2573 1998 2637 2062 2700 2126 2764 2189 2828 2253 2317 2956 2381 2445 64 64 2892 3020 3083 681 3147 3211 3275 3338 3402 3460 3530 3593 3657 3721 64 682 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 683 442! 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 684 5050 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 685 5691 5754 5817 5881 5944 0007 6071 6134 6197 6201 63 686 6324 6387 6451 6514 6577 0041 6704 6767 68.30 6894 63 687 6957 7020 7083 7146 7210 7273 7336 7399 7402 7525 63 688 7588 7652 7715 7778 7841 7904 7907 8030 8093 81.56 63 689 690 8219 838849 8282 8345 8408 9038 8471 8534 9104 8597 8660 9289 8723 9352 8786 63 63 8912 8975 9101 9227 9415 691 9478 9541 9604 9667 9729 9792 9855 9918 9981 ..43 63 692 840106 0169 0232 0294 0357 0420 0482 0545 0008 0671 63 693 0733 0796 0859 0921 0984 1040 1109 1172 1234 1297 63 694 1359 1422 1485 1547 1610 1672 1735 1797 1800 1922 63 695 1985 2047 2110 2172 2235 2297 2300 2422 2484 2547 62 696 2609 2672 2734 2796 2859 2921 2983 3046' 3108 3170 62 697 3233 3295 3357 3420 3482 3544 3000 3669 3731 3793 62 698 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 62 699 4477 4539 4601 4604 4726 4788 4850 4912 4974 .5036' 6*i 1 N. 1 !l|2|3|4|6|6|7| _8 1 .?» 1 -iT! 12 LOGARITHMS OF NUMBERS. N. 1 |l|2|3|4|oit>|V;8!9|D. 700 845098 5160.5222 5284 534 6 1 5408 .5470| 5532 5.594 |5656i 621 701 5718 5780, 5842 5904 5966 6028 609C 6151 6213 6275 62 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 703 6955 7017 7079 7141 7202 7264 732T> 7388 7449 7511 62 704 7573 7634 7696 7758 781.1' 7881 7943 8004 8066 8128 62 705 8189 8251 8312 8374 843.V8497 8559 8620 8682 8743 62 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 935S 61 707 9419 9.481 9542 9604 9665 9726 9788 9849 9911 9972 61 708 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 61 709 710 0646 0707 1320 0769 0830 1442 0891 1503 0952 1564 1014 1075 1136 1747 1197 61 61 851258 1381 1625 1686 1809 711 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 3090 3150 3211 3272 3333 3394 3455 3516 .3577 3637 61 7J4 3898 3759 3820 3881 3941 4002 40(53 4124 4185 4245 61 7i5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 716 4913 4974 5034 5095 5156 5216 .5277 5337 5398 5459 61 717 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 718 6124 6185 6245 6306 6366 6427 6487 6.548 6608 6668 60 719 720 6729 67891 6850 6910 7513 6970 7574 ~031 7634 7091 7694 7152 7212 7815 7272 7875 60 60 857332 7393 7453 7755 721 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 723 9138 9198 9258 9318 9379 9439 9499 95.59 9619 9679 60 724 9739 9799 9859 9918 9978 . .38 ..98 .1.58 .218 .278 60 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 60 726 0937 0996 10561 1116 1176 1236 1295 1355 1415 1475 60 727 1534 1594 1654 1714 1773 1833 1893 19.52 2012 2072 60 728 2131 2191 2251 2310 2370 24.30 2489 2549 2608 2668 60 729 730 2728 2787 2847 3442 2906 2966 3025 3085 3080 3144 3739 3204 3799 3263 60 59 863323 3382 3501 3561 3620 3858 731 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 732 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 733 5104 5163 5222 5282 5341 5400 5459 .5519 5578 5637 59 734 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 735 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 730 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 59 738 8056 8115 8174 8233 8292 8350 8409 8468 8527 8.586 59 739 740 8644 8703 9290 8762 8821 9408 8879 9466 89.38 95/5" 8997 9584 9056 9114 9701 9173 9760 59 59 869232 9349 9642 741 9818 9877 9935 9994 ..53 .111 .170 .228 .287 .,345 59 742 870404 0462 0521 0579 0638 0696 0755 0813 0872 0930 h8 743 0989 1047 1106 1164 1223 1281 1339 1398 14.56 1515 58 744 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 58 745 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 58 747 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 748 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 ^8 749 750 4482 4540 4598 4656 4714 5293 4772 5351 4830 5409 4888 4945 .5003 .5.582 58 58 875061 5119 5177 5235 ,5466 5524 751 5640 5698 5756 5813 5871 5929 .5987 6045 6102 6160 58 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 753 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 58 754 7371 7429 7487 7544 7602 7659 7717 VVV4 7832 7889 58 755 7947 8004 8062 8119 8177 82.34 8292 8349 8407 8464 57 756 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 757 9096 9153 9211 9268 9325 9383 9440 9497 9,555 9612 57 758 9669 9726 9784 9841 9898 9956 ..13 ..70 .127 .185 57 759 880242 0299! 03561 04131 0471 0528 0585 0642 0699 07.56 57 _Nd 1 1 1 2 i 3 1 4 1 5 i 6 1 7 1 8 i 9 1 D. ! LOGARITHMS OF NUMBERS. 13 N. ' 1 1 1 2 1 3 I 4 1 5 1 6 1 7 1 8 1 9 1 D. ( 760" 880814 0871 0928 09851 1042 1099 1156 1213 1271! 1328 671 761 1385 1442 1499 1556J 1613 1670 1727 1784 1841 1 1898 57 762 1955 2012 2069 2126 2183 2240 2297 23.54 241112468 57 763 2525 2581 2638 2695! 2752 2809 2866 2923 2980 3037 57 76 i 3093 3150 3207 3264 3321 3377 3434 .3491 3548 3605 57 765 3661 3718 3775 3832, 3888 3945 4002 40.59 4115 4172 57 | 766 4229 4285 4342 4399:4455 4512 4569 4625 4682: 4739 57 767 479') 4852 4909 4965; 5022 5078 5135 5192 5248 i .5305 57 768 5361 5418 5474 5531 15587 5644 5700 5757 581315870 57 769 5926 5983 6039 6096 6152 6209 6265 6321 6378! 6434 56 770 886491 6547 6604 6660 6716 6773 6829 6885 6942' 6998 56 771 7054 7111 7167 72231 7280 7336 7392 7449 7505: 7561 56 772 7617 7674 7730 77861 7842 7898 7955 8011 8067 8123 56 773 8179 8236 8292 8348! 8404 8460 8516 8573 8629 8685 56 774 8741 8797 8853 8909' 8965 9021 9077 9134 9190, 9246 56 775 9302 9358 9414 9470 9526 9582 9638 9694 97501 9806 56 776 9862 9918 9974 ..30 ..86 .141 .197 .253 .309 .365 56 777 890421 0477 0533 0589 0645 0700 0756 0812 0868 0924 56 778 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 56 779 780 1537 1593 2150 1649 2206 17051 1760 226212317 1816 2373 1872 2429 1928 2484 1983 2540 2039 2595 56 56 892095 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 56 782 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 56 783 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 55 784 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 55 785 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 55 786 5423 5478 5533 5588 5644 5699 5754 5809 .5864 5920 55 787 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 55 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 789 790 7077 7132 7682 7187 7737 7242 7792 7297 7847 7352 7908 7407 7957 7462 8012 7517 8067 7572 8122 55 55 897627 791 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 55 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 793 9273; 9328 9383 9437 9492 9547 9G02 9656 9711 9766 55 794 982110875 9930 9985 ..39 ..94 .149 .203 .258 .312 55 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 55 79G 0913i 0968 1022 1077 1131 1186 1240 1295 1349 1404 55 797 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 54 799 800 2547 2601 3144 2655 3199 2710 3253 2764 3307 2818 3361 2873 3416 2927 3470 2981 3524 3036 3578 54 54 903090 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 802 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 803 4716 4770 4824 4878 4932 4986 5040 5094 5148 .5202 54 804 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 805 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 806 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 54 907 6874 6927 6981 7035 7089 7143 7680 7196 7250 7304 7358 54 808 7411' 7405 7519 7573 7626 7734 7787 7841 7895 54 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 810 908485 8539 8592 8646 8699 8753 8807 8860 89141 8967 54 811 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 54 812 955619610 9663 9716 9770 9823 9877 9930 9984 ..37 53 813 910091 0144 0197 0251 0304 0358 0411 0464 0518 0.571 53 814 0624 0678 0731 0784 0838 0891 l0944 0998 1051 1104 53 815 1158 1211 1264 1317 1371 1424 1 1477 15.30 1584 1637 53 816 1690 1743 1797 1850 1903 1956 I2OO9 2063 2116 2169 53 817 2222 2275 2328 2381 2435 2488 12541 2594 2647 2700 53 818 2753 2806 ! 2859 2913 2966 3019 3072 3125 3178 3231 53 819 3284 3337 1 3390 34431 3496 3549 1 3602 3655 3708' 3761 53 N _ 1 1 1 1 * 1 3 1 4 1 5 1 6 I 7 1 8 1 U 1 D.l 14 LOGARITHMS OF NUMBERS. N. 1 1 ! 2 1 3 1 4 1 5 1 6 1 7 I 8 1 9 1 D. 820 913814,3867 3920 3973 4026 4079 4132 4184 4237 4290 63 821 43431 4396 4449 4502 4555 4608 4660 4713 4766 4819 63 822 4872 4925 4977 6030 .5083 5136 5189 .5241 6294 .5347 53 823 5400 5453 5505 6558 .5611 .5664 ,5716 6769 .5822 5875 63 824 5927 5980 6033 6085 61.38 6191 6243 6296 6349 6401 53 825 6t54 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 S2B 6980, 7033 7085 71.38 7190 7243 7295 7348 7400 7453 53 827 750C 7558 7611 7663 7716 7768 7820 7873 7925 7978 52 828 80301 8083 8135 8188 8240 8293 8345 8397 8450 8.502 62 829 830 8555 8607 8659 8712 8764 928- 8816 9.340 8869 9392 8921 9444 8973 9496 9026 9.549 62 62 919078 9130 9183 9235 831 9001 9653 9706 9758 9810 9862 9914 9967 ..19 .71 62 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 62 833 0645 0697 0749 0801 0853 0900 09.58 1010 1062 1114 52 834 1166 1218 1270 1.322 1374 1426 1478 1.530 1.582 1634 52 835 1686 1738 1790 1842 1894 1946 1998 2060 2102 2154 62 836 2206 2258 2310 2362 2414 2466 25! 8 2570 2622 2674 62 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 62 838 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 62 839 840 3762 924279 3814 3865 4383 3917 4434 3969 4021 4072 4124 4641 4176 4228 4744 52 62 4331 4486 4538 4589 4693 841 4796 4848 4899 4951 5003 5054 5106 5157 5209 6261 62 842 6312 6364 5415 6467 .5518 .5570 5621 6673 6725 6776 52 843 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 51 844 6342 6394 6445 6497 6548 6600 6651 6702 67.54 6805 61 845 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 51 846 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 61 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 61 848 8396 8447 8498 8549 8601 8652 8703 8764 8805 8867 51 849 850 8908 9294 1 9 8959 9470 9010 9061 9112 9163 9215 9725 9266 9317 9368 61 51 9521 957^ 9623 9674 9776 9827 9879 851 9930 9981 ..32 ..83 .134 .185 .236 .287 .338 .389 61 852 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 61 853 0949 1000 1051 1102 11.53 1204 12.54 1305 1356 1407 51 854 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 855 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 51 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 61 857 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 61 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 51 859 860 3993 4044 4094 4145 4650 4195 4700 4246 4751 4296 4347 4852 4397 4448 61 60 934498 4549 4599 4801 4902 49.53 801 6003 5054 5104 5154 6205 5255 .5.306 5356 5406 .5457 60 862 5507 5558 5608 5658 5709 5759 6809 6860 5910 5960 50 863 6011 6061 6111 0162 6212 6262 6313 6363 6413 6463 60 864 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 60 885 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 60 866 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 60 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 60 868 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 60 1869 870 9020 939519 9070 9569 9120 9170 9669 9220 9270 9769 9320 9819 9369 9419 9918 9469 60 50 9619 9719 9869 9968 871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 50 872 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 50 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 50 874 1511 1561 1611 1660 1710 1760 1809 1869 1909 1958 50 875 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 50 876 2504 2554 2603 2663 2702 2752 2801 2851 2901 2950 50 877 3000 3049 3099 3148 3198 3247 3297 3346 3396 344 5 49 878 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 49 879 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 49 I 1 I 2 I 3 1 7 I 8 '9 ( D. 1 LOGARITHMS OF NUMBERS. 15 880" |l|2|3|4|5|6|7|8|9lD. 1 944483 45321 4581 46311 4680 4729 4779 482814877 4927 49 881 4976 5025 5074 5124 5173 5222 5272 .5321 .5370 ,5419 49 882 5469 5518 5567 5616 5665 5715 5764 5813 5862 5912 49 883 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 49 884 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 49 885 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 886 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 88,53 49 889 890 8902 8951 8999 9048 9536 9097 9146 9634 9195 9683 9244 9731 9292 9780 9,341 9829 49 '49 949390 9439 9488 9585 891 9878 9926 9975 ..24 ..73 .121 .170 .219 .207 .316 49 892 950365 0414 0462 0511 0560 0608 0057 0706 0754 0803 49 893 0851 0900 0949 0997 1046 1095 1143 1192 1240 12«9 49 894 1338 1386 1435 1483 1532 1.580 1629 1677 1726 1775 19 805 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 48 896 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 48 89? 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 48 898 3276 3325 3373 3421 3470 .3518 3566 3615 3663 3711 48 899 900 3760 3808 4291 3856 3905 4387 3953 4435 4001 4049 4532 4098 4580 4146 4628 4194 4677 48 "48 954243 4339 4484 901 4725 4773 4821 4869 4918 4966 .5014 5062 5110 51.58 48 902 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 903 5688 57361 5784 5832 5880 5928 .5976 6024 6072 6120 48 904 6168 6216 6265 6313 6361 6409 6457 6505 65.53 6601 48 305 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 907 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 48 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 909 910 8564 8612 8659 8707 8755 9232 8803 9280 8850 9328 8898 9375 8946 9423 8994 48 48 959041 9089 9137 9185 9471 911 9518 9566 9614 9661 9709 9757 9804 98.52 9900 9947 48 912 9995 ..42 ..90 .138 .185 .2.33 .280 .328 .376 .423 48 913 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 48 914 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 47 915 1421 1469 1516 1563 1611 16.58 1V06 1753 1801 1848 47 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 917 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 918 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 47 919 920 3316 3363 3835 3410 3882 3457 3929 3504 3977 3552 4024 3.'^99 4071 3646 4118 3693 4165 3741 4212 47 47 963788 921 42b0 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 922 4731 4778 4825 4872 4919 4966 .5013 5061 5108 51.55 47 923 5202 5249 5296 5343 5390 .5437 5484 5531 5578 5625 47 924 5672 5719 5766 5813 .5860 5907 5954 6001 6048 6095 47 925 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 47 926 6611 6658 6705 6752 679^^ 6845 6892 6939 6986 7033 47 927 7080 7127 7173 7220 7267 "314 7361 7408 74,54 7,501 47 928 7548 7595 7642 7688 7735 /782 7829 7875 7922 7969 47 929 930 8016 8062 8530 8109 8576 8156 8203 86/0 8249 8716 8296 8763 8343 8810 8390 8856 8436 8903 47 47 968483 8623 931 8950 8996 9043 9090 91.36 9183 9229 9276 9323 9369 47 932 9U6 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 933 9882 9928 9975 ..21 ..68 .114 .101 207 .2.54 .300 47 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 46 935 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 40 936 1276 1322 1369 1415 1461 1508 1.5.54 1601 1647 1693 46 937 1740 1786 1832 187S 1925 1971 2018 2064 2110 21.57 46 938 2203 2249 22951 23421 2388 2434 2481 2527 2573 2619 46 939 26661 2712' 2758' 2804' 2851 2897' 2943 2989' 30.35' 3082' 46 | "nT 1 1 1 1 2 3 1 4 1 5 1 6 ! 7 1 8 1 9 1 D. j 16 LOGARITHMS OF NUMBERS. N. |l|2j3|4|5|6|7|8|9|D. 1 940 973128 3174 3220 3266 33131 3359 3405 3451 3497 3543 46 941 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 46 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 943 4512 4558 4604 4650 4696 4742 4788 48.34 4880 4926 46 944 4972 5018 5064 5110 5156 5202 5248 5294 .5.340 5386 46 945 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 946 5891 5937 5983 6029 6075 6121 6167 6212 62.58 6304 46 947 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 948 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 949 950 7266 977724 7312 7358 7815 7403 7449 7906 7495 7541 7998 7586 7632 8089 7678 81.35 46 "46 7769 7861 7952 8043 951 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 45 956 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 45 957 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 958 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 45 959 960 1819 982271 1864 2316 1909 2362 1954 2407 2000 2045 2090 2135 2181 2226 2678 45 45 2452 2497 2543 2588 2633 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 45 962 3175 32201 32651 3310 3356 3401 3446 3491 3536 3581 45 963 3626 36711371613762 3807 3852 3897 3942 3987 4032 45 961 4077 41221416714212 4257 4302 4347 4392 4437 4^182 45 965 4527 4572 4617 4062 4707 4752 4797 4842 4887 4932 45 966 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 967 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 45 968 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 969 '970 6324 6369 6817 6413 6458 6906 6503 6951 6548 6996 6593 6637 7085 6682 6727 45 45 986772 6861 7040 7130 7175 971 7219 7264 7309 7353 7398 7443 7488 7.532 7577 7622 45 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 4,^ 973 8113 8157 8202 8247 8291 83.36 8381 8425 8470 8514 45 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 975 9005 9049 9094 9138 9183 9227 9272 9316 9.361 9405 45 976 9450 9494 9539 9583 9628 9672 9717 9701 9806 9850 44 977 9895 9939 9983 ..28 ..72 .117 .161 .206 .2.50 .294 44 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 44 979 984) 0783 991226 0827 0871 1315 0916 1359 0960 1004 1448 1049 1093 1137 1580 1182 1625 44 44 1270 1403 1492 1536 981 1669 1713 1758 1802 1846 1890 1935 1979 202'< 2067 44 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2.509 44 983 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 4317 4361 4405 4449 4493 4.537 4.581 4625 4669 4713 44 988 4757 4801 4845 4889 4933 4977 5021 .5065 5108 5152 44 989 990 5196 5240 5679 5284 5328 .5372 5416 5854 5460 5504 5942 5547 5591 6030 44 44 995635 5723 5767 .5811 .5898 5986 991 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 994 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 995 7823 7867 7910 7954 7998 8041 8085 8129 8172 8216 44 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 997 8695 8739 8782 8826 8869 8913 89.56 9000 9043 9087 44 998 9131 9174 9218 926] 9305 9348 9392 9435 9479 9.522 44 999 9565 9609 9652 9606 9739 9783 9826 987019913 9957 43 "157 1 1 1 1 2 I 3 1 4 1 5 1 6 1 7 1 8 1 9 1 D. 1 ^dr. ?/-/n^- LOGARITHMIC SINES AND TANGENTS, EVERY DEGREE AND MINUTE OF THE QUADRANT. The minutes in the left-hand column of each page, increas- ing downward, belong to the degrees at the top ; and those increasing upward, in the right-hand column, belong to the degrees below. <^/.^fC^^^^ - 5?^ /9??f^ib^-''/kJ^ 18 0°. LOGAKITHMIC M 1 Sine 1 D. Cosine 1 D. Tang. 1 U. Cotang. !_ ~o" 0.000000 , 10.000000 1 0.000000 Inniiiie. 60 59 1 6.403726 501717 000000 00 6.463726 501717 13.536274 2 764756 293485 000000 00 764756 293483 235244 58 3 940817 208231 000000 00 940847 208231 059153 57 4 7.065786 161517 000000 00 7.065786 161517 12.934214 56 6 162696 131968 000000 00 162696 131969 837304 55 6 241877 111575 9.999999 01 241878 111578 758122 54 7 308824 96653 999999 01 308825 99653 691175 53 8 366816 85254 999999 01 366817 85254 633183 52 9 417968 76263 999999 01 417970 76263 582030 51 10 11 463725 7.505118 68988 999998 01 01 463727 68988 536273 12.494880 50 49 62981 9.999998 7.5Col20 62981 12 542906 57936 999997 01 542909 57933 457091 48 13 577668 53641 999997 01 677672 63612 422328 47 14 609853 49938 999996 01 609857 49939 390143 46 15 039816 46714 999996 01 639820 46715 360180 45 16 667845 43881 999995 01 667849 43882 332151 44 17 694173 41372 999995 01 694179 41373 30.5821 43 18 718997 39135 999994 01 719003 39136 280997 42 19 742477 37127 999993 01 742484 37128 257c 16 41 20 21 764754 7.785943 353.15 999993 9.999992 01 01 764761 7.785951 35136 33673 235239' 40 33672 12.214049 39 22 806146 32175 999991 01 806155 32176 19.3845 38 23 825451 30805 999990 01 825460 30806 174540 37 24 843934 29547 999989 02 843944 29549 156056 36 25 861662 28388 999988 02 861674 28390 138326 35 26 878695 27317 999988 02 878708 27318 121292 34 27 895085 26323 999987 02 895099 2C325 104901 33 28 910879 25399 999986 02 910894 25401 089106 32 29 926119 24538 999985 02 926134 24'>40 073866 31 30 31 940842 7.955082 23733 999983 02 02 940858 7.955100 23735 22981 059142 30 29 22980 9.999982 12.044900 32 968870 22273 999981 02 968889 22275 031111 28 33 982233 21608 999980 02 982253 21610 017747 27 •SA 995198 20981 999979 02 995219 2)983 004781 26 35 8.007787 20390 999977 02 8.007809 2.)392 11.992191 25 30 020021 19831 999976 02 020045 19883 979955 24 37 031919 19302 999975 02 031945 19305 S68U55 23 38 043501 18801 999973 02 043527 18803 956473 22 39 054781 18325 999972 02 054809 18327 945191 21 40 065776 17872 999971 02 065806 17874 934194 20 41 8.076500 17441 9.999969 02 8.076531 17444 11.923469 19 42 086965 17031 999968 02 086997 17034 913003 18 43 097183 16639 999966 02 097217 16642 902783 17 44 107167 16265 999964 03 107202 16268 892797 16 45 116926 15908 999963 03 116963 15910 883037 15 46 126471 15566 999961 03 126510 15568 873490 14 47 135810 15238 999959 03 135851 15241 864149 13 48 144953 14924 999958 03 144996 14927 855004 12 49 153907 14622 999956 03 153952 14627 846048 11 50 51 162681 8.171280 14333 14054 999954 03 03 162727 14336 14057 837273 11.828672 10 9 9.999952 9.171328 52 179713 13786 999950 03 179763 13790 820237 8 53 187985 13529 999948 03 188036 13532 811964 7 54 196102 13280 999946 03 196156 13284 803844 6 55 204070 13041 999944 03 204126 13044 V95874 5 56 211895 12810 999942 04 211953 12814 7fe8047 4 57 219581 12587 999940 04 219641 12590 780359 3 58 227134 12372 999938 04 227195 12376 772803 2 59 234557 12164 999936 04 234621 12168 765379 1 60 241855 11963 9999341 04 241921 11967 75S07S 3 Corvine j 1 «"•« 1 Cutang. j 1 Tang. j Ki. | SINES AND TANGENTS. 1°. 19 Ml Sine 1 D. Cosine | D. Tang. { D. | Ootnng. i 8.241855 11963 9.999934 04 8.241921 11967 11.758079 60 1 249033 11768 999932 04 249102 11772 750898 59 2 256094 11.580 999929 04 256165 11584 743835 58 3 263042 11398 999927 04 263115 11402 736H85 57 4 269881 11221 999925 04 269956 11225 730044 56 5 276614 11050 999922 04 276691 11054 723309 55 6 283243 10883 999920 04 283323 10887 716677 54 7 289773 10721 999918 04 289850 10726 710144 53 8 296207 10565 999915 04 296292 10570 703708 52 9 302546 10413 999913 04 302034 10418 697366 51 10 11 308794 8.314954 10266 999910 9.999907 04 04 308884 8.315046 10270 691116 50 49 10122 10126 11.684954 12 321027 9982 999905 04 321122 9987 678878 48 13 327016 9847 999902 04 327114 9851 672886 47 14 332924 9714 999899 05 333025 9719 666975 46 15 338753 9586 999897 05 338856 9590 661144 45 16 344504 9460 999894 05 344610 9465 655390 44 17 350181 9338 999891 05 350289 9343 649711 43 18 355783 9219 999888 05 355895 9224 644105 42 19 361315 9103 999885 05 361430 9108 638570 41 20 21 366777 8990 999882 05 05 366895 8.372292 8995 633105 40 39 8.372171 8880 9.999879 8885 11.627708 22 377499 8772 999876 05 377622 8777 622378 38 23 382762 8667 999873 05 382889 8672 617111 37 24 387962 8564 999870 05 388092 8570 611908 36 25 393101 8464 999867 05 393234 8470 006766 35 26 398179 8366 999864 05 398315 8371 601685 34 27 403199 8271 999861 05 403338 8276 596662 33 28 408161 8177 999858 05 408304 8182 591696 32 29 413068 8086 939854 05 413213 8091 586787 31 30 417919 7996 999851 06 418068 8002 581932 30 31 8.422717 7909 9.999848 06 8.422869 7914 11.577131 29 32 427462 7823 999844 06 427618 7830 572382 28 33 432156 7740 999841 06 432315 7745 567685 27 34 436800 7657 999838 06 436962 7663 663038 26 35 441394 7577 999834 06 441560 7583 658440 25 36 445941 7499 999831 06 446110 7505 .553890 24 37 450440 7422 999827 06 450613 7428 549387 23 38 454893 7346 999823 06 455070 7352 644930 22 39 459301 7273 999820 06 459481 7279 540519 21 40 463665 7200 999816 06 463849 7206 536151 20 41 8.467985 7129 9.999812 06 8.468172 7135 11.531828 19 42 472263 7060 999809 06 472454 7066 527546 18 43 478498 6991 999805 06 476693 6998 523307 17 44 480693 6924 999801 06 480892 6931 619108 16 45 484848 6859 999797 07 485050 6865 614950 15 46 488963 6794 999793 07 489170 6801 610830 14 47 493040 6731 999790 07 493250 6738 606750 13 48 497078 6669 999788 07 497293 6676 502707 12 49 601080 6608 999782 07 501298 6615 498702 11 60 605045 6548 999778 07 505267 6555 494733 10 51 8.508974 6489 9.999774 07 8.. 509200 6496 11.490800 9 62 612867 6431 999769 07 613098 6439 486902 Sj 53 516726 6375 999765 07 616961 6382 483039 7 64 620551 6319 999701 07 520790 6326 479210 6 55 524343 6264 999757 07 624586 6272 475414 5 56 628102 6211 999753 07 628349 6218 471651 4 67 631828 6158 999748 07 532080 6165 467920 3 58 635523 6106 999744 07 535779 6113 464221 2 59 639186 6055 999740 07 539447 6062 460553 1| 60^ 542819! 6004 999735' 07 543081' 6012 456916 ol CO...U. j 1 Sine 1 1 Cotang. 1 1 Tang m 83° 20 2°. LOGARITHMIC , M. Sine — a-- Cosine | D. 1 Tane. j D. 1 rotane. 1 ] 8.542819 6004 9.999735 07 18.543084 6012 11.456916 60 1 546422 5955 999731 07 646691 .5962 453309 59 2 549995 5906 999726 07 550268 5914 449732 58 3 553539 5858 999722 08 653817 5866 446183 57 4 557054 .5811 999717 08 557336 5819 442664 56 5 560540 5765 999713 08 560828 5773 439172 65 6 563999 5719 999708 08 564291 5727 435709 54 7 567431 5674 999704 08 567727 5682 432273 63 8 570836 5630 999699 08 571137 5638 428863 52 9 574214 5587 9i)9694 08 574520 5595 425480 51 10 11 577566 8.580892 5544 999689 08 08 677877 8.581208 5552 422123 50 49 5502 9.999685 5510 11.418792 12 584193 5460 999680 08 584514 ,5468 415486 48 13 687469 5419 999675 08 687795 .5427 412205 47 14 590721 5379 999670 08 591051 5387 408919 46 15 593948 5339 999665 08 594283 5347 405717 45 16 597152 5300 999660 08 597492 5.308 402508 44 | 17 600332 5261 999655 08 600677 5270 399323 43 18 603489 5223 999650 08 603839 6232 396161 42 19 606623 5186 999645 09 606978 5194 393022 41 20 21 609734 8.612823 5149 999640 09 09 610094 5158 5121 389906 40 39 5112 9.999035 8.613189 11.386811 22 615891 5076 999629 09 616262 5085 383738 38 23 618937 .5041 999624 09 619313 5050 380687 37 24 621962 5006 999619 09 622343 .5015 377657 36 25 624965 4972 999614 09 625352 4981 374648 35 20 627948 4938 999608 09 628340 4947 371660 34 27 630911 4904 999603 09 631308 49J3 368692 33 28 633854 4871 999597 09 634256 4880 365744 32 29 636776 4839 999592 09 637184 4848 362816 31 30 639680 4806 999586 09 640093 4816 .359907 30 31" 8.642563 4775 9.999581 09 8.642982 4784 11.357018 29 32 645428 4743 999575 09 645853 4753 354147 28 33 648274 4712 999570 09 648704 4722 351296 27 34 651102 4682 999564 09 651537 4691 348463 26 35 653911 4652 999558 10 654352 4661 345648 25 36 656702 4622 999553 10 657149 4631 342851 24 37 659475 4592 999547 10 65992S 4602 340072 23 38 662230 4563 999541 10 662689 4573 337311 22 39 664968 4535 999535 10 665433 4544 334567 21 40 41 667689 4506 4479 999529 9.999524 10 10 668160 4526 331840 20 19 8.670393 8.670870 4488 11.329130 42 673080 4451 999518 10 673563 4461 326437 18 43 675751 4424 999512 10 676239 4434 323761 17 44 678405 4397 999506 10 678900 4417 ,321100 16 45 681043 4370 999500 10 681544 4380 318456 15 46 683665 4344 999493 10 684172 4354 315828 14 47 686272 4318 999487 10 686784 4328 313216 13 48 688863 4292 999481 10 689381 4303 310619' 12 49 691438 4267 999475 10 691963 4277 308037 11 50 693998 4242 999469 10 694529 4252 305471 10 51 8 696543 4217 9.999463 11 8.697081 4228 11.302919 9 52 699073 4192 999456 11 099617 4203 300383 8 53 701589 4168 999450 11 702139 4179 297861 7 54 704090 4144 999443 11 704646 4155 295354 6 55 706577 4121 999437 11 707140 4132 292860 5 56 709049 4097 999431 11 709618 4108 290^^82 4 57 711507 4074 999424 11 712083 4085 287917 3 58 713952 4051 999418 11 714534 4062 285465 2 59 716383 4029 999411 11 716972 4040 283028 I 60 718800 4006 999404 11 719396 4017 280604 ~ Cosine | 1 Sine 1 Cotang. 1 1 Tang. |M.| SINES AND TANGENTS. 3°. 21 jrj Slno 1 n. 1 Cosine | D. | Tansr. _I^_. CofHnir. 1 8 718800 4006 9.999404 11 8.719.396 40 i 7 11.280604 60 721204 3984 99939S 11 721806 3995 278194 .59 2 723595 3962 999391 11 724204 3974 275795 58 3 725972 3941 999384 11 726588 .3952 273412 57 4 728337 3919 999378 11 728959 3930 271041 56 5 730688 3898 999371 11 731317 3909 268683 55 6 733027 3877 999364 12 733663 3889 2663.37 54 7 735354 3857 999357 12 735996 3868 264004 53 8 737607 3836 999350 12 738317 3848 261683 52 9 739969 3816 999343 12 740626 3827 259374 51 10 742259 8.744536 3796 999336 9.999329 12 12 742922 3807 3787 257078 50 49 J I 3776 8.745207 11.2.54793 12 746802 3756 999322 12 747479 3768 2.52521 48 13 749055 3737 999315 12 749740 3749 250260 47 .4 751297 3717 999308 12 751989 3729 248011 46 15 753528 3698 999301 12 754227 3710 245773 45 16 755747 3679 999294 12 756453 3692 243547 44 17 757955 3661 999286 12 758668 3673 241332 43 18 760151 3642 999279 12 760872 36.55 239128 42 19 762337 3624 999272 12 763065 3636 236935 41 20 21 764511 8.766675 3606 999265 12 12 765246 8.767417 3618 234754 40 ,39 3588 9.999257 3600 11.232.583 22 768828 3570 999250 13 769578 3583 230422 38 23 770970 3553 999242 13 771727 3565 228273 37 !;4 773101 3535 999235 13 77.3866 3548 226134 36 25 775223 3518 999227 13 775995 3.531 224005 35 26 777333 3501 999220 13 778114 3514 221886 34 27 779434 3484 999212 13 780222 3497 219778 33 28 781524 3467 999205 13 782320 3480 217680 32 29 783605 3451 999197 13 784408 3464 215.592 31 30 31 785675 8.787736 3431 999189 13 786486 3447 3431 213514 11.211446 30 29 3418 9.999181 13 8.788.5.54 32 789787 3402 999174 13 790613 3414 209387 28 33 791828 3386 999166 13 792662 3399 207338 27 34 793859 3370 999158 13 794701 3383 205299 26 35 795881 3354 999150 13 796731 3368 203269 25 36 797894 3339 999142 13 798752 3352 201248 24 37 799897 3323 999134 13 800763 3337 199237 23 38 801892 3308 999126 13 802765 3322 1972.35 22 39 803876 3293 999118 13 804758 3307 195242 21 40 41 805852 8.807819 3278 999110 13 13 806742 8.808717 3292 1932.58 20 19 3263 9.999102 3278 11.191283 42 809777 3249 999094 14 810683 3262 189317 18 43 811726 3234 999086 14 812641 3248 1873.59 17 44 813667 3219 999077 14 814.589 3233 18.5411 16 45 815599 3205 999069 14 816.529 3219 183471 15 46 817522 3191 999061 14 818461 3205 181.539! 14 47 819436 3177 999053 14 820384 3191 179616; 13 48 821343 3163 999044 14 822298 3177 177702 12 49 823240 3149 999036 14 824205 3163 175795 11 50 825130 3135 999027 14 826103 31.50 17.3897 10 51 8.827011 3122 9.999019 )4 8.827992 3136 11.172008 9 52 828884 3108 999010 14 829874 3123 170126 8 53 830749 3095 999002 14 831748 3110 1682.52 7 54 832607 3082 998993 14 8.33613 3096 166.387 6 55 834456 3069 998984 14 83.5471 3083 164529 5 56 83*^297 3056 998976 14 837.321 3070 162679 4 57 838130 3043 9989671 15 839163 3057 160837 3 58 839956 3030 998958 15 840998 3045 159002 2 59 841774 3017 998950 15 842825 3032 1.57175 I 60 843585 3000 998941 15 844644 3019 15.53.56 Cosine 1 1 Sine 1 Cotang. 1 1 Tang. j M. 1 MP, 22 4°. LOGARITHMIC M.| Sine n. 1 Cosine 1 D. | Tang. 1 D. 1 (-'"tang. 1 1 8.843585 .3005 9.998941 15 8.844644 3019 11.155356 60 1 845387 2992 998932 15 846455 3007 153.545 59 2 847183 2980 998923 15 848260 2995 151740 58 3 848971 2967 998914 15 850057 2982 149943 57 4 850751 2955 998905 15 851846 2970 148154 56 6 852525 2943 998890 15 853628 2958 146372 55 6 85429 1 2931 998887 15 855403 2946 144597 54 7 856049 2919 998878 15 857171 2935 142829 53 8 85780 1 2907 998869 15 858932 2923 141068 52 9 859546 289G 998860 15 860686 2911 139314 51 10 11 861283 8.863014 288 1 998851 15 9.998841 15 862433 2900 2888 137.367 11.1.35827 50 49 2873 8.864173 12 864738 2861 998832 15 865906 2877 134094 18 13 866455 2850 998823 16 867632 2866 132368 47 14 868165 2839 998813 16 869351 2854 130649 46 15 869868 2828 " 998804 16 871064 2843 1289.36 45 ir. 871565 2817 998795 16 872770 2832 127230 44 17 873255 2806 998785 16 874469 2821 12.5531 43 18 874938 2795 998776 16 876162 2811 123838 42 19 876615 2786 998766 16 877849 2800 122151 41 20 21 878285 2773 2763 998757 9.998747 16 16 879529 2789 120471 40 39 8.879949 8.881202 2779 11.118798 22 881607 2752 998738 16 882869 2768 117131 38 23 883258 2742 998728 16 884530 2758 115470 37 24 884903 2731 998718 16 886185 2747 113815 36 25 886542 2721 998708 16 887833 2737 112167 35 26 888174 2711 998699 16 889476 2727 110524 34 27 889801 2700 998689 16 891112 2717 108888 33 28 891421 2690 998679 16 892742 2707 107258 32 29 893035 2680 998669 17 894366 2097 105634 31 30 31 894643 8.896246 2670 2660 998659 17 17 895984 8.897596 2687 2677 104016 30 29 9.998649 11.102404 32 897842 2651 998639 17 899203 2667 100797 28 33 899432 2641 998629 17 900803 2658 099197 27 34 901017 2631 998619 17 902398 2648 097602 26 35 902596 2622 998609 17 903987 2638 096013 25 36 904169 2612 998599 17 905570 2629 094430 24 37 905736 2603 998589 17 907147 2620 092853 23 38 907297 2593 998578 17 908719 2610 091281 22 39 908853 2584 998568 17 910285 2601 089715 21 40 41 910404 8.911949 2575 998558 9.998548 17 17 911846 2592 088154 20 19 2566 8.913401 2583 11.086599 42 913488 2556 998537 17 914951 2574 085049 18 43 915022 2547 998527 17 916495 2565 083505 17 44 916550 2538 998516 18 918034 2556 081966 16 45 918073 2529 998506 18 919568 2.547 080432 15 46 919591 2520 998495 18 921096 2538 078904 14 47 921103 2512 998485 18 922619 2530 077381 13 48 922610 2503 998474 18 924136 2521 075864 12 49 924112 2494 998464 18 925649 2512 074351 11 50 925609 2486 998453 18 927156 2.503 072844 10 51 8.927100 24-^7 9.998442 18 8.928658 2495 11.071342 9 bZ 928587 2469 998431 18 930155 2486 069845 8 53 930068 2460 998421 18 931647 2478 068353 7 54 93 1 54 t 2452 998410 18 933134 2470 066866 6 55 933015 2443 998399 18 934616 2461 065384 5 56 934481 2435 998388 18 936093 2453 063907 4 57 935942 2427 998377 18 937565 2445 062435 3 58 937398 2419 998366 18 999032 2437 060968 2 59 938850 2411 998355 18 940494 2430 059506 I 60 940296 2403 098344 18 941952 2421 058048 Coi>i;ie 1 Sn.e 1 1 C.tang. j 1 Tauii. } M. 86° SINES AND TANGENTS. 5°. 23 E «ine D. Cosine 1 D. j Ta..^. 1 D. 1 Cofniig 1 1 8.940296 2403 9 . 998344 19 8.941952 2421 11. 0580481 60 1 94173S 2394 998333 n 943404 2413 056596 59 2 943174 2387 998322 19 944S52 2405 055148 58 3 944606 2379 998311 19 946295 2397 053705 57 4 946034 2371 998300 19 947734 2390 052266 56 5 947456 2363 998289 19 949168 2382 0508321 55 1 6 948874 2355 998277 19 950597 2374 049403 54 7 950287 2348 998266 19 952021 2366 047979 53 8 951696 2340 998255 19 953441 2360 046559 52 9 953100 2332 993243 19 954856 2351 045144 51 10 954499 2325 993232 19 956267 2344 043733 50 11 8.955894 2317 9.998220 19 8.957674 2337 11.042326 49 12 9572S4 2310 998209 19 959075 2329 040925 48 13 958670 2302 998197 19 960473 2323 039527 47 U 960052 2295 998186 19 961866 2314 038134 46 15 961429 2288 993174 19 963255 2307 036745 45 16 962801 2280 993163 19 964639 2300 035361 44 17 964170 2273 998151 19 966019 2293 033981 43 18 965534 2266 998139 20 967394 2286 032606 42 I'l 966893 2259 998128 20 968766 2279 031234 41 20 21 968249 8.969600 2252 998116 9.998104 20 20 970133 8 971496 2271 029867 40 39 2244 2265 11.028504 22 970947 2238 998092 20 972855 2257 027145 38 23 972289 2231 998080 20 974209 2251 025791 37 24 973628 2224 993068 20 975560 2244 024440 36 25 974962 2217 993056 20 976906 2237 023094 35 26 976293 2210 998044 20 978248 2230 021752 34 27 977619 2203 993032 20 979586 2223 020414 33 2S 978941 2197 993020 20 930921 2217 019079 32 29 980259 2190 998008 20 932251 2210 017749 31 30 981573 2183 997996 20 933577 2204 016423 30 1 31 8.982883 2177 9.997934 20 8.984899 2197 11.015101 29 32 984189 2170 997972 20 986217 2191 013783 28 33 935491 2163 997959 20 987532 2184 012468 27 31 986789 2157 997947 20 988842 2178 011158 26 35 9S8083 2150 997935 21 990149 2171 009851 25 30 989374 2144 997922 21 991451 2165 008549 24 37 990660 2138 997910 21 992750 2158 007250 23 38 991943 2131 997897 21 994045 2152 005955 22 39 993222 2125 997885 21 995337 2146 004663 21 40 994497 2119 997872 21 996624 2140 003376 20 il 8 995768 2112 9.997860 21 8.997908 2134 11.002092 19 43 997036 2106 997847 21 999188 2127 000812 18 43 998299 2100 997835 21 9.000465 2121 10.999535 17 41 9995ft0 2094 997322 21 001738 2115 998262 16 45 J 0003.6 2087 997809 21 003007 2109 996993 15 46 002069 2082 997797 21 004272 2103 995728 14 47 003318 2076 997784 21 005534 2097 994466 13 IS 004563 2070 997771 21 006792 2091 993208 12 19 005805 2064 997758 21 008047 2085 991953 H 50 007044 2058 997745 21 009298 2080 990702 m 51 9.003278 2052 9.997732 21 9.010546 20T4 10.939454 9 52 009510 2046 997719 21 011790 2068 938210 8 53 010737 2040 997706 21 013031 2062 986969 7 54 011962 2034 997693 22 014268 2056 985732 6 55 013182 2029 997680 22 015502 2051 984498 5 56 014400 2023 9976C7 22 0J6732 2045 983268 4 57 5613 2017 997654 22 017959 2040 982041 3 58 016824 2012 997641 22 019183 2033 980817 2 59 018031 2006 997628 22 020403 2028 979597 1 60 019235 2030 997614 22 021620 2023 978380 iH CDs'iiie 1 1 SiMe 1 1 Cotang. 1 Tang. 1 M. ) MP 24 6°. LOGAUITHMIC ^ Brno 1 D. Cosine | D. Tang 1 D. 1 Cntang. 1 ] M 9.019235 2U00 9.997614 22 9.021620 2023 10.978380 60 1 020435 1995 997601 22 022834 2017 977166 59 2 021632 1989 997588 22 024044 2011 975956 58 3 022825 1984 997574 22 02.5251 2006 974749 57 4 024016 1978 997501 22 026455 2000 973.545 56 6 025203 1973 997547 22 027655 1995 972345 55 6 026386 1967 997534 23 028852 1990 971148 54 7 027567 1962 997520 23 030046 1985 9699.54 53 8 028744 1957 997.507 23 031237 1979 968763 52 9 029918 1951 997493 23 032425 1974 967575 51 10 11 031089, 1947 997480 23 23 033609 9.034791 1969 1964 966391 50 49 9.0322o7{ 1941 9.997466 10.965209 12 0334211 1936 997452 23 035969 1958 964031 48 13 034582 1930 997139 23 0.37144 19.53 9628.56 47 14 035741 1925 997425 23 038316 1948 961684 46 15 036896 1920 997411 23 039485 1943 960515 45 16 038048 1915 997397 23 040651 1938 959349 44 17 039197 1910 997383 23 041813 19.33 9.58187 43 18 040342 1905 997369 23 042973 1928 957027 42 19 0414S5 1899 997355 23 044130 1923 95.5870 41 20 21 04262,") 9.043762 1894 997341 9.997327 23 24 045284 9.0464.34 1918 1913 9.54716 10.9.53.566 40 39 1889 22 044895 1884 997313 24 047582 1908 9.52418 38 23 046026 1879 997299 24 048727 1903 951273 37 24 047154 1875 997285 24 049869 1898 950131 36 25 048279 1870 997271 24 051008 1893 948992 35 26 049400 1865 997257 24 0.52144 1889 947856 34 27 050519 1860 997242 24 053277 1884 946723 33 28 051635 1855 997228 24 054407 1879 94.5593 32 29 052749 1850 997214 24 055535 1874 944465 31 30 31 053859 054966 1845 997199 9.997185 24 24 056659 9.0.57781 1870 94334 1 30 29 1841 1865 10.942219 32 056071 1836 997170 24 058900 1869 941100 28 33 057172 1831 997156 24 060016 1855 939984 27 34 058271 1827 997141 24 061130 1851 938870 26 35 059307 1822 997127 24 062240 1846 937760 26 36 060460 1817 997112 24 063348 1842 936652 24 37 061561 1813 997098 24 064453 1837 935547 23 38 062639 1808 997083 25 06.5556 1833 934444 22 39 063724 1804 997068 25 066655 1828 933345 21 40 41 064806 1799 997053 25 25 067752 1824 1819 93^>248 20 19 9.065885 1794 9.997039 9.068846 10.931154 42 066962 1790 997024 25 069938 1815 930062 18 43 068036 1786 997009 25 071027 1810 928973 17 44 069107 1781 996994 25 072113 1806 927887 16 45 070176 1777 996979 25 073197 1802 926803 15 46 071242 1772 996964 25 074278 1797 925722 14 47 072.306 1768 996949 25 075356 1793 924644 13 48 073366 1763 996934 25 076432 1789 923568 12 49 074424 1759 996919 25 077505 1784 922495 11 50 51 075480 9.076533 1755 996904 9.996889 25 25 078576 1780 1776 921424 10 9 1750 9.079644 10.9203,56 52 077583 1746 996874 25 080710 1772 919290 8 53 078631 1742 996858 25 081773 1767 918227 7 64 079676 1738 996843 25 082833 1763 917167 6 55 080719 1733 996828 25 1 083891 17.59 916109 5 56 081759 1729 996812 26 1 084947 1 1755 91.5053 4 57 082797 1725 996797 26 08600( 1751 914000 3 58 083832 1721 996782 26 087050 1 1747 9129.50 2 59 084864 1717 996766 26 088098 1743 911902 60 085894 1713 .996751 26 089144 1738 910856 ._ Cosine | Bine 1 j Co arig. 1 1 Tang. j M. ! SINES AND TANGENTS. f70 25 M. Sii.^ 1 n. 1 fWine 1 n. , Tanp. 1 D. 1 Coiaiip. j I 9.085894 1713 9.996751 26 9.089144 1738 10.910856 60 1 086922 1709 996735 26 090187 1734 909813 59 2 087947 1704 996720 26 091228 1730 908772 58 3 088970 1700 996704 26 092266 1727 907734 57 4 089090 1696 996688 26 093.302 1722 906698 56 5 091008 1692 996673 26 094336 1719 905664 55 ( 6 092024 1688 996657 26 09.5367 1715 904633 54 7 093037 1684 996641 26 096395 1711 903605 .531 8 094047 1680 996625 26 097422 1707 902578 52 9 095056 1676 996610 26 098446 1703 901.5.54 51 10 006062 1673 996594 26 099468 1699 900532 50 11 9.097065 1668 9.996578 27 9.100487 1695 10.899513 49 12 098066 1665 996562 27 101504 1691 898496 48 13 099065 1661 996.546 27 102519 1687 897481 47 14 100062 1657 996530 27 103.532 1684 896468 46 15 101056 16.53 996514 27 104.542 1680 89.5458 45 16 102048 1649 996498 27 10.5.5.50 1676 894450 44 17 103037 1645 996482 27 106556 1672 893444 43 18 104025 1641 996465 27 107559 1669 892441 42 19 105010 16.38 996449 27 108560 1665 891440 41 20 105992 1634 996433 27 1095.59 1661 890441 40 21 9.106973 1630 9.996417 27 9.110.5.56 16.58 10.889444 39 22 107951 1627 996400 27 111.551 1654 888449 38 23 108927 1623 996384 27 112,543 16.50 887457 37 24 109901 1619 996368 27 11.3533 1646 886467 36 25 1.0873 1616 996351 27 114.521 1643 88.5479 35 26 lil842 1612 996335 27 115.507 1639 884493 34 27 112809 1608 996318 27 116491 1636 883.509 33 28 113774 1605 996302 28 117472 1632 882528 32 29 114737 1601 996285 28 118452 1629 881.548 31 30 1 1.5698 1.597 996269 28 119429 1625 880571 30 31 9.116656 1.594 9.996252 28 9.120404 1622 10.879596 29 32 117613 1.590 9962.35 28 121377 1618 878623 28 33 118567 1.587 996219 28 122348 1615 877652 27 34 119519 1583 996202 28 123317 1611 876683 26 35 120469 1.580 996185 28 124284 1607 875716 25 36 121417 1.576 996168 28 125249 1604 874751 24 37 122362 1573 996151 28 126211 1^1 873789 23 38 123306 1569 996134 28 127172 1.597 872828 22 39 124248 1.566 996117 28 1281.30 1.594 871870 21 40 41 125187 1562 1559 996100 9.996083 28 29 129087 9.130041 1.591 870913 20 19 9.126125 1587 10.8699.59 42 127060 1.556 996066 29 130994 1584 869006 18 43 127993 1552 996049 29 131944 1.581 S68056 17 44 128925 1.549 9960.32 29 132893 1577 867107 16 45 129854 1545 996015 29 133839 1.574 866161 15 46 130781 1542 99.5998 29 1.34784 1571 86.5216 14 47 131706 1539 995980 29 135726 1.567 864274 13 4S 132630 1,535 99.5963 29 136667 1564 863333 12 49 133551 1.532 995946 29 • 137605 1561 862395 11 60 134470 1529 995928 29 138.542 1.5.58 861458 12 f>T 9.135387 1525 9.995911 29 9.139476 1.555 10.860.524 9 52 136303 1522 99.5894 29 140409 1551 8.59.591 8 53 137216 1519 99.5876 29 141340 1548 858660 7 54 138128 1516 995859 29 142269 1545 8.57731 6 55 1390.37 1512 99.5841 29 143196 1.542 856804 6 56 139944 1509 99.5823 29 144121 1539 855879 4 57 140850 1.506 995806 29 14.5044 1535 8.54956 3 58 141754 1.503 995788 29 145966 1532 854034 2 59 142655 1.500 99.5771 29 146885 1529 853115 1 60 1435.55 1496 995753 29 147803 1.526 8.52197 1 <%.si..e j siiic j 1 Louina. Tiuig (RTJ 26 LOGARITHMIC M Sine 1 D. Cosine 1 D Tang. 1 D. Coiang. n T 9.143555 1496 9.995753 30 9.147803 1.526 10.8.521971 00 1 144453 1493 995735 30 148718 1523 851282 59 2 145349 1490 995717 30 149632 1.520 8.50368 68 3 140243 1487 995699 30 150.544 1517 849456 57 4 147136 1484 995681 30 151454 1514 848.546 56 5 148026 1481 995664 30 152363 1511 847637 55 G 148915 1478 995646 30 153269 1.508 846731 54 T 149802 1475 995628 30 154174 1.505 845826 53 8 150H86 1472 995610 30 15.5077 1.502 844923 52 9 151569 1469 995591 30 15.5978 1499 844022 21 10 152451 1466 995573 30 1.56877 1496 843123 50 11 9 153330 1463 9.995555 30 9.157775 1493 10.842225 49 12 154208 1460 995537 30 1.58671 1490 841329 48 13 155083 1457 995519 30 159.565 1487 840435 47 14 155957 1454 995501 31 160457 1484 839.543 46 15 156830 1451 995482 31 161347 1481 838653 45 16 157700 1448 995464 31 162236 1479 837764 44 17 158569 1445 995446 31 163123 1476 836877 43 18 159435 1442 995427 31 164008 1473 835992 42 19 160301 1439 995409 31 164892 1470 835108 41 20 161164 1436 995390 31 165774 1467 834226 40 21 9.162025 1433 9.995372 31 9.166654 1464 10.833346 .39 22 162885 1430 995353 31 167532 1461 832468 38 23 163743 1427 995334 31 168409 1458 831591 37 24 164600 1424 995316 31 169284 14.55 830716 36 25 165454 1422 995297 31 1701.57 1453 829843 35 26 166307 1419 995278 31 171029 1450 828971 34 27 167159 1416 995260 31 171899 1447 828101 33 28 168008 1413 995241 32 172767 1144 827233 32 29 168850 1410 995222 32 173634 1442 826366 31 30 31 169702 9.170547 1407 995203 32 32 174499 1439 82.5501 30 29 1405 9.995184 9.17.5362 1436 10.824638 32 171389 1402 995165 32 176224 1433 823776 28 33 172230 1399 995146 32 177084 1431 822916 27 34 173070 1396 995127 32 177942 1428 822058 26 35 173908 1394 995108 32 178799 1425 821201 25 36 174744 1391 995089 32 1796.55 1423 820.345 24 37 17557S 1388 99.5070 32 180508 1420 819492 23 38 176411 1386 99.5051 32 181360 1417 818640 22 39 177242 1383 99.5032 32 182211 1415 817789 21 40 41 178072 1380 1377 995013 32 32 183059 1412 816941 20 19 9.178900 9.994993 9.183907 1409 10.810093 42 179726 1374 994974 32 1847.52 1407 815248 18 43 180551 1372 994955 32 18.5.597 1404 814403 17 44 181374 1369 994935 32 186439 1402 813.561 16 45 182196 1366 994916 33 187280 1399 812720 15 46 183016 1364 994896 33 188120 1396 811880 14 47 183834 1361 994877 33 1889.58 1393 811042! ni 48 184651 1359 994857 33 189794 1391 810206 U 49 185466 1356 99483^ 33 190629 1389 809371 50 186280 1353 904818 33 191462 1386 808538 10 51 9.187092 1351 9.994798 33 9.192294 1.384 10.807706 9 52 187903 1348 994779 33 193124 1381 80C876 8 53 188712 1346 994759 33 193953 1379 806047 7 54 189519 1343 994739 33 194780 1376 805220 6 65 190325 1341 994719 33 195606 1374 804394 5 66 191130 1338 994700 33 196430 1371 803570 4 67 191933 1336 994680 33 1972.53 1369 802747 3 58 192734 1333 994660 33 198074 1366 801926 2 59 193534 1330 994640 33 198894 1364 801106 7 60 194332 1328 994620 33 199713 1361 800287 ~1 Cosine i Sine 1 Couna. Tan,. L_^L 81° SINES AND TANGENTS. 9^ 27 M. gi"« 1 D. 1 Cosine | D. Tanpr. d' Cotang. j 1 9.194332 1328 9.994620 33 9.199713 1361 10. 8002871 60 1 1 195129 J 326 994G00 33 200529 1359 7994V 1 59 2 195925 1323 994.580 33 201345 1356 798655 58 3 196719 1321 994560 34 202159 13,54 797841 57 4 197511 1318 994540 34 202971 1352 797029 56 5 198.302 1316 994519 34 203782 1349 796218 55 6 199091 1313 994499 34 204592 1347 795408 54 7 199879 1311 994479 34 205400 1345 794600 53 8 200666 1308 994459 34 206207 1342 793793 52 9 201451 1306 994438 34 207013 1340 792987 51 IC 202234 1304 1301 994418 34 34 207817 1338 1335 792183 50 49 9.203017 9.994397 9.208619 10.791381 12 . 203797 1299 994377 34 209420 1333 790580 48 13 204577 1296 994357 34 210220 1331 789780 47 14 205364 1294 994336 34 211018 1.328 788982 46 15 206131 1292 994316 34 211815 1326 788185 45 16 206906 1289 994295 34 212611 1324 787389 44 17 207679 1287 994274 35 213405 1321 786595 43 18 208452 1285 994254 35 214198 1319 78.5802 42 19 209222 1282 994233 35 214989 1317 785011 41 20 21 209992 9.210760 1280 994212 35 35 21.5780 9.216568 1315 784220 40 39 1278 9.994191 1312 10.783432 22 211526 1275 994171 35 2173.56 1310 782644 38 23 212291 1-273 994150 35 218142 1308 7818.58 37 24 21.30.'>5 1271 994129 35 218926 1305 781074 36 25 213818 1268 994108 35 219710 1303 780290 35 26 214.')79 1266 994087 35 220492 1301 779.508 34 27 215338 1264 994066 35 221272 1299 778728 33 28 216097 1261 994045 35 222052 1297 777948 32 29 216854 1259 994024 o5 222830 1294 777170 31 30 31 217609 9.218363 12.57 1255 994003 35 35 223606 9.224382 1292 776394 30 29 9.993981 1290 10.775618 32 219116 12.53 993960 35 2251.56 1288 774844 28 33 219868 1250 993939 35 225929 1286 774071 27 34 220618 1248 993918 35 226700 1284 773300 26 35 221367 1246 993896 36 227471 1281 772529 25 36 222115 1244 993875 36 228239 1279 771761 24 37 222861 1242 993854 36 229007 1277 770993 23 38 223606 1239 993832 36 229773 1275 770227 22 39 224349 1237 993811 36 230539 1273 769461 21 40 41 225092 9 225833 1235 1233 993789 9.99.3768 36 36 231302 1271 768698 20 19 9.232065 1269 10.767935 42 226573 1231 993746 36 232826 1267 767174 18 43 227311 1228 993725 36 233586 1265 766414 17 44 228048 122C 993703 36 234.345 1262 765655 16 45 228784 1224 993681 36 235103 1260 764897 15 46 229518 1222 993660 36 235859 1258 764141 14 47 2302.52 1220 993638 36 236614 1256 763386 13 48 230984 1218 99.3616 36 237368 1254 762632 12 49 231714 1216 99.3594 37 238120 1252 761880 11 50 232444 1214 993572 37 238872 12.50 761128 10 51 9.233172 1212 9.993.5.50 37 9.239622 1248 10.760378 9 52 233899 1209 993.528 37 240371 1246 759629 8 53 234625 1207 993.506 37 241118 1244 7.58882 7 54 235.349 1205 993484 37 241865 1242 758135 6 55 236073 1203 993462 37 242610 1240 757390 f> 56 236795 1201 993440 37 243354 1238 766646 4 57 237515 1199 993418 37 244097 1236 755903 3 58 238235 1197 993396 37 244839 1234 7.55161 2 59 238953 1195 993374 37 245579 1232 754421 I 60 239670 1193 993351 37 246319 1230 75.3681 LJ Cosine 1 Sine I Cotaiig. Tang. |M.j 80= <:^^^^-^-'\^ 28 10°. LOGARITHMIC M. Sine i D. Cosit)e 1 D. 1 Taiic 1 D. 1 Cotang. j 1 9.239670 1193 9.993351 37 9.246319 1230 10 753681 60 1 240386 1191 993329 37 247057 1228 7.52943 59 2 241101 1189 993307 37 247794i 1226 7.52206 58 3 241S14 1187 993285 37 248530 1224 751470 57 4 242526 1185 993262 37 249264 1222 7.50736 56 5 243237 1183 993240 37 249998 1220 750002 55 6 243947 118i 993217 38 2507.30 1218 749270 54 7 244656 1179 993195 38 251461 1217 748539 53 8 245363 1177 993172 38 2.52191 1215 747809 52 ! 9 246069 1175 993149 38 2.52920 1213 747080 51 10 U 246775 9.247478 1173 1171 993127 9.993104 38 38 253648 9.254374 1211 1209 746352 50 49 10.74.5626 12 248181 1169 993081 38 2.55100 1207 744900 48 13 248883 1167 993059 38 255824 1205 744 lt6 47 14 249583 1165 993036 38 256.547 1203 743453 46 15 250282 1163 993013 38 257269 1201 742731 45 16 2509 SO 1161 992990 38 257990 1200 742010 44 17 251677 1159 992967 38 258710 1198 741290 43 18 252373 1158 992944 38 259429 1190 740571 42 19 253067 1156 992921 38 260146 1194 739854 41 20 21 253761 1154 1152 992898 9.992875 38 38 260863 9.261578 1192 739137 40 39 9.254453 1190 10.738422 22 255144 1150 992852 38 262292 1189 737708 38 23 255834 1148 992829 39 263005 1187 736995 37 24 256523 1146 992S06 39 263717 1185 736283 36 25 257211 1144 992783 39 264428 1183 735572 35 26 257898 1142 992759 39 265138 1181 734862 34 27 258583 1141 99273G 39 26.5847 1179 734153 33 28 259268 1139 992713 39 266555 1178 733445 32 29 259951 1137 992690 39 267261 1176 732739 31 30 260633 1135 992666 39 267967 1174 732033 30 31 9.261314 1133 9.992643 39 9.268671 1172 M)T731329 29 32 261994 1131 992619 39 269.375 1170 730625 28 33 262673 1130 992596 39 270077 1169 729923 27 34 263351 1128 992572 39 270779 1167 729221 26 35 264027 1126 992549 39 271479 1165 ♦728521 25 36 264703 1124 992525 39 272178 1164 727822 24 37 265377 1122 992501 39 272876 1162 727124 23 38 266051 1120 992478 40 273573 1160 726427 22 39 266723 1119 992454 40 274269 11.58 725731 21 40 41 267395 9.268065 1117 992430 9.992406 40 40 274964 1157 725036 20 19 1115 9.2756,58 11.55 10.724342 42 268734 1113 992382 40 276351 1153 723649 18 43 269402 1111 992359 40 277043 1151 722957 17 44 270069 1110 992335 40 277734 1150 722266 lr6 45 270735 1108 992311 40 278424 1148 721576 15 46 271400 1106 992287 40 279113 1147 720887 14 17 272064 1105 992263 40 279801 1145 720199 13 18 272726 1103 992239 40 280488 1143 719512 12 49 273388 1101 992214 40 281174 1141 718826 11 50 274049 1099 992190 40 281858 1140 718142 10 51 9.274708 1098 9.992166 40 0.282.542 1138 10.7174.58 *9 52 275367 1096 992142 40 283225 11.36 716775 8 53 276024 1094 992117 41 283907 1135 716093 7 54 276681 1092 992093 41 284588 1133 71.5412 G 55 277337 1091 992069 41 285268 1131 714732 5 56 277991 1089 992044 41 285947 1130 714053 4 57 278644 1087 992020 41 286624 1128 713376 3 58 279297 1086 991996 41 287301 1126 712699 2 59 279948 1084 991971 41 287977 1125 712023 1 60 280599 1082 991947 41 288652 1123 71 1348 n C\»iii« 1 i 81... 1 Col a 111.' 1 1 Tang. 1 M w SINES AND TANGENTS. 11' 29 M. 1 Sine 1 n. I Cwiiw 1 1). 1 Tanc. 1 D. 1 C.»tang. 1 "F 9.280599 1082 9.991947 41 9.2886,52 1123 10. 7113481 Wl 1 281248 1081 991922 41 2893?'' 1122 710674 59 2 281897 1079 991897 41 "^9999 1120 710001 58 3 282514 1077 991873 A ' 290671 1118 709329 57 4 283190 1076 9'»lS?^8 41 291342 1117 708658 56 5 283836 1074 99lff23 41 292013 1115 707987 55 6 284480 1072 99W99 41 292682 1114 707318 54 7 285124 1071 991-74 42 293350 1112 706650' 53 8 285766 1069 991^749 42 294017 1111 705983; V2 9 286408 1067 991724 42 294684 1109 705316! 51 IC 287048 1066 99'1699 42 295349 1107 704651 50 11 9.287687 1064 9.991674 42 9.296013 1106 10. 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' 991397 42 30326 1 1089 696739 38 23 2952S6 -^991372 43 303914 1087 696086 37 24 295913 1043 991346 43 304567 1086 695433 36 25 296539 1042 991321 43 305218 1084 694782 35 26 297164 1040 991295 43 305869 1083 694131 34 27 297788 1039 991270 43 306519 1081 693481 33 28 298412 1037 991244 43 307168 1080 692832 32 29 299034 1036 991218 43 307815 1078 692185 31 3f) 299655 1034 991193 43 308463 1077 691537 30 31 9.300276 1032 9.991167 43 9.309109 1075 10.690891 29 32 300895 1031 991141 43 309754 1074 690246 28 33 301514 1029 991115 43 ^310398 1073 6S9802 27 34 302132 1028 991090 43 811042 1071 6889.':8 26 35 302748 1026 991064 43 311685 1070 688315 25 36 303364 1025 991038 43 312327 1068 687673 24 37 303979 1023 991012 43 312967 1067 687033 23 38 304593 1022 990986 43 313608 1005 686392 22 39 305207 1020 990960 43 314247 1064 685753 21 40 41 305819 9.306430 1019 990934 9.990908 44 44 314885 9.31.5523 1062 685115 20 19 1017 1061 10.684477 42 307041 1016 990882 44 316159 1060 683841 18 43 307650 1014 990855 44 316795 10.58 683205 17 44 308259 1013 990829 44 3174.30 10.57 682570 16 45 308867 1011 990803 44 318064 10.55 681936 15 46 309474 1)10 990777 44 318697 1054 681303 It 47 3.10080 1008 990750 44 319329 1053 680071 13 1 48 310685 1007 990724 44 319961 1051 680039 12 49 311289 1005 990697 44 320592 1050 679408 11 50 311893 1004 990671 44 321222 1048 67877 S -0 51 i). 3 12495 1003 9.990644 44 9.321851 1047 10 67814 9 9 52 313097 1001 990618 U 322479 1045 677521 8 53 313698 1000 99059144 ^ 99056^14 •^ 990538 44 323106 1044 676894 7 54 314297 998 323733 1043 676267 6 55 314897 997 324358 1041 675642 5 36 315495 996 990511 45 324983 1040 67.5017 4 57 316092 994 990485 45 325607 10.39 674393 3 58 316689 993 990458 45 .326231 1037 673769 2 59 3172^4 991 990431 45 326853 1036 673147 1 60 317879 990 990404 45 327475 1035 672525 Cosiiu Sine [ I Cotaiiu. Ta..«. ( 78' 30 ^.€ ^'tco/t LOGARITHMIC M_ Sisie ! !> Cosine | 0. 1 'I'anp. I). 1 Cntang. | 9.317879" 990 9.990404 45 9.327474 1035 10.672526 60 318473 988 990378 45 328095 1033 671905 59 2 319066 987 990351 45 328715 1032 671285 58 3 319658 986 990324 45 329334 1030 670666 57 4 320249 984 990297 45 329953 1029 670047 56 ft 320840 983 990270 45 330570 1028 669430 55 6 321430 982 990243 45 331187 1026 668813 54 7 322019 980 990215 45 331803 1025 668197 53 8 322607 979 990188 45 332418 1024 667582 52 9 323 94 977 990161 45 333033 1023 G66967 51 10 323780 976 990134 45 333646 1021 666354 50 11 9.324366 975 9.990107 46 9.334259 1020 10.665741 49 12 324950 973 990079 46 334871 1019 665129 48 13 325534 972 990052 46 335482 1017 664518 47 14 326117 970 990025 46 336093 1016 663907 46 15 326700 969 98'J997 46 336702 1015 663298 45 16 327281 968 989970 46 337311 013 662689 44 17 327862 966 989942 46 337919 1012 662081 43 18 328442 965 989915 46 338527 1011 661473 42 19 329021 964 989887 46 339.L33 1010 660867 41 20 21 -329599 962 959^0 "4f^ 46 9.340344 1008 1007 660261 12 9.330176 961 9.989832 10.659656 39 22 330753 960 989804 46 340948 1006 659052 38 23 331329 958 989777 46 341552 1004 658448 37 24 331903 957 989749 47 342155 1003 657845 36 25 332478 956 989721 47 342757 1002 657243 35 26 333051 954 989693 47 343358 1000 656642 34 27 333624 953 989605 47 343958 999 656042 33 28 334195 952 989637 47 344558 998 655442 32 29 334766 950 989609 47 345157 997 C54843 31 30 335337 949 989582 47 345756 996 654245 30 31 9.335906 948 9.989553 47 9.. 346353 994 10.653647 29 32 336475 946 989525 47 346949 993 653051 28 33 337043 945 989497 47 347545 992 652455 27 34 337610 944 989169 47 .348141 991 651859 26 35 338176 943 989441 47 348735 990 651265 25 36 338742 941 989413 47 349329 988 650671 24 37 339306 940 989384 47 349922 987 650078 23 38 339871 939 989356 47 350514 986 649480 22 39 340434 937 989328 47 351106 985 648894 21 40 41 340996 936 989300 9.989271 47 47 351697 9.352287 983 982 648303 20 9.341558 935 10.647713 19 42 342119 934 989243 47 352876 981 647124 18 43 342679 932 989214 47 358465 980 646535 17 44 343239 931 989186 47 354053 979 645947 16 45 343797 930 989157 47 354640 977 645360 15 46 344355 929 989128 48 355227 976 G44773 14 47 344912 927 989100 48 355813 975 644187 13 48 345469 926 989071 48 356398 974 643602 12 49 346024 925 989042 48 356982 973 643018 11 50 51 346579 924 989014 9.988985 48 48 357566 9.358149 971 970 642434 10.641851 10 9.347134 92? 9 52 347687 921 988956 48 358731 969 641269 8 53 348240 920 988927 48 359313 968 640687 7 54 348792 919 988898 48 359893 967 640107 6 55 349343 917 988869 48 860474 966 1 639526 5 66 349893 916 988840 48 361053 965 638947 4 57 350443 915 988811 49 361632 963 638368 3 68 350992 914 988782 49 362210 962 637790 2 59 351540 913 988753 49 362787 961 637213 1 60^ 352088 911 988724 49 363364 960 636636 Cosiiic nzz 1 Sine 1 1 Cutaiig. Tang ) M. SINES AND TANGENTS. 13°. 31 M. time 1 D. 1 Cosine 1 I). Tane. 1 D. 1 Ootang. 1 j 9.352088 911 9.988724 49 9.363364 960 10.6,366361 60 \ 1 352635 910 988695 49 363940 959 636060 59 2 353181 909 988666 49 364515 958 6354851 58 I 3 353726 908 988636 49 365090 957 634910 .57 4 ,3542'i 1 907 988607 49 365664 955 634336 56 1 5 354815 905 988578 49 366237 954 633763 55 6 355358 904 988548 49 366810 953 633190 54 7 355901 903 988519 49 367382 952 632618 53 8 356443 902 988489 49 367953 951 632047 52 9 356984 901 988460 49 368524 950 631476 51 10 11 357524 899 988430 9.988401 49 49 369094 949 630906 50 49 9.358064 898 9.369663 948 10.6.30337 12 358603 897 988371 49 370232 946 629768 48 13 359141 896 988342 49 370799 945 629201 17 14 359678 895 988312 50 371367 944 628633 46 15 360215 893 988282 50 371933 943 623067 45 16 360752 892 988252 50 372499 942 627.501 44 17 361287 891 988223 50 373064 941 626936 43 18 361822 890 988193 50 373629 940 626371 42 19 362356 889 988163 50 374193 939 625807 41 20 362889 888 988133 50 374756 938 625244 40 21 9.363422 887 9.988103 50' 9.375319 937 10.624681 39 22 363954 885 988073 50 375881 935 624119 38 23 364485 884 988043 50 376442 934 623558 37 24 365016 883 088013 50 377003 933 622997 36 25 365546 882 987983 50 377583 932 622437 35 26 366975 881 987953 50 378122 931 621878 34 27 386604 880 937922 50 378681 930 621319 33 28 367131 879 987892 50 379239 929 620761 32 29 387659 877 987862 50 379797 928 620203 31 30 31 368185 876 987832 9.987801 51 51 380354 927 619646 30 29 9.368711 875 9.380910 926 10.619090 32 369236 874 987771 51 381466 925 018.534 28 33 369761 873 987740 51 382020 924 617980 27 34 370285 872 987710 51 382575 923 617425 26 35 370808 871 987679 51 383129 922 616871 25 36 371330 870 987649 51 383682 921 616318 24 37 371852 869 987618 51 384234 920 615766 23 38 372373 867 987588 51 384786 919 615214 22 39 372894 866 987557 51 385,337 918 614663 21 40 41 373414 865 987526 9.987496 51 51 385888 917 614112 20 19 9 373933 864 9.386438 915 10.61.3.562 42 374452 863 987465 51 ,386987 914 613013 18 43 374970 862 987434 51 387.536 913 612464! 17 44 375487 861 987403 52 388084 912 6119161 16 45 376003 860 987372 52 388631 911 611369, 15 46 376519 859 987341 52 389178 910 610822 14 47 377035 858 997310 52 389724 909 610276 13 48 377549 857 987279 52 390270 908 609730 12 49 378003 856 987248 52 390815 907 609185 11 50 378577 854 987217 52 391360 906 608640 10 51 9.379089 853 9.987186 52 9.391903 905 10.608097 9 52 379601 852 987155 52 392447 904 607553 8 53 380113 851 987124 52 392989 903 6070111 7 54 380624 850 987092 52 393531 902 606469! 6 55 381134 849 987061 52 394«73 901 605927J 5 56 381643 848 987030 52 394614 900 6053861 4 57 382152 847 986998 52 395154 899 604846 3 58 382(561 846 986967 52 395694 898 604306 2 59 383168 «45 986936 52 .396233 897 603767 1 60 383675 814 986904 52 39677ll 896 603^29 Li C(»^iille 1 Sine 1 ! C'olniis;. 1 iang. jAI. 76» 32 14°. LOGARITHMIC 'iM.I Sine 1 D. 1 Cosine 1 1). Tang. 1 D. ! Colang. { 1 9.3S3675I 844 9.986904 .52 9.396771 896 10.603229 60 I 384182 843 986873 53 397309 896 602691 59 2 384687 842 986841 .53 397846 895 6021.54 58 3 385192 841 986809 53 398383 894 601617 57 4 385697 840 986778 53 398919 893 601081 56 5 386201 839 986746 53 399455 892 600545 55 6 386704 838 986714 53 399990 891 600010 54 7 387207 837 986683 53 400524 890 599476 .531 8 387709 836 986651 53 4010.58 889 598942 .521 9 388210 835 086619 53 401591 888 598409 51 10 11 388711 834 986587 9.986555 53 53 402124 687 597876 10.. 597344 50 49 9.389211 833 •9.4026.56 886 12 389711 832 986.523 53 403187 885 .596813 48 13 390210 831 986491 53 403718 884 596282 47 14 390708 830 986459 53 404249 883 59.5751 46 15 391206 828 986427 53 404778 882 595222 45 16 391703 827 986396 53 405308 881 694692 44 17 392199 826 986363 54 405836 880 594164 43 18 392695 825 986331 54 406304 879 693636 42 19 393191 824 986299 54 406892 878 593108 41 20 21 393685 9.394179 823 986260 9.986234 54 54 407419 9.40 7945 877 876 592.581 10.5920.55 40 39 822 22 394673 821 986202 54 408471 875 591529 38 23 395166 820 986169 54 408997 874 591003 37 24 395658 819 986137 54 409521 874 .590479 36 25 396150 818 986104 54 410045 873 5899.55 35 28 396641 817 986072 54 410569 872 589431 34 27 397132 817 986039 54 411092 871 .588908 33 28 397621 816 986007 54 411615 870 588.385 32 29 398111 815 985974 54 412137 869 587863 31 30 398600 814 985942 54 4126.58 868 587342 30 31 9.399088 813 9.985909 55 9.413179 867 10.. 586821 29 32 399575 812 985876 55 413699 866 .586301 28 33 400062 811 985843 55 414219 865 585781 27 34 400549 810 985811 55 414738 864 58.5262 26 35 401035 809 985778 55 41.5257 864 684743 25 36 401520 808 985745 55 415775 863 584225 24 37 402005 807 985712 55 410293 862 683707 23 38 402489 806 985679 55 416810 861 .583190 22 33 402972 805 985646 55 417326 860 582674 21 40 41 403455 804 98.5613 55 55 417842 859 .582158 20 19 9.403938 803 9.985.580 9.4183.58 8.58 10.. 58 1642 42 404420 802 98.5547 55 418873 857 681127 18 43 404901 801 985514 55 419387 856 680613 17 44 40.5382 800 985480 55 419901 855 680099 16 45 40.5862 40G341 799 985447 55 420415 855 679.585 15 46 798 985414 56 420927 854 679073 14 47 406820 797 985380 56 421440 853 678560 13 48 407299 796 98.5347 56 421952 852 578048 12 49 407777 795 985314 56 422463 851 677537 11 50 51 408254 9.408731 794 985280 9.98.5247 56 56 422974 9.423484 850 .577026 10.576516 10 9 794 849 52 409207 793 985213 56 423993 848 676007 8 53 409682 792 985180 56 424503 848 675497 7 54 410157 410632 791 985146 56 42.5011 847 674989 6 55 ' 790 985113 56 425519 846 674481 6 56 411106 789 98.5079 56 426027 845 573973 4 57 411.579 788 985045 56 4265.34 844 573466 3 58 4120i32 787 985011 56 427041 843 672959 2 59 412524 786 984978 58 427547 843 672453 1 60 412996 785 9S4944 58 428052 842 .571948 Sine Cotaiig. I I Tanf I M. 75° SINES AND TANGENTS. 15' 33 M.l Sine 1 !)• 1 Cosine 1 D. 1 Taiii,'. 1 D. 1 Cotang. 1 1 9.412996 785 9.984944 |57 9.428052 842 10.571948 -60' 1 413467 784 984910 57 428557 841 571443 59 2 413938 783 984876 57 429062 840 570938 58 3 414408 783 984842 57 429566 839 570434 57 4 414878 782 984808 57 430070 838 569930 56 5 415347 781 984774 57 430573 838 569427 55 6 415815 780 984740 57 431075 837 568925 54 7 416283 779 984706 57 431577 836 .568423 53 8 416751 778 984672 57 432079 835 567921 52 9 417217 777 9846371 57 432580 8.34 567420 51 lU 11 417684 9.418150 770 775 984603J 57 433080 833 .566920 10.. 566420 50 49 9.984.569 57 9.433580 832 12 418615 774 984535 57 434080 832 56.59V0 48 13 419079 773 984500 57 434579 831 56.5421 47 14 419544 773 984466 57 435078 830 564922 46 15 420007 772 984432 58 435576 829 564424 45 16 420470 771 984397 58 436073 828 563927 44 17 420033 770 984363 58 436570 828 563430 43 18 421395 769 984.328 58 437067 827 562933 42 19 421857 768 984294 58 437563 826 662437 41 20 422318 767 984259 58 438059 825 561941 40 21 9 422778 767 9.984224 58 9.438.5.54 824 10.561446 39 22 423238 766 984190 58 439048 823 560952 38 23 423697 765 9841.55 58 439.543 823 560457 37 •24 424156 764 984120 58 440036 822 659964 36 25 424615 763 9S4085 58 440529 821 659471 35 26 425073 762 984050 58 441022 820 558978 34 27 425530 761 984015 58 441514 819 558486 33 28 425987 760 983981 58 442006 819 557994 32 29 426443 760 983946 58 442497 818 557503 31 30 31 420899 759 983911 58 442988 9.443479 817 6.57012 30 29 9.427354 758 9.983875 .58 816 10.556.521 32 427809 757 983840 59 443968 816 556032 28 33 428263 756 983805 59 444458 815 555542 27 34 428717 755 983770 59 444947 814 555053 26 35 429170 754 983735 59 445435 813 5.54565 25 36 429623 753 983700 59 445923 812 5.54077 24 37 430075 752 983664 59 446411 812 .553589 23 38 4.30527 752 983629 59 446898 811 5.53102 22 39 430978 751 983.594 59 447384 810 552616 21 40 41 431429 750 9835.58 9.983.523 59 59 447870 809 5.52130 20 19 9.431879 749 9.448.3.'>6 809 10.551644 42 432329 749 983487 59 448841 808 651159 18 43 432778 748 98.3452 59 449326 807 550674 17 44 433226 747 9834161 59 449810 806 550190 16 45 433675 746 983381 59 450294 806 549706 15 46 434122 745 983345 .59 450777 805 549223 14 47 434569 744 983309 59 451260 804 548740 13 48 43.5016 744 983273! 60 451743 803 548257 12 49 435462 743 983238! (U) 452225 802 547775 11 50 51 435^08 9.4363.53 742 983202 60 9.983166 60 452706 802 547294 JO "9 741 9.453187 801 10.. 5468 13 52 436798 740 983130 60 4.53668 800 546332 8 53 437242 740 983094 60 454143 799 64.5852 7 54 437686 739 983058 60 4.54628 799 545372 6 55 4.38129 738 98.3022 60 4.55107 798 544893 5 56 438572 737 982986 60 455586 797 544414 4 57 4.39014 736 982950 60 456064 796 543936 3 58 439456 736 982914 60 456542 796 6434.58 2 59 439897 735 982878 60 4.57019 795 642981 1 60 440338 734 9828421 60 1 457496 794 642504 1 Cosine | 1 Sine 1 , Clang. 1 1 Tan. ,11.1 34 16°. LOGARITHMIC »M Si.ir D. Cosine D. Tnnp D. Cotang. 1 , "F 9.440338 734 9.982842 60 9.457496 ~y9i~ 10.542504 60 1 440778 733 982805 60 457973 793 542027 59 2 441218 732 982769 61 458449 793 541551 58 3 441658 731 982733 61 458925 792 641075 57 4 442096 731 982696 61 459400 791 540600 56 5 442535 730 982660 61 459875 790 540125 55 6 442973 729 982624 61 460349 790 639651 64 7 443410 728 982587 61 460323 789 539177 63 8 443847 727 982551 61 461297 788 638703 52 9 444284 727 982514 61 461770 788 538230 51 10 u 444720 726 982477 61 61 462242 9.462714 787 786 637758 10.537286 50 49 9.445155 725 9.982441 12 445590 724 982404 61 463186 785 636814 48 IH 446025 723 982367 61 463658 785 636342 47 14 446459 723 982331 61 464129 784 635871 46 15 446893 722 982294 01 464599 783 635401 45 16 447326 721 982257 61 465069 783 634931 44 17 447759 720 982220 62 465539 782 634461 43 18 448191 720 982183 62 466008 781 533992 42 19 448623 719 982146 62 466476 780 533524 41 20 21 449054 718 982109 62 62 466945 780 533055 10.532587 40 39 9.449485 717 9.982072 9.467413 779 22 449915 716 982035 62 467880 778 532120 38 23 450345 716 981998 62 468347 778 531653 37 24 450775 715 981901 62 468814 777 631186 36 25 451204 714 981924 62 469280 776 530720 35 26 451632 713 981886 62 469746 775 530254 34 27 452060 713 981849 62 470211 775 629789 33 28 452488 712 981812 62 470676 774 629324 32 29 4529 15 711 981774 62 471141 773 628859 31 30 3f 453342 710 981737 9.981699 62 63 471605 773 528395 10.627932 30 29 9.453768 710 9.472068 772 32 454194 709 981662 63 472532 771 627468 28 33 4rvi6l9 708 981625 63 472995 771 627005 27 34 4:-5044 707 981587 63 473457 770 . 526543 26 35 455469 707 981549 63 473919 769 626081 25 36 455893 706 981512 63 474381 769 625619 24 37 456316 705 981474 63 474842 768 525158 23 38 456739 704 981436 63 475303 767 624697 22 39 457162 704 981399 63 475763 767 524237 21 40 41 457584 703 981361 9.981323 63 63 476223 9.476683 766 765 523777 20 19 9.458006 702 10.523317 42 458427 701 981285 63 477142 765 622858 18 43 458848 701 981247 63 477601 764 522399 17 44 459268 700 981209 63 478059 763 621941 16 45 459688 699 981171 63 478517 763 521483 15 46 460108 698 981133 64 478975 762 621025 14 47 460527 698 981095 64 479432 761 620568 13 48 460946 697 981057 64 479889 761 520111 12 49 461364 696 981019 64 480345 760 519655 11 50 51 461782 695 980981 9.980942 64 64 480801 759 519199 10.518743 10 9 9.462199 695 9.481257 759 52 462616 694 980904 64 481712 758 618288 8 53 463032 693 980866 64 482167 757 617833 7 54 463448 693 980827 64 482621 757 617379 6 55 463864 692 980789 64 483075 756 516925 5 56 464279 691 980760 64 483529 755 616471 4 57 464694 690 980712 64 483982 755 616018 3 58 465108 690 980673 64 484435 754 515565 2 59 465522 689 980635 64 484887 753 515113 1 60 465935 688 98059b 64 485339 753 514661 • Cosine Sine ( Colang ' Tang, j M 73° SINES AND TANGENTS. 17°. 35 ■s- Slew 1 n Cosine | D. 1 Tang. D. Coiang. 1 1 9.465935 688 9.980596 64 9.4853.39 755 10.514661 60 I 466348 688 980558 64 485791 752 614209 i9 2 466761 687 980519 65 486242 751 6137.58 68 3 467173 686 980480 65 486693 751 613.307 57 4 467585 685 980442 65 487143 7.50 6128,57 66 6 467996 685 980403 65 487593 749 612407 65 6 468407 684 980364 65 488043 749 611957 54 7 468817 683 980325 65 488492 748 611.508 63 8 469227 683 980286 65 488941 747 5110.59 52 9 469637 682 980247 65 489390 747 610610 51 10 11 470046 9.470455 681 680 980208 9.980169 65 65 489838 746 510162 60 49 9.490286 746 10.509714 12 470863 680 980130 65 490733 745 609267 48 13 471271 679 980091 65 491180 744 608820 47 14 471679 678 9800.52 65 491627 744 608373 46 16 472086 678 980012 65 492073 743 607927 45 16 472492 677 979973 65 492519 743 607481 44 17 472898 676 979934 66 492965 742 607035 43 18 473304 676 979895 66 49.3410 741 506590 42 19 473710 675 979855 66 493854 740 606143 41 20 474115 674 979816 66 494299 740 505701 40 21 9.474519 674 9.979776 66 9.494743 740 10.50.5257 39 22 474923 673 979737 66 495186 739 604814 38 23 475327 672 979697 66 495630 738 604370 37 24 475730 672 979658 66 496073 737 503927 36 25 476133 671 979618 66 496515 737 503485 35 26 476536 670 979579 66 496957 736 503043 34 27 476938 669 979539 66 497399 736 602601 33 28 477340 669 979499 66 497841 735 6021.59 32 29 477741 668 979459 66 498282 734 601718 31 30 478142 667 979420 66 498722 7.34 501278 30 31 9.478542 667 9.979380 66 9.499163 733 10.500837 29 32 478942 666 979340 66 499603 733 500397 28 33 479342 665 979.300 67 500042 732 4999.58 27 34 479741 665 979260 67 500481 731 499519 26 35 480140 664 979220 67 500920 731 499080 25 36 480539 663 979180 67 501359 730 498641 24 37 480937 663 979140 67 501797 730 498203 23 38 481334 662 979100 67 502235 729 497765 22 39 481731 661 979059 67 502672 728 49732S 21 40 41 482128 661 979019 9.978979 67 67 503109 9.503546 728 496891 10.4964.54 20 19 9 482525 660 727 42 482921 659 978939 67 503982 727 496018 18 43 483316 659 978898 67 .504418 726 495582 17 44 483712 658 978858 67 604854 725 495146 16 45 484107 657 978817 67 505289 725 494711 15 46 484501 657 978777 67 505724 724 494276 14 47 484895 656 978736 67 5061.59 724 493841 13 48 485289 655 978696 68 506593 723 493407 12 49 485682 655 978655 68 607027 722 492973 11 50 486075 654 978615 68 507460 722 492.540 10 61 9.486467 653 9.978574 68 9.. 507893 721 10.492107 '9 52 486860 653 9/8533 68 .508326 721 491674 8 53 487251 652 978493 68 .508759 720 491241 7 64 487643 651 978452 68 509191 719 490809 6 55 488034 651 978411 68 .509622 719 490378 6 56 488424 650 978370 68 510054 718 489946 4 67 488814 650 9783^:9 66 510485 718 489515 3 58 489204 649 978288 68 610916 717 489084 2 59 489593 648 978247 68 611346 716 4886.54 1 60 489982 648 978206 68 511776 716 488221' "" Cociiiie 1 Sine 1 Cotaiig. i '1 arij: : W. 72» 36 18' LOGARITHMIC _M.| Sine D. CVysine 1 I). Tang. D Cotanff. ] 9.489982 648 9.978206 68 9.511776 716 10.488224 60 I 490371 648 978165 68 512206 716 487794 59 2 490759 647 978124 68 512635 715 487365 58 3 491147 646 978083 69 513064 714 480936 57 4 491535 646 978042 69 513493 714 480507 56 5 491922 645 978001 69 513921 713 486079 55 6 492303 644 977959 69 514349 713 485651 54 7 492695 644 977918 69 514777 712 485223 53 8 493081 643 977877 69 51.5204 712 484796 52 9 493466 642 977835 69 51.5631 711 484369 51 10 493851 9 494236 042 977794 69 69 516057 9.516484 710 483943 10.48:^516 50 49 |11 641 9.9777.52 7i0 12 494621 641 977711 69 516910 709 483090 48 13 495005 640 977669 69 517:3:15 709 482665 47 14 495388 639 977628 69 617761 708 482239 46 15 495772 639 977586 69 518185 708 481815 45 16 496154 638 977544 70 518610 707 481390 44 17 496537 637 977503 70 519034 706 480966 43 18 496919 637 977461 70 519458 706 480542 42 19 497301 636 977419 70 519882 705 480118 41 20 21 497682 636 977377 70 70 520305 705 479695 10.479272 40 39 9.498064 635 9.977335 9.520728 704 22 498444 634 977293 70 521151 703 478849 38 23 498825 634 977251 70 521573 703 478427 37 24 499204 633 977209 70 521995 703 478005 30 25 499584 632 977167 70 522417 702 477583 35 26 499963 632 977125 70 522838 702 477162 34 27 500342 631 977083 70 523259 701 476741 33 28 .500721 631 977041 70 523680 701 476320 32 29 501099 630 976999 70 524100 700 475900 31 30 31 .501476 629 976957 70 70 524520 699 475480 30 29 9.501854 629 9.976914 9.524939 699 10.475001 32 502231 628 976872 71 525359 698 474641 28 33 502607 628 976830 71 525778 698 474222 27 34 502984 627 976787 71 526197 697 473803 26 35 503360 626 976745 71 526615 697 47:3385 25 36 503735 626 976702 71 527033 696 472967 24 37 504110 625 976660 71 527451 696 472549 23 38 504485 625 976617 71 527868 695 472132 22 39 504860 624 976574 71 528285 695 471715 21 40 41 505234 623 976532 9.976489 71 71 528702 694 471298 0.470881 20 19 9.505608 623 9.529119 693 42 505981 622 976446 71 .529535 693 470465 18 43 506354 622 976404 71 529950 693 470050 17 44 506727 621 976361 71 530366 692 469634 16 45 507099 620 976318 71 530781 691 469219 15 46 507471 620 976275 71 531196 691 468804 14 47 507843 619 976232 72 531611 690 468389 13 48 508214 619 976189 72 532025 690 467975 12 49 508585 618 976146 72 532439 689 467561 11 50 51 508956 618 976103 9.976060 72 72 532853 689 467147 10.466734 10 9 9.509326 617 9.533266 688 52 509696 6]6 976017 72 533679 688 466321 8 53 510065 616 975974 72 534092 687 465908 7 54 6104.'^4 615 975930 72 534504 687 465496 6 55 510803 615 975887 72 534916 686 465084 6 56 511172 614 975844 72 635328 686 464672 4 57 511540 613 975800 72 535739 685 464261 3 58 511907 613 975757 72 536150 685 463850 2 59 612275 612 975714 72 536561 684 463439 1 60 612642 612 975670 72 536972 684 4613028 ■^ Cosine Sine ( [ Co-ang. 1 1 ''^-«- i*M 71° SINES AND TANGENTS. 19''. 37 "jL mne D. Cosine | D. Tang. "~irn Coiariff. n "o" 9 512642 612 9.975670 73 9.536972 684 10.463028 60 1 513009 6il 97.5627 73 637.382 683 462618 59 2 513375 611 97.5583 73 637792 683 462208 58 3 613741 610 975539 73 638202 682 461798 57 4 514107 609 97.5';96 73 638611 682 461389 56 5 514472 609 975452 73 539020 681 460980 55 6 514837 608 975408 73 539429 681 460.571 54 7 515202 608 975365 73 539837 680 460163 53 8 515506 607 97.5321 73 .540245 680 4.59755 52 9 515930 607 975277 73 640653 679 459347 51 H) 11 516294 9.516657 606 605 975233 9.975189 73 73 .'i4l061 H79 458939 10.4.58532 50 49 9.541468 678 12 517020 605 975145 73 641875 678 458125 48 13 517382 604 975101 73 .542281 677 457719 47 14 517745 604 97.5057 73 642688 677 4573'^ 46 15 518107 603 97.5013 73 643094 676 456906 45 16 618468 603 974969 74 643499 676 456.501 44 17 618829 602 974925 7^1 643905 675 456095 43 IS 619190 601 974880 74 644310 675 45.5690 42 I^ 519551 601 974836 74 644715 674 455285 41 20 519911 600 974792 74 645119 674 4.54881 40 21 9.520271 600 9.974748 74 9.-545524 673 10.4.54476 39 22 520631 599 974703 74 545928 673 454072 38 23 520990 599 974659 74 546331 672 453669 37 24 521349 598 974614 74' 646736 672 453265 36 25 521707 598 974570 74 647138 671 452862 35 26 522066 .597 974525 74 547540 671 452460 34 27 522424 596 974481 74 647943 670 452057 33 28 522781 596 974436 74 648345 670 4516.55 32 29 5231.38 595 974391 74 548747 669 451253 31 30 31 523495 9.523852 595 974347 75 75 649149 669 450851 30 29 594 9.974302 9.549550 668 10.4.504.50 32 524208 594 974257 75 549951 668 450049 28 33 524564 593 974212 75 550352 667 449648 27 34 524920 593 974167 75 550752 667 449248 26 35 525275 592 974122 75 551152 666 448848 25 36 525630 591 974077 75 651552 666 448448 24 37 52.5984 .591 974032 75 551952 665 448048 23 38 526339 590 970987 75 552351 665 447649 22 39 .526693 590 973942 75 552750 665 447250 21 40 41 527046 .589 973897 75 75 5.53149 9.5.53548 664 446851 20 19 9.527400 589 9.973852 664 10.4464.52 42 527753 588 973807 75 553946 663 446054 18 43 528105 588 973761 75 5.54344 663 445656 17 44 628458 .587 973716 76 6,54741 662 445259 16 45 .528810 587 973671 76 .555139 662 444861 15 46 529161 .586 973625 76 555536 661 444464 14 47 529513 586 973.580 76 555933 661 444067 13 48 529864 585 973535 76 656329 660 44.3671 12 49 530215 685 973489 76 556725 660 443275 11 50 51 530565 584 973444 76 76 557121 6.59 442879 10.442433 10 9 9.530915 584 9.973398 9.557517 6.59 62 531265 583 973352 76 5,57913 659 442087 8 53 531614 682 973307 76 558308 6.58 441692 7 54 531963 582 97326 1 76 658702 6.58 441298 6 55 .532312 .581 973215 76 559097 657 440903 6 56 632661 581 973169 76 6.59491 657 440509 4 57 .533009 .580 973124 76 659885 656 440115 3 58 533357 580 973078 76 560279 6.56 439721 2 59 533704 579 973032 77 560673 655 439327 I l60 534052 578 972986 77 .561066 6.55 438934 U n Cosiiiti 1 Sine 1 Coiang. i j Tanc. I M."| TOP 38 20°. LOGARITHMIC M. Sine 1 D. 1 (.^osiiie 1 D. 1 Tang. 1 D. 1 ^"*""?-. 1 1 ~o" 9.534052 578 9.972986 77 9.561066 655 10.438934 -60- 1 534399 577 972940 77 561459 654 438541 59 2 634745 577 972894 77 561851 654 438149 58 3 635092 577 972848 77 562244 653 437756 57 4 535438 576 972802 77 562636 653 437364 56 5 635783 576 972755 77 563028 653 436972 55 6 636129 675 972709 77 .563419 652 436581 54 7 536474 574 972663 77 56381 1 652 438189 53 8 636818 674 972617 77 564202 651 435798 52 9 .537163 673 972570 77 664592 651 43.5408 •^1 10 11 537507 9.537851 573 972524 9.972478 77 77 564983 650 435017 50 49 572 9.565373 650 10.434627 12 538194 672 972431 78 565763 649 434237 48 13 538538 671 972385 78 5661.53 649 433847 47 14 538880 571 972338 78 666542 649 433458 46 15 639223 570 972291 78 566932 648 433068 45 16 639565 570 972245 78 567320 648 432680 44 17 639907 569 972198 78 567709 647 432291 43 18 540249 569 972151 78 568098 647 431902 42 19 540590 668 972105 79 568486 646 431514 41 20 §1 540931 9.541272 .568 972058 9.972011 78 78 568873 9.569261 646 431127 40 39 567 645 10.430739 22 541613 ,567 971964 78 569648 645 430352 38 23 541953 566 971917 78 570035 645 429965 37 24 .542293 566 971870 78 570422 644 429578 36 25 542632 565 971823 78 570809 644 429191 35 26 542971 565 971776 78 571195 643 428805 34 27 543310 564 971729 79 571581 643 428419 33 28 543649 564 971682 79 671967 642 428033 32 29 543987 563 971635 79 572352 642 427648 31 30 31 544325 563 971588 9.971540 79 79 572738 9.573123 642 641 427262 10.426877 30 29 9.544663 562 32 545000 562 971493 79 573507 641 426493 28 33 64533S 561 971446 79 573892 640 426108 27 34 545674 561 971398 79 574276 640 425724 26 35 646011 560 971.351 79 574660 639 425340 25 36 .546347 660 971303 79 575044 639 424956 24 37 646683 659 971256 79 575427 639 424573 23 38 547019 .559 971208 79 575810 638 424190 22 39 547354 558 971161 79 576193 638 423807 21 40 41 547689 9.548024 658 971113 9.971066 79 80 676576 637 423424 10.423041 20 19 657 9.576958 637 42 548359 557 971018 80 577341 636 422659 18 43 548693 556 970970 80 577723 636 422277 .7 44 549027 556 970922 80 578104 636 421896 16 45 549360 555 970874 80 578486 635 421514 15 46 649693 555 970827 80 578867 635 421133 14 47 550026 654 970779 80 579248 634 420752 13 48 650359 654 970731 80 579629 634 420371 12 49 550692 653 970683 80 580009 634 419991 11 50 551024 553 970635 80 580389 633 419611 10 51 9.551356 552 9.970586 80 9.580769 633 10.419231 9 52 551687 552 970538 80 581149 632 418851 8 53 552018 552 970490 80 581528 632 418472 7 54 552349 551 970442 80 581907 632 418093 6 55 552680 551 970394 80 582286 631 417714 5 56 553010 550 970345 81 682665 631 417335 4 57 553341 550 970297 81 583043 630 416957 3 58 553670 549 970249 81 583422 630 416578 2 59 554000 649 970200 81 583800 629 416200 I 60 554329 548 970152 81 584177 629 415823 0.»iiie 1 Sine 1 Coiaijg. 1 Tan, - ^^j SINES AND TANGENTS. 2V 39 M. Sine T D. 1 Cosine 1 I). 1 Tanp. 1 D. 1 Cotang. 1 1 "U" "9".6i:.4329 548 9.970152 81 1 9.584177 629 10.416823 60 1 554658 648 970103 81 584555 629 41.5445 59 3 554987 547 970055 81 584932 628 415068 58 3 555315 547 970006 81 585309 628 414691 57 4 555643 546 969957 81 585686 627 414314 56 5 655971 546 969909 81 586062 627 413938 55 1 6 556299 645 969860 81 586439 627 413.561 54 7 656626 545 969811 81 586815 626 413165 63 8 656953 644 969762 81 687190 626 412810 52 9 557280 544 969714 81 687566 625 412434 51 10 ll 557606 643 969665 81 82 587941 625 625 412059 10,411684 50 49 9.557932 643 9.969616 9.588316 12 558258 543 969567 82 588691 624 411309 48 13 558583 542 969518 82 589066 624 410934 47 14 558909 542 969469 82 589440 623 410560 46 15 659234 541 969420 82 589814 623 410186 45 16 559558 541 969370 82 590188 623 409812 44 17 559883 640 969321 82 690562 622 409438 43 18 660207 640 969272 82 590935 69,2 409065 42 19 660631 639 969223 82 591308 622 408692 41 20 21 560855 639 969173 82 82 591681 9.592054 621 408319 10.407946 40 39 9.561178 638 9.969124 621 22 661501 538 969075 82 592426 620 407574 38 23 561824 637 969025 82 592798 620 407202 37 24 562146 637 968976 82 593170 619 406829 36 25 562468 536 968926 83 593542 619 406458 . 35 26 662790 536 968877 83 593914 618 406086 34 27 663112 636 968827 83 694285 618 405715 33 28 663433 635 968777 83 694656 618 405344 32 29 663755 635 968728 83 690027 617 404973 31 30 664075 634 968678 83 695398 617 404602 30 31 9.564396 534 9.968628 83 9.595768 617 10.404232 29 32 664716 633 968578 83 696138 616 403862 28 33 565036 533 968528 83 596508 616 403492 27 34 565356 532 968479 83 696878 616 403122 26 35 565076 632 968429 83 697247 615 402753 25 36 665995 631 968379 83 697616 615 402384 24 37 566314 531 968329 83 697985 615 402015 23 38 666632 631 968278 83 598354 614 401646 22 39 566951 630 968228 84 698722 614 401278 21 40 41 567269 530 968178 9.968128 84 84 599091 613 613 400909 20 19 9.667587 529 9.599459 10.400641 42 667904 529 968078 84 599827 613 400173 18 43 568222 528 968027 84 600194 612 399806 17 44 568539 628 967977 84 600562 612 399438 16 45 568856 528 967927 84 600929 611 399071 15 40 569172 527 967876 84 601296 611 398704 14 47 669488 527 967826 84 601602 611 398338 13 4^ 669804 626 967775 84 602029 610 397971 12 49 670120 526 967725 84 602395 610 397605 11 50 570435 625 967674 84 602761 610 397239 10 51 9.670751 525 9.967624 84 9.603127 609 10.396873 9 62 671066 624 967573 84 603493 609 396.507 8 53 57138( 624 967522 85 003858 609 396142 7 54 671695 523 967471 85 604223 608 395777 6 65 572003 523 967421 85 604588 608 395412 5 66 572323 623 967370 85 604953 607 39.5047 4 57 67263f 622 967319 85 605317 607 394683 3 58 5729.5C ► 522 967269 85 605682 607 394318 2 59 57326? \ 521 967217 85 606046 606 393954 1 60 57357? 521 967166 86 606410 606 393590 Cosine 1 1 ^"- 1 1 Culaiig. 1 'I'aiig. 1 M. 1 68° 40 22°. LOGARITHMIC M. 1 Sine 1 D. 1 Cfwine 1 D. 1 Tan^. n. Coiang. 1 ] U. 573575 521 9.967166 85 9.606410 600 10.393590 60 1 573888 520 967115 85 606773 606 393227 59 2 574200 520 967064 85 607137 605 392863 58 3 574512 519 967013 85 607500 605 392500 57 4 574824 519 966961 85 607863 604 392137 56 5 575136 619 966910 85 608225 604 391775 56 6 575447 518 966859 85 608588 604 391412 64 7 575758 518 966808 85 608950 603 391050 53 8 576069 517 966756 86 609312 603 390088 52 9 576379 517 966705 86 609674 603 390326 51 10 576689 516 966653 86 610036 602 389964 5C 11 9.576999 516 9.966602 86 9.610397 602 10.389603 49 12 577309 516 966550 86 610759 602 389241 48 13 677618 516 966499 86 611120 601 388880 47 14 577927 615 966447 86 611480 601 388,520 40 15 578236 514 966395 86 611841 601 388159 45 10 578.545 514 966344 86 612201 600 .387799 44 r 578853 513 966292 86 612561 600 387439 43 lo .579162 513 966240 86 612921 600 387079 42 19 579470 613 966188 86 613281 599 386719 41 2(» 21 579777 612 966136 9.966085 86 87 613641 599 386359 10.380000 40 39 9.. 580085 612 9.614000 598 22 580392 511 966033 87 614.3,59 598 38.5641 38 23 580699 611 965981 87 6U718 598 385282 37 24 581005 511 965928 87 615077 597 384923 36 25 581312 510 965876 87 615435 597 384565 35 26 681618 510 965824 87 61.5793 .597 384207 34 27 681924 509 965772 87 616151 596 383849 33 28 582229 609 965720 87 616.509 596 383491 32 29 682535 509 965668 87 616867 596 383133 31 30 31 582840 508 96,5615 9.965563 87 87 617224 595 382776 10.382418 30 29 9.583145 608 9.617.582 595 32 683449 607 96.5511 87 617939 595 382061 28 33 5837,54 507 96.5458 87 618295 594 381705 27 34 684058 506 96.5406 87 618652 594 381348 26 35 684361 .506 9653.53 88 619008 594 380992 25 36 584 G65 506 96.5.301 88 619364 593 380636 24 37 584968 505 965248 88 619721 593 380279 23 38 685272 .505 965195 88 620076 593 379924 22 39 585574 604 965143 88 620432 592 379568 21 40 685877 .504 965090 88 620787 592 379213 20 41 9.. 586 179 503 9.96.5037 88 9.621142 692 10.3788,58 19 42 586482 ,503 964984 88 621497 691 378,503 18 43 586783 503 964931 88 6218.52 591 378148 17 44 587085 502 964879 88 622207 590 377793 16 45 687386 602 964826 88 622,561 590 377439 15 46 687688 501 964773 88 622915 590 377085 14 47 6^798 y 501 964719 88 623269 589 376731 13 48 688289 501 964666 89 623623 689 376377 12 49 688590 500 964613 89 623976 689 376024 11 50 51 588890 ft.. 589 190 600 964560 89 89 624330 588 375670 10.37,5317 10 9 499 9.964507 9.624683 588 52 689489 499 964454 89 62.5036 588 374964 8 53 689789 499 964400 89 625,388 587 374612 7 54 590088 498 964347 89 625741 587 374259 6 55 690387 498 964294 89 626093 587 373907 5 56 590686 497 964240 89 626445 586 373555 4 57 690984 497 964187 89 626797 586 373203 3 58 691282 497 964133 89 627149 .586 372851 2 59 691580 496 964080 89 627501 685 372499 1 60 .591878 496 964026 '89 627852 585 372 1 48 . Cosine 1 1 Sine 1 Coti.iig. 1 1 Tang. j M. j 67° SINES AND TANGENTS. 23°. 41 M. 1 Sine I). 1 Cosine | D. 1 Tana. D. 1 C<.tang. 1 1 9.591878 "496" 9.964026 89 9.627852 .585 10.. 372 148 "00 1 592176 495 963972 89 628203 585 371797 59 2 592473 495 963919 89 6285.54 .585 371446 68 3 592770 495 963865 90 628905 584 371095 57 4 593067 494 963811 90 6292.55 584 370745 56 5 593363 494 963757 90 629606 .583 370.394 56 6 593659 493 963704 90 629956 583 370044 54 7 593955 493 963650, 90 630306 .583 369694 53 8 594251 493 9635961 90 630656 583 309.344 52 9 594547 492 963542' 90 631005 .582 368995 51 10 Ti 594842 9.595137 492 963488 90 90 631355 582 368645 10., 368296 50 49 491 9.963434 9.631704 .582 12 595432 491 963379 90 632053 .581 367947 48 13 59572'" 491 963325 90 632401 581 367599 47 14 596021 490 963271 90 632750 581 3672.50 46 15 596315 490 963217 90 633098 580 366902 45 16 696609 489 963163 90 633447 .580 366.5.53 44 17 696903 489 963108 91 633795 580 366205 43 18 597196 489 963054 91 634143 679 36.5857 42 19 697490 488 962999 91 634490 579 36.5510 41 20 21 597783 488 962945 91 91 634838 9.6.35185 579 365162 10.364816 40 39 9.. 598075 487 9.962890 578 22 698368 487 962836 91 635532 578 364468 38 23 598660 487 962781 91 63.5879 578 364121 37 24 698952 486 902727 91 63G226 577 363774 36 25 599244 486 962672 91 636572 577 363428 35 26 699536 485 962617 91 636919 677 363081 34 27 699827 485 962562 91 637265 577 362735 33 28 600118 485 962508 91 637611 676 362389 32 29 600409 484 962453 91 6379.56 576 362044 31 30 31 600700 9.600990 484 962398 92 92 638302 9.638647 576 361698 10.3613.53 30 29 484 9.962.343 575 32 601280 483 962288 92 638992 575 361008 28 33 601570 483 962233 92 639337 575 360663 27 34 601860 482 962178 92 639682 574 360318 26 35 602150 482 962123 92 640027 574 359973 25 36 602439 482 962067 92 640371 574 359629 24 37 602728 481 962012 92 640716 573 359284 23 38 603017 481 961957 92 641060 573 358940 22 39 603305 481 961902 92 641404 573 358596 21 40 41 603594 9.603882 480 961846 92 92 641747 572 358253 20 19 480 9.961791 9.642091 572 10.3.57909 42 604170 479 9617.35 92 642434 572 357566 18 43 604457 479 961680 92 642777 572 357223 17 44 604745 479 961624 93 643120 571 356880 16 45 605032 478 961.569 93 64.3463 571 356.537 15 46 605319 478 961513 93 643806 671 3.56194 14 47 605606 478 9614.58 93 644148 670 3558.52 13 48 605892 477 961402 93 644490 670 35.5510 12 49 606179 477 961346 93 644832 670 3.55168 11 50 51 606465 476 961290 93 93 645174 609 3.54826 10.3.54484 10 9 9.606751 476 9.961235 9.64.5516 569 52 607036 476 961179 93 645857 .569 3.54143 8 53 607322 475 961123 93 646199 569 353801 7 54 607607 475 961067 93 646.540 568 353460 6 55 607892 474 961011 93 646881 568 3.53119 6 56 608177 474 960955 93 647222 568 352778 4 67 608461 474 960899 93 647562 567 352438 3 58 608745 473 960843 94 647903 567 352097 2 59 6090291 473 960786 94 648243 567 3517.57 1 60 ' 6093131 478 960730 94 648583 .566 351417 .U i'osiiie 1 Sine 1 1 Cotang. Tang. 1 J\l. 1 42 24°. LOGARITHMIC M.| Sine D. Cosine 1 D. TflMff. D. Cotang. n 9.609313 473 9.960730 94 9.648583 566 10.351417" -60 1 609597 472 960674 94 648923 566 351077 69 2 609880 472 960618 94 649263 566 350737 58 3 610164 472 960561 94 649602 566 350398 57 4 010447 471 9G0505 94 649942 565 350058 56 5 610729 471 960448 94 650281 565 349719 65 6 611012 470 960392 94 650020 565 349380 54 7 611294 470 960335 94 650959 564 349041 53 8 611576 470 960279 94 651297 564 348703 52 9 611858 469 960222 94 651636 564 348364 51 10 11 612140 .469 960165 9.960109 94 95 651974 563 348026 10.347688 50 49 9.612421 469 9.652312 563 12 612702 468 960052 95 652650 563 347350 48 13 612983 468 959995 95 652988 563 347012 47 14 613264 467 959938 95 653326 562 346674 46 15 613545 467 959882 95 653663 562 346837 45 Ifi 613825 467 959825 95 654000 562 346000 44 17 614105 466 959768 95 654337 561 345663 43 18 614385 466 95971 1 95 654674 561 345326 42 19 614665 466 959654 95 655011 561 344989 41 20 21 614944 465 959596 95 95 655348 561 560 344652 40 39 9.615223 465 9.959539 9.655684 10.344316 22 615502 465 959482 95 656020 560 343980 38 23 615781 464 959425 95 656356 560 343644 37 24 616060 464 959368 95 656692 559 343308 36 25 616338 464 959310 96 657028 559 342972 35 26 016616 463 959253 96 657364 559 342636 34 27 616894 463 959195 96 657699 559 342301 33 28 617172 462 959138 96 658034 558 341966 32 29 617450 462 959081 96 658369 558 341631 31 30 31 617727 462 959023 9.958965 96 96 658704 558 341296 30 29 9.618004 461 9.659039 558 10.340961 32 618281 461 958908 96 659373 557 340027 28 33 618558 461 958850 96 659708 .557 340292 27 34 618834 460 958792 96 660042 557 339958 26 35 619110 460 ' 958734 96 660376 557 339624 25 36_ ,619386 619G?;2 460 958677 96 660710 556 339290 24 37 459 958619 96 661043 556 338957 23 38 619938 459 958561 96 661377 556 338623 22 39 620213 459 958503 97 661710 555 338290 21 40 41 620488 9.620763 458 958445 9.958387 97 97 662043 555 .337957 10.337624 20 19 • 458 9 662376 555 42 621038 457 958329 97 662709 654 337291 18 43 621313 457 958271 97 663042 554 336958 17 44 621587 457 958213 97 663375 554 336625 16 45 621861 456 958154 97 663707 554 336293 15 46 622135 456 958096 97 664039 553 335961 14 47 622409 456 958038 97 664371 553 335629 13 48 622682 455 957979 97 664703 553 335297 12 49 622956 455 957921 97 665035 553 334965 11 50 51 623229 455 957863 97 97 665366 9.665697 552 334634 10.334303 10 9 9.623502 454 9.957804 552 52 623774 454 957746 98 666029 652 333971 8 63 624047 454 957687 98 666360 551 333640 7 54 624319 453 957628 98 666691 651 333309 6 65 624591 453 957570 98 667021 551 332979 6 56 624863 453 957511 98 667352 551 332648 4 57 625135 452 957452 98 667682 550 332318 3 58 625406 452 957393 98 668013 650 331987 2 69 625677 452 9573.35 98 668343 550 331657 1 60 625948 451 957276; 98 668672 550 331328 u 1 Cosine 1 Sine 1 Cotang. 1 1 'lang 1 M."j 61^ SINES AND TANGENTS. 25°. E Sine D. Cosine | D. 1 Tang. P. 1 Cotang 1 ' 9.625948 451 1 9.957276 98 9.668673 5.50 10.331327|60I 1 626219 451 957217 98 669002 549 330998 59/ 2 626490 451 1 957158 98 669332 549 3306G8 58 3 626760 450 1 957099 98 669661 549 330339 57 4 627030 45C 957040 98 669991 548 330009 56 5 627300 450 1 956981 98 670320 548 329680 55 6 627570 449 956921 99 670649 548 329351 54 7 627840 449 956862 99 670977 548 329023 53 8 628109 449 956803 99 671306 547 328694 52 9 628378 448 956744 99 671634 547 328366 51 10 11 628647 9.628916 448 447 956684 99 99 671963 547 328037 50 49 9.956625 9.672291 547 10.327709 12 629185 447 956566 99 672619 546 327381 48 13 629453 447 956506 99 672947 546 327053 47 14 629721 446 956447 99 673274 646 326726 46 15 629999 446 956387 99 673602 546 326398 45 16 630257 446 956327 99 673929 545 326071 44 17 18 tmn 446 956268 99 100 674257 674584 545 545 325743 325416 43 42 445 956208 19 631059 445 956148 100 674910 544 325090 41 20 21 631326 9.631593 445 956089 100 100 675237 9.675564 544 544 324763 40 39 444 9.956029 10.324436 22 631859 444 955969 100 675890 544 324110 38 23 632125 444 955909 100 676216 543 323784 37 24 632392 443 955849 100 676543 543 323457 36 25 632658 443 955789 100 676869 543 323131 35 26 632923 443 955729 100 677194 543 322806 34 27 633189 442 955069 100 677520 542 322480 33 28 633454 442 955609 100 677846 542 322154 32 29 633719 442 955548 100 678171 .542 321829 31 30 633984 441 955488 100 678496 542 321504 30 31 9.634249 441 9.955428 101 9.678821 641 10. .321 179 29 32 634514 440 955368 lOl 679146 541 320854 28 33 634778 440 955307 101 679471 541 320529 27 34 635042 440 955247 101 679795 541 320205 26 35 635306 439 955186 101 »?80120 540 319880 25 36 635570 439 955126 101 O80444 540 319556 24 37 635834 439 955065 101 680768 540 319232 23 38 636097 438 955005 101 681092 540 318908 22 39 636360 438 954944 101 681416 539 318584 21 40 41 636623 438 954883 9 954823 101 101 681740 639 539 318260 20 19 9.638886 437 9.682063 10.317937 42 637148 437 954762 101 682387 539 317613 18 43 63741 1 437 954701 101 682710 538 317290 17 44 637673 437 954640 101 683033 538 316i)67 16 15 637935 436 954579 101 683356 538 316044 15 46 638197 436 954518 102 683679 538 316321 14 47 638458 436 954457 102 684001 537 315999 13 48 638720 435 954396 102 684324 537 315676 12 49 638981 435 954335 102 684046 637 315354 '1 50 639242 435 954274 102 684968 537 315032 10 51 9.639503 434 9.954213 102 9.685290 536 10.3147101 9| 52 639764 434 954152 102 685612 536 314388; 8 1 53 640024 434 954090 102 685934 536 314066 7 54 640284 433 954029 102 686255 536 313745 6 55 640544 433 953968 102 686577 535 313423 5 56 640804 433 953906 102 686898 535 313102 4 57 641064 432 953845 102 687219 535 312781 3 68 641324 432 953783 102 687540 535 312460 2 59 641584 432 953722 103 6878 a 1 534 312139 1 60 641842 431 953660 103 688182 534 311818 t Corfiiie Sine 1 Cuiang. Tang. ,M.J 44 26°. LOGARITHMIC M. Sine D. Cosine | \). 1 Tn... D j Cotaiig. r T 9.641842 642101 431 9.9.53660 103 9.688182 534 1 10.311818 60 1 431 953599 103 688502 534 311498 59 2 642360 , 431 9.53537 103 688823 634 311177 68 3 642618 430 953475 103 689143 533 310857 67 4 642877 430 9.53413 103 689463 533 310.537 66 5 643135 430 953352 103 689783 633 310217 55 6 643393 430 953290 103 690103 633 309897 64 7 643650 429 953228 103 690423 533 309577 53 8 643908 429 9.53166 103 690742 •532 3092.58 52 9 644165 429 9,53104 103 691062 532 308938 5] 10 11 644423 428 953042 9.9.52980 103 104 691381 9.691700 532 308619 10.308300 50 49 9.644680 428 531 12 644936 428 952918 104 692019 631 307981 48 13 645193 427 9528.55 104 G92338 631 307662 47 14 645450 427 9.52793 104 692656 531 307344 46 15 645706 427 952731 104 692975 .531 307025 45 16 645962 426 952669 104 693293 530 306707 44 17 646218 426 952606 104 69.3612 5,30 306388 43 18 646474 426 952.544 104 693930 630 306070 42 19 646729 425 9.52481 104 694248 630 305752 41 20 21 646984 425 9,52419 9.9,523.56 104 104 694566 629 305434 40 39 9.647240 425 9.694883 529 10.3D5117 22 647494 424 952294 104 69,5201 629 304799 38 23 647749 424 9,52231 104 69,5518 629 304482 37 24 648004 424 9.52168 105 695836 529 304164 36 25 648258 424 952106 105 6961,53 628 303847 35 26 .648512 423 952043 105 696470 528 303530 34 27 648766 423 951980 105 696787 628 303213 33 28 649020 423 951917 105 697103 628 302897 32 29 649274 122 951854 105 697420 627 302580 31 30 649527 422 951791 105 697736 527 302264 30 31 9.649781 422 9.951728 105 9.6980.53 627 10.301947 29 32 650034 422 951665 105 698369 527 301631 28 33 650287 421 951602 105 698685 526 301315 27 34 650539 421 951.539 105 699001 526 300999 26 35 650792 421 951476 105 699316 526 300684 26 36 651044 420 951412 105 699632 526 300368 24 37 651297 420 951349 106 699947 526 300053 23 38 651549 420 951286 106 700263 525 299737 22 39 651800 419 951222 106 700578 525 299422 21 40 41 652052 9.652304 419 9511.59 106 106 700893 .525 299107 20 19 419 9.951096 9.701208 524 10.298792 42 652.555 418 951032 106 701.523 524 208477 18 43 652806 418 950968 106 701837 524 298163 17 44 653057 418 950905 106 702152 624 297848 16 45 653.308 418 950841 106 702466 624 297534 15 46 653558 417 950778 106 702780 623 297220 14 47 653808 417 950714 106 703095 623 296905 13 48 6.54059 417 950650 106 703409 .523 296.59 J 12 49 6.54309 416 9.50586 106 703723 .523 296277 11 5) 6.545.58 41G 950522 107 704036 522 295964 10 51 if 354808 416 9.9,50458 107 9. 704.35 J 522 10.29.56.50 9 52 655058 416 9,50394 107 704663 522 295337 8 53 6,5.5307 415 950330 107 704977 .522 295023 7 54 655556 415 9.50266 107 705290 522 294710 6 55 65.5805 415 950202 107 705603 .521 294397 5 o6 6560.54 414 9,50138 107 705916 .521 294084 4 57 656302 414 950074 107 706228 .521 293772 3 58 6.56.551 414 950010 107 706541 .521 293459 2 59 656799 413 949945 107 706854 .521 293146 1 60 657047 413 949881 107 707166 520 292834 Zi Cosine S..e 1 , Col an?. , 'J anj!. ) U. SINES AND TANGENTS. 27 45 _M_ Si lie T> Coslno 1 n 1 T;mg. 1 n. 1 Coiang. 1 *^^ 0.657047 413 9.949881 107 19.707166 520 10.292834 60 1 657295 413 949816 107 707478 520 292.522 59 2 657542 412 949752 107 707790 620 292210 58 3 657790 412 949688 108 708102 520 291898 67 4 658037 412 949623 108 708414 519 291.586 56 5 658284 412 949558 108 708726 519 291274 55 6 658531 411 949494 108 709037 519 2909G3 54 7 658778 411 949129 108 709349 519 290651 53 8 659025 411 949364 108 709660 519 290340 52 9 659271 410 949300 108 709971 518 290029 51 10 11 659517 9.659763 410 949235 108 108 710282 518 289718 10.289407 50 49 410 9.949170 9.710.593 518 12 060009 409 949105 108 710904 518 289096 48 13 660255 409 949040 108 711215 518 288785 47 14 660501 409 948975 108 711.525 517 288475 46 15 660746 409 948910 108 711836 617 288164 45 16 660991 408 948845 108 712146 517 2878.54 44 17 661236 408 948780 109 7124.56 517 287544 43 18 661481 408 948715 109 712766 516 287234 42 19 661726 407 948650 109 713076 516 286924 41 20 21 661970 9.662214 407 407 948584 9.948519 109 109 713386 516 2866 14 10.286304 40 39 9.713696 516 22 662459 407 948454 109 714005 516 28.5995 38 23 662703 406 948388 109 714314 515 285GS6 37 24 662946 406 948323 109 714624 5!5 28.5376 36 25 663190 406 948257 109 714933 515 28.5067 35 26 663133 405 948192 109 715242 515 2847,58 34 27 663677 405 948126 109 71,5551 514 2844^19 33 28 663920 405 948060 109 715860 514 284140 32 29 664163 405 947995 110 716168 514 283832 31 30 31 664406 9.664648 404 947929 110 110 716477 514 283523 10.28.3215 30 29 404 9.947863 9.716785 514 32 664891 404 947797 110 717093 513 282907 28 33 665133 403 947731 110 717401 513 282599 27 34 665375 403 947665 110 717709 513 28229 1 26 35 665617 403 947600 110 718017 513 281983 25 36 665859 402 947533 110 718325 513 281675 24 37 666100 402 947467 110 718633 512 281367 23 38 666342 402 947401 110 718940 512 281060 22 39 660583 402 947335 110 719248 512 280752 21 40 41 666824 401 947269 9.947203 110 110 719.5.55 512 280445 20 19 9.667065 401 9.719862 512 10.280138 42 667305 401 947136 111 720169 511 279831 18 43 667546 401 947070 111 720476 511 279524 17 44 667786 400 947004 111 720783 511 279217 16 45 668027 400 946937 111 721089 511 278911 15 46 668267 400 946871 111 721396 611 278604 14 47 668506 399 946804 111 721702 510 278298 13 48 668746 399 946738 111 722009 610 277991 12 49 668986 399 946671 111 722315 610 277685 11 50 51 669225 399 946604 9.946538 111 111 722621 610 277379 10 9 9.669464 398 9.722927 610 10 277073 52 669703 398 946471 111 723232 509 276768 8 53 669942 398 946404 111 723538 509 276462 7 54 670181 397 946337 111 723844 509 2761.56 6 65 670419 397 946270 112 724149 509 27.5851 5 56 6706.58 397 946203 112 724454 ,509 275546 4 57 670896 397 946136 112 724759 508 27524 1 3 58 671134 396 946069 112 725065 608 274935 2 59 671372 396 946002 112 725369 508 274631 I 00 671609 396 945935 112 725674 508 274326 n Cosine Sine 1 1 Coiang. 1 Tani. |M H'i* ►.^c n ^/ ^y^^ 28°. LOGARITHMIC M. J Sine 1 n. 1 Cosine | D 1 Taim. 1 r» Ci)taii2. 1 1 "o" 9.671609 1 396 9.945935 112 9.72.5674 508 10.274326 60 1 671847 1 395 945868 112 725979 508 274021 59 2 672084 395 945800 112 726284 507 273716 58 3 672321 395 945733 112 726588 507 273412 57 4 672558 395 945666 112 726392 507 273108 56 ft 672795 394 945598 112 727197 507 272803 55 6 673032 394 945531 112 727501 607 272499 54 7 673268 304 945464 113 727805 606 272195 53 8 673505 394 945396 113 728109 506 271891 52 9 673741 393 945328 113 728412 606 271588 51 TO 673977 393 945261 113 728716 506 271284 50 if 9.674213 393 9.945193 113 9.729020 506 10.270980 49 ]2 674448 392 945125 113 729323 605 270677 48 13 674684 .392 945058 113 729626 505 270374 47 14 674919 392 944990 113 729929 605 270071 16 15 675155 392 94^1922 113 730233 505 269767 15 16 675390 391 944854 113 730535 605 269465 44 17 675624 391 944786 113 730838 504 269162 43 18 675859 391 944718 113 731141 504 268859 42 19 676094 391 944650 113 731444 604 268556 41 20 21 676328 9.676562 390 944582 9.944514 114 114 731746 9.732048 504 504 2682.54 40 .39 390 10.267952 22 676796 390 944446 114 732351 603 267649 38 23 677030 390 944377 114 732653 603 267317 37 24 677264 389 944309 114 7329.55 503 267045 36 25 677498 389 944241 114 733257 503 206743 35 26 677731 389 944172 114 733558 603 266442 34 27 677964 388 944104 114 733860 602 266140 33 28 678197 388 944036 114 734162 602 265838 32 29 678430 388 943967 114 734463 602 265,537 31 30 31 678683 9.678895 388 943899 114 114 734764 603 265236 10.264931 30 29 387 9.943830 9.73.5066 602 32 679128 387 943761 114 735367 602 264633 28 33 6793G0 387 943693 115 735668 601 264332 27 | 34 679592 387 943624 115 735969 501 264031 26 35 679824 386 943555 115 736269 501 263731 25 36 680056 386 943486 115 736570 601 263430 24 37 680288 386 943417 115 736871 601 263129 23 38 680519 385 943348 115 737171 600 262829 22 39 680750 385 943279 115 737471 600 262529 21 40 680982 385 943210 115 737771 600 262229 20 | 41 9.681213 385 9.943'41 115 9.738071 600 10.261929 19 42 681443 384 943072 115 7.38371 600 261629 18 43 681674 384 943003 115 738671 499 261329 17 44 681905 384 942934 115 738971 499 261029 16 45 682135 384 942^61 115 7.39271 499 260729 15 46 682365 383 942.95 116 739570 499 260430 14 47 682595 383 942726 116 739870 499 260130 13 48 682825 383 942656 116 740169 499 259831 12 49 683055 .383 942587 116 740468 498 259532 1 1 60 51 683284 382 942517 116 116 740767 9.741066 498 498 259233 10.258934 10 1 9.683514 382 9.942448 52 683743 382 942378 116 741365 498 258635 8 53 683972 3S2 942308 116 741664 498 258336 7 54 684201 381 9422.39 116 741962 407 2580.38 6 55 684430 381 942169 116 742261 497 257739 5 56 684658 381 942099 116 7425.59 497 25744 1 4 57 684887 380 942029 116 742858 497 2.57142 3 58 ««5115 380 941959 116 7431.56 497 256844 2| 59 685343 380 941889 117 7434.54 497 256546 1 60 6^5571 380 941819 117 743752 496 256248 1 Cosine | 1 Sitin j 1 Cutang. 1 Tang. 1 M. ) 610 SINES AND TANGENTS. 29^ 47 «J Sine D. Cosine j D. | Tang. I D. Cotang. 1 j 9.685571 380 9.941819 117 9.743752 496 10.256248 60 I 685799 379 941749 117 744050 496 2.55950 59 2 686027 379 941679 117 744348 496 255652 58 3 686254 379 941609 117 744645 496 255355 57 4 686482 379 941539 117 744943 496 255057 66 5 686709 378 941469 117 745240 496 254760 55 6 686936 378 941398 117 745538 495 254462 54 7 687163 378 941328 117 745835 495 254165 53 8 6S7389 378 941258 117 746132 495 2.53868 52 9 687616 377 941187 117 746429 495 253571 51 10 687843 377 941117 117 746726 495 253274 50 11 9.688069 377 9.941046 118 9.747023 494 10.252977 19 12 688295 377 940975 118 747319 494 25268 1 48 13 688521 376 940905 118 747616 494 252384 4-^ 14 688747 376 940834 118 747913 494 252087 46 15 688972 376 940763 118 748209 494 251791 45 16 689198 376 940693 118 748505 493 251495 44 17 689423 375 940622 118 748801 493 251199 43 18 689648 375 940551 118 749097 493 250903 42 19 689873 375 940480 118 749393 493 250607 41 20 21 690098 9.690323 375 374 940409 9.940338 118 118 749689 493 2,50311 40 39 9.749985 493 10.2.50015 22 690548 374 940267 118 750281 492 249719 38 23 690772 374 940196 118 750576 492 249424 37 24 69099G 374 940125 119 750872 492 249128 36 25 691220 373 940054 119 751167 492 248833 35 26 691444 373 939982 119 751462 492 248538 34 27 691668 373 939911 119 751757 492 248243 33 2S 691892 373 939840 119 752052 491 247948 32 t29 692115 372 939768 119 752347 491 247653 31 £0 692339 372 939697 113 752642 491 247358 ?,0 31 9.692562 372 9.939625 119 9.752937 491 10.247063 29 32 692785 .371 9395.54 119 75.3231 491 246769 28 33 693008 371 939482 119 753526 491 246474 27 34 693231 371 939410 119 753820 490 246180 26 35 693453 371 939.339 119 754115 490 245885 25 36 693676 370 939267 120 754409 490 245591 24 37 693898 370 939195 120 754703 490 245297 23 38 694120 370 9.39123 120 754997 490 245003 22 39 694342 370 939052 120 755291 490 244709 21 40 694564 369 93S980 120 755585 489 244415 20 41 9.694786 369 9.938908 120 9.7.55878 489 10.244122 19 42 695007 309 938836 120 756172 489 243828 18 43 695229 369 938763 120 756465 489 243535 17 44 695450 368 938691 120 756759 489 243241 16 45 695671 368 938619 120 757052 489 242948 15 46 695892 368 938547 120 757345 488 242655 14 47 696113 368 938475 120 757638 488 242362 13 48 696334 367 938402 121 757931 488 242069 12 49 696554 367 938330 121 758224 488 241776 11 50 51 696775 1 367 938258 9.938185 121 121 758517 9.758810 488 488 241483 10.241190 10 '9 9.696995 367 62 697215 366 938113 121 759102 487 240898 » 53 697435 366 938040 121 759395 487 240605 7 54 697654 366 D37967 121 759687 487 240313 6 65 697874 1 366 937895 121 759979 487 240021 5 56 69809'! e 365 937822 121 760272 487 239728 4 57 698312 t 365 9377491 121 760564 487 239436 3 58 698532 5 365 937676, 121 760856 480 239 1 44 2 59 698751 365 937604 121 761148 486 23bfe52 1 60 698970 ) 364 937,53 ll 121 761439 486 238561 1 Cosine 1 1 Sine 1 1 Colaiig. ==r=^ 1 Tang. 15 &r 48 30°. LOGARITHMIC M Sine D. 1 Cosine 1 D. | Tang. 1 D. Cotnnj?. 1 n 9.698970 364 9.937531 121 9.761439 486 10.238561 (iO 1 699 189 364 937458 122 761731 486 238269 59 3 699407 364 937385 122 762023 486 237977 58 3 699626 364 937312 122 762314 486 237686 'Si 4 699844 363 937238 122 762606 485 237394 .')6 5 7000B2 363 937165 122 937092 122 762897 485 237103 55 6 700280 363 763188 485 236812 54 7 700498 363 937019 122 763479 485 236521 53 8 700716 363 936946 122 763770 485 236230 52 9 700933 362 936872 122 764061 485 235939 51 10 11 701151 9.701368 362 936799 9.936725 122 122 764352 484 235648 10.235357 ?0 49 362 9.764643 484 12 701585 362 936652 123 764933 484 235067 48 13 701802 361 936578 123 765224 484 234776 47 14 702019 361 936505 123 765514 484 234486 46 15 702236 361 936431 123 765805 484 234195 45 16 702452 361 936357 123 766095 484 233905 44 17 702669 360 936284 123 766385 483 233615 43 18 702885 360 936210 123 766675 483 233325 42 19 703101 360 936136 123 766965 483 233035 41 20 21 703317 9.703533 360 359 93M002 9.935988 123 123 767255 483 232745 40 39 9.767545 483 10.232455 22 703749 359 935914 123 767834 483 232166 38 23 703964 359 935840 123 768124 482 231876 37 24 704179 359 935766 124 768413 482 231587 36 25 704395 359 935G92 124 768703 482 231297 35 26 704G10 358 935618 124 768992 482 231008 34 27 704825 358 935543 124 769281 482 230719 33 28 705040 358 935469 124 769570 482 230430 32 29 705254 358 935395 124 769860 481 230140 31 30 705469 357 935320 124 770148 481 229852 30 31 9.705683 357 9.935246 124 9.770437 481 10.229563 29 32 705898 357 935171 124 770726 481 229274 28 33 706112 357 935097 124 771015 481 228985 27 34 706326 358 935022 124 77 1303 481 228697 26 35 706539 356 934948 124 771592 481 228408 25 36 706753 356 934873 124 771880 480 228120 24 37 706967 356 934798 125 772168 480 227832 23 38 707180 355 934723 125 772457 480 227543 22 39 707393 355 934649 125 772745 480 227255 21 40 41 707606 9.707819 355 355 934574 9.934499 125 125 773033 480 226967 20 19 9.773321 480 10.226079 12 708032 354 934424 125 773608 479 226392 18 43 708245 354 934349 125 773896 479 226104 17 44 708458 354 934274 125 774184 479 225816 16 45 708670 354 934199 125 774471 479 225529 15 46 708882 353 934123 125 774759 479 225241 14 47 709094 353 934048 125 775040 479 224954 13 48 709306 353 933973 125 775333 479 224667 12 49 709518 353 933898 126 775621 478 224379 11 60 51 709730 353 933822 9.933747 126 125 775908 478 224092 10 9 9.709941 352 9.776195 478 10.223805 52 710153 352 933671 126 776482 478 223518 8 53 710364 352 933596 126 776769 478 223231 7 54 710575 352 933520 126 777055 478 222945 6 55 710786 351 933445 126 777342 478 222658 5 56 •710997 351 933369 126 777628 477 222372 4 57 711208 351 933293 126 777915 477 222085 3 58 711419 351 933217 126 77820t 477 221799 2 59 711629 350 933141 126 778487 477 221512 1 60 711839 1 350 933066 126 778774 477 221226 1 Cosine 1 Si..e 1 Cotana. 1 j Taiig. 1 M. 69» SINES AND TANGENTS. 31' 49 M." Sine Q- (V)sine | D. Tang. _2^ Cotans. 1 1 "T 9 711839 350 9.933066 126 9.778774 477 10.221226 60 1 712050 350 932990 127 779060 477 220940 59 2 712260 350 932914 127 779346 176 220654 58 3 712469 349 932838 127 779632 476 220368 57 4 712679 349 932762 127 779918 476 220082 56 5 712889 349 932685 127 780203 476 219797 55 6 713098 349 932609 127 780489 476 219511 54 7 713308 349 932533 127 780775 476 219225 53 8 713517 348 932457 127 781060 476 218940 52 9 713726 348 932380 127 781346 475 2186.54 51 10 713935 348 932304 127 781631 475 218369 50 11 9.714144 348 9.932228 127 9.781916 475 10.218084 49 12 714352 .347 932151 127 782201 475 217799 48 13 714561 347 932075 128 782486 475 217514 47 14 714769 347 931998 128 782771 475 217229 46 15 714978 347 931921 128 783056 475 216944 45 16 715186 347 931845 128 783341 475 216659 44 17 715394 346 931768 128 783626 474 216374 43 18 715602 346 931691 128 783910 474 216090 42 19 715809 346 931614 128 784195 474 215805 41 20 21 716017 9.716224 346 931537 128 128 784479 474 215521 40 39 345 9.931460 9.784764 474 10.2152.36 22 716432 345 931383 128 785048 474 214952 38 23 716639 345 931306 128 785332 473 214668 37 24 716846 345 931229 129 785616 473 214384 36 20 717053 345 931152 129 785900 473 214100 35 26 717259 344 931075 129 786184 473 213816 34 27 717466 344 930998 129 786468 473 213532 33 28 717673 344 930921 129 786752 473 . 213248 32 29 717879 344 930843 129 787036 473 212964 31 30 31 718085 343 930766 129 129 787319 472 212681 30 29 9.718291 343 9.930688 9.787603 472 10.212397 32 718497 343 930611 129 787886 472 212114 28 33 718703 343 930533 129 788170 472 211830 27 34 718909 343 930456 129 788453 472 211547 26 35 719114 342 930378 129 788736 472 211264 25 36 719320 342 930300 130 789019 472 210981 24 37 719525 342 930223 130 789302 471 210698 23 38 719730 342 930145 130 789585 471 210415 22 39 719935 341 930067 130 789868 471 210132 21 40 41 720140 9.720345 341 341 929989 9.929911 130 130 790151 471 209849 20' 19 9.790433 471 10.209567 42 720549 341 929833 130 790716 471 209284 18 43 720754 340 929755 130 790999 471 209001 17 44 720958 340 929677 130 791281 471 208719 16 45 721162 340 929599 130 791563 470 208437 15 46 721366 340 929521 130 791846 470 208154 14 47 721570 340 929442 130 792128 470 207872 13 48 721774 339 929364 131 792410 470 207590 12 49 721978 339 929286 131 792692 470 207308 11 50 51 722181 9.722385 339 929207 131 131 792974 470 207026 10 9 339 9.929129 9.793256 470 10.206744 52 722588 339 929050 131 793538 469 206462 8 53 722791 338 928972 131 793819 469 206181' 7 54 722994 338 928893 131 794I0I 469 205899 6 55 723197 338 928815 131 794383 469 205617 5 56 723400 338 928736 131 794664 469 205336 4 57 723603 337 928657 131 794945 469 20.5055 3 58 723805 337 928578 131 795227 469 204773 2 59 724007 337 928499 131 795508 468 204492 1 60 724210 337 928420 131 795789 468 204211 1 Citeiiic 1 Sine 1 Coiang. Thus. 2! 68° 50 32°. LOGARITHMIC M. [ Sine D. Cosine | D. 1 Tanjr. 1 j>. 1 Cotang. 1 9.724210 337 9.928420 132 9.795789 468 10.204211,60 1 724412 337 928342 132 796070 468 203930 59 2 724614 336 928263 132 796351 468 .203649 58 3 724816 336 928183 132 796632 468 20.3368 57 4 725017 335 928104 132 796913 468 203087 5b 5 725219 336 928025 132 797194 468 202806' 55 1 6 725420 335 927946 132 797475 468 202525 54 7 725622 335 927867 132 797755 468 202245 53 8 725823 335 927787 132 798036 467 201964 52 9 726024 335 927708 132 798316 467 201684 51 10 II 726225 335 927629 132 132 798596 467 20 1404 50 49 9.726426 334 9.927549 9.798877 467 .0.201123 12 726626 334 927470 133 799157 467 200843 48 13 726827 334 927390 1.33 799437 467 200563 47 14 727027 334 927310 133 799717 467 200283 46 15 727228 334 927231 133 799997 466 200003 45 16 727428 333 927151 133 800277 466 199723 44 17 727628 333 927071 133 800557 466 199443 43 18 727828 333 926991 133 800836 466 199164 42 19 728027 333 926911 133 801116 466 198884 41 20 728227 333 926831 133 801396 466 198604 40 21 9.728427 332 9.926751 133 9.801675 466 10.198325 39 22 728626 332 926671 133 801955 466 198045 38 23 728825 332 926591 133 802234 465 197766 37 24 729024 332 926511 1.34 802513 465 197487 36 25 729223 331 926431 134 802792 465 197208 35 26 729422 .331 926351 134 803072 465 196928 34 27 729621 331 926270 134 803351 465 196649 33 28 729820 331 926190 134 803630 465 196370 32 29 730018 330 926110 134 803908 465 196092 31 30 31 730216 9.730415 330 330 926029 9.925949 134 134 804187 9.804466 465 464 19.5813 30 29 10.195534 32 730613 330 925868 134 804745 464 195255 28 33 730811 330 925788 134 805023 464 194977 27 34 731009 329 925707 134 805302 464 194698 26 35 731206 329 925626 134 805580 464 194420 25 36 731404 329 925545 135 805859 464 194141 24 37 731602 329 925465 135 806137 464 193863 23 38 731799 329 925384 135 806415 463 193585 22 39 731996 328 925303 1.35 806693 463 193307" 21 40 732193 328 925222 135 806971 463 193029 20 41 9.732390 328 9.925141 135 9.807249 463 10.. 92751 19 42 732587 328 925060 135 807527 463 192473 18 43 732784 328 924979 135 807805 463 192195 17 44 732980 327 924897 135 808083 463 191917 16 45 733177 327 924816 135 808361 463 191639 15 46 733373 T§§i69 327 924735 136 808638 462 19i362 14 191084^ 47 327 92^654 136 808916 462 48 733765 327 924572 136 809193 462 190807 12 49 733961 326 924491 136 809471 462 190529' 11 50 734157 326 924409 136 809748 462 190252 K 51 9.734353 326 9.924328 136 9.810025 462 10.189975 9 52 734549 326 924246 136 810302 462 189698' 8 53 734744 325 924 ir4 136 810580 462 1894201 7 64 734939 325 924083 136 810857 462 189143 ^ 55 735135 325 924001 136 811134 461 188866 6 56 735330 325 923919 136 811410 461 188590 4 57 735525 325 923837 136 811687 461 188313 3 58 735719 321 923755 137 811964 461 188036 2 69 735914 324 923673 137 812241 461 187759 1 60 736109 324 92359 1 137 812517 461 187483 «= Cosine Sine 1 1 Cotang. 1 Tang. ;M.J SINES AND TANGENTS. 33°. 61 M. Sine 0. Cogine 1 D. 1 Tang. D. Cotairg. 1 9.736109 324 9.92.3,591 137 9.812517 461 10.187482 60 i 736303 324 923509 137 812794 461 187206 59 2 73G498 324 923427 137 81.3070 461 186930 68 3 736692 323 923.345 137 813347 460 186653 57 4 736880 323 923263 137 813623 460 186377 56 5 737080 323 923.-81 137 813899 460 186101 55 6 737274 323 923098 137 814175 460 185825 54 7 737467 323 923016 137 814452 460 18.5.548 53 8 737661 322 922933 137 814728 460 18.5272 52 9 737855 322 922851 137 815004 460 184996 51 10 11 738048 322 922768 9.922686 138 138 815279 460 184721 50 49 9.738241 322 9.815555 459 10.184445 12 738434 322 922603 138 815831 459 184169 48 13 738627 321 922520 138 816107 459 183893 47 14 738820 321 922438 1.38 816382 459 183618 46 15 739013 ,321 922355 138 816658 459 183342 45 16 739206 321 922272 138 816933 459 183067 44 17 739398 321 922189 138 817209 459 182791 43 18 739590 .320 922106 1.38 817484 459 182516 42 19 739783 320 922023 138 8177,'59 459 182241 41 20 739975 320 921940 138 818035 458 181965 40 21 9.740167 320 9.921857 139 9.818310 458 10.181690 39 22 740359 320 921774 139 818585 458 181415 38 23 740550 319 921691 139 818860 458 181140 37 24 740742 319 921607 1.39 819135 458 180865 36 25 740934 319 921.524 139 819410 458 180.590 35 26 741125 319 .921441 139 819684 458 180316 34 27 741316 319 921357 139 819959 458 180041 33 28 741508 318 921274 139 820234 458 179766 32 29 741699 318 921190 139 820508 457 179492 31 30 31 741889 318 921107 9.921023 139 139 820783 9.821057 457 179217 30 29 9.742080 318 457 10.178943 32 742271 318 920939 140 8213.32 457 178668 28 33 742462 317 920856 140 821606 457 178394 27 34 742652 317 920772 140 821880 457 178120 26 35 742842 317 920688 140 822154 457 177846 25 36 743033 317 920604 140 822429 457 177571 24 37 743223 317 920520 140 822703 4.57 177297 23 38 743413 316 920436 140 822977 456 177023 22 39 743602 316 920352 140 823250 456 176750 21 40 743792 316 920268 140 823524 456 176476 20 41 9.74.3982 316 9.930184 140 9.823798 456 10.176202 19 42 744171 316 920099 140 824072 4.56 175928 18 43 744361 315 920015 140 824345 456 1756.55 17 44 744550 315 919931 141 824619 456 17.5.381 16 45 744739 315 919846 141 824893 456 175107 15 46 744928 315 919762 141 825166 456 174834 14 47 745117 315 919677 141 825439 455 174.561 13 4^ 745306 314 919593 141 825713 455 174287 12 49 74.5494 314 919508 141 82.5986 455 174014 11 50 745683 314 919424 141 826259 455 173741 10 51 9.74.5871 314 9.919339 141 9.826.532 455 IQ. 173468 9 52 746059 314 9192.54 141 826805 455 173195 8 53 746248 313 919169 141 827078 4.55 172922 7 54 746436 313 919085 141 827351 455 172649 6 55 746624 313 919000 141 827624 455 172376 5 56 746812 313 918915 142 827897 4.54 172103 4 57 746999 313 9188.30 142 828170 454 171830 3 58 747187 312 918745 142 82S442 454 171.5.58 2 59 747374 312 9186.59 142 828715 454 171285 1 60 747562 312 918.574 142 828987 454 171013 ~ Cosine nine 1 CoVmg. 1 Tang. 1 M. | 52 34°. LOGARITHMIC M. I Sine D. I Cosine | D. | Tang. \ I I'otang. I 3.' 306 .306 306 305 305 305 305 305 304 304 '304 304 304 304 303 303 303 303 303 302 302 302 302 302 301 301 301 301 301 301 9.918574 142 9.828987 454 918489 142 829260 454 918404 142 829532 454 918318 142 829805 454 918233 142 830077 454 918147 142 830349 453 918062 142 830621 453 917976 143 830893 453 917891 143 831165 453 917805 143 831437 453 917719 143 831709 453 9.917634 143 9.831981 453 917548 143 832253 453 917462 143 832525 453 917376 143 832796 453 917290 143 833068 452 917204 143 833339 452 917118 144 8.33611 452 917032 144 833882 452 916946 144 834154 452 916859 9.916773 144 144 834425 452 9.834696 452 916687 144 834967 452 916600 144 835238 452 916514 144 835.509 452 916427 144 835780 451 916341 144 838051 451 916254 144 836322 451 916167 145 836593 451 916081 145 836864 451 915994 145 8371.34 451 9.915907 145 9.837405 451 915820 145 837675 451 915733 145 837946 451 915646 145 838216 451 915559 145 838487 450 915472 145 838757 450 915385 145 839027 450 915297 145 839297 450 915210 145 839568 450 915123 146 8398,38 450 9.915035 146 9.840108 450 914948 146 840378 450 914860 146 840647 450 914773 146 840917 449 914685 146 841187 449 914598 146 841457 449 914510 146 841726 449 914422 146 841.996 449 914334 146 842266 449 914246 147 842535 449 9.914158 147 9.842805 449 914070 147 843074 449 91.3982 147 843343 449 913894 147 843612 449 913806 147 843882 448 913718 147 844151 448 91.3630 147 844420 448 913541 147 844689 448 9134.53 147 844958 448 913365 147 84.5227 448 10.171013 170740 170463 170195 169923 169651 169379 169107 168835 168583 168291 10.1f.8019 167747 167475 167204 166932 166661 166389 166118 165846 165575 10.165304 165033 164762 164491 164220 163949 163678 163407 163136 162886 107162595 162325 162054 161784 161513 161243 160973 160703 160432 : 60 162 10. Sine Cuiang. 1.59802 159622 1.59353 159083 1.58813 1.58.543 158274 1.58004 1.57734 157465 10.1.57195 1.56926 156657 1.56388 1.56118 15.5849 15.5580 155311 155042 1547731 — ^n^.~ 65° SINES AND TANGENTS. 35°. 63 M. Pl!.e 1 D. 1 Cosine 1 D. 1 Tane. D. ColtniB. ] 'o~ 9.Vr)8.'>9J 301 9.913365 147 9.84.5227 448 10 154773| 60 i 7.58772 300 913276 147 84.5496 448 1.54.504 59 2 758952 300 913187 148 845764 448 1.54236 58 3 759132 300 913099 148 8460p 448 153967 57 4 759312 300 913010 148 S463IJ2 448 I53G98 56 5 759492 300 912922 148 846570 447 153430 55 6 759672 299 912833 148 846839 447 1.53161 54 7 759852 299 1 9127441 148 847107 447 152893 53 8 760031 299 9)26.55 148 847376 447 152624 52 9 760211 299 912566 148 847644 447 152356 51 10 760390 299 912477 148 847913 447 152087 50 11 9.760.569 298 3.912388 148 9.848181 447 10.151819 49 12 760748 298 912299 149 848449 447 151.551 48 13 760927 298 912210 149 848717 447 151283 47 14 761106 298 912121 149 848986 447 151014 46 15 761285 298 912031 149 849254 447 1.50746 45 16 761464 298 911942 149 849522 447 150478 44 17 761642 297 911853 149 849790 446 1.50210 43 18 761821 297 911763 149 850058 446 149942 42 19 761999 297 911674 149 850325 446 149675 41 20 21 762177 297 911584 9.911495 149 149 8.50593 446 149407 40 39 9.762356 297 9.8.50861 446 10.149139 22 762534 296 911405 149 851129 446 148871 38 23 762712 296 911315 150 851396 446 148604 37 24 762889 296 911226 150 851664 446 148336 36 2.5 763067 296 911136 150 851931 446 148069 35 26 763245 296 911046 150 852199 446 147801 34 27 763422 296 910956 1.50 852466 446 147,534 33 28 763600 295 910866 150 852733 445 147267 32 29 763777 295 910776 150 853001 445 146999 31 iSO 763954 295 910686 150 150 8.53268 445 146732 30 29 IsT 9.764131 295 9.910596 9.853.535 445 10.146465 32 764308 295 910.506 150 853802 445 146198 28 33 764485 294 910415 150 854069 445 145931 27 34 764662 294 910325 151 854336 445 14.5681 26 35 764838 294 910235 151 854603 445 145397 25 36 76.5015 294 910144 151 854870 445 145130 24 37 76519] 294 9100.54 151 855137 445 144863 23 38 765367 294 909963 151 855404 445 144.596 22 39 76.5544 293 909873 151 855671 444 144.329 21 40 41 785720 293 293 909782 9.909691 151 151 855938 444 144062 10.143796 20 19 9.765896 9.8.56204 444 42 766072 203 909601 151 856471 444 143529 18 43 766247 293 909510 151 8.56737 444 143263 17 44 766423 293 909419 151 857004 444 1429961 16 I'i 766598 292 909328 1.52 857270 444 142730 15 46 766774 292 909237 1.52 857.537 444 142463 14 17 766949 292 909146 1.52 857803 444 142197 13 48 767124 292 909055 1.52 8.58069 444 141931 12 49 767300 292 908964 152 858336 444 141664 11 50 61 767475 291 908873 9.908781 1.52 152 858602 443 141398 10 9 9.767649 291 9.858868 443 10.141132 62 767824 291 908690 152 8.59134 443 140866 8 53 767999 291 908.599 1.52 859400 443 140600 7 64 768173 291 908507 152 8.59666 443 140334 6 65 768348 290 908416 153 859932 443 140068 5 56 768.522 290 908324 153 860198 443 1.39802 4 57 768697 290 908233 1.53 860464 443 139536 3 58 768871 290 908141 1.53 860730 443 139270 2 59 769045 290 908049 153 860995 443 139005 1 60 769219 290 907958' 1.53 861261 443 138739 Cosine 1 Sine 1 1 Colung. 1 1 'I'ang. yr 64" 64 36°. LOGARITHMIC M. Sine 1 D. Cosine | T). i 'lans. D 1 Cotanp. j j 0| 9.769219 290 9.907958 153 9.861261 443 10.138739 60 1 769393 289 907866 153 861527 443 138473 59 2 769506 289 907774 153 861792 442 138208 58 3 769740 289 907682 1.53 862058 442 137942 57 4 769913 289 907590 153 862323 442 137677 56 5 770087 289 907498 153 802.589 442 137411 55 6 770260 288 907400 153 862854 442 137146 54 7 770433 288 907314 154 863119 442 136881 53 8 770606 288 907222 154 863385 442 136615 52 9 770779 288 907129 154 8636.50 442 136350 51 10 • 770952 288 907037 154 863915 442 136085 50 11 9.771125 288 9 906945 154 9.864180 442 10.13.5820 49 12 771298 287 906852 154 864445 442 13.5555 48 13 771470 287 906760 154 864710 442 13.5290 47 14 771643 287 906667 154 864975 441 13.5025 46 15 771815 287 906575 154 865240 441 134760 45 16 771987 287 906482 154 865505 441 134495 44 17 772159 287 906389 155 865770 441 134230 43 18 7.72331 286 906296 155 866035 441 133965 42 19 772503 286 906204 155 866300 441 133700 41 20 21 772675 9.772847 286 286 906111 9.906018 155 155 866564 9.866829 441 133436 10.133171 40 39 441 22 773018 286 905925 155 867094 441 132906 38 23 773190 286 905832 155 867358 441 132642 37 24 773361 285 905739 155 867623 441 132377 36 25 773533 285 905645 155 867887 441 132113 35 26 773704 285 905552 155 868152 440 131848 34 27 773875 285 905459 155 868416 440 131.584 33 28 774046 285 905366 156 868680 440 131320 32 29 771217 285 905272 156 868945 440 131055 31 30 31 774388 284 905179 156 156 869209 9.869473 440 440 1.30791 10.130527 30 29 9.774558 284 9.905085 32 774729 284 904992 156 869737 440 130263 28 33 774899 284 904898 156 870001 440 129999 27 34 775070 284 904804 156 870265 440 129735 26 35 775240 284 904711 156 870529 440 129471 25 36 775410 283 904617 156 870793 440 129207 24 37 775580 283 904523 156 871057 440 128943 23 38 775750 283 904429 157 871321 440 128679 22 39 775920 283 904335 157 871585 440 128415 21 40 41 776090 283 904241 157 157 871849 9.872112 439 439 128151 10.127888 20 19 9.776259 283 9.904147 42 776429 282 904053 157 872376 439 127624 18 43 776598 282 903959 157 872640 439 127360 17 44 776768 282 903864 157 872903 439 '27097 16 45 776937 282 903770 .157 873167 439 126833 15 46 7 77106 282 903676 157 873430 439 126570 14 47 777275 281 903581 157 873694 439 126306 13 48 777444 281 903487 157 873957 439 126043 12 49 777613 281 903392 158 874220 439 125780 11 50 51 777781 9.777950 281 281 903298 158 158 874484 439 439- 12.5516 10.1252.53 10 9 9.903203 9.874747 52 778119 281 903108 158 875010 439 124990 8 53 778287 280 903014 1.58 87.5273 438 124727 7 54 778455 280 902919 158 875536 438 12'1464 6 55 778624 280 902824 158 875800 438 124200 5 56 778792 280 902729 1.58 876063 438 123937 4 57 778960 280 902634 1.58 876326 438 123674 3 58 779128 280 902539 1.59 876.589 438 123411 2 59 779295 279 902444 159 876851 438 123149 1 60 779463 279 902349 1 1.59 877114 438 122888 _0 J Ctisine 1 Sine [ Cotaiig. 1 'J'-«- I'M SINES AND TANGENTS. 37* 65 M. Sine _0_i Cosine 1 D. Tang. n. Cotang. ' ~-T 9. 779463 279 9.902349 159 9-877114 438 10.1228S6 60 1 779631 279 902253 159 877377 438 122623 59 2 779798 279 902158 159 877640 438 122360 58 3 779966 279 902063 159 877903 438 122097 57 4 780133 279 901967 159 878165 438 121835 56 5 7SD.100 278 901872 159 878428 438 121572 55 6 7804r>7 278 901776 159 878691 438 121309 54 7 780C3A 27S 901681 159 878953 437 121047 .53 8 780801 278 901585 159 879216 437 120784 52 9 780968 278 901490 159 879478 437 120522 61 10 781^34 278 901394 160 879741 437 120259 50 11 9.781301 - 2V7 9.901298 160 9.880003 437 10.119997 49 I'Z 781468 277 901202 160 880265 437 119735 48 13 781634 277 901106 160 880528 437 119472 47 14 781800 277 901010 160 880790 437 119210 46 15 781966 277 900914 100 881052 437 118948 45 16 782132 277 900818 16-) 881314 437 118686 44 17 782298 276 900722 160 881576 437 118424 43 18 782464 276 900626 160 881839 437 118161 42 19 782630 276 900529 160 882101 437 117899 41 20 782796 276 900433 161 882363 436 1 17637 40 21 9.782961 276 9.900337 161 9.882625 436 10.117375 39 22 783127 276 900240 161 882887 436 117113 38 23 783292 275 900144 161 883148 436 116852 37 24 783458 275 900047 161 883410 436 116590 36 25 783623 275 899951 161 883672 436 116328 35 26 783788 275 899854 161 883934 436 116066 34 27 783953 275 899757 161 884196 436 11.5804 33 28 784118 275 899660 161 884457 436 115.543 32 29 784282 274 899564 161 884719 436 115281 31 30 31 784447 274 899467 9.899370 162 162 884980 436 11.5020 30 29 9.784612 274 9.885242 436 10.114758 32 784776 274 899273 162 885503 436 114497 28 33 784941 274 899176 162 885765 436 114235 27 34 785105 274 899078 162 886026 436 11.3974 26 35 785269 273 898981 162 886288 436 113712 25 36 785433 273 898884 162 886549 435 113451 24 37 785597 273 898787 162 886810 435 113190 23 38 785761 273 898689 162 887072 435 112928 22 39 785925 273 898592 162 887333 435 112667 21 40 786089 273 898494 163 887594 435 112406 20 41 9.786252 272 9.898397 163 9.887855 435 10.112145 19 42 7864 16 272 898299 163 888116 435 111884 18 43 786579 272 898202 163 888377 435 111623 17 44 786742 272 898104 163 888639 435 111361 16 45 786906 272 898006 163 888900 435 111100 15 4f 787069 272 897908 163 889160 435 1 10840 14 47 787232 271 897810 163 889421 435 110,579 13 48 787395 271 897712 163 889682 435 110318 12 4£- 787557 271 897614 163 889943 435 110057 11 50 787720 271 897516 163 890204 434 109796 10 51 19.787883 271 9.897418 164 9.890465 434 10.109535 9 52 788045 271 897320 164 890725 434 109275 8 53 788208 271 897222 164 890986 434 109014 7 54 788370 270 897123 164 891247 434 108753 6 55 788532 270 897025 164 891.')07 434 108493 5 56 788694 270 896926 164 891768 434 108232 4 57 788856 270 896828 164 892028 434 107972 3 58 789018 270 896729 164 892289 434 107711 3 59 789180 270 896631 164 892.549 434 107451 1 60 789342 269 896532 164 892810 434 107190 Cosine , Sine , Coiang. j Tang. 1 j 62P 56 38°. LOGARITHMIC Vl' Sine I). Cosine | D. Tanc. D. Cotans. 1 u 9.789342 269 9.896532 164 9.892810 434 10.107190,60 1 789r)04 269 896433 165 893070 434 1069.30 59 2 789665 269 896335 165 893331 434 106669 58 3 789827 269 896236 165 893591 4.34 106409 57 4 789988 269 896137 165 893851 434 106149 56 5 790149 269 89603S 165 894111 434 10.5889 55 6 790310 268 895939 165 894371 434 10.5629 54 7 790471 268 895840 165 894632 433 10.5368 .53' 8 790632 268 895741 105 894892 433 105108 52, 9 790793 268 895641 165 895152 433 104848] 51 10 11 790954 268 268 895542 165 166 895412 9.895672 433 104588 50 10.104328 49 9.791115 9.895443 433 12 791275 267 895343 166 89.5032 433 104068 481 13 791436 267 895244 166 896192 433 103808 47 14 791596 267 895145 166 896452 433 103548 46, 15 791757 267 895045 166 896712 433 103288 45 16 791917 267 894945 166 895971 433 103029 44 17 792077 267 894846 166 897231 433 102769 43 18 792237 266 894746 166 897491 433 102509 42 19 792397 266 894646 166 897751 433 102249 41 20 21 792507 9.792716 266 894546 9.894446 166 167 898010 4^3 101990 40 39 266 9.898270 433 10.101730 22 792876 266 894346 167 898530 433 101470 38 23 793035 266 894246 167 898789 433 101211 37 24 793195 265 894146 167 899049 432 100951 36 25 793354 265 894046 167 899308 432 100692 35 26 793514 265 893946 167 899568 432 100432 34 27 793673 265 893846 167 899827 432 100173 33 28 793832 205 893745 167 900086 432 099914 32 29 793991 265 893645 167 900346 432 099654 31 30 31 794150 264 893544 9.893444 167 168 900605 9.900864 432 432 099395 30 29 9.794308 264 10.099136 32 794467 264 893343 168 901124 432 098876 28 33 794626 264 893243 168 901383 432 098617 27 34 794V84 264 893142 168 901642 432 098358 26 35 794942 264 893041 168 901901 432 093099 25 36 795101 264 892940 168 902160 432 097840 24 37 795259 263 892839 168 902419 432 097.581 23 38 795417 263 892739 168 902679 432 097321 22 39 795575 263 892638 168 902938 432 097062 21 40 795733 263 892536 168 903197 431 096803 20 41 9.795891 263 9.892435 169 9.903455 431 10.096545 19 42 796049 263 892334 169 903714 431 096286 18 43 796206 263 892233 169 903973 431 096027 17 44 796364 262 892132 169 904232 431 095768 16 45 7 96521 262 892030 169 904491 431 095.509 15 46 796679 262 891929 169 904750 431 095250 14 47 796836 262 891827 !69 905008 431 094992 13 48 796993 262 891726 169 90.5267 431 094733 12 49 797150 261 891624 169 90.5.526 431 094474 11 50 797307 261 891523 170 905784 431 094216 10 51 9.797464 261 9.891421 170 9.906043 431 10.0939.57 *9 52 797621 261 891319 170 906302 431 093698 8 53 797777 261 891217 170 906560 431 093440 7 54 797934 261 891115 170 906819 431 093181 6 65 798091 261 891013 170 907077 431 092923 3 56 798247 261 890911 170 907336 431 092664 4 57 798403 260 890809 170 907594 431 092406 3 58 798560 260 890707 170 907852 431 092148 2 59 798716 260 890605 170 908 1 1 1 430 091889 A 60 798872 260 890503 170 908369 430 091631 « Cosine Sine j 1 Coiang. 1 Tang. |M.| 61° SINES AND TANGENTS. 39"^. 57 M. Sine 1 n. 1 Csiim 1 1). 1 'I'niiE. 1 D. 1 Ctans;. | ; T" 9.7988721 200 9.890.503 170 9 . 908369 430 10.091631 60 1 799028 260 890400 171 908628 430 091.372 59 2 799184 260 890298 171 908886 430 091114 58 3 799339 259 890195 171 909144 430 09085G 57 4 799495 259 890093 171 909402 430 090598 56 5 799651 259 889990 171 909660 , 430 090340 55 6 799806 259 889888 171 909918 430 090082 54 7 799932 259 889785 171 910177 4.30 089823 53 8 800117 259 889H82 171 910435 430 089565 52 q 800272 258 8895/0 171 910693 430 089307 51 10 11 800427 9.800582 258 889477 9.889374 171 "172 910951 430 089049 50 49 258 9.911209 430 10.088791 |12 800737 258 889271 172 911467 430 088533 48 13 800892 258 889168 172 911724 430 088276 4? 14 801047 258 889064 172 911982 430 088018 46 15 801201 258 888961 172 912240 430 087760 45 16 801356 257 888858 172 912498 430 087502 44 17 801511 257 888755 172 9127.56 430 087244 43 18 801665 257 888651 172 913014 429 086986 42 19 801819 257 888,548 172 913271 429 086729 41 20 21 801973 9.802128 257 257 888444 9.888341 173 173 91.3529 429 086471 40 39 9.913787 429 10.086213 22 802282 256 888237 173 914044 429 085956 38 23 802136 256 888134 173 914302 429 085698 37 24 802589 256 888030 173 914.560 429 08.5440 36 25 802743 256 887926 173 914817 429 085183 35 26 802897 256 887822 173 91.^075 429 084925 34 27 803050 256 887718 173 915332 429 084668 33 28 803204 256 887614 173 91.5590 429 084410 32 29 803357 255 887510 173 915847 429 084153 31 30 803511 255 887406 174 916104 429 083896 30 31 9.803664 255 9.887302 174 9.916362 429 10.08.36.38 29 32 803817 255 887198 174 916619 429 083.381 28 33 803970 255 887093 174 916877 429 083123 27 34 804123 255 886989 174 917134 429 082866 26 3b 804276 254 886885 174 917391 429 082609 25 36 804428 254 886780 174 917648 429 082352 24 37 804581 254 886676 174 917905 429 082095 23 38 804734 254 886571 174 918163 428 081837 22 39 804886 254 886466 174 918420 428 081.580 21 40 41 805039 9.805191 2.54 886362 9.886257 175 175 918677 9.9189.34 428 428 081323 10.081066 20 19 2.54 42 805343 253 886152 175 919191 428 080809 18 43 805495 253 886047 175 919448 428 080552 17 44 605647 253 885942 175 919705 428 08029.'^ 16 45 805799 253 885837 175 919962 428 080038 15 46 805951 253 885732 175 920219 428 079781 14 47 806103 253 88.5627 175 920476 428 079524 13 48 806254 253 885522 175 920733 428 079267 12 49 806406 252 885416 175 920990 428 079010 1 1 50 51 806557 9.806709 252 88.5311 9.88.5205 176 176 921247 428 078753 10 9 252 9.921503 428 10.078497 62 806860 252 885100 176 921760 428 078240 8 53 807011 252 884994 176 922017 428 077983 7 54 807163 252 884889 176 922274 428 077726 6 55 807314 252 884783 176 922530 428 077470 5 56 807465 251 884677 176 922787 428 077213 4 57 807615 251 884572 176 923044 428 076056 3 58 807766 251 884466 176 923300 428 076700 2 59 807917 251 884360 176 92.3.557 427 076443 I 60 808067 251 884254 177 923813 427 076 1S7 1 Ct)siiie 1 Sin« 1 Cotanj?. irz Th>,«. 1 M 1 6QP 58 40°. LOGARITHmC M. S'ne D.- 1 Co.^ine | D. Ta.ig. n. Cotaiip. 1 9.808067 251 9.884254 177 9.923813 427 10.076187 wT 1 808218 251 884148 177 924070 427 0759-^0 59 ^ 808368 251 884042 177 '924327 427 075673 5^! 3 808519 250 883936 177 924583 427 075417 57 •♦ 4 808669 250 883829 177 924840 427 075160 5b 5 808819 250 883723 177 925096 427 074904 55 6 808969 250 883617 177 925352 427 074648 54 "r 809119 250 883510 177 . 925609 427 074391 5gr 8 809269 250 883404 177 925865 427 074135 52 9 809419 249 883297 178 926122 427 073S78 51i ml 809569 249 883191 178 926378 -427 073622 50 11 9.809718 249 9.883084 178 9.926634 427 10.073366 49 12 809868 249 882977 178 92^' 890 427 073110 481 13 810017 249 . 882f^l 178 927147 427 072853 % 14 810167 249 882764 178 927403 427 07^597 15 810316 248 882657 178 927659 427 072341 45 16 810165 248 882^50 17» 927915 427 072085 44 17 810614 248 882^43 178 928171 427 "071829 43 ^ 18 810763 248 88;3{336 179 928427 427 071573 42 19 810012 248 88^229 882121 179 928683 427 071317 41 20 21 811061 248 179 179 928940 9.929196 427 427 071060 10.070804 40 39 9.811210 248 9.88^ ;014 22 811358 •247 .88 9p7 1:79 929452 427 070548 If 23 811507 247 7V9 179 929708 427 070292 24 811655 247 6 2 179 929964 426 070036 36 25 811804 247 ^88 5!4 179 930220 426 069780 35 26 811952 24? V88 4 7 179 930475 426 069525 34 27 812100 247 ,88 13)9 179 930731 426 069269 33 28 812248 247 88 aRi 180 93a987 426 069013 32 29 812396 246 ;'88 153 180 931243 426 068757 31 30 812544 246 .88 #46 180 931499 426 OOS.'JOl 30 3i 9.812692 246 9.88 >638 180 9.931755 426. 10.068245 29 32 8i^4a , 246 ^8 mi> ^-m 932010 426 067990 28 33 812988 88( r/22 i^i 932266 426 . 067734 27 34 813135 . 88( 613 186 932522 426 067478 26 35 813283 2^6 88( 505 180 9:?d778 426 067222 25 36 813430 24& 881397 180 933033 426 066967 24 37 8 1 3578 245 88fK89 181 933289 426 066711 23 38 813725 245 ^ '88«h80 181 933545 426 066455 22 39 813872 ^^72 181 933800 426 066200 21 40 814019 «45 879^63 181 934056 426 065944 20 41 9.814166 245 9.879J55 181 9.934311 i 426 ^ 426 10.065689 19 42 814313 245 87pi46 l81 934567 065433 18 43 814460 244 87l9q37 181 934823 426 065177 17 44 814607 244 . J87 9.^9 181 935078 426 064922 16 45 814753 24-t 87HfeO 181' 935333 4M 064667 15 46 r'&i^oo 244 ' 87 )31 1 181 935589 426 0644 1 1 14 47 815046 m v.8792»2 182 #358^14 936100 426 064156 13 48 §l5|f93 , 879093 182 426 06390(» 12 49 615S39 i^ '878944 182 936355 426 063645 11 50 S(l548i 87i^87fe 182 936610 426 063390 10 51 9.8\^1 243 ^-878766 182 9.936866 425 10.063134 9 52 81^8 243 / ^^8T8S5 i 182 183 f. 937121 425 062879 8 53 ^ 815924 243 8^-^ ' 937376 9376^ 425 062624 7 54 • 81606^ 816215 243 -^'S^-W 182 425 062368 6 55 243 8783I1 182 937887 425 062113 5 56 816361 •fj43 8*'82l9! 183 938142 425 061858 4 57 816507 242' ._-^78l09 183 938398 425 061602 3 58 816652 877999, 183 938653 425 061347 2 59 816798 242 877890! 183 938908 425 061092 1 60 816943 242 877780 183 939163 425 060837 _^ Oosiiiu 1 Sine I C#ang. 1 Ta.ig. |M.j 49P *•- SINES AND TANGENTS. 41*'. 59 ^ ^ Sine ^ "%. Ctisine 1 D. | Tan^r. "W CofanK. 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