I J) 4 *?< SHORT AND COMPREHENSIVE COURSE OF • GEOMETRY AND TRIGONOMETRY; DESIGNED FOR GENERAL USE IN SCHOOLS AND COLLEGES. BY ANDREW H. BAKER, A.M., Ph.D. NEW YORK: P. O'SHEA, PUBLISHER, 37 BARCLAY STREET. I878. Copyright, 1878, by Patrick O'Shea. Electrotyped by SMITH & McDOUGAL, 82 Beekman St., New York. PREFACE. GEOMETRY, like every other science, has but few principles, which, if systematically arranged and thoroughly developed, may be readily comprehended, and indelibly impressed upon the mind. Plane Geometry may be said to begin and end with the circle. The angles formed at the center by the radii should be treated of first; next, inscribed angles, and an inscribed triangle; then, a hexagon and an equilateral triangle ; inscribed and circumscribed squares; regular and irregular polygons; and finally return to the circle. In the demonstrations of the propositions derived from these figures, every principle of Plane Geometry is devel- oped ; and Solid Geometry has no distinct principles. In teaching Solid Geometry, I much prefer the use of blocks. The beginner, at least, will be greatly benefited by having a material object to inspect and compute, until he becomes thor- oughly acquainted with all its properties; after which, he may employ his imagination as he likes, and conceive figures of every shape and form. The system of object teaching favors this method. In preparing this treatise, I have aimed especially at simplicity and brevity. The former, that it may be within the grasp of every student; the latter, that the memory may not be over- burdened, and too much time occupied in acquiring a thorough knowledge of the science. Although I have stated that beginners derive benefits from material figures, I do not thereby wish to intimate that Geometry 4 PEEFACE. presents no opportunity for the exercise of the imagination ; whilst, in truth, no other science presents so wide a field for the exercise of this faculty. We pierce the most distant points of the celestial concave with straight lines, and with arcs of great cir- cles ; measure and compute the distance and size of the farthest stars, thereby rendering what appeared imaginary, matters of fact. Simplicity and brevity are not only important, but they are absolutely necessary, in order that mankind generally may acquire a thorough knowledge of pure mathematics ; after which, any branch of applied mathematics may be pursued with ease and advantage. The design of the Trigonometry is the same as of the Geometry. "With these impressions, I dedicate this volume to the Amer- ican Youth ; and if it prove that I have plucked a few thorns from the rugged path of science, and strewn a few flowers therein, I shall not regret the arduous task. Author. ELEMENTS OF GEOMETRY. BOOK I. DEFINITIONS. 1. Elementary Geometry treats of the properties, rela- tions, and measurement of magnitudes. 2. Magnitudes have one, two, or three dimensions ; as, A line has only one dimension, viz., length. A surface has two, length and breadth. And a solid has three, length, breadth, and thickness. 3. Plane Geometry takes its name from the plane, as each figure is upon one plane. Rem. — As every figure, and every part of it, is on the same plane, it is not necessary to repeat " on the same plane." 4. A Mathematical Plane is a surface of indefinite extent, such that if a straight edge or rule ba applied to it, the edge or rule will coincide with it, in every position. 5. Lines are of two classes, straight and curved. A Straight Line has everywhere the same direction, or it may be said to have two directions, exactly opposite each other, from any point in the line. A Curved Line, or simply a Curve, constantly changes its direction. 6. Surfaces are of two classes, plane and curved. A Plane Surface corresponds to the mathematical plane, or a portion of it, and it may have any position whatever ; that is, it may be horizontal or vertical, or it may be oblique. A Curved Surface is such that if a straight rule be applied to it, the rule will not coincide with it in every position; as the surface of a sphere or of a cylinder. b ELEMENTS OF GEOMETKY, 7. A Point has position only ; as, any particular place in a line or plane, and the extremities of lines, are called points. 8. A Circle is a portion of a plane bounded by a curved line, every point of which is equally distant from a point within called the center. 9. The curved line is called the Circumference, and any part of it an Arc. 10. A Polygon is a portion of a plane bounded by straight lines called sides. A polygon of three sides is called a Triangle. A polygon of four sides is called a Quadrilateral. A Z_ J Q A polygon of five sides is quadrilateral. called a Pentagon. A polygon of six sides is called a Hexagon, etc. 11. The divergence of any two sid^s from their point of intersection is called an Angle of the polygon ; and the number of angles will always be the same as the number of sides of the polygon. J 12. A triangle having two equal sides is called an Isosceles triangle. A triangle having three equal sides is called an Equilateral triangle. A triangle having all its sides unequal is called a Scalene triangle. 13. Two lines are Parallel when they are everywhere equally distant, and hence will never meet. 14. A quadrilateral having its opposite sides respectively parallel is called a Parallelogram. 15. A quadrilateral having only two sides parallel is called a Trapezoid. 16. A Regular Polygon has all its sides and angles respectively equal. 17. A regular quadrilateral is termed a Square. 18. A circle, a polygon, etc., are termed Geometrical Figures. BOOK I. 7 19. An Angle may be designated by the letter at its ver- tex, or by three letters, the letter at the vertex occupying the middle place, and the letters at the extremities of its sides hold- ing the first and last places'; thus, the angle A in the triangle ABC is designated angle BAC. 20. When the angles of a quadrilateral are right angles, and the opposite sides respectively equal, it is termed a Rectangle. 21. The circumference of a circle is divided into 360 equal parts, called Degrees, and if radii be drawn to each point marking the degrees, there will be 360 angles, each of one degree. GEOMETRICAL TERMS. 1. An Axiom is a self-evident truth. 2. A Theorem is a truth which requires a demonstration. 3. A Problem is a question which requires a solution. 4. Axioms, Theorems, and Problems are Propositions. 5. A Corollary is an obvious consequence of one or more propositions, or of a definition. 6. A Scholium is a remark upon something which pre- cedes. GENERAL AXIOMS. 1. Magnitudes which are equal to the same magnitude are equal to each other. 2. If equals be added to equals, the sums will be equal. 3. If equals be subtracted from equals, the remainders will be equal. 4. If equals be added to unequals, the sums will be unequal. 5. If equals be subtracted from unequals, the remainders will be unequal. 6. If equals be multiplied by equals, the products will be equal. 7. If equals be divided by equals, the quotients will be equal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. 10. Like powers and like roots of equals are equal. 8 ELEMENTS OF GEOMETRY. SPECIAL AXIOMS. 1. A straight line is the shortest distance between two points. 2. Between two points only one straight line can be drawn. 3. Two fixed points through which a line passes determine its direction. Cor. 1.— Two straight lines, having two points common, form one and the same straight line. Cor. 2. — Two straight lines intersect at but one point. 4. Tw r o straight lines, starting from the same point and taking the same direction, form one and the same straight line. 5. Two straight lines, starting from the same point and taking different directions, form an angle. 6. Two straight lines starting from different points and taking the same direction, are parallel. 7. If two lines are each parallel to a third, they will be parallel to each other. 8. Only one perpendicular can be drawn to a straight line, either from a point without the line, or from a point on the line. Describe a circle, and show the relation of its -properties, and also that of a straight line touching it at one point. Take a string of any definite length, say six inches, attach a pin to one end, and a chalk point to the other end. Fasten the pin at any point in a plane as a center, and, with the string at full stretch, revolve the chalk point around the center, until it reaches the point from which it started ; thus, let CA represent the string, C the center-pin, and A the chalk- point. ABDE is the chalk-line made by the revolution of CA; every point in the line ABDE will be at the distance CA from the center; the curved line ABDE is the circumference of the circle ; the portion of the plane enclosed by it is the circle ; and CA is the radius. BOOK I / Def. 1. — Any straight line, as AB, pass- ing through the center and terminating on the circumference, is a Diameter. Cor. — A diameter is twice the radius. Def. 2. — Any straight line as DE, touch- ing at but one point as F, is a Tangent to the circumference. Rem. — A circumference can be described with a pair of dividers; the distance between the points is the Radius of the circle. Two radii drawn from the center of a circle to its circumfer- ence, form an angle of as many degrees as is contained in the arc intercepted by its sides. Def. 1. — When the angle is 90 degrees, it is called a Right Angle. Def. 2. — When the angle is less than 90 degrees, it is called an Acute Angle. Def. 3. — When the angle is greater than 90 degrees, it is called an Obtuse Angle. Def. 4. — The Complement of an angle is the difference between the angle and 90 degrees. Cor. — If the sum of two angles is 90 degrees, the one is the complement of the other. Def. 5.— The Supplement of an angle is the difference between the angle and 180 degrees. Cor. — If the sum of two angles is 180 degrees, the one is the supplement of the other. 10 ELEMENTS OF GEOMETRY. THEOREM I. Tit e _ diameter of a circle bisects the circle and its cir- cumference. Let AB be the diameter of the circle ADBE. Revolve the part ADB upon AB as an axis, until it falls upon AEB. The arc ADB will coincide with the arc AEB; otherwise some points in the circumference would be un- equally distant from the center of the circle ; hence the part of the circle ADB is equal to the part AEB; and the arc ADB is equal to the arc AEB. Cor. — Each one of the two equal parts of the circle is a semi- circle, and the corresponding arcs are semi-circumferences. THEOREM II. An angle at the center of the circle is measured by the arc intercepted by its sides. Let C be the center and AB the diameter of a circle, A'B' a two-pointed needle, with a pivot at the center C about which it revolves. Since A'B' passes through the center and terminates in the circumference, it is a diameter, and in every position bisects the circle and its circumference -(Theorem 1) ; hence, as A' is moved towards E, B' moves towards D; the arcs AA' and BB' are constantly equal ; and the radii CA and CA', also CB and CB', make equal angles at the center C. When A' reaches E, 90 degrees from A; B' will be at D, 90 degrees from B ; and there will be four equal angles at C, each 90 degrees ; and the diameters are said to be at right angles, or perpendicular to each other. As the arc AA' increases by one, two, etc., degrees, so also the angle ACA' increases by the same number of degrees. BOOK I. 11 THEOKEM III. If one straight line intersect another straight line, the sum of any two adjacent angles will be equal to tiro right angles. * Let the two straight lines intersect at C, then with C as a center and any radius describe a circumference cutting the lines at A, E, B and D. Since AB is a diameter, the two angles ACE and ECB will be meas- ured by the sum of the two arcs AE and EB, which is equal to a semi-circumference or two right angles. So also the two angles ACD and BCD, and since DE is a diameter, the sum of ACD and ACE is equal to two right angles, also the sum of ECB and BCD. Cor. 1. — Vertical angles are equal, as each one is the supple- ment of the same angle; thus, ACD and ECD is each the supple- ment of ACE, or its equal BCD. Cor. 2. — The sum of all the angles at a point on each side of a straight line is equal to two right angles ; and the sum of all the angles around a point is equal to four right angles. Cor. 3. — Equal arcs have equal radii and are like parts of equal circumferences. Cor. 4. — Equal angles have equal arcs, and equal arcs have equal chords, the radii being equal. Scho. — If several circumferences, with different radii, be de- scribed from the same center, the circumferences will be parallel. THEI1EM IV. TJie diameter of a circle is greater than any other chord. Let AB be the diameter and B# a chord of a circle. Draw the radius CP, which is equal to CA; 6fr is less than the sum of BC and Ct>, (Special Axiom 1); but BC + CD = AC +CB = AB ; therefore DB is less than AB, or AB > DB. 12 ELEMENTS OF GEOMETRY. THEOREM V. ^j^-J"/ two straight lines meet a third line, making any two angles which are similarly situated with regard, to the two lines, and on the same side of the third line equal, then will the two lines be parallel. The converse is also true. If the two lines CD and EF meet AB, laking the angles AGD and AHF equal, ^hen will CD and EF be parallel. CD and EF may be regarded as starting different points G and H, and as they p make the angles AGD and AHF equal, they take the same direction and are therefore parallel. (Special Axiom 6.) The converse is necessarily true. Cor. 1. — The same is true, when the equal angles are right angles ; hence, two lines per- pendicular to a third are parallel. Cor. 2. — Since the angles marked i and i are equal, and their vertical angles are also equal, hence four of these angles are equal; and as each of the remaining four is supple- mentary to one of these, they are consequently equal. Cor. 3. — If one of these angles is acute, four will be acute, and the other four will be obtuse ; but if one is a right angle, all will be right angles. Def. 1. — Angles similarly situated are called corresponding angles. Cor. 1. — If two parallels are cut by a third line, the corresponding angles will be equal. Cor. 2. — The interior angles on the same side are supplementary. C— ^ ^/ 2/1 Cor. 2. Cor. 8. Cor. 4. Cor. 3. — The alternate exterior angles are equal. Cor. 4. — The alternate interior angles are equal. Schc. — In the above figures the same numbers indicate pairs. BOOK I. 13 THEOREM VI. Two angles, having their sides parallel and lying in the same or in opposite directions, are equal. Let AB and DE be parallel, also BC and EF, and lying in the same direction ; then will the angles ABC and DEF be equal. Produce DE to H, cutting BC in G. Since the parallels are cut by DH, the corresponding angles DEF and DGC are equal ; and as BC cuts the parallels DH and AB, the angles DGC aj^d ABC are corresponding angles and hence are equal ; therefore, the angle ABC is equal to the angle DEF. (Ax. 1.) The angles DGC and BGH are vertical angles, therefore equal. Consequently the angles ABC and BGH are equal. x THEOREM VII. Two angles, having their sides 'respectively perpendicu- lar, are equal or supplementary. \ <• Let the sides of the angle EAD be respec- * fively perpendicular to the sides of the angle BAC; and also the sides of EAD' perpendicular to the sides of BAC. 1st. The angles BAD and CAE are right angles ; from each take the angle CAD, and there remains the angle BAC equal to the d' angle DAE. 2d. The angle EAD' is supplementary to EAD ; so also of its equal BAC. 14 ELEMENTS OF GEOMETEY, THEOREM VIII. If, in a circle, two diameters be drawn at right angles, and several chords be drawn -parallel to one of the diame- ters, and at the extremities of the other diameter, lines be drawn parallel to the chords, 1st. Tlie chords will be bisected by the perpendicular diameter. 2d. The lines at the extremities of the same diameter, will be tangents to the circumference. 3d. Any two parallels will intercept equal arcs of the circumference. » Since DE is a diameter, DAE is a semi- circle, and if it be revolved upon DE as an axis, until it fall upon DBE, the two semi- circles will coincide; and since all the angles made with DE by the diameter AB and each line parallel to AB are right angles, all the parts of the lines of the semicircle DAE will fall upon and coincide with those of the other semicircle; that is, CA with CB; FL with LG; and HK with Kl ; also MD with DN and PE with EQ; therefore ; M —^ r / L \« A ( IB P — ^»- C J . ^ Q 1st. The chords are bisected. 2d. The two lines MN and PQ can only touch the circumfer- ence at D and E respectively ; for, at these points, the straight lines and the curves may be regarded as starting, and taking different directions, for the straight lines are parallel to the chords, whilst the curves approach and intersect them. 3d. As the one half of each chord falls upon its other half, and the one side of each tangent falls upon its other side, so also the intercepted arcs respectively fall upon and coincide with each other, and hence are equal. Cor. 1. — A radius perpendicular to a chord bisects the chord and also its arc. BOOK I. 15 ■ Co )ok. 2. — A tangent is perpendicular to a radius at its extremity. Cor. 3. — A line perpendicular to a chord at its middle point, passes through the center of the circle. Scho. — Observe that a tangent touches the circumference at but one point, and a chord intersects the circumference at two points, each end passing away in opposite directions ; hence a straight line can only intersect a circumference at two points. THEOREM IX. If a perpendicular be erected at the middle point of a straight line, every point in the perpendicular is equally distant from the extremities of the line. Let PC be perpendicular to AB at its middle point C; then will any point in the line PC be equally distant from A and B. Take any point in the perpendicular PC as D, and draw AD and BD ; and let the part ACD be revolved on DC as an axis, until it fall upon the plane of BCD; since both angles at C are right angles, CA will take the direction of CB, and as AC is equal to CB, the point A will fall upon the point B, and CA will coincide with CB, and AD must fall upon and coincide with DB. (Special Ax. 2.} Cor. 1. — As two points determine the direction of aline; any straight line which has two points equally distant from the extremities of another line, is perpendicular to the latter at its middle point. Cor. 2. — With D as a center and DA as a radius, a circum ference may be described which will pass through the points A and B, and AB becomes a chord of the circumference ; and as a straight line cannot intersect a circumference at more than two points, there can be only two points in the line AB equally distant from the point D. x 16 ELEMENTS OF GEOMETRY. > THEOREM X A -perpendicular is the shortest distance from, the center of a circle to a chord, or from a point to a line. Let AB and EF be two parallel chords, and draw CD perpendicular to AB ; it will also be perpendicular to EF. At the point D, the extremity of the radius CD, draw GH perpendicular to CD, and it will be parallel to the chords AB and EF. 1st. The perpendicular CK is less than _ CA or CB, each of which is equal to CD,, of which CK is a part. CI is less than CE or CF for the same reason. As the chord departs from the center and consequently diminishes, the perpendicular approaches the radius in length, but can never equal it whilst the chord has any definite length. 2d. CD is less than any oblique line drawn from the point C to GH; for any oblique line as CL will terminate without the circumference, and consequently be greater than the radius. Cor. 1. — A perpendicular is the shortest distance from a point to a line, and also between two parallels. Cor. 2. — The farther distant from the center, the less the chord. Cor. 3. — The less the chord the less the arc, and conse- quently the less the opposite angle. is ' BOOK I. 17 r PROBLEM I. To bisect a given line. Let AB be the given line ; then with A and B as centers, and a radius greater than the half of AB, describe arcs above and below the line AB, intersecting at D' and E, and join D' and E, cutting AB in C, which will be the Al middle point. (Th. 9, Cor. 1.) Sch. 1. — The intersections may both be made on the same side of AB, as at D' and D, by taking different radii. Sch. 2. — As the radius becomes, as it were, an oblique line, whilst one-half of AB is a perpendicular, it must, of course, be greater than one-half of AB. PROBLEM II. From a point without a line, to draw a perpendicular to the line. Let P be a point without the line CD. With P as a center, and a radius greater than the shortest distance to CD, which would be a perpendicular, draw an arc cut- ting CD in A and B; then with A and B as centers and a radius greater than one-half of AB, describe arcs intersecting at E ; then will P and E be two points equally distant from A and B, and hence PE is perpendicular to AB or to CD. (Th. 9, Cor. 1.) BVP PROBLEM»III. At a point in a line, to erect a perpendicular to the line. Let P be a point in the line CD; then, with P as a center and a radius PA, cut CD in two points, A and B ; and with A and B respectively as centers, and a radius greater ^-\ than one-half of AB, describe arcs intersect- ing at E, and join PE. It will be perpendicular to CD at the point P. X 18 ELEMENTS OF GEOMETRY PROBLEM IV. To bisect a given angle. Let BAC be the given angle. Then with A as a center and a radius that will cut the sides AB and* AC, .draw the arc DE and its chord ; then with D and E, respec- tively, as centers and a radius* greater than the half of DE draw arcs intersecting at F, and join AF. The two points A and F will be equally distant from D and E ; hence the line AF will bisect the chord DE, its arc, and hence the angle A. \ . PROBLEM V. From a point without a line, to draw a parallel to the line. -k Let P be a point without the line CB. 'From P draw PA perpendicular to CB (Prob. 2), and from P draw a perpendicular to AP. Then will PQ be parallel to CB. (Th. 5.) PROBLEM VI. \ \ To find the center of a given circle. Draw any two chords, as AB and BC, to the gjven circle, and pass perpendiculars through their middle points; both perpen- diculars will pass through the center of the circle. (Th. 8, Cor. 3.) The point of their intersection 0, which is the only common point, is the center of the circle. Scho. — It is not necessary that the chords be consecutive, but they must not be parallel, as then there would be but one per- pendicular and the same perpendicular would pass through the middle points of both chords. BOOK I. '19 ^ f PROBLEM VII. To circumscribe a circle about a triangle. Let ABC be the given triangle. Pass the perpendiculars DO and EO through the middle points of any two sides of the triangle, as AC and CB; their point of intersection will be the center of a circle of which AC and CB are chords. Cor. 1. — A circumference can be passed through any three points not in the same straight line; but if the three points are in the same straight line, only one perpen- dicular can be drawn, and hence no solution. Cor. 2. — Each of the three points is equally distant from the center, but only two points in the same straight line can be equally distant from a point without the line. (Th. 9, Cor. 2.) BOOK II Def. — An Inscribed Angle has its vertex in the circum- ference of a circle of which its sides are chords. THEOREM I. An inscribed angle is measured by one-half the arc intercepted by its sides. FIRST CASK. SECOND CASE. THIRD CASE. There are three cases : 1st. When one of its sides is a diameter; as, the angle BAD has one side AB a diameter. Through the center C draw EF parallel to the chord AD ; then will the angles BAD and BCF be equal, as they are corresponding angles ; but BCF and ACE are equal, as they are vertical angles — they are both angles at the center, measured respectively by the arcs FB and AE, which arcs are consequently equal. Therefore arc AE is equal to arc FB, also equal to arc DF; hence FB is one-half of BD. There- fore the angle A is measured by £ arc BD, which is the arc inter- cepted by its sides. 2d. When the center of the circle is without the angle, as the angle BAD. Here angle EAD is measured by £ arc DE, And " BAE « " " \ " BE. By subtraction, " BAD " * " £ " BD. 3d. When the center is within the triangle ; as BAD. Here angle BAE is measured by J arc BE, And " DAE " " " \ " DE. By addition, " BAD" " "i" BD. V- I u BOOK II 21 THEOREM I. — Continued Cor. 1. — All the angles inscribed in the same segment are equal; since the angles A, B, and C are measured each by the half of the same arc DwE, they are equal. Cor. 2. — An angle inscribed in a semicircle is a right angle, as BAD. I Cor. 3. — An angle inscribed in a segment greater than a semicircle is acute, as BAC ; and an angle inscribed in a segment less than a semicircle is obtuse, as BDC. Cor. 2. Cor. 4. Cor. 4. — The opposite angles of an inscribed quadrilateral are supplementary; as, Angle A measured by | arc BCD. " C " " " " DAB. The sum of the two arcs is the circumference ; hence half their sum is the measurement of two right angles. Cor. 5. — If the extremities of the chords forming an inscribed angle be joined by a straight line, an inscribed triangle is formed. As in the angle A join B and C ; then the triangle ABC has each of its angles inscribed, and is therefore an inscribed triangle. 22 ELEMENTS OF GEOMETRY THEOREM II. ( TJie sum of the three angles of any triangle is equal to two right angles. Let ABC be an inscribed triangle ; then will angles A + B + C equal two right angles ; as, The angle A is measured by J arc BC, u u g u u n « « %Q f a a Q n a a a a ad The sum of the three arcs is a circumference, one-half of which measures the angles and is the measure of two right angles ; hence, the sum of the angles of an inscribed triangle is equal to two right angles; but, as a circumference may be passed through all the vertices of any triangle and the triangle become inscribed, (Book I, Problem 7), it follows that the sum of the three angles of any triangle is two right angles. Cor. 1. — If the triangle is isosceles, 'two of its angles will be equal. Cor. 2. — If the triangle is equilateral, all the angles will be equal, each 60 degrees. Cor. 3. — If the triangle is scalene, all the angles will be unequal. Cor. 4. — As an inscribed angle is measured by half the arc intercepted by its sides, and the greater the arc the greater the chord, hence the greatest angle is opposite the greatest chordWid the next to the greatest angle opposite the chord next to the longest, and the smallest angle opposite the shortest chord ; con- sequently, in any triangle the greatest angle is opposite the longest side, the next to the greatest angle opposite the next to the longest side ; and the smallest angle opposite the shortest side. Scho. 1. — If a triangle has two sides and the included angle given, the three vertices of the triangle are fixed, and the triangle determined. Scho. 2. — One side and the two adjacent angles fix the three vertices, but if one of the given angles be the opposite angle, the other adjacent angle is the supplement of the sum of the two given angles. Scho. 3. — The three sides of a triangle also determine the triangle. BOOK II ;>:; Coe. — Two triangles, each having the three parts named in either of the scholia respectively equal, are equal in all their parts. Rem. — The sum of the angles of a triangle may be determined by means of the parallels, as follows : Let ABC be any triangle. At C draw DE parallel to AB. The angles marked 1 and 1 and 2 and 2 are respectively alternate inte- rior angles, and consequently respectively equal ; hence, the three angles of the triangle are equal to all the angles at a point on one side of a straight line, which is two right angles. (Book I, Th. 3, Cor. 2.) THEOREM III If one side of a triangle is produced in one direction, the exterior angle formed is equal to the sum of the two interior angles not adjacent. Let ABC be a triangle. Produce BC to D, forming the exterior angle ACD. From C draw CE parallel to BA ; then will the angles marked 1 and 1 be cor- responding angles and equal, and the angles marked 2 and 2 be alternate' interior angles and equal: hence, the exterior angle ACD is equal- to the sum of ABC and BAC, the two interior angles not adjacent. THEOREM IV Every -point in the line which bisects an angle is equally distant from each side of the angle. Let AP bisect the angle A, and revolve the part CAP on AP as an axis ; AC will fall upon and coincide with AB, since angle CAP is equal to angle PAD. From any point as P draw PD perpendicular to AB ; it will be the shortest distance to AB, and also to AC, which coincides with AB. s. 24: ELEMENTS OF GEOMETRY THEOREM V. If from a point without a line a perpendicular be drawn to the line, and oblique lines to different points of the line : 1st. The perpendicular will be shorter than any oblique line. 2d. Any two oblique lines at equal distances from the foot of the perpendicular will be equal. Sd. TJie farther from the foot of the perpendicular, the greater the oblique line. 1st. In the triangle ABD, the angle B is a right angle ; hence it is greater than the angle D. Therefore the side AB opposite the angle D, is less than the side AD opposite the angle B. 2d. The triangles ABD and ABC have each two sides and the included angle respectively equal ; hence the triangles are equal, and AC equal to AD. 3d. In the triangle ACE, the angle ACE is obtuse, conse- quently greater than the angle AEC; therefore the side AE is greater than the side AC. THEOREM VI. The' sum of all the angles of any quadrilateral is equal to four right angles. Let A BCD be any quadrilateral. Draw the diagonal DB, dividing the quadrilateral into two triangles. All the angles of the two triangles make up precisely the angles of the quadrilateral ; but, the sum of all the angles of the two triangles is four right angles. Hence, the sum of all the angles of any quadrilateral is four right angles. BOOK II, 25 THEOREM VII The sum of all the angles of any -polygon is equal to two right angles taken as many times as the -polygon has sides, minus four right angles. Let ABCDE be any polygon. Take any point within the polygon, as 0, and from it draw lines to the extremities of all the sides. The number of triangles will be equal to the number of sides of the polygon. The sum of the angles of each triangle is two right angles ; hence, the sum of all the angles of the triangles which make up the polygon, is two right' angles taken as many times as the polygon has sides; but all the angles at 0, which equal four right angles, belong to the triangles, but not to the polygon, and must be deducted from the sum of all the angles of the triangles, and the difference will be the angles of the polygon; therefore, the sum of all the angles of any polygon is equal to two right angles taken as many times as the polygon has sides minus four right angles. THEOREM VIII. If each side of a polygon is prolonged, the sum of all the exterior angles thus formed will be equal to four right angles* At each vertex of the polygon, the sum of the interior and exterior angles is two right angles ; hence, the sum of all the in- terior and exterior angles is equal to two right angles taken as many times as the polygon has sides, which sum is four right angles more than the sum of all the interior angles. (Th. 8.) Therefore, the sum of all the exterior angles is four right angles. 2 X G) ELEMENTS OF GEOMETRY. THEOREM IX The opposite sides and opposite angles of a parallelo- gram are respectively equal. Let A BC D be a parallelogram. Draw the diagonal DB. Since AB and DC are parallels cut by DB, the angles 1 and 1 are alternate and equal; and since AD and BC are parallels cut by DB, the angles 2 and 2 are alternate and equal. The triangles ABD and BCD have the side BD common and the adjacent angles equal; hence the triangles are equal. Therefore, AB is equal to DC, and AD equal to BC. The angles 3 and 3 are equal, and the sums of the same angles at B and D are equal. THEOREM X. If the opposite sides of a quadrilateral are respectively equal, it will be a parallelogram* Let AB equal DC, and AD equal BC. Draw the diagonal DB. The triangles ABD and BCD have their three sides respectively equal; hence the triangles are equal, and the angles opposite the equal sides equal, that is, 1 equals 1, and 2 equals 2, and these are respectively alternate angles; hence, the opposite sides are parallel, and the quadrilateral is a parallelogram. THEOREM XI. If two opposite sides of a quadrilateral are equal and parallel, the figure will be a parallelogram. Let AB and DC be equal aud paral- lel, and draw the diagonal DB. Since AB and DC are parallel, the alternate angles 1 and 1 are equal, and the triangles ABD and DCB have re- spectively two sides and an included angle equal, and are there- fore equal, which makes the other sides equal and parallel, ami the opposite angles of the figure equal; hence it is a par- allelogram. BOOK II, THEOREM XII 27 The diagonals of a -parallelogram mutually bisect each other. The triangles ABE and DCE have a side and two adjacent angles respec- tively equal ; hence the side AE oppo- site the angle 2, is equal to EC opposite the angle 2 in the other triangle, and DE is equal to EB for the same reason. THEOREM XIII.* An angle formed by a tangent and a chord is measured by one- half the intercepted arc. The angle BAD, formed by the tangent AD and the chord AB, is measured by J- arc Aw?B. From B draw the chord BE parallel to the tangent AD, then the angles BAD and ABE are alternate interior angles and consequently equal. The angle ABE is inscribed, and is measured by £ arc A?iE, which is equal to the arc AmB, as they are intercejrod by two parallels; consequently the angle BAE is measured by \ arc kmB. THEOREM XIV. An angle formed by two chords intersecting ivithin the circle, is measured by one-half the sum of the intercepted arcs. Let AB and DE be two chords intersect- ing at C. ■ ^ From A draw the flV AF parallel to DE, then will the anglel^AF and BCE be corresponding angles, and equal ; the angle BAF is inscribed, and is measured by J- arc BEF; but the arc FE is equal to the arc AD, x (therefore arc BEF is equal to the sum of the arcs EB and AD ; j consequently the angle BCE, or its equal ACD, is measured by J the sum of the arcs included by its sides. S v ; 28 ELEMENTS OF GEOMETRY THEOREM XV. An angle formed by two secants meeting without th4 circle, is measured by one-half the difference of the inter- cepted arcs. The angle A is formed by the two secants AB and AD. From C draw CE parallel to AD; BAD and BCE are corresponding angles. The angle BCE is measured by \ arc BE = BD — CF; therefore, the angle A, formed by two secants meeting without the circle, is measured by one-half the difference of the intercepted arcs.. THEOREM XVI. An angle formed by a tangent and a secant meeting without a circle, is measured by one-half the difference of the intercepted arcs. The angle BAD is formed, by a tangent and a secant meeting at A. From C draw CE parallel to AB; then the angles BAD and DCE are corresponding and equal ; but the angle DCE is inscribed, and meas- ured by \ arc DE, and DE = DB — BE, or its equal BC; parallels intercept equal arcs, there- fore the angle A is measured by £ arc DE = £ arc (DB — €C). THEOREM XVII. An angle formed by two tangents meeting without a circle, is measured by one-half the difference of the inter- cepted arcs. The angle A is formed by two tangents meeting without the cir- cle. At C draw CD parallel to AB, then the angles BAC and DCE will be corresponding and equal ; but the angle DCE is formed by a tangent and a chord, and hence is measured BOOK II 29 by 4 arc CmD. (Th. 14.) Arc DmC = (arc BrDmC — arc BrD) and arc BrD = arc BnC ; therefore, an angle formed by two tangents intersecting without a circle is measured by one- half the difference of the intercepted arcs. THEOEEM XVIII. The side of a regular hexagon is equal to the radius of the circumscribed circle. Describe a circumference and make the chord AB equal to the radius of the circle. Draw the radii CA and CB. CAB will be an equilateral triangle, each side equal to the radius of the circle, and each angle equal to 60 degrees ; hence, AB is a chord of an arc of sixty degrees, which is contained exactly six times in the circumference, and is therefore a side of a regular hexagon. PROBLEM I To construct an angle equal to a given angle. Let A be the given angle. With A as a center and a radius that will cut both sides, describe the arc CD ; then, with the same radius and B as a center, describe the arc EF, making it equal to CD, and draw BF ; and EBF will be the required angle. •PROBLEM II.. Two sides and the included angle given, to construct a triangle. Make the angle A equal to the given angle, and on one side lay off AB equal to one of the given sides, and on the other AC equal to the other given side. Draw BC, and ABC will be the required triangle. Cor. — Two triangles having two sides and the included angle respectively equal, are equal in all their parts. 30 ELEMENTS OF GEOMETRY PROBLEM III. One side and the two adjacent angles given, to construct the triangle. Make AB equal to the given side. At A construct an angle equal to one of the given angles, and at B an angle equal to the other given augle ; the intersection C of the lines forming these angles will be the vertex of the third angle, and ABC will be the required triangle. Cor. — Two triangles having each a side and the two adjacent angles respectively equal, are equal in all their parts. PROBLEM IV. To construct a triangle, having given the three sides. Make AB equal to one of the given sides. Then with A as a center, and a radius equal to one of the given sides, describe an arc ; and with B as a center and the other given side, describe an arc intersecting the other arc at C and draw AC and BC, then ABC will be the required triangle. Cor. — Two triangles having their three sides respectively equal, are equal in all their parts. Scho. — The sum of any two sides of a triangle must be greater than the third side. PROBLEM V. Two sides and an angle opposite one of them given, to construct a triangle. \ Make the angle A equal to the given angle, and make AC equal to one of the given sides; then with C as a center and a radius equal to the other given side, draw the arc BB', and draw CB and CB'. In this case there are two triangles. BOOK II. 31 Scho. — The second side must be equal to or greater than the perpendicular from C to AB. If it is equal, there will be one right-angled triangle ; if it be greater than the perpendicular and less than CA, there will be two triangles; but if it be less than the perpendicular, there will be no triangle. PROBLEM VI. Form an equilateral triangle. Describe a circle, and apply the radius six times to the circumference, and draw the chords; the result is a hexagon. Join the alternate vertices, and the result is ABC, an equilateral triangle. PROBLEM VII. To construct a regular polygon of eight sides. Describe a circumference, aud divide it into eight equal parts. Draw chords to the equal arcs ; they will be the sides of the polygon. Draw radii from the extremities of the sides to the center of the circle ; there will be as many isosceles triangles as the polygon has sides. The angles at the center are equal, having equal arcs; and each angle of the polygon is composed of two equal angles of the isosceles triangles ; hence, all the angles of the polygon are equal ; and the sides being also equal, the polygon is regular. Cor. — A regular polygon of any number of sides may be constructed by dividing the circumference into as many equal parts as there are sides. Rem. — The circumference will be divided into eight equal parts by applying the chord of an arc of 45°. 32 ELEMENTS OF GEOMETRY, PROBLEM VIII. To draw a tangent to the circumference at any -point on it. Let C be the center of a given circle, and A any point in its circumference. Draw the radius CA, and from A draw AB perpen- dicular to the radius CA; then AB will be the tangent required. (Book 1, Th. 8, Cor. 2.) PROBLEM IX. From a point without the circle, to draw a tangent to the circle. Let C be the center of the given cir- cle, and A the point without the circle from which the tangent is to be drawn. Join the point A and the center C and bisect AC in ; then, with as a cen- ter and the radius OC describe a cir- cumference ; the points B and B', the intersections of the two circumferences, will be the points of tangency, AB and AB' the tangents, as each is a perpendicular to a radius at its extremity; the angles ABC and AB'C being each inscribed in a semicircle. PROBLEM X. On a straight line, to construct a segment that shall contain a given angle. Let AB be the given line. At B make the angle ABD equal to the given angle. Draw BO perpendicular to BD, and at C, the middle point of AB, erect a perpendic- ular intersecting BO at ; then, with as a center and radius OB, describe a circum- ference to which DB is a tangent and AB a chord, and the angle ABD is measured by \ arc AmB; so also every angle, as E, E', inscribed in the segment AEB. BOOK II. PROBLEM XI. To inscribe a circle in a given triangle. Bisect any two angles as A and B by the straight lines AD and BD, and as every point in each bisecting line is equally distant from the sides of the angle, hence the point of intersection D will be equally distant from the three sides, and DE, DF and DG will be radii of the inscribed circle. PROBLEM XII. Draw a common tangent to two external circles of different radii. Let C and C be the centers of two circles which are external. With C as a center and a radius equal to the difference of the radii of the circles, describe a small circumference, and from the point C draw a tangent CA' to this small circumference ; from the center C draw a radius through the point of tangency A , and extend it to A in the circumference of the large circle ; draw C'B parallel to CA and join AB, which will be the required tangent. Cor. — A second tangent B'D may always be drawn. PROBLEM XIII. To draw a tangent to two external circumferences of different radii, the tangent passing between the circles <$nd touching at points on the opposite sides of the cir- cumferences. Let CA and C'B be the radii of the given circles. With C as a center and a radius equal to the sum of the two given radii, draw a circumference. From C draw CD and CD' tangents to the large circle; then draw the radii CD and CD', cutting the circumference of the smaller given circle in A and A', which will be the points of tangency. From C draw C'B parallel to CA; and C'B' parallel to CA', and join AB and A'B', and they will be the required tangents. BOOK III. PROPORTIONS DEFINITION". When two quantities, each having the form of a fraction, that is, each having a numerator and a denominator, are equal to each other, an equation may be formed of them ; and they may be arranged proportionally. In order to show whether the ratio is increasing or decreasing, the denominators should be made the antecedents and the numerators consequents; still, they are in proportion when taken in an inverse order, that is, the numera- tors as antecedents and the denominators as consequents, but the ratios will be inverted. Thus, B A = D " c' Then will A : B : : C : D, and B 3-A : : D : C. This proportion solution; thus, is made 10 5 " very _ 12 6 sim By reduction, 2 1 " 2 end 1:2: : 1 : 2. The same proportion as 5 : 10 : : 6 : 12, or 2:1: : 2 : 1. and 10 : 5 :: 12 : C. simple by an arithmetical The equation is true if the fractions are inverted ; thus, 56 a l ■* O 1 o -. 15=5' and 2 = 2 ; •'• * sl :: *- L Proportions are much used in Geometry, and should therefore be carefully studied. Instead of two equal ratios there may be many, in which case they are termed continued proportions ; as, B_D_F_H _ K A ~ C ~ E : _ G ~~ I etc. BOOK III. 35 Which may be rendered, A : B :: C : D :: E : F :: G : H :: I : K, etc. This is read: as A is to B, so is C to D, so is E to F, so is G to H, so is I to K. The antecedent and consequent form a couplet, and in a continued proportion any two couplets, may be taken to form a proportion of four terms, which is always considered a propor- tion, and the first and last terms are called extremes, and the second and third the means. THEOEEM I. If four quantities are proportional, the product of the means equals that of the extremes. If A : B : : C : D, a B D then ^ = -. Clearing of fractions or multiplying both members by A and C (Gen. Ax. 6), BC = AD. Cor. 1. — B : A : : D : C ; that is, if four quantities are in proportion, they are also in proportion by inversion. Cor. 2. — They are also in proportion by alternation ; thus, v- = p- Multiplying both members by = , ■£= = p" r ; reducing, C D j — = ; therefore, A : C : : B : D, and again by inversion, C : A :: D : B. THEOREM II. A mean proportional between two quantities is equal to the square root of their product . Let B be a mean proportional between A and C ; as, A : B : : B : C. The product of the means is equal to that of the extremes; thus, B 2 = A x C. Extracting the root of both members, B = VA~xC. 36 ELEMENTS OF GEOMETRY. THEOREM III. If the product of two quantities equals the product of two other quantities, either of the two forming a product may be made the means, and the other two the extremes of a proportion. Let B x C = A x D; divide by A x C, then B_x^ _ A_x^D _ B _ D AxC"AxC~A"C and A : B :: C : D, (1) or C : D :: A : B. (2) In the first proportion, A and D are the extremes, and B and C the means ; in the second, B and C are the extremes, and A and D the means. THEOREM IV. If four quantities are proportional, they will also be proportional by composition and division. If jf = £ , then A - + l = - + l, and --1 = ^-1. Reducing to improper fractions, B + A D + C , B-A and A C ' A C ' and A : B + A :: C : D + C; also, A : B— A :: C : D— C. By alternation, A : C :: B + A : D + C; also, A : C :: B— A : D— C. C_D + C . C_ D-C A^BTA 5 aUK> ' A"B^A" Gen. Ax. 1, ?±£ _ ^-C . B + A : D + C :: B-A : D-C. D + A b — A By alternation, B + A : B — A :: D + C : D — C. BOOK III. 37 THEOREM V. Like powers and like roots of proportional quantities are proportional, B D B 2 D 2 Squaring both sides, -r = ~ ; then ^ = r2 > B n D n B» D« and x- ss 7T-, and — r = — r- (Gen. Ax. 10.) A* C' 1 ' A» C» A 2 : B 2 :: C 2 : D 2 , and A» : B n :: C : D», and A» : B» :: O : D». THEOREM VI Any equimultiple of one couplet will he proportional to the other couplet or to any equimultiple of it. This depends upon the principle that multiplying both numerator and denominator of a fraction by the same quantity does not change its value. A _ c , ana m £-Q- n Q THEOREM VII. If the corresponding terms of two proportions be mul- tiplied, their products will be proportional. hence (Gen. Ax. 6), A : B : : C : D E : F : : G : H B A " D F E ~ H = G ; BF AE = DH " CG* AE : BF : : CG : DH 38 ELEMENTS OF GEOMETRY. THEOREM VIII. In a series of proportions, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. A : B :: C : D :: E : F :: G : H, etc. AD = BC AF = BE AH = BG AB = BA . A(B + D + F-fH) = B(A + C+E + G) A : B :: A + C + E + G : B + D + F + H. Cor. 1. — If any two proportions have an equal ratio, then the other terms are proportional. Cor. 2. — The same is true if the antecedents are the same in two proportions. BOOK IT THEOKEM I. The area of a rectangle is equal to the product of its base arid altitude. There may be three cases : 1st. When the base and altitude are composed of units of the same denomina- tion ; then it is evident that there will be as many square units for every unit in alti- tude as there are units in the base ; and for every additional unit in altitude as many more square units ; hence the area will be the product of the base and altitude. 2d. If there be a fraction in one or both the dimensions, the common denominator will be the denomination of the unit of measure ; hence, the product of the base and altitude will give the area, in units of the same denomination. 3d. If the dimensions are incommensurable, the unit of measure will be an infinitesimal. . THEOREM II. The area of a parallelogram is equal to the product of its base and altitude. Let ABCD be a parallelogram, AB its *» base, BE its altitude, its area = AB x BE. Construct the rectangle ABEF; its area = AB x BE. FE = AB and DC = AB, .-. FE ALl - JB = DC, and taking from each DE, there remains FD = EC; hence the triangles ADF and BCE are equal, having all their sides equal. In changing the parallelogram into the rectangle, we have added and subtracted the same area; hence the parallelogram is equal to the rectangle. .*. the area of the parallelogram is AB x BE, product of base and altitude. 4:0 ELEMENTS OF GEOMETBT, THEOREM III. The area of a triangle is equal to one-half the pro- duct of the base and altitude. Area ABC = }(ABx CE). Let ABC be the given triangle, AB its base, and EC its altitude. Construct a parallelogram on AB as one of its sides and BC as another, draw AD parallel to BC and CD parallel to AB; then will ABCD be a parallelogram. The triangles ABC and ACD will have their three sides respec- tively equal ; hence the triangles are equal and each is one-half of the parallelogram ABCD ; and as the area of the parallelogram is AB x CE, that of the triangle is \ (AB x CE) ; therefore, the area of a triangle is equal to one-half the product of the base and altitude. Coe. 1. — Rectangles, parallelograms and triangles are to each other as the products of their bases and altitudes respectively. Cor. 2. — If the bases are equal, they are to each other as their altitudes. Cor. 3. — If the altitudes are equal, they are to each other as their bases. THEOREM IV. The area of a trapezoid is equal to the product of its altitude and half the sum of its parallel bases. , c Let ABCD be a trapezoid, DE its alti- \ \ tude, and AB and DC its parallel bases. \ \ Draw the diagonal DB, dividing the trape- \] B zoid into two triangles whose common altitude is DE and their bases AB and DC The area of the triangle ABD = $ (AB x DE), " " " " " BCD = j-(DC x DE), By addition, area of ABCD = DE J- (AB + DC). That is, the area of a trapezoid is equal to the product of its altitude and \ the sum of its bases. BOOK 41 THEOREM V. The square described on the sum of two lines, is equiv- alent to the sum of the squares of the lines, increased by twice the rectangle of the lines. E I ^D ACDE'is the square described on the sum of AB and BC; and corresponds to the algebraic formula {a + b) 2 = a 2 + 2ab -f- b 2 , in which AB = a and BC = b. Cor. — If the lines are equal there will be four equal squares. Let AB = 1 and BC = 1 ; then the square of two is four times the square of one. ab ft a H a* ab THEOREM VI. The square described on the difference of two lines, is equivalent to the sum of the squares of the lines, dimin- ished by twice the rectangle of the lines. AB = a, KB = b, a b e d _j c (a — b) 2 = a 2 - 2ab + b 2 , ABCD as a\ EDFG = b\ BCIK = ab, and EIFH = ab, F G H THEOREM VII. The rectangle contained by the sum and difference of two lines, is equivalent to the difference of their squares. AB = a, and LB = BK = b, a + b = AK, and a — b = AE = AL, (a + b) x (a — b) = a 2 — b 2 , ABCD = a 2 , and FHGC = b 2 , A L The rectangle EFGD = recti BKIH = b (a — b), ABHE = = a (a — b), By addition, AKIE = = (a + b) {a f; h B K b). 42 ELEMENTS OF GEOMETEY. THEOREM VIII. The square described on the hypothenuse of a right- angled triangle, is equivalent to the sujti of the squares of the other two sides. Let ABC be a triangle, right-angled at A, then will BC 2 = AB 2 + AC 2 , Construct a square on each side of the triangle. From A draw a perpendicular to BC and extend it to ED, and draw AE, AD, IC and BF. The triangles ABE and IBC have two sides respectively equal, viz., AB =3 Bl and BC =3 BE, being respectively sides of the same square, and the included angles equal ; that is, ABE = I BC, as each one is composed of the angle ABC and a right angle ; hence, triangle ABE = triangle IBC; but triangle ABE is one- half the rectangle BELK, having the same base and altitude BE and BK ; and the triangle IBC is one-half the square ABIH = IB x AB = AB 2 ; therefore, AB 2 = rectangle BELK. By the same process, we prove the triangle BCF = ACD, and the square ACFG == rect. CDLK ; therefore, BC 2 = AB 2 + AC 2 . And by transposing, Cor. 1. BC 2 - AB 2 = AC 2 , and BC 2 - AC 2 = AB 2 . Cor. 2. — The square described on the diagonal of a square is double the square described on the side, as the sides are equal ; hence, the square of diag. : sq. of side : : 2:1, and diag. : side : : a/2 : 1. Since AB 2 = rect. BELK, and AC 2 = rect. CDLK, the resulting proportion AB 2 : AC 2 :: BK : KC; that is, the squares of the sides are proportional to their adjacent segments Cor. 3. of the hypothenuse. And BC 2 : AB 2 BC : BK, and BC* AC :: BC : KC ; that is, the square of the hypothenuse is to the square of either side as the hypothenuse is to the segment adjacent to the side. Scho. — Observe, that if the right angle A be diminished, the sides about it remaining the same, the third side BC will be dimin- ished; and if the angle A be increased, BC will be increased ; in the first case the square of BC will be less, and in the second greater than the sum of the other two ; hence, the right-angled triangle is the only one in which the square of one side is equiv- alent to the sum of the squares of the two. BOOK IV. 43 THEOREM IX. In any triangle, the square of a side opposite an acute angle is equivalent to the sum of the squares of the two other sides, minus twice the rectangle of the base, and the distance from the acute angle to the foot of the per- pendicular let fall from the vertical angle on the base, or the base produced. In the triangle ABC the side AB is oppo- site the acute angle C ; hence, AB 2 = AC 2 + BC 2 - 2BC x CD ; the perpendicular falling on the base, BD = BC - DC. Squaring both members, _BD 2 = BC 2 + DC 2 - 2B.C x DC, and by adding AD 2 to each member, BD 2 + AD 2 = BC 2 + DC 2 + AD 2 - 2BC x DC; and by Theorem 8, AB 2 = BC 2 + AC 2 - 2BC x DC. The same process will give the same result, when the perpendicular falls upon the base produced. THEOEEM X. In an obtuse- angled triangle, the square of the side opposite the obtuse angle is equivalent to the sum of the squares of the two other sidles, plus twice the rectangle of the base and the distance of the obtuse angle from the foot of the perpendicular let fall from the vertical angle on the base produced. Enunciation, AC 2 = AB 2 + BC 2 + 2BC x BD, CD = BC+ BD; by squaring, CD 2 = BC 2 + BD 2 + 2BC x BD; A D B adding AD 2 to each side, CD 2 + AD 2 = BC 2 + BD 2 + AD 2 + 2BC x BD. Theorem 8. AC 2 = BC 2 + AB 2 + 2BC x BD. u ELEMENTS OF GEOMETRY, THEOEEM XI. If from the vertex of any angle of a triangle, a line be drawn to the middle point of the opposite side, then twice the square of the bisecting line, plus twice the square of half the bisected side, will be equal to the sum of the squares of the two other sides. From the vertex A of the triangle ' ABC draw AD to the middle point of BC ; then will D 2AD 2 + 2BD 2 = AB 2 + AC 2 . In the triangle ADC., the side AC is opposite the obtuse angle ADC. AC 2 = AD 2 + DC 2 + 2DC x DE. (1) And in the triangle AB D the side AB is opposite the acute angle AD B. AB 2 ■= AD 2 + BD 2 - 2BD x DE. (2) By adding equations (1) and (2), and observing that BD = DC, AB 2 + AC 2 = 2AD 2 + 2BD 2 . Cor. — The sum of the squares of all the sides of a parallelo- gram, is equivalent to the sum of the squares of the diagonals. D _^ Since the diagonals mutually bisect each other, DC 2 + BC 2 By addition, AB (Th. 8, Cor. 2.) 2CE 2 + 2DE 2 , AB 2 + AD 2 = 2AE 2 + 2DE 2 . + DC 2 + AD 2 + BC 2 = 4AE 2 + 4DE 2 = AC* BD 2 . THEOEEM XII. If a line be drawn parallel to one of the sides of a triangle cutting the other sides, it will divide them pro- portionally. Draw DE parallel to BC, and draw BE and DC; then the triangles DEB and DEC have the same base DE and the same altitude, as both their vertices are in the line BC, parallel to DE ; hence, they are equivalent. The triangles ADE and BDE having the same altitude, as they have a common vertex E, are to each other as their bases ; hence, ADE : BDE :: AD : BD. BOOK IV. 45 The triangles ADE and DEC have a common vertex D ; hence, ADE : DEC : : AE : EC ; but triangle DEB = triangle DEC, and the two proportions have an equal ratio ; .-. AD : BD :: AE : EC, and by composition, Cor. 1. AD + BD : BD :: AE + EC : EC; that is, AB : BD :: AC : EC, and AD + BD : AD :: AE + EC : AE, that is, AB : AD :: AC : AE. Cor. 2. parallel to will be cut — If any number of a side of a triangle, proportionally. lines be drawn the other sides i o Ax \f \d \ Cor. 3. — If any number of lines be cut by the parallels, they will be cut proportionally. THEOREM XIII, A line which bisects an angle of a triangle, divides the opposite side into segments proportional to the adja- cent sides. Let AD bisect the angle A ; then BD : DC :: AB : AC. (1) From C draw a line parallel to DA, inter- secting BA produced in E ; then BD : DC ::' AB : AE. (2) The angle CAD = ACE, alternate; CAD = BAD, bisected; and BAD = BEC, corresponding; .*. ACE = AEC, and the triangle AEC is isosceles ; hence, side AE =z AC, and (2), BD : DC :: AB : AC = AE. 46 ELEMENTS OF GEOMETRY. THEOREM XIV. Triangles which are mutually equiangular have the sides opposite the equal angles respectively proportional, and hence the triangles are called similar. AB : DE :: AC : DF :: BC : EF. The triangles ABC and DEF are mutually equiangular ; that is, A = D> E = B, and F = C. Place DEF on ABC, the point D on A, the side DE on AE', and DF will fall on AC, since angle D = angle A. Since the angles 2 and 2 are equal, E'F' is parallel to BC; and AB : AC : : AE' : AF', or, AB : AC :: DE : DF; and by alternation, AB : DE :: AC : DF. By placing F on C, we obtain the proportion, AC : DF :: BC : EF. Cor. — Two triangles having two angles respectively equal are similar. Rem. — The sides opposite the equal angles are called homol- ogous. THEOREM XV. The figure formed by joining the middle points of any quadrilateral by straight lines is a parallelogram. Let ABCD be any quadrilateral. Join the middle points of the sides, and draw the diagonals. BE : BF : : EA : FC AE : AH : : EB : HD DH : DG : : HA GC CF : CG : : FB GD. It follows that GF is parallel to DB, and HE is also parallel to DB. .-. GF and HE are parallel ; so also EF and HG are parallel. BOOK IV. 47 THEOREM XVI. Two triangles which have their sides respectively pro- portional are similar. Since the sides are respectively proportional, then AB : DE :: AC : DF :: BC : EF (1) Make AE' = DE, and draw E'F' parallel to BC ; then will AB : AE' :: AC : AF' :: BC : E'F'; (2) but AE' = DE. The proportions (1) and (2) have an equal ratio, hence the other ratios must be the same ; hence AF' == DF and E'F' = EF; the triangle AE'F' = DEF; but the triangles AE'F' and ABC are equiangular, as E'F' is parallel to BC ; and as they are equiangular they are similar. Cor. 1. — If two triangles have each an equal angle included by proportional sides, they are similar. Cor. 2. — Two triangles which have their sides respectively parallel or perpendicular to each other are similar. 48 ELEMENTS OF GEOMETRY. THEOREM XVII. Regular polygons of the same number of sides are similar figures. Construct a regular polygon, as in Problem VII, Book II, and let a smaller one be placed upon it, the angles being the same in both polygons ; they will also be the same in the isosceles triangles; consequently the sides AB and ab, also BC and be, etc., will be parallel ; hence the proportions, AB : ab :: R : r, BC : be :: R : r, etc. .-. the triangles are similar; and as each polygon is composed of an equal number of similar triangles, the polygons are similar. Cor. 1. — It is evident that a circumference may be inscribed in the polygon, as the perpendicular OP, which is called the apothegm of the polygon is the distance from the center to each side. Cor. 2. — As the equal sides of the isosceles triangles become radii of the circumscribed circle, a circle may be passed through all the vertices. Cor. 3. — Circles are similar figures,. Def. — Two polygons which are mutually equiangular and have their corresponding sides proportional, are similar. BOOK IV. 49 THEOREM XVIII. In a right-angled triangle, if aline be drawn from t-he right angle perpendicular to the hypothenuse, it will divide the given triangle into two triangles, simi- lar to the given triangle and similar to each other. Let ABC be right-angled at C, and C D perpendicular to the hypothenuse AB. The triangles ABC and ADC have the angle A common, and each has a right angle; they are therefore similar. And for the same reason ABC and BCD are similar; consequently, ADC and BCD are similar. Cor. 1. Since ABC and ADC are similar, AB : AC :: AC : AD, AB x AD = AC 2 . . (1) Since ABC and BCD are similar, AB : BC :: BC : BD, AB x BD = BC 2 . (2) Since ADC and BCD are similar, AD : DC :: DC : BD; (3) whence, DC 2 = AD x BD. Adding (1) and (2), AB 2 = AC 2 + BC 2 . From proportions (1) and (2), the result is that the square of the hypothenuse is equivalent to the sum of the squares of the other sides ; and from (3), that the perpendicular is a mean pro- portional between the segments of the hypothenuse. Cor. 2. If from any point in the cir- cumference of a circle a perpendicular be drawn to the diameter, it will be a mean proportional between the segments of the diameter. Let C be the point in the circumfer- ence from which is drawn the perpendic- ular CD to the diameter AB; by_drawing AC and CB, ABC becomes right-angled at C; hence, CD 2 = AD x BD. (3) 3 50 ELEMENTS OF GEOMETRY, THEOREM XIX. If two chords intersect each other in a circle, their seg- ments are reciprocally proportional. The triangles ACE and BCD are simi- lar; the angle E = angle B, and A = D, respectively measured by \ the same arc ; hence, AC : DC :: CE : BC. Cor. AC x BC = DC x CE, the product of the segments of the one chord equal to the product of the segments of the other chord. THEOREM XX. If from a point without a circle, two secants be drawn terminating in the concave arc, the whole secants will be reciprocally proportional to their external segments. In the similar triangles ACE and BCD, AC : BC :: CE : DC. Cor. AC x DC = BC x CE. THEOREM XXI. If from a point ivithout the circle a tangent and a secant be drawn, the tangent will be a mean proportional between the whole secant and its external segment. The similar triangles ABC and ACD give the following pro- portion : CB : AC :: hence, CB xCD AC : CD; :AC 2 . BOOK IV. 51 THEOREM XXII. i"/ a line be drawn bisecting an angle of a triangle and intersecting the opposite side, the rectangle of the sides about the bisected angle equals the rectangle of the seg~ ments of the third side plus the square of the bisecting line. Circumscribe a circle about the given triangle ABC, and bisect the angle C and extend the bisecting line to the circumfer- ence of the circle and draw BE. The triangles ADC and BCE are similar; hence, AC : CE :: CD : BC; AC x BC = CE x CD; but CE = CD + DE, and (CD + DE) x CD = DE x CD + CD 2 , and DE x DC = AD x DB. AC x BC = AD x BD + CD 2 . THEOREM XXIII. Two triangles, having an angle in each equal, are to each other as the rectangles of the sides containing the equal angles. Let the triangles ABC and ADE have the angle A common; then will ABC : ADE :: AB x AC : AD x AE. Draw BE; then the triangles ABE and ADE have the same altitude, and hence ABE : ADE :: AB : AD; the triangles ABC and ABE have the same altitude, ABC : ABE :: AC : AE. Multiplying the proportions and observing that ABE is common to antecedent and consequent, ABC : ADE :: AB x AC : AD x AE. 52 ELEMENTS OF GEOMETKY. THEOREM XXIV. Similar triangles are to each other as the squares of their homologous sides. Let ABC and DEF be similar triangles ; angle A = angle D, ABC : DEF :: AB x AC : DE x DF. (1) AB : DE :: AC : DF. (2) Multiplying this proportion by the identical pro- portion, DF :: AC : DF, AC AB x AC Since the 1st and 4th have equal ratios DE x DF : : AC 2 : DF 2 - (3) (4) ABC : DEF :: AC 2 : DP and as the homologous sides are proportional, so also the triangles are to each other as AB 2 : DE 2 and BC 2 : EF 2 - Ooe, 1. — The areas of regular polygons are to each other as the squares of the radii of the inscribed or circumscribed circle. Cor. 2. — The areas of circles are to each other as the squares of their radii, or the squares of the diameters. GENERAL COROLLARIES. 1. The perimeters of similar polygons are to each other as their homologous sides, or as their corresponding diagonals. 2. The perimeters of regular polygons of the same number of sides are to each other as the radii of the inscribed or circum- scribed circles. 3. The circumferences of circles are to each other as their radii or diameters. BOOK IV, 53 g f PROBLEM I. To divide a given line into five equal parts* Let AB be the given line. From A draw an indefinite line AH, making any angle with AB, and on it lay off the same distance five times. Join the last point C with B, and from each point draw lines parallel to CB; then AB will be divided into five equal parts. (Th. 12, Cor. 2.) PROBLEM II. To divide a given line into parts proportional to several given lines. Let AB be the given line, abed Draw AH an indefinite line, and on it lay off the several given lines a, b, c, d, and join the last point with B, and from each point draw lines parallel to this line; then will the parts Aa', ab', b'c, and c'B be proportional to the given lines a, b, c, and d. ab e PROBLEM III. To find a fourth proportional to three given lines. From any point, as A, draw two indefinite lines AH and AY ; on AY lay off a, and on AH lay off b and join ab; on AY lay off c and draw cd parallel to ab ; bd will be a fourth proportional. PROBLEM IV. To construct a mean proportional to two given lines. On an indefinite line lay off AB and BC, equal respectively to the given lines. On AC describe a semi-circumference, and at B erect the perpendicular BD, which will be a mean proportional between AB and BC. (Th. 18, Cor. 2.) 54 ELEMENTS OF GEOMETRY, PROBLEM V. To construct a triangle equivalent to a given -polygon. Let ABCDE be the given polygon. From A draw the diagonals AD and AC ; then from E and B draw EF and BG, respectively par- allel to the diagonals AD and AC, intersect- ing the base produced ; then AFG will be the required triangle. c PEOBLEM VI. To inscribe a square in a circle and circumscHbe a square about a circle. Draw two diameters at right angles and join their extremities, and we have an inscribed square, as each side is a chord of ninety de- grees, and each angle is measured by one-half a semi-circumference. At each extremity of the perpendicular diameters draw tangents to the circumference and we have the circumscribed square. Cor. 1. — The circumscribed square has double the area of the inscribed, as it has eight equal triangles, whilst the inscribed has only four of the equal triangles ; hence, area of cir. sq. : area of ins. sq. :: 2:1, and side of cir. sq. : side of ins. sq. :: \/2 : 1; same result as in Th. 8, Cor 2. Scho. — The side of the circumscribed square is the same as the diagonal of the inscribed. Cor. 2,— In the triangle CBC, right- angled at B, we have CC 72 - CB 2 =s CB 2 . CC = 2, and CB as 1. 4 - 1 = CB 2 , and CB 2 = 3, CB = V3. That is, TJie side of an equilateral triangle : Radius : : \/3 : 1. BOOK IV. 55 PROBLEM VII. To divide a given line into extreme and mean ratio ; that is, into two such parts that the greater part shall be a mean proportional between the whole line and the less part. Let AB be the given line. At B erect a perpendicular BC equal to -JAB ; and with C as a center and radius CB, describe a circum- ference. From A draw AF through the center and terminating in the concave arc, and with A as a center and AD as radius, draw the arc DE, making AE equal to AD ; then DF = AB, and (Theorem 21) AF : AB :: AB : AD, by inversion AB : AF :: AD : AB, and by division AB : AF - AB :: AD : AB - AD; that is, AB : AD :: AD : EB, orAB : AE :: AE : EB. PROBLEM VIII. To construct a square equivalent to a given triangle. A mean proportional between the BAe *- l altitudb. base and half the altitude of the tri- angle will be a side of the square. Let B = Base and A = \ Altitude. DC is a mean proportional between base and one-half altitude. * PROBLEM IX. To construct a square equivalent to two given squares. Construct a right angle. On one of a b the sides of the angle lay off a distance equal to a side of one of the squares ; and on the other side of the angle a distance equal to a side of the other square, and draw the hypothenuse ; it will be a side of the required square. Rem. — By this principle the side of a square equivalent to any number of squares may be found. Cor. — By making the longer side the hypothenuse, the third side will be the side of a square equal to the difference of two squares. 56 ELEMENTS OF GEOMETRY. Rem. 1. — If similar polygons be constructed on the three sides of a right-angled triangle, the given sides being homologous, the polygon constructed on the hypothenuse will be equivalent to the sum of the two others. Rem. 2. — To construct a square equivalent to a given poly- gon, reduce the polygon to an equivalent triangle and find a mean proportional between the base and half the altitude of the triangle. PROBLEM X. To construct a polygon, similar to a given polygon, on a given side homologous to one of the sides of the given polygon. Let ABCDEF be the given polygon and AB' a side of the required polygon homologous to AB. Lay off AB' on AB, and from A draw all the diagonals. Draw B'C parallel to BC to the first diagonal ; then from one diagonal to another draw sides parallel to the opposite side of the given poly- gon. AB'C'D'E'F' will be the required polygon. (Th. 17, Cor. 4.) Cor. — To construct a regular polygon, having one of the sides given: First construct a regular polygon of the proper number of sides; then find a fourth proportional to the side of the constructed polygon, the side of the required polygon, and the radius of the circumscribed circle of the constructed polygon ; the fourth proportional will be the radius of the circumscribed circle of the required polygon. PROBLEM XI. To extract the square root of a quantity, or, what is the same thing, to find the side of a square equivalent to a given surface. The surface of a square is found by squaring a side; thus, 3x3 = 9, that is, 3 in length and 3 in breadth. 9 is the surface of which we wish to find a side of a square equivalent ; and, as 3 x 3 = 9, it is evident that 3 is the square root of 9 ; so also 4 of 16, 5 of 25, 6 of 36, etc. ; but when the number is large it is not so easily found. BOOK IV. 5? Let us take an algebraic binomial, as (a + b) 2 = a 2 + 2ab -\- b 2 , and exhibit it geometrically. The divisors must be such as to render the quotient a root (a + b). a) a 2 + 2ab + V i (a + b ab b 2 a 2 ab 2a + b ) + 2ab + b 2 + 2ab + W> Next take a trinomial ; as a ) a 2 + 2ab + b 2 + 2ac+2bc+c 2 ( a + 5 + c 2a + b ) 2«6 + £ 2 2^ + 52 2a + 2£-fc) -f-2ffc + 2fc+c* + 2«c + 2^4-6' 2 M ?H- ,•2 a& /; 2 ^ « 2 aZ> ■3 Cor. 1. — The first term of the roots is obtained the same as that of finding the root of a monomial; the divisor and root are the same, as the first surface a 2 is a square ; after that we have rectangles, the breadth of which is the root, and our divisor must be the whole length of the rectangles. Cor. 2. — Each successive divisor is double the root already found, plus the next term of the root. Rem. — In Arithmetic we pursue the same course, except that the squares are not so entire and separate as in Algebra and Geometry; hence, in general, we take the nearest root, the largest figure of which the square is less than the given number, and we point off the figures in periods of two each, beginning at the unit's place. The reason of pointing off in periods is shown by the increase of the numbers in squares ; thus, 11 11 121 9 _9 81 99 99 9801 The increase of one figure in the side makes two in the sur- face ; it will always be this, and never more or less, as is shown by taking the smallest and the largest digits. 58 ELEMENTS OF GEOMETRY. PROBLEM XII. To find the circumference of a circle whose radius is unity. With 1 as a radius describe a circumfer- ence, and inscribe in it a regular hexagon, each side of which will be unity. Take any side, as AB, and bisect it in D and its arc in C, and draw the chord CB, which will be the side of a regular polygon of double the number of sides. The triangle ODB is right- angled at D ; hence, OD = Vo B 2 - 61? = Vl^i == l-y/3, and CB = Vj + (l and CD = 1— |V3, Let C represent a side of the first polygon, and iV3)2. c a side of the polygon of double the number of sides ; in each successive com- putation, after c is found, make it C in the next, and continue this process until the difference between C and c has no appre- ciable value ; then this value of C multiplied by the number of sides will give 6.2832, which is the approximate length of the circumference when the radius is 1 and the diameter 2. When the diameter is 1, the circumference is 3.1416, which number is represented by tt ; hence, nd or 2nr represents circum- ference. PROBLEM XIII. The area of a regular -polygon is equal to its perimeter multiplied by one-half the radius of the inscribed circle. Let ABCDEF be a regular inscribed hexagonal polygon, and OK the radius of the inscribed circle. The polygon is com- posed of six triangles, each having a side of the polygon for its base and OK for its altitude ; hence the area of the polygon is its perimeter multiplied by £OK; that is, \ the radius of the inscribed circle. Coe. — When the number of sides of the polygon is indefinitely increased, it becomes a circle, and the radius of the inscribed circle, which has been increasing as the number of sides increased, is now the radius of the circle, and the perimeter of the polygon is the circumference of the circle ; hence, the area of a circle is equal to its circumference multiplied by one-half its radius. BOOK V. THEOEEM I. When the distance between the centers of two circles is greater than the sum of their radii, they are external, and the straight line joining their centers will be the shortest distance between the center of either circle and the circum- ference of the other ; and if this line be extended to bite concave arcs of both circles, it will be greater than between any two other points in the circumference. Let C and C be the centers of two circles external to each other; C B is the shortest distance from C to any point in the circumference C ; for, let the tangents DE and D'E' be drawn at B and A', they will be perpendicular to C'C ; and as a perpendic- ular is the shortest distance from a point to a line, C'B is the shortest distance from C to the tangent DE, and any other line from C to the circumference is oblique to the line DE, and must go beyond it before it can reach the circumference. 2. C'A is longer than any other line drawn from C to the circumference C, as CF. Draw the chord BF; BF is less than AB a diameter; hence, C'B -r BF < C'A, and CT CF. 60 ELEMENTS OF GEOMETRY. THEOEEM II. When the distance between the centers is equal to the sum of the radii, they are tangents externally , and the straight line joining their centers passes through the point of tangency. They must touch on the line joining their centers, as CD + DC = CC; let D be this point, and through D draw AB per- pendicular to CC; it will be a common tangent to both circles as it is perpendic- ular to each radius at its extremity. Or, CC is the shortest distance between the centers, and CD is the shortest distance from C to AB ; CD is also the shortest distance from C to AB ; therefore, the line between the centers of the two circles passes through their point of tangency. THEOEEM III. When the distance between the centers is less than the sum and greater than the difference of the radii, they intersect each other, and the line joining their centers is perpendicular to their common chord at its middle point. CC is the shortest distance between C and C; hence, CD and CD must both be perpendicular to A B at its middle point, and CC must be a straight line. THEOEEM IV. When the distance between the centers is equal to the difference of the radii, the smaller circle is tangent internally to the larger one, and the line joinijig their centers extended passes through the point of ta:igency. C'A is the shortest distance from the center C to the circumference C ; therefore, A is the point of tangency. BOOK V 61 THEOEEM V. When the distance between the centers is less than the difference of the radii, the smaller is wholly within the larger, and the nearest and the fartJiest points in the circumference of the one circle, from the center of the other circle, is in the extensions of the line joining their centers. C'B is the nearest distance and C'A is the farthest in the circumference C from the center C (Th. 1.) Cor. — When they are concentric, their circumferences are parallel. Gen. Cor. — The line joining the centers passes through the points of taiigency, the middle point of the common chord, the nearest and the farthest point of the circumference. THEOREM VI If on one side of a given polygon another polygon be constructed within the given polygon, the perimeter of the interior polygon will be less than that of the given polygon. Produce each side of the interior poly- gon until it meets a side of the exterior. Then ke is less than A Be, If < beCf, eg < cfDg, and dE < dgE. .-. AeCDE AD ; hence the angle ASC > ASD ; therefore the sum of the two plane angles BSC and ASC is greater than the third angle ASB. 4 n ELEMENTS OF GEOMETRY THEOREM II. The sum of all the angles which form a polyedral angle is less than four right angles. Let S be the vertex of a polyedral angle, and pass a plane cutting the planes and forming the polygon ABCDE. From any point as within the poly- gon, draw lines to the extremities of all its sides ; there will be as many triangles as faces forming the polyedral angle. At each vertex of the polygon there is a triedral angle, formed by one plane angle of the polygon and two in the faces of the polyedral angle S; the one angle in the polygon is less than the sum of the two others (Th. 1); but as the number of the tri- angles in the. polygon is the same as of the plane angles forming the polyedral angle S, hence the sum of all the angles of the tri- angles in the polygon is equal to the sum of all the angles of the triangles forming the polyedral angle S. And as the sums of all the angles of the polygon are less than the sums of all the angles on the faces at the points A, B, C, etc., at which triedral angles are formed, on the faces there are two angles at each vertex, whilst in the polygon there is but one, the remaining angles of the triangles of the polygon must be greater than the sum of the angles forming the polyedral angle at S. The sum of all the angles at is four right angles; hence the sum of all the plane angles forming the polyedral angle S is less than four right angles. BOOK* VIII. 75 THEOREM III. The surface generated by the revolution of a regular semi-polygon about the diameter of the circumscribed circle as an axis, is equal to the circumference of the inscribed circle multiplied by this axis. Let ABCD be one-half of a regular polygon, which being revolved about AD the diameter of the circumscribed circle, as an axis, the sur- face generated by the perimeter is 2ixO«x AD. The triangles AB& and CDc will generate equal cones, and the rectangle BbCc will gen- erate a cylinder. The surface generated by AB = 2/txex AB, " BC = 2n x nO x BC, " " " " CD = 2^x«'m'xCD. The triangles Oam and AB# are similar; consequently, AB :Oa :: 1Kb : am\ .'. ABxam = Oaxfcb\ hence also, a'm x CC = Oa' x cD. Surface generated by AB = 2n x Oa x kb, " " « BC = 2nxOnxbc, " CD = 2n X Oa'xcD. Oa — On = Oa', each equal to the radius of the inscribed circle. By addition, Whole surface == 2 A B. Join the vertices. A, B, and C, with 0, the center of the sphere, and a triedral angle is formed, the arcs of whose facial angles are the sides of the spherical triangle; but (Th. 1) "the sum of any two of the plane angles which form a triedral 80 ELEMENTS OF GEOMETRY angle is greater than a third ; hence, the sum of any two sides of a spherical triangle is greater than the third side. THEOREM X. TJie sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be a spherical triangle. Pro- duce AB and AC until they meet in D. The arcs ABD and ACD are semi-circum- ferences, since two great circles always bisect each other. In the triangle BCD, the sum of the two sides CD and BD is greater than the third side BC ; hence the sum of the three sides AB, AC, and BC is less than the circumference of a great circle. Rem. — The sides of a spherical triangle are arcs which cor- respond with and measure the facial angles of a triedral angle, the vertex of which is at the center of the sphere. THEOREM XI. The sum of all the sides of a spherical polygon is less than the circumference of a great circle. Let ABCDE be a spherical polygon. Produce AB and DC until they meet in G ; also produce AE and CD until they meet in F. BC is less than BG + CG, and DE is less than EF-f-FD; hence the sum of the sides of the polygon is less than the sum of the sides of the triangle AFG ; and the sum of the sides of the triangle is less than the circumference of a great circle (Th. 10) ; much more then is the sum of all the sides of the polygon less than the circumference of a great circle. BOOK VIII 81 THEOKEM XII. If from the vertices of the angles of a spherical tri- angle as poles, with a distance between the points of the dividers eqaal to a quadrant, arcs be drawn forming another spherical triangle, the vertices of this triangle will be respectively the poles of the sides of the first triangle. Let ABC be a spherical triangle, and then with each vertex as a center and a distance between the points of the dividers equal to ninety degrees, describe the polar triangle DEF. First with A as a pole describe the arc EF, with B as a pole describe DF, and with C as a pole describe DE; in each case the distance between the points of the dividers being 90 degrees. Since AE =: 90° and CE = 90°, E is the pole of AC ; and since BD = 90° and DC = 90°, D is the pole of BC ; and as BF = 90° and AF = 90°, F is the pole of AB. THEOREM XIII. Any angle in one of two polar triangles is measured by a semi- circumference minus the side opposite of the other triangle. Let ABC and DEF be triangles polar to each other ; produce the sides of ABC until they meet those of DEF; A is the pole of the arc GH by which the angle A is meas- ured, E is the pole of KH, and F is the pole of LG; hence, EH = 90° and FG == 90°; hence, GH = 180°— EF; that is, the angle A is measured by a semi-circumference minus the side opposite in its polar triangle ; so also in regard to each of the other angles. Cor. to 10th and 13th Th. — The sum of the three angles of a spherical triangle is less than six right angles and greater than two. 82 ELEMENTS OF GEOMETRY. Cor. 2. — A spherical triangle ma) 7 have two or even three right angles, or as many obtuse angles ; when it has three right angles it is called the trirectangular triangle, whose surface is equal to one-eighth the surface of the sphere. Cor. 3. — The sum of the three angles of a spherical triangle is not a constant quantity, but varies between two and six right angles, never reaching either of these limits. Rem. — The excess of the sum of the angles of a spherical triangle over two right angles is called the spherical excess. THEOREM XIV. A lune is to the surface of a sphere as the are lohich measures its angle is to the circumference of a great circle. Surface, of lune : 8 trirectangular tri. : : angle A : 4 ; .-. 4L = Ax8T, and L = A x 2T. Cor. — The surface of a lune is equal to its angle multiplied by twice the trirectangular triangle. Rem. — As the right angle is the unit, the angle will be indicated by a fraction ; as fjj-, which is 43 degrees. THEOREM XV. The area of a, spherical triangle is equal to its spher- ical excess multiplied by the trirectangular triangle. Let ABC be the spherical triangle. The triangles ADE and AFG form a lune ?= angle A x 2T; the triangles BGH and BID form a lune = angle B x 2T; the triangles CFI and CEH form a lune = angle C x 2T. By addition we get 2(A+B + C)T = 2 area of ABC + 4T, and (A + B -f C) T = area ABC + 2T, and (A + B + C) T — 2T = area ABC. Area ABC = (A + B + C - 2) T. BOOK VIII. 83 THEOREM XVI. The area of a spherical polygon is equal to its spher- ical excess multiplied by the trirectangiilar triangle. Joining A and C and A and D by arcs of great circles, we divide the polygon into tri- angles. The sum of all the angles of the triangles is equal to the sum of all the angles of the polygon ; hence, the area of the polygon is equal to the sum of the areas of the triangles. Let S = Sum of all the angles, and n = Number of the sides of the polygon, (n — 2) = " " triangles. Area of polygon = [S — (n — 2) 2] T; Area of ABCDE = (S - %n + 4) T. , APPLICATION OF LOGARITHMS. A Logarithm is the exponent or power of some number, which is called the base of the logarithms. The base of the logarithms in common use is 10 ; hence, 10° = -fg- = 1 ; therefore, the logarithm of 1 is 0. 10 1 = 10; " « * " 10 " 1. 10 2 = 100 ; " " " " 100 " 2. 10 3 = 1000 ; " " " " 1000 " 3. The integral number of the logarithm is called the Char- acteristic, and when positive is one less than the number of integral figures in the number of which it is the logarithm. The logarithm of any number betwe en 1 and 10 mu st be between and 1 ; the logarithm of any number between 10 and 100 must be between 1 and 2 ; that is, the logarithm of any number hptwpp.n 1 and 10 is a fracti on, and the logarithm of any number between 10 and 100 is 1 -f- a fraction. The calculations of the fractions is made by an algebraic process, and logarithmic tables are formed to facilitate trigono- metrical computations. Observe also that .1 = — = 10 _1 ; that is, the logarithm of .1 is — 1 ; and .01 =^ = 10" 2 ; " " " « ".01 "-2. .001 = ~ = 10~ 3 ; " " « " " .001 " — 3. Rem. — The characteristic of the logarithm of a fraction is negative and corresponds to the number of decimals ; hence, the logarithm of any number between 1 and .1 is between and — 1, and is put — 1 -f- a fraction. As logarithms are exponents, they can only be used as such ; that is, for Multiplication and Division, for Involution and Evolution; thus, a 1 x a 1 = a 2 ; a 2 x « 3 = a 5 ; a 1 x b 2 = ab 2 \ a 2 x b 2 = a % & ; a 1 x a 1 x a 1 = « 8 ; a s -^ a 2 = a 1 ; V« 2 X Vfl — Va? = fl*5 VaM = ab^' y V«W = (!) and R p x R sin~P ' and sine P = — r— h = cosine B, and from 2 and 2, p x R _ l_x_ R sin P (2) (3) (3) and PD: PH :: PC: PB; that is, sine B : b : : R : h ; sine Bx/t b - and t. b x R a = — - — 5" sm B (1) (2) sine B = 6xR = cosine P, (3) and jB _ b_ ; sin B ; (4) (4) sin B ' sin P .-. sin P : sin B :: p : b. Alternating 1st and 2d proportions, sin P : R :: p : h, and sin B : R :: b : h. The triangles PFE and PH B are similar ; hence the proportion, EF : BH :: PF : PH ; that is, tang P : p :: R : b; hence, p = — L , and b = -^^ R — -> b y analogy. (5) Tang P = ~^ = cot B, and tang B = -— = cot P, an., (6) and Rxb tang B' and b == Rxp tang P (5) 90 PLANE TRIGONOMETKY. Eem. — In a right-angled triangle, either acute angle is a com- plement of the other ; hence, the sine of the one is the cosine of the other, and the tangent of the one is the cotangent of the other. As these formulas are general principles of right-angled triangles ; they must be fixed indelibly on the mind of the student. Rem. — The small letters represent the sides opposite the angles having corresponding large letters. FORMTJlAS. 1. Either side is equal to the sine of the opposite angle mul- tiplied by the hypothenuse and divided by the radius. 2. The hypothenuse is equal to the radius multiplied by either side, and divided by the sine of the angle opposite this side. 3. The sine of either acute angle is equal to the opposite side multiplied by the radius and divided by the hypothenuse. 4. Either side, including the hypothenuse, is to any other side as the sine of the angle opposite the former side is to the sine of the angle opposite the latter side. 5. Either side is equal to the tangent of its opposite angle multiplied by the other side and divided by the radius ; or, Either side is equal to the radius multiplied by the other side and divided by the tangent of the angle opposite this other side. 6. The tangent of either acute angle is equal to its opposite side, multiplied by the radius and divided by the other side. EXAMPLES. 1. Given k = 43, B = 25°, to find;?, b, and P. p = 90° — 25° = 65°. sin P ^ If Formula 1, p = ~ ; .*. log j p=logsin65° + log43— log R, K log siu of 65° =s 9.957276 log 43 — 10 = 1.63346 .8 p = 38.971, No. corres. 1.590744 Formula 1, b -as „ — ; .*. log. b = log sin 25°+ log 43— log R, log sin 25° = 9.625948 log 43-10 = 1.633468 b = 18.172, No. corres. to log, 1.259416 2. Given h = 624, P = 48°, to find;?, b, and B. B = 42°, p ±= 463.723, and b = 417.538, Ana. PLANE TRIGONOMETfiY. 91 3. Given b = 535, B = 65° 15', to find P,p, and k. P = 24° 45', p ss 246.738, and h = 589.114, Ans. 4. Given b = 47, P = 35°, to find -B, p, and A. B == 55°, p = 32.91, and h = 57.376, <4ns. 5; Given./? = 275, B = 58°, to find P, b, and li. P b= 32°, b s= 440,92, and A = 518.946, ^«* 6. Given p = 15, J = 25, to find P, B, and h. (Use for- mula 6.) 7* = 29.05, B = 59° 23' 3 ", and P = 30° 36' 57", Ans. PROBLEM II. To show that the sines of the angles, in any plane triangle, are respeetively proportional to their opposite sides. Let ABC be any plane triangle, and • from any vertex as B draw a perpendic- ular BD to the opposite side AC, making two right-angled triangles, ABD and BCD; from which we have in the tri- angle ABD, BD triangle BCD, BD = sin A x c , . ,, g, — , and in the smCxff R . sin A x c = sin C x a ; hence, sin A : sin C :: a : e, and by inversion, sin C : sin A :: c \ a, and sin B : sin A : : b : a, and sin A : sin B : : a : b; hence, in any triangle, the sines of the angles are respectively proportional to their opposite sides. 1. Given A EXAM PLES. 55°, B = 51°, c = 143, to find C, a, and b. C ss 74°, a = 121.86, and b = 151.61, Ans. 2. Given A = 60, c = 54, and A == 26°, to find B, C, and b. C = 23° 10' 13", B = 130° 45' 47", and b = 103.667, Ans. Rem. — If c is greater than a there is two triangles; thus, c. = 60, a = 54, and A == 26°. Observe, that the angle AC'B is the supplement of ACB. C = 29° 8' 56", C = 150° 51' 4", B = 124° 51' 4", B' *= 3° .8' 56", b = 101.089, and b' = 6.7665, Ans. 92 PLANE TRIGONOMETRY / PROBLEM III. To show by Diagram No. 2 certain relations of the functions of angles, when the radius is unity. In the triangle CEB, BE 2 + CE 2 = CB 2 ; that is, sin 2 C + cos 2 C = R 2 = 1, sin 2 C = 1 — cos 2 C, and cos 2 C = 1 — sin 2 C. 0) (2) (3) The triangles CEB and CAT are similar, and also CFB and CDT' are similar. .-. CE : CA :: EB : AT, cos C : 1 : : sin C : tang C, cos C x tang C = sin C, (4) , ^ sin C /rN tan e c = ^ s -c ; (5 > tang C x cot C .% CF : CD :: FB : DT', sin C : 1 : : cos C : cot C, sin C x cot C = cos C, cos C sin C* cot C (6) (?) sin C cos C .. cos C sin C tan C = cot C (8) (9) cot C 1 tang C Rem. — In the above diagram, C represents any angle ; hence the relations apply to all angles. SYNOPSIS OF ^HE FORMULAS, cin2 I cosH= 1. (i) sin 2 =1 — cos 2 . ' m cos 2 = 1 — sin 2 . (3) sin = cos x tang. (4) sin tang = ° cos (5) cos == sin x cot. (6) cos cot = - — sm (?) *"•* = m (8) COt = • tang (9) PLANE TRIGONOMETRY 93 PROBLEM IV. To show the relations of the functions of a right- angled triangle with the sides, when the radius is unity. The triangles PDC and PHB are similar, and give the proportions, CD : BH and PD : PH sin P : p sin B : b PC : PB, PC : PB. 1 : A, 1 : lu p = sin P x h, sin P sin P = ?, h V b — sin B x h, (1) _b_ sin B' b h = sin B _ h (3) (3) From (2), — -£-= = - — =, .-. sin P : sin B :: p : b. v ' sin P sin B The triangles PFE and PHB are similar; BH : EF :: PH : PF, p : tang P : : b : 1 ; hence, and by a similar process, p = tang P x b, b = tangP' and b = tang B x p, b P ~ tangB' tang P = | = cot B, tang B = - = cot P. P (4) (5) 94 PLANE TRIGONOMETRY. ENUNCIATION OF FORMULAS. 1. Either side is equal to the product of the sine of the oppo- site angle and the hypothenuse. 2. The hypothenuse is equal to either side, divided by the sine of the angle opposite that side. 3. The sine of either acute angle is equal to its opposite side divided by the hypothenuse. 4. Either side is equal to the product of the tangent of its opposite angle and the other side ; or, either side is equal to the other side divided by the tangent of the angle opposite that other side. 5. The tangent of either angle is equal to its opposite side divided by the other side. Rem. — In computing these formulas by logarithms, when the formula consists of the product of two numbers, 10 must be subtracted from the sum of the logarithms ; and when it is fractional, 10 must be added to the differ- ence of the logarithms. PROBLEM V. To find the sine and cosine of the sum and difference of two arcs, whose sines and cosines are known. Let ACB and BCD be the two angles whose sines and cosines are known. Let the angle ACB be designated angle A. B. « " BCD i( a Make BCE = BCD; then DK = sin (A + B), and EM = sin (A — B). CK = -- cos (A + B), CM = : COS (A-B). and The triangles DFG and GHE are equal and similar to CLG, whose sides are respectively perpendicular to those of DFG and GHE. the angle FDG = angle LCG = angle A. PLANE TRIGONOMETRY. 95 In the triangle DFG, DF = DG x cos A, and in the triangle CLG, GL = CG x sin A. DF = sin B x cos A, and GL = cos B x sin A, GL + DF= DK = sin (A+B) = sin Ax cos B 4- cos A xsin B ) & G L — DF = EM = sin (A— B) = sin A x cos B — cos A x sin B \ $ By addition, sin (A + B) 4- sin (A — B) = 2 sin A x cos B, and sin (A+B) — sin (A — B) ±= 2 cos A x sin B. Put A + B = M, and A - B = N ; then A = £(M + N), and B = £(M-N); then sin M -f- sin N = 2 sin|(M + N) x cos -J-(M — N), (1) and sin M — sin N =2 cos£(M + N) x sin£(M — N); (2) dividing (1) by (2), and reducing by tang = — , cos sinM + sinN sin£(M + N) xcos£(M — N)' tang|(M + N) sin M — sin N — cos J(M -i-N) xsin £(M — N) — tangi(M — N) (3) In the triangle CLG, CL =s CG x cos A = cos B x cos A. " " DFG, FG = DG x sin A = sin B x sin A. By subtraction and addition, CL — FG = CK = cos M = cos A x cos B — sin A x sin B ) S> CL + FG = CM — cos N = cos A x cos B -f- sin A x sin B \ 1 Cor. — By addition and subtraction, cos M + cos N = 2 cos -J- (M + N) x cos i (M — N), (4) cos M — cos N = — 2 sin |(M + N) x sin £(M — N) ; (5) dividing (5) by (4), cos M — cos N — sin \ (M -f N) sin J(M — N) cos M + cos N ~ cos \ (M + N) cos J (M — N) = - tang i (M + N) x tang \ (M - N). Observe that the tangent has a negative sine, which is correct, as the numerator of the first member of the equation is negative, the cosine N being greater than the cos M ; a small angle has a larger cosine than a large angle. PLAXE TRIGONOMETRY. EXAM PLES. 1. To find the sine and cosine of 30°, 60°, and 45°, when the radius is 1. The sine of 30° is half the chord of 60°, the chord of 60° is radius = 1 ; hence, sine of 30° = £, and cosine of 60° = £. t sin 30° = cos 60° = J. Cosine of 30° = VT^J = V% = W$'> sme of 60° = .f\/3; and cos 30° = sin G0° = |\/3. When the angle is 45°, the sine and cosine will be equal, and as sin 2 -f cos 2 = 1, that is, f-'-K-J == 1. sin 2 45° = i and sin 45° = Vi = W% = cos 45°, and sin 45° = cos 45° = Ja/2. 2. To find the sine, cosine, tangent, and cotangent of every arc from 1° to 90°. The semi -circumference, when the radius is 1, is 3.1415926535 ; which being divided by 10800, the number of minutes it contains, gives the length of 1', equal to .0002908882, which in so small an arc does not differ materially from the sine of 1', and may be regarded as such; and cos 1' = Vl~ sin*!' = .9999999577. By taking the formula from Prob. 5, sin (A -f B) -+- sin (A — B) = 2 sin A x cos B, by transposition, sin (A -h B) = 2 sin A x cos B — sin (A — B), and making B = 1', and A = 1', 2', 3', 4', etc., in succession, sin 1' = .0002908882; sin 2' = 2 sin V x cos 1' - sin = .0005817764; sin 3' = 2 sin 2' x cos 1' — sin 1' = .0008726646. PLANE TRIGONOMETRY. 97 By continuing this process, we can get the sines of every arc from 1' to 90° ; and taking them in an inverse order, we have the cosines of every arc from 1' to 90° ; then, as tang = — , we can get the tangents of every arc from 1' to 90° ; and by taking them in an inverse order we have the cotangents of every arc from 1' to 90°. These will form a table of natural sines, cosines, tangents and cotangents. The logarithms of these numbers, with the addition of 10 to to each logarithm, forms the table of logarithmic sines, cosines, etc. In the table the radius is taken as ten billions, whose log- arithm is 10 ; and as the functions are proportional to the radii, hence the natural sines, etc., must be multiplied by this number, which is done by adding the logarithms of the natural sine, etc., and of the radius. PROBLEM VI. Two sides and the included angle of a triangle given, to find the other angles. By Problem 2, a : b : : sin A : sin B ; by composition and division, a + b : a — b : : sin A + sin B : sin A — sin B, a + b sin A + sin B a — b ~~ sin A — sin B By Problem 5, Formula (3), sin M -f sin N _ tang £ (M -f N) # sin M — sin N ~ tang i (M — N) ' a + b _ sin A + sin B _ tang \ (A -f B) # a — b ~~ sin A — sin B ~~ tang £ (A — B) ' hence the proportion, a + b : a — b :: tang J(A + B) : tang £ (A — B )- Knowing the sum and difference of two angles, we easily find the angles. 5 98 PLANE TRIGONOMETRY. PROBLEM VII. To find the area of a triangle, having given two sides and the included angle. The angle A and the sides b and c given. area ABC = \pc. In the triangle ADC, p = sin A x o ; area of ABC = |5 x c x sin A. PROBLEM VIII 7/* from the vertex of any angle of^a triangle a line be drawn perpendicular to the opposite side, produced if necessary, then will the sum of the segments of the opposite side he to the sum of the other two sides as the difference of those sides is to the difference of the seg- ments. p 2 = a 2 — n 2 = c 2 — m 2 , m 2 — n 2 =r c 2 — a 2 , (m + n) {m — n) = (c + a) (c — a), m -f- n : c -\- a :: c — a : m— n. PROBLEM IX. If from the half sum of the three sides of a triangle, each side be subtracted separately, then the square root of tlie continued product of the half sum and the tliree remainders will be the area of the triangle. *- c 2 - c 2 f nv a 2 — n 2 , and a< =z m 2 — V) 2 m 2 — n 2 = c 2 — a 2 , (m + n) (m — ri) = c 2 — a 2 ; PLANE TRIGONOMETRY. 99 and as m + n = b, b (m — n) = c 2 — a 2 , & — a 2 m — n = — r — , m + n = b, 2m = b + & m = b ' b 2 + & + a 2 A 2 2 2 . (P + P — tfY and p 2 = c 2 — m 2 = c 2 — I -^ ) , 46V- (y + c 2 -a 2 ) 2 4J 2 + $2 + g2 _ ^2) [2&c — (b 2 + c 2 — a 2 )] ib 2 " [(F+ "^-fl 2 ] x [fl 2 -"^^) 2 ] 4£ 2 /(b+c + a) (b + c—a)(a + b—c)(a + c—b) = Y 4^ area of ABC = \pb /(b + c + ajJb-\-c^~a)(a + b—c) {a + c—b) [& = y w x V 4 ; b 2 is canceled, and the two 4's factored, area of ABC - - /(P+c+dH b+c-<^ ± b-41a+c^ L = , /& + <> + «) 2 2 2 EXAMPLES. Two sides and the included angle given. 1. Given a = 75, 5 = 90, and C == 20°, to find A, B, and e. b + a : b — a : : tang. J (B -f A) : tang. £ (B — A) ; 165 : 15 : : tang. 80° : tang. B — A. tang, t (B — A) = log 15 + log tang. 80° — log 165, ■L (B + A) = 80° 00' 00", j(B — A) = 27 16 27. B = 107° 16' 27", A = 52° 43' 33". 100 PLANE TRIGONOMETRY. Sin A : sin C : : a : c, log c = log sin C + log a — log sin A. Arith. comp. 165 = 7.782516 log 15 = 1.176091 log tang. 80° = 10.7536 81 tang. \ (B-A) = 27° 16' 27" = 9.712288 Arith. comp. sin 52° 43' 33" = 0.099225 log sin C 20° 9.534052 log a 75 1.87506 1 c = 32.235 1.508338 When the three sides are given. 2. Given a = 237, I = 495, and C = 327. m -f n : c + a :: c — a : m—n; m 4- n = h, log {m — n) = log (c + a) +log (c — a) — log I. m + n = 495 m — n = 102.546 m =298^773, and n = 196.227. In the triangle ABD, c : m :: R : sin ABD ; log sin ABD = log R + log m — log c. Arith. comp. 495 log 564 log 90 (m - n) = 102.546 Arith. comp. 327 log 298.773 logR sin ABD = cos A 23° 59' = 7.305395 = 2.751279 = 1.954243 2.010917 7.485452 2.475342 10.0000 00 9.960794 PLANE TRIGONOMETRY. 101 In the triangle BDC, a : n :: R : cos C; log cos C = log n + log R — log a. A = 23° 59' 00" C = 34° 6' 36" 180 — 58° 5' 36" = 121° 54' 24" = B. Arith. comp. 237 = 7.625252 log 196.227 = 2.292759 log R 10.000 000 cos C 34° 6' 36" 9.918011 PKOBLEM. A side and two adjacent angles given, also two sides and an angle opposite one of them. EXAM PLES. 1. Given A = 32°, a = 40, and b = 50, to find B, C, and c. ( B = 41° 28' 59", C = 106° 31' 1", and c = 72,368. Ans. | B _ 138 o 31 , j-,r c = 9 o 2g , g9 „ „ c _ 12#436# In this case there are two triangles. 2. Given a = 450, b = 540, and C == 80°, to find A, B, and c. A = 43° 49', B = 56° 11', and c = 640.08. 3. Given a = 40, b = 34, and c = 25 yards, to find the angles. A = 83° 53' 16", B = 57° 41' 24", and C = 38° 25' 20". 4. Given b = 306, c = 274, and B = 78° 13', to find A, C, and a. A = 40° 33', C s= 61° 14', and a = 203.2. 5. Given B = 100°, a - 280.3, and c = 304, to find A, C, and b. A = 38° 3' 3", C = 41° 56' 57", and b = 447.856. 6. Find the area of a triangle having two sides equal to 30 and 40 ft. respectively, and the included angle 28° 57', Ans. 290.427 sq. ft. 7. Find the area of a triangle whose sides are respectively 30, 40, and 50 rods. Ans. 3 acres 3 rods. 8. What is the area of a triangle, whose base is 50 rods and altitude 30 rods ? Ans. 4 acres 2 rods 30 perches. 102 PLANE TEIGONOMETEY, PRACTICAL PROBLEMS. 1. Find the distance AC across a deep river, having given AB = 500 yards, the angle BAC = 74° 14', and the angle ABC = 49° 23'. sin C : sin B : : c : h ; log b = log sin B + log c — log sin C = 577.8 yards. 2. Given AC = 735 yards, BC = 840, and the angle C = 55° 40', to find AB. Two sides and the included angle. AB == 741. 3. Given AB = 600 yards, and the adja- cent angles A = 57° 35' and B = 64° 51', to find the angle C and the sides AC and BC. AC = 643.49 yards. BC = 600.11 " 4. Find the height of D a point on a mountain above a horizontal plane. The angle of elevation at B, a point at the foot of the mountain, is 27° 29' ; and at A distant from B 975 yards, in a direct line from B, and in the plane DBA, is 15° 36'. DC = 587.61 yards. 5. Wishing to know the distance between two inaccessible objects C and D, I measured a line, AB = 339 feet, from both ends of which the objects were visible ; I found the angles BAD = 100°, BAC = 36° 30', ABC = 121°, and ABD = 49°; find the distance DC. DC = 697J feet. ---^ PLAtfE TRIGONOMETRY. ioa 6. Wishing to know the dis- tance between two inaccessible objects, A and B, and finding no place from which both could be seen, two points C and D, 200 yards distant, were found; from the former point A could be seen, and from the latter B ; from C, a distance of 200 yards were measured to a point F, from which A could be seen ; and from D the same distance was measured to E, from which B could be seen, and the following angles taken, viz., ACD = 53° 20', ACF = 54° 31', AFC = 83° 00', Find the distance AB. BDC = 156° 25', BDE = 54° 30', BED = 88° 30'. Ans. AB — 345.467 yards. 7. The distance between three points A, B, and C, are known, viz., and AB AC BC 800 yds., 600 " 400 " All are visible from a distant point P at which the angles are measured, A PC = 33° 45', Find AP, BP, and CP and BPC = 22° 30'. Ans. AP = 710.193 yds., BP = 934.291 " CP = 1042.522 " SPHERICAL TRIGONOMETRY. Spherical Trigonometry treats of spherical triangles, the sides of which are arcs of great circles, each less than 180°, and the angles are diedral angles, formed by the planes of the great circles ; each angle is less than two right angles. Napier's Five Circular Parts form the basis for the analysis of the functions of right-angled spherical triangles. The two sides about the right angle, and the complements of the hypothenuse and of the two oblique angles are the five circular parts. The spherical triangle ABC is right-angled at A. The sides b and c, and the complements of the hypothenuse a and of the angles B and C are the five circular part3. In taking any three of these parts, they will either be found to be adjacent to each other, or one of them will be separated from both the others. When they are adjacent, the one lying between the others is called the middle part, and when they are not adjacent, the one separated from both the others is the middle part and the others are opposite. Let ABC be a spherical tri- angle, right-angled at A, the center of the sphere. Draw CD perpendicular to OA, and DE per- pendicular to OB, and join CE. As the angle A is a right angle, the angle CDE is also a right angle, as CD is perpendicular to the SPHERICAL TRIGONOMETRY. 105 plane ABO in which DE is drawn perpendicular to OB, a line of the plane; hence CE is perpendicular to OB (Th. 3, Bk. 6), and CED = B. 1. sin b = sin a x sin B = cos comp. a x cos comp. B. 2. sin c = sin a x sin C = cos comp. a x cos comp. C. 3. cos B = cos b x sin C = cos b x cos comp. C = sin comp. B. 4. cos C s= cos c x sin B = cos c x cos comp. B = sin comp. C. 5. cos a = cos b x cos c = cos b x cos c = sin comp. a. By Prob. 4, in the triangle CED, CD = CE x sin B, sin b = sin a x sin B. (1) No. 2 is derived in the same way, by making B the vertex instead of C. d DE DE cos B = ^ = - CE sin a DE = cos b x sin a. sin c = sin a x sin C. (2) In the triangle OED, DE = OD x sin DOE, D cos b x sin a x sin C D . - /oX .-. cos B = s = cos B x sin C. (3) sin a v ' In No. 4, cos C is found like cos B, by making B the vertex. In the triangle ODE, OE = ODx cos DOE; that is, cos a — cos b x cos c, (5) From these five formulas, five others may be derived ; thus, ^ . , . . _ sin c x cos C sin c cos C 1. sin b = sin a x sin B = — — ^ == x - — =; sin C x cos c cos c sin C == tang c x cot C = sin b. . . _ sin b x cos B sin b cos B 2. sin c = sin a x sin C = - — =-- r = 7 x -. — 5 sin B x cos J cos 6 sin B = tang b x cot B = sin c. 106 SPHERICAL TRIGONOMETRY. ~ 7 . ~ cos a x sin c cos a sin c 3. cos B = cos b x sm C = — : = - x cos c x sm a sin a cos c = tang c x cot a = sin comp. B. . ^ . _> cos a x sin b sin # cos a 4. cos C =s cos c x sin B = -, r — = T x - — cos b x sin a cos sm a = tang 5 x cot a = sin comp. C. - , cos B x cos C cos B cos C 5. cos a = cos o x cos c = -* — ^ : — = = -s — => x -s — ^ sin C x sm B sm B sm C = cot B x Cot C = sin comp. a. From the first five formulas : The sine of the middle part is equal to the produot of the sines of the opposite parts. From the second : The sine of the middle part equals the product of the tangents of the adjacent parts. Rem. — Observe that the cosine of an angle is equal to the sine of the complement, and the cotangent is equal to the tangent of the complement. THE SPECIES OF THE FUNCTIONS OF ANGLES OR ARCS. As the functions of an arc and of its supplement are lines of equal length, there is a distinction necessary, in order that we may know whether the arc is greater or less than 90° ; hence the minus sign is given to the co"sine, the tangent, and the cotangent when the arc is greater than 90°, or terminates in the second quadrant. Two arcs are said to be of the same species when they are both less or both greater than 90°, and of different species when the one is greater and the other less than 90° 1. From the 3d and 4th formulas of circular parts, . n cos B , . _ cos C sm C = r i and sin B = cos b cos c As the sines of C and B are both positive, hence the cosines of each oblique angle must have the same sign as the cosines of SPHERICAL TRIGONOMETRY. 107 the opposite sides ; consequently, the oblique angles and their opposite sides are of the same species. 2. When the hypothenuse is less than 90°, the other two sides and their opposite angles are of the same species ; for, as cos a = cos b x cos c, and when a is less than 90° its cosine is positive; hence the cosines of b and c have like signs, that is, b and c are of the same species. But when a is greater than 90°, its cosine is negative ; hence the cosines of b and c have different signs ; that is, b and c are of different species. By these two rules the nature of each result is determined, except when an oblique angle and the opposite side are given, to find the other parts. Let ABC be right-angled at A ; and B and b be known. 1st. If the sine of b is greater than the sine B, there can be no solution ; for, as sin a = -: — ~>1, sin B which is impossible. 2d. If sine b = sin B, then sin a = . -» = 1 ; hence, the sin B vertex B is the pole of the opposite side b, and a and c are each 90°. 3d. If sine b is less than sine B, when B is less than 90°, there will be two solutions, as shown in the above figure; as ABC and AB'C both fulfill the conditions. When B is greater than 90° ; then, in order that sin b < sin B, the side b must be greater than the angle B ; when the result will be the same as above, and a and c in the one triangle will be complements of the same letters in the other triangle. 108 SPHEEICAL TRIGONOMETRY. EXAMPLES. 1. Given a = 86° 51', and B = 18° 3' 32", to find b, c, and C. 1. sin b = sin a x sin B, 5. cos c = cos a -f- cos b ; 4. cos C = cos c x sin B. log sin B = 18° 3' 32" = 9.491354 cos c = 86° 41' 14" = 8.761826 C _ 10 = 88° 58' 25" = 8.253180 = cos C. log sin a = 86° 51' = 9.999343 log sin b = 18° 3' 32" = 9^491354 b — 10 = 18° 1' 50" = 9.490697 = sin b. log cos a = 86° 51' = 8.739969 log cos b = 18° 1' 50" = 9.978 143 c + 10 = 86° 41' 14" = 9.761826 = cos c. 2. Given b = 155° 27' 54", and c = 29° 46' 8", to find a, B, and C. Ans. a = 142° 9' 13", B = 137° 24' 21", and C = 54° 1' 16". 3. Given B = 47° 13' 43", and C = 126° 40' 24", to find a, b, and c. Ans. a = 133° 32' 26", b = 32° 8' 56", and c = 144° 27' 3". Eem. — As the formulas are constructed with unity as radius, if logarithms are used, when the formula is a product, 10 must be subtracted, but when a quotient, 10 must be added. A spherical triangle which has one of its sides a quadrant, is called a Quadrantal Triangle, and is readily solved by passing to its polar triangle, which will be right-angled, solving it, and returning to the quadrantal triangle. The supplement of any side of a triangle is equal to the oppo- site angle of the polar triangle, and the supplement of any angle is equal to the opposite side of the polar triangle. The return is effected in the same way as each triangle is polar to the other. SPHERICAL TRIGONOMETRY, 109 PKOBLEM I. To show that the sines of the sides of a spherical trian- gle are respectively proportional to their opposite angles. Let ABC be any oblique-angled triangle. From either vertex, as A, draw an arc of a great circle perpendicu- lar to the opposite side ; then will the triangles ABD and ADCbe right- angled at D, and sin V = sin c x sin B ; and sin V = sin b x sin C. and sin b x sin C = sin c x sin B; sin b : sin c : : sin B : sin C. In like manner, sin a : sin b : : sin A : sin B ; and sin a : sin c : : sin A : sin C. The result is the same when the perpendicular falls on the opposite side produced. In the triangle ABD and in the triangle ACD, sin V = sin c x sin B ; sin V = sin b x sin C. .*. sin b x sin C = sin c x sin B ; and sin b : sin c : : sin B : sin C, etc. 110 SPHERICAL TRIGONOMETRY. PROBLEM II. In an oblique-angled spherical triangle, if from the vertex of either angle an arc be drawn perpendicular to the opposite side, dividing it into two segments, find these segments. Let ABC be any oblique-angled spher- ical triangle. From either vertex, as C, draw CD an arc of a great circle perpendicular to the opposite side ; then, from 5th formula of Napier, (s -f- 8* =; c). In the triangle ACD and in the triangle BCD, cos b cos a = cos p x cos cos b cos p x cos s' and cos« cos a cos p x cos s cos b : : cos s cos p x cos s ; _ cos s' ~ cos s ' cos s' : and by composition and division, cos a — cos b : cos a -f cos b : : cos a — cos b cos s — cos s' : cos s -f cos s'. cos s — cos s' COS -|- cos s' * cos a + cos b and from Prob 5, Plane Trig., cos M — cos N sin -| (M -j- N) x sin -J- (M — N) cos M -f- cos N cos a — cos b cos a -f- cos b cos i (M + N) x cos J (M — N) = — tang. J(M + N)x tang. J (M - N). = — tang. I (a + b) x tang. £ (a — b) ; cos 5 — cos s -, = — tang. £ (s + s 1 ) x tang. J (5 — s'). and cos s -+- cos s And tang. £ (5 + «') x tang. £ (s—s') — tang. £ (« + b) x tang. J (a— b) ; .*. tang £ (s -f- 5') : tang. £ {a -\- b) :: tang. i(a— b) : tang. £ (s— »'). KBIT. — 5 and *' being determined, in each right-angled triangle are known two sides and an angle opposite one of them. SPHERICAL TEIGONOMETEY, 111 PROBLEM III. When two sides and the included angle are given, to find the other parts. Let ABC be an oblique spherical tri- angle, a, c, and B given. From A draw AD an arc of a great circle perpendicular to the opposite side BC, and in the triangle ABD, and and sin^ = sin c x sin B; . _ APk cos B sin BAD = : aD sin BD = sin c x sin BAD. DC = a — BD; cos o = cos p x cos DC ; and sin C == sin CAD sin p m sin b 9 sin DC sin b angle A s= angle BAD + angle CAD ; hence, b, A, and C are determined. PROBLEM IY. When a side and the two adjacent angles are given. Let B, C, and a be given ; then in the polar triangle, and I = 180 — B, c — 180 — c, A = 180 — «; that is, two sides and the included angle known. Solve the polar triangle by Problem 3, and return to the original triangle 112 SPHERICAL TRIGONOMETRY. PKOBLEM V. Wlien two sides and an angle opposite one of them is given. Let B, b, and c be given, and from the angle A, opposite the unknown side a, draw an arc of a great circle perpendicular to it. In the triangle ABD, sin jp = sin c x sin B, . DAr . cos B sin BAD = , cosp and sin BD = sin c x sin BAD. In the triangle ADC, and cos DC = sin CAD = cos b cos p' sin DC sin b 9 angle A = angle BAD + angle CAD ; hence all the parts are determined. Eem. — When the three sides are given, the angles are found by this problem, after having found the segments of one side by Problem 2. PEOBLEM VI. When two angles and a side opposite one of them is given. Let B, C, and c be given; then, in the polar triangle, b = 180 - B, c = 180 - C, and C = 180 — c; the same as in Problem 5, and must be solved accordingly, and then return to the original triangle. SPHERICAL TRIGONOMETRY. 113 PROBLEM VII. To find the area of a spherical polygon. When the angles are not given, find them by the foregoing problems ; then from Geometry, Book 8, The area of a spherical triangle is equal to the product of its spherical excess and the trirectangular triangle, and the same for any polygon, EXAMPLES. 1. What is the area of a spherical triangle on the surface of a sphere whose diameter is 20 feet ; the angles of the triangle arc A = 130° ; B = 110° ; and C = 165°. 130 . ... , , ,. . Wx 3.1416 i surface of trirectangular triangle = 5 110 o 165 = 157.08 405 157.08 x| = 392.7 sq. ft., Ans. 180 -— =s f, spherical excess. 2. What is the area of a spherical polygon of five sides on a sphere whose diameter is 40 feet, and the sum of the angles of the polygon is 660°. 40 2 x 3.1416 4 g x g = 40 x 5 x 1.0472 = 209.44 sq. ft., Ans. 3. Find the area of a spherical polygon of eight sides, on a sphere 30 feet in diameter, and each angle of the polygon being 150 degrees. 302x3.1416 4 _. ln 1KfV _ Q g — X o = 30 x 10 x 1.5708 = 471.24 sq. ft., Ans. in SPHERICAL TRIGONOMETRY. PROBLEM VIII. To find the shortest distance, on the surface of the earth, between two places whose latitudes and longitudes are known. Rem.— The shortest distance between two points on the sur- face of the earth is measured on the arc of a great circle joining the points. EXAMPLES. 1. The latitude of New York City is 40° 48' ; its longitude 3° east; the latitude of San Francisco is 37° 45' north, and its longitude 45° 40' west. What is the distance between them ? The radius of the earth is 39G2 miles, making 69.15 miles to a degree. Ans. 37° 18' 46" == 2580.18 miles. Let this figure represent a hemi- sphere; NS a meridian passing through Washington ; EQ, equator. The point C represents New York, and B' San Francisco ; the point B is at the North Pole; BC and BB' are the colatitudes of New York and San Francisco, and the angle B the difference of longitude of C and B'. From C draw CA perpendicular to BB' ; then in the triangle BB'C, angle B = 48° 40', the side a = 49° 12', and BB' == 52° 15; and as sin b = sin a x sin B = 34° 38' 23" and cos c 37° 25' 14" cos a _ cos b c =& 52° 15' — 37° 25' 14" = 14° 49' 46" ; and cos a' — cos b x cos c' = 37° 18' 46" = 2580.18 miles. 2. The latitude and longitude of New York given, also the distance from New York to San Francisco, and the latitude of the latter place, to find its longitude, 3. Given the latitude and longitude of New York, the distance to San Francisco and its longitude, to find the latitude. Rem. — The student will become more familiar with the prin- ciples by finding the different parts of the same problem, than by taking different orres. OB R Afl ok xms XJNIVERSITY SPHERICAL TRIGON 115 PROBLEM IX. To -find the hour of the day ; the altitude of the sun, its declination and the latitude of the observer being given. The spherical triangle of which we know the three sides are in the celestial concave. Its vertices are the sun, the zenith of the observer, and the Celestial Pole, or the point in the heavens pierced by the axis of the earth, perpendicular to the equator. The arc of the great circle joining the sun and the pole is the codeclination of the sun, when the sun and the observer are both on the same side of the equator; when they are on different sides of the equator, it is the sum of the declination and 90°. The Coaltitude of the sun is the arc of the great circle joining the sun and the zenith of the observer; and the Colatitude of the observer is the arc joining the zenith and the pole. zc EXAMPLE. In latitude 36° 40' the declination of the sun is 12° 20' N., and its altitude 30° 30'. What is the hour of the day ? Ans. Either 7h. 56m. 2 sec. a.m., or 4 h. 3 m. 58 sec. p. m. In this example the three sides are given to find the angle at the pole, which is the hour angle, and being reduced from degrees, etc., to hours, minutes, etc., by dividing by 15, gives either the time before or after 12 m. The angle having its vertex at the pole, one side of which extends from the pole to the sun, the other to the zenith. The sun, the zenith, and the celes- tial pole N are the vertices of the triangle BCB' ; the three sides are given. Draw CA perpendicular to BB' ; then find the segments of c, 8, and tf; and then the angle B, which reduce to hours, etc., and it is either so long before or after 12 o'clock m. 116 SPHERICAL TRIGONOMETRY. PROBLEM X. To find the length of the day at any place, the latitude and declination of the sun being known. Let NS be the meridian at which the sun reaches the horizon when it is on the equator ; that is, when it rises at 6 o'clock. When the sun has a declina- tion north, it will be at s on the ecliptic, instead of being at C on the same meri- dian at 6 o'clock. It has already passed the distance Bs above the horizon, and the time taken for this passage is in. the same proportion to 24 hours, that this arc AC is to 360 degrees. The angle is AN C, and is measured by the arc AC. In the triangle ABC, right angled at A, AB is equal to the declination of the sun ; the angle ACB = ECH is the coaltitude of the place. EXAM PLE. What is the length of the day in latitude 40° 30' north, when the declination of the sun is 12° 50' ? T will be the position of the traveler. sin l = cot C x fcang. c ; .\ log tang. comp. C 40° 30' = 9.931499 log tang, c - 10 12° 50' = 9.3575 66 sin b = sin C 11° 13' 10" 9.289065 Time before 6 o'clock that the sun rises and of course the same time after 6 it sets. h. min. sec. 11° 13' 10" = 44 53 2 Hence, 1 29 46 12 Length of day, 13 29 46 SPHERICAL TRIGONOMETRY. 117 Twice the time of the sun passing from the horizon to the meridian NS must be added to 12 hours to get the length of the day. Rem. — As a traveler goes north, starting at the equator, for every degree that he travels, the south pole recedes one degree ; therefore, the angle HCS measures his latitude, and HCE is his colatitude. This is the same as the north pole rising a degree for every degree he travels ; hence, the altitude of the north pole is his latitude. TA B L E, 1<> COMTAINI.VO THE LOGARITHMS OF NUMBERS FROM 1 TO 10,000. Hf 6> n it NUMBERS FROM 1 TO 100 AND THEIR LOGARITHMS, WITH THEIR INDICES. t u0« » 9 [ No. Logarithm. Na Logarithm. No. Logarithm. No. Logarithm. No. Loga ithm. !1- 0000000 21 1-322219 41 1-612784 61 1-785330 81 1908485 i 0-301030 22 1-342423 42 1-623249 62 1-792392 82 1-913814 3 477121 23 1-361728 43 1-633468 63 1*799341 83 1-919D78 1-924279 ) 4 0*003060 24 1 -3802 11 44 1-643453 64 1-800180 84 .*«* 1 5 0-698970 25 1-397940 45 1-653-213 65 1-812913 85 1-929419 1-934498 > 6 0778151 26 1-414973 46 1-662V58* 66 1819544 86 ( 7 Q-845M8 27 1-431304 47 1-672098 67 1-826075 87 1-939519 / • ? 8. 0-903090 28 1-447158 48 1-681241 68 1-832509 88 1-944483 ( 9 954243 29 1-462398 49 1-690 193 69 1-838849 89 1-949390 1-954243 1-959041 i 10 1-000000 30 1-477121 50 1-698970 7J 1-845098 90 ) U 1041393 31 1-491362 51 1-707570 71 1851258 91 ) 12 1079181 32 1-505150 52 1-716003 72 1-857332 92 1-963788 ) 13 1113943 33 1 518.314 53 1-724276 73 1-863323 93 1-968483 ; W 1 140128 34 1-531479 54 1*732394 74 1669232 94 1973128 i 15 1176091 35 1-544068 55 1-740363 75 1.875061 95 1-977724 I w 1-204120 36 1-556303 56 1-748188 76 1-880814 96 1-982271 17 1-2:10449 37 1-568202 57 1-755875 77 1-886491 97 1-986772 ( 18 1-255273 38 1-579784 58 1-763428 78 1-892095 98 1-991226 ( I9 1-278754 39 1-591065 59 1-770852 79 1-897627 99 1-995635 > 20 1-301030 40 1-602060 60 1-778151 80 1-903090 100 2000000 t* J * Note. — In the following part of the Table, the Indices are omitted, as they can be very easily supplied. Thus, the index of the logarithm of every integer number, consisting only of one number, is 0; of two figures, 1; of three figures, 2; of four figures, 3: being always a unit less than the number of figures contained in the integer number. The index to the logarithm of every number above 100, in the following part of the Table, is omitted ; yet, in the operation, it must be prefixed, according to this remark of 600 is 2-77815, and that of 6000 is 377815, so that the logarithm ind so of the rest. ISO LOGARITHM No. | | 1 | 3 | 3 | 4: | I 6 | 100 000000 1 4321 s 8600 3 012837 4 7033 s 021189 G 5306 7 9384 8 033424 9 7426 110 041393 1 5323 2 9218 3 053078 4 6905 5 060698 4458 7 8186 8 071882 5547 120 079181 1 082785 2 6360 3 9905 41093422 5 6910 6 100371 7 3804 8 7210 9 110590 130 113943 1 7271 2 120574 3 3852 4 7105 5 130334 6 3539 7 6721 8 9879 9 143015 140 146128 1 9219 1 152288 3 5336 4 8362 5 161368 f) 4353 7 7317 B 170262 9 3186 ISO 176091 1 8977 2 181844 3 4691 4 7521 5 190332 6 3125 7 5900 8! 8657 9 201397 000434 4751 9026 013259 7451 021603 5715 041787 5714 9606 053463 7286 061075 4832 8557 072250 5912 079543 083144 6716 090258 3772 7257 100715 4146 7549 110926 114277 7603 120903 4178 7429 130655 3858 7037 140194 3327 146438 9527 152594 5640 8664 161667 4650 7613 170555 3478 176381 9264 182129 4975 7803 190612 3403 6176 8932 201670 000868 5181 9451 013680 7868 022016 6125 030195 4227 8223 042182 6105 9993 053846 7666 061452 5206 079904 083503 7071 090611 4122 7604 101059 4487 7888 111263 114611 7934 121231 4504 7753 130977 4177 7354 140508 176670 9552 182415 5259 8084 190892 3681 6453 9206 201943 001301 5609 9876 014100 8284 0224^8 6533 030600 4628 8620 042576 6495 050380 4230 8046 061829 5580 9298 072985 6640 080266 3861 7426 090963 4471 7951 101403 114944 8265 121560 4830 8076 131298 4496 7671 140822 3951 147058 150142 3205 6246 9266 162266 5244 8203 17114! 4060 176959 9839 182700 5542 8366 191171 35)51) 6729 9481 202216 001734 6038 010300 4521 8700 022841 6942 031004 5029! 9017 002166 ( 6466 010724 ( 4940! 9116! 023252 ( 7350 1 031408 ( 5430 9414 043362 i 7275 051153 4996 043755 766-1 051538 5378 9185 062358 080626 4219 7781 091315 4820 8298 101747 5169 8565 111934 115278 8595 121888 5156 8399 131619 4814 7987 141136 4263 147367 150449 3510 6549 P567 162164 5541 8497 171434 4351 177248 180126 2985 5825 8647 191451 4237 7005 9755 202488 080987 4576 8136 091667 5169 8644 102091 5510 8903 112270 131939 5133 8303 141450 4574 147676 150756 3815 6852 177536 180413 3270 6108 8928 191730 4514 7281 200029 2761 081347 4934 8490 092018 5518 8990 102434 5851 9241 112605 115943 9256 122544 5806 9045 132260 5451 8618 141763 4885 t 8 | 003029 003461 7321 7748 011570 011993 5779 6197 9947 020361 024075 4486 8164 8571 032216 032619 6230 6629 040207 040602 044148 044540 8053 8442 051924 052309 5760 6142 9563 9942 063333 063709 7071 7443 070776 071145 4451 4816 8094 8457 081707 082067 5291 5647 8845 9198 092370 092721 5866 6215 9335 9681 102777 103119 6191 6531 9579 9916 112940 113275 116276 11G608 9586 9915 122871 123198 6131 6456 9368 9690 132580 132900 5769 6086 8934 9249 142076 142389 5196 5507 148294 148603 151370 151676 4424 4728 7457 7759 1604(59 160769 3460 3758 6430 6726 9380 • 9674 172311 172603 5222 5512 178113 178401 180986 181272 3839 4123 6674 6956 9490 9771 192289 192567 5069 5346 7832 8107 200577 200850 3305 3577 9 | DiffA 8978 033021 7028 040998 044932 393 ; 8830 390 > 052694 1 386 6524 i 383 0603201379 40831376 7815 ! 373 0715141370 5182J366 8819 i 363 1300121323 I | 1 I 7 | 8 | 9 |D,ff. OF NUMBERS. 121 no. ; o | 1 | a T^^M^sn^sn^ls^TonTuiffl 160 1 204120 204391 204063 204934 205204 205475 205746 206016 206286 1 206556 | 271 \ 1 0826 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 ) 2 9515 9783 210051 210319 210586 210853 211121 211388 211654 211921 267) 3 212188 212454 2720 2986 3252 3518 3783 4049 4314 4579 266 S 4 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 S 5 7484 7747 8010 8273 8536 . 8798 9060 9323 9585 9840 262 ) 6 220108 220370 220631 220892 221153 221414 221675 221936 222196 222450 261 ( 7 27 lu 2976 3230 3496 3755 4015 4274 4533 4792 5051 259 ( 8 5309 5568 5826 6084 6342 6000 6858 7115 7372 7630 258 ( > 9 7887 8144 8400 8657 8913 9170 9426 9682 9938 230193 256 J •170 230449 230704 230960 231215 231470 231724 231979 232234 232488 232742 255; ' 1 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 > 2 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 252) 3 8046 8297 8548 8799 9049 9299 9550 9800 240050 240300 250) 4 240549 240799 241048 241297 241546 241795 242044 212293 2541 2790 249 ) 5 3038 3280 3534 3782 4030 4277 4525 4772 5019 5266 248) > 6 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246) > 7 7973 8219 8464 8709 8954 9198 9443 9087 9932 250176 245) > 8 250420 250664 250908 251151 251395 251638 251881 252125 252368 2610 243 > > 9 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 j > 183 255273 255514 255755 255996 256237 256477 256718 256958 257198 257439 241 > 1 7079 7918 8158 8398 8637 8877 9116 9355 9594 9833 239.' 2 200071 260310 260548 260787 261025 261263 261501 261739 261976 262214 238? 3 2451 2688 2925 3102 3399 3636 3873 4109 4346 4582 237? ' 4 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 ( 1 5 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234) > 6 9513 9746 9980 270213 270446 270679 270912 271144 271377 271609 233) i 7 271842 272074 272306 2538 2770 3001 3233 3464 3696 3927 232) f 8 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 230 > 1 9 6402 6692 6921 7151 7380 7609 7838 8067 8296 8525 229) ! 190 278754 278982 279211 279439 279667 279895 280123 280351 280578 280806 228{ 1 281033 281261 281488 281715 281942 282169 2396 2622 2849 3075 227 ( 1 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 ( 3 5557 5782 6007 6232 G456 6681 6905 7130 7354 7578 225 ( 7802 8026 8249 8473 8696 8920 9143 9366 9539 9812 223 ( 290035 290257 390480 290702 290925 291147 291369 291591 291813 292034 222 ( ( 6 2256 2478 2099 2920 3141 3363 3584 3804 4025 4246 221 ( ( 7 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220/ 8 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219) 218? 9 8853 9071 9289 9507 9725 9943 300161 300378 300595 300813 >2O0 301030 301247 301464 301681 301898 302114 302331 302547 302764 302980 217 ( } 1 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 ( J 2 5351 5566 5781 5990 6211 6425 6639 6854 7068 7282 215 ( ) 3 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 213 < ) 4 9630 9843 310050 310268 310481 310093 310906 311118 311330 311542 212 ( > 5 311754 311966 2177 2389 2600 2812 3023 3234 3445 3656 21H ) 6 3807 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 < ) 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209? 5 8 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 i ) 9 320146 320354 320562 320769 320977 321184 321391 321598 321805 322012 207 206 S \210 322219 322426 322633 322839 323046 323252 323458 323665 323871 324077 { 1 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 { \ 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 ( S 3 8380 8583 8787 8991 9194 9398 9601 9805 330008 330211 203 ( S 4 330414 330617 330819 331022 331225 331427 331630 331832 2034 2236 202 ( { 5 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 ( ( 6 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 ( < 7 6460 6660 6860 7060 7280 7459 7659 7858 8058 82571200 ( J 8 8456 8656 8855 9054 9253 9451 9650 9849 340047 3402461199 ( 2225J198J 9 34044^ 340642 340841 341039 341237 341435 341632 341830 2028 SiJLSU, j^zjl* 122 LOGARITHMS Na| o | 1 I « 1 3 I 4 5 ! 6 | 7 ! s ; 9 DiS. ( 220 342423 342620 342817 343014 343212 343409 343606 343802 343999 344196, 197 I ( 1 4392 4589 4785 4981 5178 5374 5570 5766 59(52 6157 196 ( S 2 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 ( ( 3 8305 8500 8694 8889 9083 9278 9472 9666 98M 350054 194 ( 1989 193 ( 3916 193 { 5834 192/ ( 4 350248 350442 350636 350829 351023 351216 351410 351603 351796 S 5 2183 2375 2568 2761 2954 3147 3339 3532 3724 ( 6 4108 4301 4493 46851 4876 5068 5260 5452 5643 i 7 \ 8 6026 6217 6408 65991 6790 6981 7172 7363 7554 7744 191 / 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 : 189 ( 188) ) 9 9835 360025 360215 360404 360593 360783 360972 361161 361350 361539 }230 361728 361917 362105 362294 362482 362671 362859 363048 363236 363424 J 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188, ) 2 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187) ) 3 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 ) ; 4 9216 9401 9587 9772 9958 370143 370328 370513 370698 370883 J 85 ) 5 5 371068 371253 371437 371622 371806 1991 2175 2360 2544 2728 184 ) ) 6 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184) ) 7 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183) ) 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182) 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 S )S40 380211 380392 380573 380754 380934 381115 381296 381476 381656 381837 181 \ ( 1 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 ( / 2 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 f ( 3 5606 5785 5964 6142 6321 6499 6077 6856 7034 7212 178 < 1 ^ 7390 7568 7746 71*3 8101 8279 8456 8634 8811 8989 178 ( I 5 9166 9343 9520 9698 9875 390051 390228 390405 390582 390759 177 ( ? 6 390935 391112 391288 391464 391041 1817 1993 2169 2345 2521 176 ( 1 7 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176/ 1 8 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 175 [ 1 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 J ( 250 397940 398114 398287 398461 398634 398808 398981 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2882 2901 3039 3118 3196 3275 3353 3431 78 ) \ 4 3510 3588 3667 3745 3823 3902 3930 4058 4136 4215 78 ) ( 5 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 > 6 ( 1 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 > 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 ) \ 8 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 ) J 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 j 1 560 748188 748266 748343 748421 748498 748576 748653 748731 748808 748885 77 < ) 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 ( / 2 9736 9814 9891 9968 750045 750123 750200 750277 750354 750431 77 ( ) 3 750508 750586 750663 750740 0817 0894 0971 1048 1125 1202 77 { / 4 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 77 ( / 5 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 S 6 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 ) 7 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77. 76 76 ; > 8 4343 4425 4501 4578 4654 4730 4807 4883 4960 5036 ! 9 5112 5189 5265 5341 5417 5494 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778224 778296 778368 778441 778513 778585 778658 778730 ; 1 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 ( ) 2 9596 9669 9741 9813 9885 9957 780029 780101 780173 780245 72 / ) 3 780317 780389 780461 780533 780605 780677 0749 0821 0893 0965 72 ( ) 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 ( ? 5 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72) ) 6 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 ) } 7 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 7) ) J 8 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 ) 9 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 | (610 785330 785401 785472 785542 785615 785686 785757 785828 785899 785970 71 ( 1 I 6041 6112 6183 6254 6325 6390 6467 6538 6609 6680 71 ( 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 ( { 3 7460 .7531 7602 7673 7744 7815 7885 7956 8027 8098 71 ( 71 ( \ 4 8168 8239 8310 8381 8451 8522 8593 8663 8734 880-1 ( 5 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 n ( ( 6 9581 9651 9722 9792 9863 9933 790004 790074 790144 790215 70 ( < 7 7902«5 790356 790426 790496 790567 790637 0707 0778 0848 0918 70 ( S 8 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 ( ; 9 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 / )620 792392 792462 792532 792602 792672 792742 792812 792882 792952 793022 70 | ) 1 3092 3] 62 3231 3301 3371 3441 3511 3581 3651 3721 70 > 2 3790 3860 3930 4000 4070 4139 42J9 4279 4349 4418 70 ; 3 4488 4558 4627 4697 4767 4836 4906 4976 5045 4115 70 ) ; 4 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 } > 5 5889 5949 6019 6088 6158 6227 6297 6306 6436 6505 69 ) ; 6 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 ) ) 7 72(58 7337 7406 7475 7545 7614 7683 7752 7821 7890 69 } S 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 ) j 9 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 j $630 799341 799409 799478 799547 799616 799685 799754 799823 799892 799961 69 ) / 1 800029 800098 800167 800236 800305 800373 800442 800511 800580 800648 69 / ( 2 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 69 ) ( 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 / ( 4 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 68 ) ( 5 2774 2812 2910 2979 3047 316 3184 3252 3321 3389 68 ( 4071 68 / ( G 3457 3525 3594 3662 3730 3798 3867 3935 4003 « 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 68 / 4821 4889 4857 5025 5093 5161 5229 5297 5365 5433! 68 ) 9 5501 5569 5637 5705 5773 5841 5906 5976 COM 61121 68 } (no,| | 1 | a | 3 | 4 | 5 | c s , >-/>>. >• ^^s~^^. r^U OP NUMBERS. 129 ^nTTo |l.|a>3|4:|5|6|7|8|9| Diff. { ( 040 806180 806248 8063 Hi 306384 806451 806519 806587 806655 806723 806790 68 ( < 1 6858 6926 6994 7061 7129 7197 7264 7332 7400 7467 68 ( S 2 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 68 ( { 3 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 67 I < 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 I < 5 9560 9627 9094 9762 9829 9896 9904 810031 810098 810165 67? { G 810233 810300 810367 810434 810501 810569 810636 0703 0770 0837 67 ? 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1503 67 ( 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 ( < 9 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 67 , 650 812913 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 ^ 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 67) ? 2 4248 4314 4381 4447 4514 4581 4647 4714 • 4780 4847 67 S ; 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 66 ) ; 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 66 ) 5 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 66 ) ) 6 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 S ) 7 7565 7631 769$ 7764 7830 7896 7962 8028 8094 8160 66 ) 8 8226 8292 8356 8424 8490 8556 8622 8688 8754 8820 66 ) > 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 S I WO 819544 819610 819676 819741 819807 819873 819939 820004 820070 820136 66 ( I 1 820201 820267 820333 820399 820464 820530 820595 0661 0727 0792 66 ( ( 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 / I 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 65 / ( 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2750 65 ( ? 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 ( ) 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 65 ( ? 7 4126 4191 425G 4321 4386 4451 4516 4581 4646 4711 65 < ) 8 4776 4841 4906 4971 5030 5101 5166 5231 5296 5361 65 ( I 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 65 ( (670 826075 826140 826204 826269 826334 826399 826464 826528 890503 826658 65 J ( 1 6723 6787 6852 6917 6981 7040 7111 7175 7240 7305 65) S 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 > S 3 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 64 > S 4 6660 8724 8789 8853 8918 8982 9016 9111 9175 9239 64 ) ^ 5 9304 9368 9432 0497 9561 9625 9690 9754 9818 9882 64 ) ( 6 9947 830011 830075 830139 830204 830268 830332 830396 830460 830525 64 > ( 7 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 ) ? 8 1230 1294 1358 1422 1486 15fiQ 1614 1678 1742 1806 64 > I 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 S S680 832509 832573 832637 832700 832764 832828 832892 832956 833020 833083 64 ( > 1 3147 3211 3275 3338 3402 3466 3530 3593 3057 3721 64 < ) 2 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 / ) 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 / ) 4 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 / ) 5 5691 5754 5817 588* 5944 6007 6G71 6134 6197 6261 63 J 1 6 6324 6337 6451 6514 6577 6641 6704 6767 6830 6894 63 / i 1 6957 7020 7083 7146 7210 7273 7336 7399 7462 7SB5 63 ) 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 , ( 9 j 690 t 1 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 j 838849 838912 838975 839038 839101 839164 839227 839289 839352 839415 63 J 9478 9541 9604 9667 9729 9792 9855 9918 9981 840043 63S I 2 840106 840169 840232 840294 840357 840420 840482 840545 840608 0671 63< > 3 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 63 < ) 4 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 63 ( ) 5 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 62 ( ) 6 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 62 ( ) 7 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 ( ) 8 3855) 3918 3930 4042 4104 4166 4229 4291 4353 4415 62 ( ) 9 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 62 ( Uo, | | 9 | Di£/ 130 LOGARITHMS No. | O I 1 | I 3 | 4 | 5 7 | 8 | 9 |Di£{ '700 845098 845160 845222 845284 845346 845408 845470 845532 845594 845656 62 1 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 2 (5337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 3 C955 7017 7079 7141 7202 7264 732f 7388 7449 7511 62 4 7573 7634 7696 7758 7819 78&1 7943 8004 8066 8128 62 ' 5 8189 8251 8312 8374 8435 8559 8620 8682 8743 62 6 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 7 9419 9181 9542 9604 9665 9726 9788 9849 9911 9972 61 ' 8 850033 850095 850156 850217 850279 850340 850401 850462 850524 850585 61 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 710 851258 851320 851381 851442 851503 851564 851625 851686 851747 851809 61 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 o 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 4 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 61 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 8 6124 6185 6245 6306 6366 6427 6487 6548 6008 6668 GO 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 720 857332 857393 857453 857513 857574 857634 857694 857755 857815 857875 60 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 ro 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 GO 3 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 4 9739 9799 9859 9918 9978 860038 860098 860158 860218 8C0278 60 5 860338 860398 8G0 158 860518 860578 0037 0697 0757 0817 0877 60 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 GO 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 720 863323 863382 8*53442 863501 863561 863G20 863680 863739 863799 863858 59 < 1 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 < 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 < 3 5104 5163 5222 5282 5341 54C0 5459 5519 5578 5637 59 < 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 5 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 i 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 59 1 8 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 59 { 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 I 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869760 59 ! 1 9818 9877 9935 9994 870053 970111 870170 870228 870287 8703-15 59 2 S70404 870462 870521 870579 0638 0696 0755 0813 0872 0930 58 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 53 : 58 ) 58 ) 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 6 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 ; 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 ) 9 4482 4540 4598 4656 4714 4772 4330 4888 4945 5003 58 J 750 875061 875119 875177 875235 875293 875351 375409 875466 875524 875582 58 ( 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 ' 58 ) 3 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 i 5 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 < 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57/ 7 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 8 9609 9726 9784 9841 9898 9956 380013 880070 380127! 8801 85 57 9 880-242 880299 880356 880413 88047] 880528 * — _ 05851 0642 0(599 1 0756 57 ! 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7302 7955 8516 9077 9638 890197 0756 1314 1872 892429 2985 3540 4094 4648 5201 5754 6306 6857 7407 897957 8506 9054 9602 900149 0695 1240 1785 2329 2873 903416 3958 4499 5040 5580 6119 6658 7196 7734 8270 908807 9342 9877 910411 0944 1177 2009 2541 3072 3602 881213 1784 2354 2923 3491 4059 4625 5192 5757 6321 7449 8011 8573 9134 9094 890253 0812 1370 1928 892484 3040 3595 4150 470-1 5257 5809 6361 6912 7462 898012 8561 9109 9656 900203 0749 1295 1840 2384 2927 903470 4012 4553 5094 5634 6173 6712 7250 7787 8324 908860 9396 9930 910464 0998 1530 2063 2594 3125 3655 881271 1841 2411 2980 3548 4115 4682 5248 5813 6378 886942 7505 8067 8629 9190 9750 890309 0868 1426 1983 892540 3096 3651 4205 4759 5312 5861 6416 6967 7517 898067 8615 9164 9711 900258 0804 1349 1894 2438 2981 903524 4066 4607 5148 5688 6227 6766 7304 7841 8378 908914 9449 9984 910518 1051 1584 2116 2647 3178 3708 2468 3037 3605 4172 4739 5305 5870 6434 7561 8123 8685 9246 9806 890365 0921 1482 2039 892595 3151 3706 4261 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8652 8703 8754 8805 8857 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 929419 929470 929521 929572 929623 929674 929725 929776 929827 929879 9930 9981 930032 930083 930] 34 930185 930236 930287 930338 930389 930440 930491 0542 0592 0643 0694 0745 0796 0847 0898 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 2474 2524 2575 2026 2677 2727 2778 2829 2879 2930 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 934498 934549 934599 934650 934700 934751 934801 934852 934902 934953 5003 5054 5104 5154 5205 5255 5306 5356 5406 5457 5507 5558 5608 5658 5709 5759 5809 5860 5910 5960 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 7016 7066 7117 7167 7217 7267 7317 7367 7418 7408 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 9020 9070 9120 8170 9220 9270 9320 9369 9419 9469 932519 939569 939619 939669 939719 939769 939819 939869 939918 939968 940018 940068 940118 940168 940218 940267 940317 940307 940417 940467 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 1511 1561 1611 1660 1710 1760 1809 1S59 1909 1958 2008 2058 2107 2157 2207 2256 2306 2355 2405 2435 250-1 2354 2603 2653 2702 2752 2801 2851 2901 2950 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 No. | OP NUMBERS. 133 JnoT 0|1|»|3|4|5|6|T|8| 9|dS] )880 944483 944532 944581 944631 944680 944729 944779 944828 944877 944927 49 < $ 1 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 ( ) 2 5469 5518 5567 5616 5665 5715 5764 5813 5862 5912 49 ( ) 3 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 49 ( S 4 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 49 S 5 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 S 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 ( < 7 7924 7973 8022 8070 8119 81 68 8217 8266 8315 8364 49 I ^ 8 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 49 ( ( 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 ( ) 890 949390 949439 949488 949536 949585 949634 949683 949731 949780 949829 49 ) ) 1 9878 9926 9975 950024 950073 950121 950170 950219 950267 950316 49 ) ) 2 950365 950414 950462 0511 0560 0608 0657 0706 0754 0803 49 ) ) 3 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 49 > ) 4 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 49 > / 5 1823 1872 1920 1969 2017 2066 2114 2163 2211 22(H) 48 > S 6 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 43 > ) 7 2792 2841 2889 2938 2980 3034 3083 3131 3180 3228 48 > 8 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 48 ) 9 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 48 S 900 954243 954291 954339 954387 954435 954484 954532 954580 954628 954677 48 \ ? 1 4725 4773 4821 4869 4913 4966 5014 5062 5J10 5158 48 ( < 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 ( 1 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 ( 1 4 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 ( I 5 6649 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 < / 6 7128 7176 7224 7272 7320 7966 7416 7464 7512 7559 48 ( ? 7 WOT 7655 7703 7751 7799 7847 7894 7942 7990 8038 48 I ) 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 ( ) 9 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 J <910 959041 959089 959137 959185 959232 959280 959328 959375 959423 959471 48 ) 48 , 1 9518 9566 9614 9661 9109 9757 9804 9852 9900 9947 2 9995 960042 960090 960138 960185 960233 960281 960328 960376 960423 48 ) S 3 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 48 ) S 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 48 ) S 5 1421 1469 1516 1563 1011 1658 1706 1753 1801 1848 47 ) C 6 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 ) 7 2369 2417 2464 2511 2559 2G06 2653 2701 2748 2795 47 > 8 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 47 S 9 3316 3303 3410 3457 3504 3552 3599 3646 3693 3741 47 S 920 963788 963835 963882 963929 963977 964024 964071 964118 964165 964212 47 / 1 4860 4307 4354 4401 4448 4495 4542 4590 4637 4(584 47 / I 2 4731 4778 4825 4872 4919 4966 5013 5061 5108 5155 47 / 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 47 ; 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 47 ) 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 47 ) ) 6 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 47 ) ) 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 ; 8 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 \ {930 968483 968530 968576 968623 968670 968716 968763 968810 968856 968903 47 ( { 1 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 47 < I 2 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 < ( 3 9882 9928 9975 970021 970068 970114 970161 970207 970254 970300 47 < ( 4 970347 970393 970440 0486 0533 0579 0626 0672 0719 0765 46 ' ) 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 46^ { 6 I 7 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 < 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 46 < 5 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 ( 9 2666 2712 2758 2804 2851 28^7 2943 2989 3035 3082 46 j ,£ -^L-3~LJL~LJLJ ^^L JL XJLX~S JLs 134 LOGARITHMS, ETC. ■;m 973128 3590 4051 4512 4972 5432 5891 6350 6808 7266 977724 8181 8637 9093 9548 980003 0458 0912 1366 1819 982271 2723 3175 3026 4077 4527 4977 5426 5875 6324 970 986772 7219 7666 8113 8559 9005 9450 9895 990339 0783 991220 1669 2111 2554 2995 3436 3877 4317 4757 519(5 995035 6074 6512 6949 7386 7823 8259 8695 9131 9565 973174 973220 3636 3682 4097 4143 4558 4604 5018 5064 5478 5524 5937 5983 6396 6442 6854 6900 7312 7358 977769 977815 8226 8272 8683 8728 9138 9184 9594 9639 980049 980094 0503 0549 0957 1003 1411 1456 1864 1909 982316 982362 2769 2814 3220 3265 3671 3716 4122 4167 4572 4617 5022 5067 5471 5516 5920 5965 6369 6413 986817 986861 7264 7309 7711 7756 8157 8202 8604 8648 9049 9094 9494 9539 9939 9983 990383 990428 0827 0871 991270 991315 3713 1758 2156 2200 2598 2642 3039 3083 3480 3524 3921 3965 4361 4405 4801 4^45 5240 5284 995679 995723 6117 6161 6555 6599 6993 7037 7430 7474 7867 7910 8303 8347 8739 8782 9174 9218 9609 9652 973266 3728 4189 4650 5110 5570 6029 6488 6946 7403 977861 8317 8774 9230 9685 980140 0594 1048 1:01 1954 982407 2859 3310 3762 4212 4662 5112 5561 6010 6458 7353 7800 8247 8693 9138 9583 990028 0472 0916 991359 1802 2244 2686 3127 3568 4009 4449 4889 5328 995767 6205 6643 7080 7517 7954 8390 8826 9261 9696 73313 3774 4235 4696 5156 5616 6075 6533 6992 7449 977906 8363 8819 9275 9730 980185 0640 1093 1547 2000 982452 2904 3356 3807 4257 4707 5157 5606 6055 6503 986951 7398 7845 8291 8737 9183 9628 990072 0510 0960 991403 1846 2288 2730 3172 3613 4053 4493 4933 5372 99581] 6249 6687 7124 7561 7998 8434 8869 9305 9739 973359 3820 4281 4742 5202 5662 6121 6579 7037 7495 977952 8409 8865 9321 9776 980231 0685 1139 1592 2045 982497 2949 3401 3852 4302 4752 5202 5651 6100 6548 986996 7443 7890 8336 8782 9227 9672 990117 0561 1004 991448 1890 2333 2774 3216 3657 4097 4537 4977 5416 995854 6293 6731 7168 7605 8041 8477 8913 934H 9783 734051973451 3866 4327 4788 5248 5707 6167 6625 7083 7541 977998 8454 8911 9366 9821 980276 0730 1184 1637 2090 982543 2994 3446 3897 4347 4797 5247 5696 6144 6593 987040 7488 7934 8381 8826 9272 9717 990161 0605 1049 991492 1935 2377 2819 3260 3701 4141 4581 5021 5460 995898 6337 6774 7212 7648 8085 8521 8956 9TO2 9826 3913 4374 4834 5294 5753 6212 6671 7129 7586 978043 8500 8956 9412 9867 980322 0776 1229 1683 2135 982588 3040 3491 3942 4392 4842 5292 5741 6189 6637 987085 7532 7979 8425 8871 9316 9761 990206 0650 1093 991536 1979 2421 2863 3304 3745 4185 4625 5065 5504 995942 6380 6818 7255 7692 8129 8564 9000 9435 9870 973497 973543 3959 4005 4420 4466 4880 4926 5340 5386 5799 5845 6258 6304 6717 6763 7175 7220 7632 7678 978089 978135 8546 8591 9002 9047 9457 9503 9912 9958 980367 980412 0821 0867 1275 1320 1728 1773 2181 2226 982633 982678 3085 3130 3536 3581 3987 4032 4437 4482 4887 4932 5337 5382 5786 5830 6234 6279 6682 6727 987130 987175 7577 7622 8024 8068 8470 8514 8916 8960 9361 9405 9806 9850 990250 990294 0694 073« 1137 1182 991580 991625 2023 2067 2465 2509 2907 2951 3348 3392 3789 3833 4229 4273 4669 4713 5108 5152 5547 5591 995986 996030 6424 6468 6862 6906 7299 7343 7736 7779 8172 8216 8608 8652 9043 9087 9479 9522 9913 91)57 l^-LjLXJLJ^Ji^JL ^ •' JLJ.- LOGARITHMIC SINES AND TANGENTS, FOR EVERY DEGREE AND MINUTE THE QUADRANT. N. B. Thk minutes in the left-hand column of each pagfc ; increasing downwards, belong to the degrees at the top ; and those increasing upwards, in the right-hand column, belong to the degree below. 13t 1 (0 Degree.) A TABLE OJ * LOGARITHMIC nr | Sine 1 D. Cosine 1 D. | Tan?. 1 D. | Cotansr. 'H ( ° 0-000000 10-000000 o-oooooo 1 Infinite. 60 ; / i 6463726 501717 000000 00 0-463726 1 501717 13-536274 59? ( 2 764756 293485 000000 00 764756 293483 235244 58 ; / 3 940847 208231 000000 00 940847 208231 059153 57 ) I 4 7-065786 161517 000000 00 7-065786 161517 12-934214 56 > ( 5 162696 131968 000000 00 162696 131969 837304 55 ) ( 6 241877 111575 9-999999 01 241878 111578 758122 54 ) / V 308824 96653 999999 01 308825 99653 691175 53 ) t 8 366816 85254 999999 01 366817 85254 633183 52 ) < 9 417968 76263 999999 01 417970 76263 582030 51 ) 463725 68988 999998 01 463727 68988 536273 50 / 49 ( \ ]1 7-505118 62981 9-999998 01 7-505120 62981 12-494880 ( 12 542906 57936 999997 01 542909 57933 457091 43 ( S 13 577668 53641 999997 01 577672 53642 422328 47 ( \ u 609853 49938 999996 01 609857 49939 390143 46 ? I 15 639816 46714 999996 01 639820 46715 360180 45 ( M6 667845 43881 999995 01 667849 43882 332151 44 ( 33 982233 21608 999980 02 982253 21610 017747 27 I 34 995198 20981 999979 02 995219 20983 004781 26 ) )35 8-007787 20390 999977 02 8-007809 20392 11-992191 25 S S36 020021 19831 999976 02 020045 19833 979955 24 ) )37 031919 19302 999975 02 031945 19305 908055 23 ( 38 043501 18801 399973 02 043527 18803 956473 22 ( )39 054781 18325 999972 02 054809 18327 945191 21 ( Wo 065776 17872 999971 02 065806 17874 934194 20 ( 42 8-076500 17441 9-999969 02 8-076531 17444 11-923469 19 ) 0869C5 17031 999968 02 080997 17034 913003 18 ) }43 097183 16639 999966 02 097217 16642 902783 17 ) >44 107167 16265 999964 03 107202 16268 892797 15 ; )45 116926 15908 999963 03 116963 15910 883037 >46 126471 15566 999961 03 126510 15568 873490 14 > )47 135810 15238 999959 03 135851 15241 864149 13 ) )48 144953 14924 999958 f,3 144996 14927 855004 12) )49 153907 14622 999956 03 153952 14627 846048 J l ) )50 162681 14333 999954 03 162727 14336 837273 10 j (51 8171280 14054 9-999952 03 8-171328 14057 11-828672 9 ( (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 G I (55 204070 13041 999944 03 204126 13044 795874 5 ) (56 211895 12810 999942 04 211953 12814 788047 4 ) (57 219581 12587 999940 04 219641 12590 780359 3 > 58 50 ' 60 227134 12372 999938 04 227195 12376 772805 2 ) 234557 12164 999936 04 234621 12168 765379 1 ) 241855 11963 999934 04 241921 11967 758079 > L. Cosine ^^J ^Ji^J ~J Cotang. | ^^^^i ->*>£3L*J. 89 1 .Jegre Mi sines and tangents. (1 Degree,) 137 I m. Sine ~ D. Cosine D. Tang, D. | Cotang. 1 i o 8241855 11963 9999934 04 8241921 11967 11-758079 60 ( 1 249033 11768 999932 04 249102 11772 750898 59 ( 2 256094 11580 999929 04 256165 11584 743835 58? 3 263042 11398 999927 04 263115 11402 736885 57 { 4 269881 11221 999925 04 269956 11225 730044 56 ( ) 5 27(3614 11050 999922 04 276691 11054 723309 55 ( ) 6 283243 10883 999920 04 283323 10887 716677 54 I ) 7 289773 10721 999918 04 289856 10726 710144 53 { 52 ( 51 ( < 8 296207 10565 999915 04 296292 10570 703708 ) 9 302546 10413 999913 04 302634 10418 697366 ) 10 308794 10266 999910 04 308884 10270 691116 50 I ) H 8-314954 10122 9-999907 04 8-315046 10126 11-684954 49 S ) 12 321027 9982 999905 04 321122 9987 678878 48 ) ) 13 327016 9847 999902 04 327114 9851 672886 47 S ) 14 332924 9714 999899 05 333025 9719 666975 46 S ) 15 338753 9586 999897 05 338856 9590 601144 45 ) > 16 344504 9460 999894 05 344610 9405 655390 44 ) ) 1 ? 350181 9338 999891 05 350289 9343 649T1 1 43 ; ) I 8 355783 9219 999888 05 355895 9224 644105 42 ) ) 1 ( J 361315 9103 999885 05 361430 9108 638570 41 ) (20 360777 8990 999882 . 05 366895 8995 633105 40 S ?21 8-372171 8880' 9-999879 05 8-372292 8885 11-627708 39 I 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 > ) 2 - 5 393101 8464 999867 05 393234 8470 606766 35 ) ) 2(i 368179 8366 999864 05 398315 8371 601685 34 > ) 27 403199 8271 999861 05 403338 8276 590602 33 ) ) 28 408161 8177 999858 05 408304 8182 591696 32 ) > 29 413068 8086 999854 05 413213 8091 586787 31 > S 30 417919 7996 999851 06 418068 8002 581932 30 ) ( 31 8422717 7909 9-999848 06 8-422869 7914 11-577131 29 > ) 32 427462 '7823 999844 06 427618 7830 572382 28 ( ) 33 4:12 1 56 7740 999841 06 432315 7745 567685 27 ( ) 34 436800 7657 999838 06 436962 7663 563038 26 ( ) 35 441394 7577 999834 06 441560 7583 558440 25 ( ) 36 44.i941 7499 999831 06 446110 7505 553890 24 ( > 37 4504-10 7422 999827 06 450613 7428 549387 23 ( ) 38 45-1893 7346 999823 06 455070 7352 544930 22 ( ) 39 459301 7273 999820 06 459481 7279 540519 21 ( j 40 463665 7200 999816 06 463849 7206 536151 20 ( ( 41 8-467985 7129 9-999812 06 8-468172 7135 11531828 19 > ( 42 472263 7060 999809 06 472454 7066 527546 18 ) ( 43 476498 6991 999805 06 476693 6998 523307 17 S < 44 480693 6924 999801 06 480892 6931 519108 16 ) ( 4 5 484848 6859 999797 07 485050 6865 514950 15 ) ?46 488963 6794 999793 07 489170 6801 510830 14 ) /47 493040 6731 999790 07 493250 6738 506750 13 S /48 497078 6669 999786 07 497293 6676 51,2707 12 ) I 49 501080 6608 999782 07 501298 6615 498702 11 ) ) 50 505045 6548 999778 07 505267 6555 494733 10 \ I 51 8-5(i8974 6489 9-999774 07 8-509200 6496 11-490800 9 ) ( 52 512867 6431 999769 07 513098 6439 480902 8 / I 53 516726 6375 999765 07 516961 6382 483039 7 ) ( 54 520551 6319 999761 07 520790 6326 479210 6? / 55 524343 6264 999757 07 524586 6272 475414 5 ) ( 56 528102 6211 989753 07 528349 6218 471651 4 ) ( 57 531828 6158 999748 07 532080 6165 467920 3 ) (58 535523 6106 999744 07 535779 6113 464221 2 ) (59 539186 6055 999740 07 539447 6062 460553 1 ) (60 542819 6004 999735 07 543084 6012 456916 J Cosine | ^~^J. Sine ~-^J Cotang. L^~^ ^^Tang^^ J^i Degrees. 138 (2 Degrees.) a table OP logarithmic r»«r Sine D. Cosine D. Tan*. 1 D. Cotanj. H 1 8-542819 6004 9999735 07 8 543084 6012 11-456916 60? 546422 5955 999731 07 546691 5962 453309 59 I 549995 5906 999726 07 550268 5914 449732 58 \ 3 553539 5858 999722 08 553817 5866 446183 57 < ) 4 557054 5811 999717 08 557336 5819 442664 56 ( S 5 560540 5765 999713 08 560828 5773 439172 55 ( 54 I ) 6 563999 5719 999708 08 564291 5727 435709 ) 7 567431 5674 990704 08 567727 5682 432273 S3( 52 } ) 8 570836 5630 999699 08 571137 5638 428863 } 9 574214 5587 999694 08 574520 5595 425480 51 | < 10 577566 5544 999689 08 577877 5552 422123 50 l ! n 8-580892 5502 9-999685 08 8581208 5510 11-418792 49 j ) 12 584193 5460 999680 08 584514 5468 415486 48 < ) 13 587469 5419 999675 08 587795 5427 412205 47 < ) W 590721 5379 999670 08 591051 5387 408949 46 < ) 15 593948 5339 999665 08 594283 5347 405717 45 < ) 16 597152 5300 999660 08 597492 5308 402508 44 < ) 17 600332 5261 999655 08 600677 5270 399323 43 S ) 18 603489 5223 999650 08 603839 5232 396161 42 < ) 1!> 606623 5186 999645 09 606978 5194 393022 41 i j 20 609734 5149 999640 09 610094 5158 389906 40 s } 21 8-612823 5112 9-999635 09 8-613189 5121 11-386811 39 ) ) 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 ) ) 96 627948 4938 999608 09 628340 4947 371660 34 ) ) 27 630911 4904 999603 09 631308 4913 368692 33 ) ) 28 633854 4871 999597 09 634256 4880 365744 32 ) ) 29 636776 4839 999592 09 637184 4848 362816 31 ) j 30 639680 4806 999586 09 640093 4816 359907 30 j >31 8642563 4/75 9-999581 09 8642982 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 4C91 348413 26 / ( 35 653911 4652 999558 10 654352 4661 345648 25 ( I 36 656702 4622 999553 10 657149 4631 342851 24 ) ) 37 659475 4592 999547 10 659928 4602 340072 23 ? ) 38 662230 4563 999541 10 662689 4573 337311 22 ( I 39 664968 4535 999535 10 665433 4544 334567 21 ? ) 40 667689 4506 999529 10 668160 4526 331840 20 / i 41 8-670393 4479 9-999524 10 8-670870 4488 11-329130 19 ( < 42 673080 4451 999518 10 673.563 4461 326437 18 < 43 ( 44 I 45 675751 4424 999512 10 676239 4434 323761 8} 678405 4397 999506 10 678900 4417 321100 681043 4370 999500 10 681544 4380 318456 ( 46 683665 4344 999493 10 684172 4354 315828 S 47 686272 4318 999487 10 686784 4328 313216 13 ( 48 688863 4292 999481 10 689381 4303 310619 12 f ( 49 691438 4267 999475 10 691963 4277 308037 ii , ( 50 693998 4242 999469 10 694529 4252 305471 10 1 9 i 51 8-696543 4217 9-999463 11 8-697081 4228 11-302919 52 53 699073 4192 999456 11 699617 4203 300383 8 ( 701589 4168 999450 11 702139 4179 297861 7 . $54 704090 4144 999443 11 704646 4155 295354 ! ) 55 706577 4121 999437 11 707140 4132 292800 ) 56 709049 4097 999431 11 709618 4108 290382 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 1 ) J 60 718800 4006 999404 11 719396 4017 280604 j LwJ Cosine | ^ | ^-2^!~4 Cotang. „ 1 ^J^Ek^. J^J 87 Decrees. SINES AND TANGENTS. (3 Degrees.) 139 r»L Sine D. | Cosine D. Tang. D. Cotang. 1 ) ( ° 8-718800 4006 9-999404 11 8-719396 4017 11-280604 i 60 I ? i 721204 3984 999398 11 721806 3995 278194 59 } i 2 723595 3962 999391 11 724204 3974 275796 58 ( 3 725972 3941 999384 11 726588 3952 273412 57 ' 4 728337 3919 999378 11 728959 3930 271041 56 ) i 5 730688 3898 999371 11 731317 3909 268683 55 ) ) 6 733027 3877 9993(34 12 733663 3889 266337 54 ) ? 7 735354 3857 999357 12 735996 3868 264004 53 ) ) 8 737667 3836 999350 12 738317 3848 261683 52 ) ) 9 739969 3816 999343 12 740626 3827 259374 51 ) \ 10 742259 3796 999336 12 742922 3807 257078 50 J < U 8744536 3776 9-999329 12 8-745207 . 3787 11.254793 49 > ( 12 746802 3756 999322 12 747479 3768 252521 48 > ( I 3 749055 3737 999315 12 749740 3749 250260 47 > ( I 4 751297 3717 999308 12 751989 3729 248011 46 ) 45 ) ) 15 753528 3698 999301 12 754227 3710 245773 ) 16 755747 3679 999294 12 756453 3692 243547 44 ( I 7 757955 3661 999286 12 758668 3673 241332 43 ) 18 760151 3642 999279 12 760872 3655 239128 42 \ ) 19 762337 3624 999272 12 763065 3636 236935 41 ) 40 ', ) 20 764511 3606 999265 12 765246 3618 234754 \ 21 8-766675 3588 9-999257 12 8-767417 3600 11232583 39 ) ( 22 768828 3570 999250 13 769578 3583 230422 38 \ ( 23 770970 3553 999242 13 771727 3565 228273 37 i ( 24 773101 3535 999235 13 773866 3548 226134 36 ^ ? 25 775223 3518 999227 13 775995 3531 224005 35) ? 26 777333 3501 999220 13 778114 3514 221886 34 S ? 27 779434 3484 999212 13 780222 3497 219778 33 S ? 28 781524 3467 999205 13 782320 3480 217680 32 ) ? 29 783605 3451 999197 13 784408 3464 215592 31 ) j 30 785675 3431 999189 13 786486 3447 213514 30 S < 31 8-787736 3418 9-999181 13 8-788554 3431 11-211446 29 > ( 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 ) k 38 ( 39 801892 3308 999126 13 802765 3322 197235 22 ) 803876 3293 999198 13 804758 3307 195242 21 ) J 40 805852 3278 999110 13 806742 3292 193258 20 ) 41 8-807819 3263 9-999102 13 8-808717 3278 11191283 19 } S 42 809777 3249 999094 14 810683 3262 189317 18 > \ 43 811726 3234 999086 14 812641 3248 187359 17 > S 44 813667 3219 999077 14 814589 3233 185411 16 ) 5 45 815599 3205 999069 14 816529 3219 183471 15 ) \ 46 817522 3191 999061 14 818461 3205 181539 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 015195. 11 ) < 50 825130 3135 999027 14 826103 3150 173897 10 } } 51 8-827011 3122 9-999019 14 8-827992 3136 11172008 9 I ) 52 828884 3108 999010 14 829874 3123 170126 8 ( ) 53 830749 3095 999002 14 831748 3110 168252 7 ) 54 832607 3082 998993 14 833613 3096 166387 6 ) 55 834456 3069 998984 14 835471 3083 164529 5 ( ) 56 836297 3056 998976 14 837321 3070 162679 4 ( > 57 838130 3043 998967 15 839163 3057 160837 3 I S 58 839956 3030 998958 15 840998 3045 159002 2 I ) 59 841774 3017 998950 15 842825 3032 157175 1 > j 60 843585 3000 998941 15 844644 3019 155356 ° L~. Coame ~>*~*~ J Sine .^-^ Coiaiig. Tang. *y 86 Degrees. 140 (4 Degrees.) a TABLE OP LOGARITHMIC (TT | Sme D. Cosine D. 1 Tang. 1 D. Cotang. ^^ ) o 8843585 3005 9-998941 15 8-844644 3019 11155356 60 ; 1 845387 2992 998932 15 846455 3007 153545 59 ) 2 847183 2980 998923 15 848260 2995 151740 58 ) 3 848971 2967 998914 15 850057 2982 149943 57 ? 4 830751 2955 998905 15 851846 2970 148154 56 ; 5 832523 2943 998896 15 853628 2958 146372 55 J G 854291 2931 998887 15 855403 2946 144597 54 ) ~ 856049 2919 998878 15 857171 2935 142829 53 ) 8 837801 2907 998869 15 858932 2923 141068 52 ) 9 859546 2896 998860 15 860686 2911 139314 51 ( 10 861283 2884 998851 15 862433 2900 137567 50 } J1 8863014 2873 9-998841 15 8-864173 2888 11135827 49 ) 12 864738 2861 998832 15 865906 2877 134094 48 ) 13 866455 2850 998823 16 867632 2866 132308 47 ) 14 868165 2839 998813 16 869351 2854 130649 46 ( 15 869868 2S28 998804 16 871064 2843 128936 45 ) 16 871565 2817 998795 16 872770 2832 127230 44 ) 17 873255 2806 998785 16 874469 2821 125531 43 ( 18 874938 2795 998776 10 8761G2 2811 123S33 42 ) 19 876615 2786 998766 16 877849 2800 122151 41 > 20 878285 2773 998757 16 879529 2789 120471 40 ( 21 8879949 2763 9-998747 16 8-881202 2779 11-118798 39 ( 22 881607 2752 998738 16 882869 2768 117131 38 < 23 883238 2742 998728 16 884530 2758 115470 37 ( 24 884903 2731 998718 16 886185 2747 113815 36 ( 23 886542 2721 998708 16 887833 2737 112167 35 ( 2(5 888174 2711 998G99 16 889476 2727 110524 34 ( 27 880801 2700 998689 16 891112 2717 108888 33 } 28 891421 2690 998G79 16 892742 2707 107258 32 < 29 893035 2680 998669 17 894366 3697 105634 31 I 30 894643 2670 998659 17 895984 2687 104016 30 ) 31 8-896246 2660 9-998649 17 8-897596 2677 11-102404 29 S 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 S 3G 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 9985G8 17 910285 2601 089715 21 J 40 910404 2575 998558 17 911846 2592 088154 20 ) 41 8-911949 2566 9-998548 17 8-913401 2583 11-086599 19 ) 42 913488 2556 998537 17 914951 2574 085049 18 ) 43 915022 2547 998527 17 916495 2565 083505 17 ) 44 910550 2538 998516 18 918034 2556 081966 16 ) 43 918073 2529 998506 18 919568 2547 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 $50 924112 2494 998464 18 925049 2512 074351 11 925609 2486 998453 18 927156 2503 072844 10 >51 8-927100 2477 9-998442 18 8-928658 2495 11-071342 9 fSQ 928387 2409 998431 18 930155 2486 069845 8 l S3 ) 54 930068 2460 998421 18 931647 2478 068353 7 931544 2452 998410 18 933134 2470 006866 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 )59 937398 2419 998366 38 939032 2437 060968 2 938850 2411 998355 18 940494 2430 059506 1 ; oo 94(1296 2403 998344 13 941952 2421 058048 | Cosine Sine | Cotang. I ^ I^Ta"*^ M. _ 85 ^ Degre iS SINES AND TANGENTS. (5 Degrees.) 141 PmT Sine D. | Cosine D. Tang. 1 D. Cotang. r . J 8-940296 2403 9-998344 19 8-941952 2421 11058048 60 ) 941738 2394 998333 19 943404 2413 056596 59 ) 2 943174 2387 998322 19 944852 2405 055148 58 S \* 944606 2379 998311 19 946295 2397 053705 57 ) 946034 2371 998300 19 947734 2390 052266 56 ) s 947456 2363 998289 19 949168 3282 050832 55 ) 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 21532 998243 19 954856 2351 045144 51 ) { 10 954499 2325 998232 19 956267 2344 043733 50 S J U 8955894 2317 9-998220 19 8-957674 2337 11042326 49 j ( 12 957284 2310 998209 19 959075 2329 040925 48/ < 13 958670 2302 998197 19 960473 2323 039527 47 ) S 14 9G0052 2295 998186 19 961866 2314 038134 46) < 15 961429 2288 998174 19 963255 2307 036745 45/ ( 10 962801 2280 998163 19 964G39 2300 035361 44) s 1 7 964170 2273 998151 19 966019 2293 033981 43 ) < 18 965534 2266 998139 20 967394 2286 032606 42) ( 19 966893 2259 998128 20 968766 2279 031234 41 ) j 20 968249 2252 998116 20 970133 2271 029867 40} ) 21 8-969600 2244 9-998104 20 8-971496 2265 11028504 39 1 S 22 970947 2238 998092 20 972855 2257 027145 38, 37/ S 23 972289 2231 998080 20 974209 2251 025791 S 24 973628 2224 998068 20 975560 2244 024440 36/ ( 25 974902 2217 998056 20 970906 2237 023094 35/ S 26 976293 2210 998044 20 978248 2230 021752 34 / \ 27 977619 2203 998032 20 979586 2223 020414 33/ ( 28 978941 2197 998020 20 980921 2217 019079 32/ ( 29 980259 2190 998008 20 982251 2210 017749 31 30 J } 30 981573 2183 997996 20 983577 2204 016423 )31 > 32 8-982883 2177 9-997984 20 8-984899 2197 11015101 29 I 984189 2170 997972 20 986217 2191 013783 28 ( ) 33 985491 2163 997959 20 987532 2184 012468 27 ( ) 34 986789 2157 997947 20 988842 2178 011158 26 ( S 35 988083 2150 997935 21 990149 2171 009851 25 ( S 36 989374 2144 997922 21 991451 2165 008549 24 ( <37 ) 38 990660 2138 997910 21 992750 2158 007250 23 ( 991943 2131 997897 21 994045 2152 005955 22( ) 39 993222 2125 997885 21 995337 2146 004663 21 ( S 40 994497 2119 997872 21 996624 2140 003376 20 ( $41 8-995768 2112 9-997860 21 8-997908 2134 11-002092 19 J ) 42 997036 2106 997847 21 999188 2127 000812 18) ) 43 998299 2100 997835 21 000465 2121 10-999535 17 ) ) 44 999560 2094 997822 21 001738 2115 998262 16) ) 45 9-000816 2087 997809 21 003007 2109 996993 15 S ) 46 002069 2082 997797 21 004272 2103 995728 14 S > 47 003318 2076 997784 21 005534 2097 994466 13 ) ) 48 004563 2070 997771 21 006792 2091 993208 12) \ 49 005805 2064 997758 21 008047 2085 991953 11 ) S 50 007044 2058 997745 21 009298 2080 990702 10 j I 51 8-008278 2052 8-997732 21 8010546 2074 10-989454 9 ! ) 52 009510 2046 997719 21 011790 2068 983210 8/ ) 53 010737 2040 997706 21 013031 2062 986969 T( ) 54 011962 2034 997693 22 014268 2056 985732 6/ ) 55 013182 2029 997680 22 015502 2051 984498 5/ / 56 >57 014400 2023 997667 22 016732 2045 983268 4 ) 015613 2017 997654 22 017959 2040 982041 3/ ) 58 016824 2012 997641 22 019183 2033 980817 2 ) ) 59 018031 2006 997628 22 020403 2028 979597 1 > ) 60 1 019235 2000 997614 22 021620 2023 973380 0? Cosine I I Cotang. Tang. I M. 84 Degrees. 142 (6 Degrees.) A TABLE OF LOGARITHMIC n*. | Sine I D. | Cosine 1 D. 1 Tang. 1 D. Cotang. 1 ) o 9019235 2000 0997614 22 9-021620 2023 10-978380 60 ) i i 020435 1995 997601 22 022834 2017 977166 59 ) ) o 021632 1989 997588 22 024044 2011 975956 58 ) ! 3 022825 1984 997574 22 025251 2006 974749 57 ) I 4 024016 1978 997561 22 026455 2000 973545 56 ) / 5 025203 1973 997547 22 027655 1995 972345 55 ) / 6 026386 1967 997534 23 028852 1990 971148 54 ) ^ 7 027567 1962 997520 23 030046 1985 969954 53 S 8 028744 1957 997507 23 031237 1979 968763 52 ) S 9 029918 1951 997493 23 032425 1974 967575 51 ) ) W 031089 1947 997480 23 033609 1969 966391 50 J ( H 9032257 1941 9-997466 23 9034791 1964 10-965209 49 ) ( 12 033421 1936 997452 23 035969 1958 964031 48 ) < 13 034582 1930 997439 23 037144 1953 962856 47 ) < 14 035741 1925 997425 23 C38316 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 1933 958187 43 ) < 18 040342 1905 997369 23 C42973 1928 957027 42 ) ( 19 041485 1899 997355 23 044130 1923 955870 41 ) (20 042625 1894 997341 23 045284 1918 954716 40 ) >21 9 043762 1889 9-997327 24 9-046434 1913 10-953566 39 ( ) 22 044895 1884 997313 24 047582 1908 952418 38 ( > 23 046026 1879 997299 24 048727 1903 951273 37 < S 24 047154 1875 997285 24 049869 1898 950131 36 ( ) 25 048279 1970 997271 24 051008 1893 948992 35 ( ) 26 049400 1865 997257 24 052144 1889 947856 34 \ ) 27 050519 1860 997242 24 053277 1884 946723 33 I > 28 051635 1855 997228 24 054407 1879 945593 32 ( ) 29 052749 1850 997214 24 055535 1874 944465 31 I j 30 053859 1845 997199 24 056659 1870 943341 30 J ( 31 054966 1841 9997185 24 9057781 1865 10-942219 29 j I 32 056071 1836 997170 24 058900 1869 941100 28 ) / 33 057172 1831 997156 24 060016 1855 939984 27 ) (34 > 35 058271 1827 997141 24 061130 1851 938870 26 ) 059367 1822 997127 24 062240 1846 937700 25 ) I 36 060460 1817 997112 24 063348 1842 936652 24 ) / 37 061551 1813 997098 24 064453 1837 935547 23 ) ( 38 062639 1808 997083 25 065556 1833 934444 22 ) < 39 063724 1804 997068 25 066655 1828 833345 21 ) ( 40 064806 1799 997053 25 067752 1824 932248 20 S ( 41 9065885 1794 9997039 25 9-068846 1819 10-931154 19 j ( 42 066902 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 07UI76 1777 996979 25 073197 1802 926803 15 ) ( 46 071242 1772 996964 25 074278 1797 925722 14 ) ( 47 072306 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 075480 1755 996904 25 078576 1780 921424 10 ) ) 51 9-076533 1750 9-996P89 25 9079644 1776 10920356 9 ( ) 52 077583 1746 996874 25 080710 1772 919290 8 ( ) 53 078631 1742 996858 25 081773 1767 918227 7 < ) 54 079676 1738 996843 25 082833 1763 917167 6 { , >55 080719 1733 996828 25 083891 1759 916109 5 ( ) 56 081759 1729 996812 26 084947 1755 915053 4 { ) 57 082797 1725 996797 26 086000 1751 914000 3 { 58 083832 1721 996782 26 087050 1747 912950 2 ( ) 59 084864 1717 996766 26 088098 1743 911902 1 ( 1 60 085894 1713 996751 26 08tl44 1738 910856 / Sine | Cotang. | 83 Degrees. ^Tjuig^ UlLj sines and tangents. (7 Degrees.) 143 1 2 I 4 5 | Cosine | I Tanp. D. | Cotanff. 9-08.5894 1713 9-996751 26 9089144 1738 10-9108.56 60/ 080922 1709 996735 26 090187 1734 909813 59) 087947 1704 996720 26 091228 1730 908772 58) 088970 1700 996704 26 092206 1727 907734 57) 089990 1696 996688 26 093302 1722 906698 56) 091008 1692 996673 26 094336 1719 905664 55) 092024 1688 996657 26 095367 1715 904633 34) 093037 1684 996641 26 096395 1711 903605 53) 094047 1680 996625 26 097422 1707 902578 52) 095056 1676 996610 26 098446 1703 901554 51 ) 096062 1673 996594 26 099468 1699 900532 50) 9097065 1668 9-996578 27 9-100487 1695 10899513 49 < 098066 1665 996562 27 101504 1691 898496 48? 099065 1661 996546 27 102519 1687 897481 47? 101)062 1657 996530 27 103532 1684 896468 46/ 101056 1653 996514 27 104542 1680 895458 45? 102048 1649 996498 27 105550 1676 894450 44 ? 103037 1645 996482 27 106556 1672 893444 43? 104025 1641 996465 27 107559 1669 892441 42? 105010 1638 996449 27 108560 1665 891440 41 ' 105992 1634 996433 27 109559 1661 890441 40? 39< 9-106973 1630 9-996417 27 9110556 1658 10-889444 10T951 1627 996400 27 111551 1654 888449 38( 108927 1623 996384 27 112543 1650 887457 37 ( 109901 1619 996308 27 113533 1616 886467 36 < 110873 1616 996351 27 114521 1643 885479 35 ( 111842 1612 996335 27 115507 1639 884493 34 ( 112809 1608 996318 27 116491 1636 883509 33 ( 113774 1605 996302 28 117472 1632 882528 32 ( 114737 1601 996285 28 118452 1629 881548 31 ( 115698 1597 996209 28 119429 1625 880571 30 < 9110656 1594 9-996252 28 9120404 1622 10-879596 29 j 117613 1590. 996235 28 121377 1618 878623 28) 11S567 1587 996219 28 122348 1615 877652 27) 119519 1583 996202 28 123317 1611 876683 26) 120469 1580 996185 28 124284 1607 875716 25 S 121417 1576 996108 28 125249 1604 874751 24 S 122362 1573 996151 28 126211 1601 873789 23 J, 123306 1569 996134 28 127172 1597 872828 oo ( 124248 1566 996J17 28 128130 1594 871870 21 J 125187 1562 996100 28 129087 1591 870913 90 9126125 1559 9-996083 29 9-130041 1587 10-8J9959 869006 19) 127060 1556 996066 29 130994 1584 18) 127993 1552 996049 29 131944 1581 868056 17 ) 128925 1549 996C32 29 132893 1577 867107 16 ) 129854 1545 996015 29 133839 1574 866161 15) 14) 130781 1542 995998 29 134784 1571 865216 131706 1539 995980 29 135726 1567 864274 13) 132630 1535 995963 29 136607 1564 863333 12) 133551 1532 995946 29 137605 1561 862395 11 134470 1529 995928 29 138542 1558 861458 10 J 9135387 1525 9-99591 1 29 9139476 1555 10-860524 9 I 136303 1522 995894 29 140409 1551 859591 8 I 137216 1519 995876 29 141340 1548 858660 7 < 138128 1516 995859 29 142269 15-15 857731 6 I 139037 1512 995841 29 143196 1542 856804 5 ( 139944 1509 995823 29 144121 1539 855879 4 ? 140850 1506 995806 29 145044 1535 854956 3 ? 141754 1503 995788 29 145966 1532 K54034 2? 142655 1500 995771 29 146885 1529 853115 1 ) 143555 1496 995753 29 147803 1526 &52197 ) S2 Degrees. 144 (8 Degrees.) a TABLE OF LOGARITHMIC F"mT | Sine D. Cosine I D. Tan?. 1 D. Cotang. r^i s o 9143555 1496 9995753 30 9-147803 1526 10.852197 60 I 59 ( ( i 144453 1493 995735 30 148718 1523 851282 S 2 145349 1490 995717 30 149632 1520 850368 58 ( S 3 14(i243 1487 995699 30 150544 1517 849456 57 / \ 4 147136 1484 995681 30 151454 1514 848546 56 ( \ 5 148026 1481 995664 30 152363 1511 847637 55 I ( 6 148915 1478 995646 30 153209 1508 846731 54 I S 7 149802 1475 995628 30 154174 1505 845826 53 J S 8 130686 1472 995610 30 155077 1502 844923 52 I S 9 151569 1469 995591 30 155978 1499 844032 51 / S 10 152451 1466 995573 30 156877 1496 843123 50 { > H 9153330 1463 9-995555 30 9157775 1493 10-842225 49 I ) 13 154208 14(30 995537 30 158671 1490 841329 48 { S 13 155083 1457 995519 30 159565 1487 840435 47 ( ) I 4 155957 1454 995501 31 160457 1481 839543 46 ( ) I 5 156830 1451 995482 31 161347 1481 838653 45 I S 16 157700 1448 995464 31 162236 1479 837764 44 I S 1 7 158569 1445 995446 31 163123 1476 836877 43 ( ) 18 159435 1442 995127 31 164008 1473 835992 42 { < 19 160301 1439 995409 31 164892 1470 835108 41 ( j 20 161164 1436 995390 31 165774 1467 834226 40 ) 21 9-162025 1433 9-995372 31 9-1G6654 1464 10-833346 39 { ) 22 102885 1430 995353 31 167532 1461 832468 38 ( ) 23 163743 1427 995334 31 168409 1458 831591 37 ( ) 24 164600 1424 995316 31 169284 1455 830716 36 < ) 25 165454 1422 995297 31 170157 1453 829843 35 < > 20 106307 1419 995278 31 1710-29 1450 828971 34 ' > 27 167159 1416 995260 31 171899 1447 828101 33 ( ) 28 $29 168008 1413 995241 32 172767 1444 827233 32 < 168858 1410 995222 32 173634 1442 826366 31 ( J 30 169702 1407 995203 32 174499 1439 825501 30 ( ? 31 9170547 1405 9-995184 32 9-1753G2 1436 10824638 29 > ) 32 171389 1402 995165 32 176224 1433 823776 28 ) > 33 172230 1399 995146 32 177084 1431 822916 27 S ) 34 173070 1396 995127 32 177942 1428 822058 26 > ) 35 173908 1394 995108 32 178799 1425 821201 25 ) ) 36 174744 1391 995089 32 179055 1423 820345 24 ) ) 37 175578 1388 995070 32 180508 1420 819492 23 S ) 38 176411 1388 995051 32 181360 1417 818640 22 S ) 3!) 177242 1383 995032 32 182211 1415 817789 21 ) ) 40 178072 1380 995013 32 183059 1412 816941 20 S ) 41 9178900 1377 9-994993 32 9-183907 1409 10-816093 19 ) ) 42 17972G 1374 994974 32 184752 1407 815248 18 > ) 43 180551 1372 994955 32 185597 1404 814403 17 ) 44 181374 1369 994935 32 186439 1402 813561 16 ) ; 45 182196 1366 994916 33 187280 1399 812720 15 ) ) 46 183016 1364 994896 33 188120 1396 811880 14 ) I 47 183834 1361 994877 33 188958 1393 811042 13 ) ) 48 184651 1359 994857 33 189794 1391 810206 12 ) ) 49 185466 1356 994838 33 190629 1389 809371 11 ) / 50 ) 51 186280 1353 994818 33 191462 1386 808538 10 > 9-187092 1351 9-994798 33 9-192294 1384 10-807706 9 ( 1 52 187903 1348 994779 33 193124 1381 806870 8 I I 53 188712 1346 994759 33 193953 1379 806047 7 ( }54 1895J9 1343 994739 33 194780 1376 805220 6 I ( 55 190325 1341 994719 33 195606 1374 804394 5 I I 56 191130 1338 994700 33 196430 1371 803570 4 I >57 191933 1336 994680 33 197253 1369 802747 3 > 58 192734 1333 9946)50 33 198074 1366 801926 2 > \ 59 193534 1330 994640 33 198894 1364 801106 1 ) \ 60 194332 1328 994620 33 199713 1361 800287 > Cosine | Sine .^1 Cotang-. .~^~ ,^Tan&^ ^mJ 81 Degrees. SINES AND TANGENT! (9 Degrees.) 145 Tm. 1 Sine 1 D. | Cosine 1 D. Tan*. 1 D. Cotanff. . ( o 19-194332 1328 9-994020 33 9 199713 1361 10-800287 60 ( 1 105129 1326 994600 33 200529 1359 799471 59 ( 2 1£5925 1323 994580 33 201345 1356 798655 58 ( 3 196719 1321 994560 34 202159 1354 797841 57 ( ( 4 1 197511 1318 994540 34 202971 1352 797029 56 ( ( 5 198302 1316 994519 34 203782 1349 796218 55? ( 6 199091 1313 994499 34 204592 1347 795408 54? S 7 199879 1311 994479 34 205400 1345 794600 53? < 8 2(10666 1308 994459 34 206207 1342 793793 52? \ 9 201451 1306 994438 34 207013 1340 792987 51? ) 10 202234 1304 994418 34 207817 1338 792183 50? > H 9203017 1301 9-994397 34 9-208619 1335 10-791381 49 ( ) & 203797 1299 994377 34 200420 1333 790580 48 ( > 13 204577 1296 994357 34 210220 1331 789780 47 ( > 14 205354 1294 994336 34 211018 1328 788982 41) ( > i5 206131 1292 994316 34 211815 1326 788185 45 ( S 16 206906 1289 994295 34 212611 1324 787389 44 ( ) I 7 207679 1287 994274 35 213405 1321 786595 43 ( > 18 208452 1285 994254 35 214198 1319 785802 42 { ) I 9 209222 1282 994233 35 214989 1317 785011 41 ( S 20 209992 1280 994212 35 215780 1315 784220 40 ( ,1 22 9-210760 1278 9-994191 35 9216568 1312 10-783432 39 j 38 < 211526 1275 994171 35 217356 1310 782644 ,l 23 212291 1273 994150 35 218142 1308 781858 37 S ) 24 213055 1271 994129 35 218926 1305 781074 36 \ > 25 213818 1268 994108 35 219710 1303 780290 35$ ) 26 214579 1266 994087 35 220492 1301 779508 34 \ ) 27 215338 1264 994066 35 221272 1299 778728 33S ) 28 216097 1261 994045 35 222052 1297 777948 32 ( > 29 216854 1259 994024 35 222830 1294 777170 31 ( j 30 217609 1257 994003 35 223606 1292 776394 30 < ( 31 9-218363 1255 9-993981 35 9-224382 1290 10-775618 29) ( 32 219116 1253 993960 35 225156 1288 774844 28) ( 33 219868 1250 993939 35 225929 1286 774071 27) S 34 220018 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 S ( 38 223606 1239 993832 36 229773 1275 770227 22 ) ( 39 224349 1237 993811 36 230539 1273 769461 21 ) ( 40 225092 1235 993789 36 231302 1271 768698 20 j > 41 9225833 1233 9-993768 36 9-232065 1269 10-767935 19.! S 42 226573 1231 993746 36 232826 1267 767174 18»i > 43 227311 1228 993725 36 233586 1265 766414 17) ) 44 228048 1226 993703 36 234345 1262 765655 16 ) ) 45 228784 1224 993681 36 235103 1260 764897 15 ) ) 46 229518 1222 993660 36 235859 1258 764141 14 > 13, ) 47 230252 1220 993638 36 236614 1256 763386 >48 230984 1218 993616 36 237368 1254 762632 12 \49 231714 1216 993594 37 238120 1252 761880 11) I 50 232444 1214 993572 37 238872 1250 761128 10 j 51 9-233172 1212 • 9-993550 37 9-239622 1248 10-760378 9( >52 233899 1209 993528 37 240371 1246 759629 8 > J 53 234625 1207 993506 37 241118 1244 758882 7 > 54 235349 1205 993484 37 241865 1242 758135 6 > 55 236073 1203 993462 37 242610 1240 757390 5 ) 56 236795 1201 993440 37 243354 1238 756646 4 ) 57 237515 1199 993418 37 244097 1236 755903 3 ) 58 238235 1197 993396 37 244839 1234 755161 2 > 59 238953 1195 993374 37 245579 1232 754421 1) ) 60 239670 1193 993351 37 246319 1230 753681 ° L-^ Cosine * -*»-^ <^^- Sine | Cotanff. r J ^•^1 51 80 Degrees. tf>~ 146 (10 Degrees.; a table of logarithmic M. 1 •2 a 4 9 (i 7 8 B 10 11 19 j:j 14 15 16 17 18 19 20 31 SS S3 34 '25 96 27 2d 20 30 31 32 33 34 35 36 37 38 30 40 41 42 43 .' 44 45 4li 47 4^ 41 50 51 (52 ( 53 < 54 ( 55 < 50 ( 57 ( 58 < 51) ( 60 Sine i D- Cosine D. Tansf. 1 D- Cotang. 9239670 1193 9-903351 37 9-246319 1230 10-753681 60 240386 1191 9J3329 37 247057 1228 752943 59 241101 1189 993307 37 247794 1226 752206 58 241814 1187 993285 37 248530 1224 751470 57 242526 1185 993262 37 249264 1222 750736 56 243237 1183 993240 37 249998 1220 750002 55 243947 1181 993217 38 250730 1218 749270 54 244656 1179 993195 38 251461 1217 748539 53 245363 1177 993172 38 252191 1215 747809 52 246069 1175 993149 38 252920 1213 747080 51 246775 1173 993127 38 253648 1211 740352 50 9-247478 1171 9-993104 38 9-2.54374 1209 10-745626 49 248181 1169 993081 38 255100 1207 744900 48 248883 1167 993059 38 255824 1205 744176 47 24J583 1165 993036 38 256547 1203 743453 46 250282 1163 903013 38 257269 1201 742731 45 250980 1161 992990 38 257990 1200 742010 44 251677 1159 902967 38 258710 1198 741290 43 252373 1158 992944 38 259429 1196 740571 42 253067 1156 902921 38 260146 1194 739854 41 253761 1154 992898 38 260863 1192 739137 40 9254453 1152 9-902875 38 9-261578 1190 10-738422 39 255144 1150 992852 38 262292 1189 7377u8 38 255834 1148 992829 30 263005 1187 736995 37 256523 1146 902806 39 263717 1185 736283 36 257211 1144 9J2783 39 264428 1183 735572 35 257898 1142 992759 39 265138 1181 734802 34 258583 1141 992736 39 265847 1179 734153 33 259268 1139 992713 39 260555 1178 733445 32 259951 1137 992690 39 267261 1176 732739 31 260633 1135 992o66 39 267967 1174 732033 30 9-261314 1133 9-992643 39 9-268671 1172 10731329 29 261994 1131 992619 39 2U9375 1170 730625 28 262673 1130 992596 39 270077 1169 729923 27 263351 1128 992572 39 270779 1167 729221 26 264027 1126 992549 39 271479 1165 728521 25 204703 1124 992525 39 272178 1164 727822 24 265377 1122 992501 39 272876 1162 727124 23 266051 1120 9J2478 40 273573 1160 726427 22 266723 1119 9J2454 40 274269 1158 725731 21 267395 1117 992430 40 274i>64 1157 725036 20 9-268065 1115 9-992406 40 9-275658 1155 10-724342 19 268734 1113 99;J382 40 276351 1153 723649 18 269402 1111 992359 40 277043 1151 722957 17 270069 1110 992335 40 277734 1150 722266 16 270735 1108 992311 40 278424 1148 721576 15 271400 1106 992287 40 279113 1147 720887 14 < 272064 1105 992263 40 279801 1145 720199 13 < 272726 1103 992239 40 280488 1143 719512 12 < 273388 1101 992214 40 281174 1141 718826 11 274049 1199 992190 40 281858 1140 718142 10 < 9274708 1098 9-992166 40 9-282542 1138 10-717458 9 275367 1096 992142 40 283225 1136 716775 8 ( 276024 1094 992117 41 283907 1135 7160"3 7 . 276681 1092 992093 41 284588 1133 715412 6 277337 1001 992069 41 285268 1131 714732 5 277991 1689 992044 41 285947 1130 714053 4 278644 1087 992020 41 286624 1128 713376 3 279297 1086 991996 41 287301 1126 712609 2 279948 1084 991971 41 287977 1125 712023 1 286599 1082 991947 41 288652 1123 711348 • 70 Degrees. SINES AND TANGENTS. (11 Degrees.) U7 Pm^ | Sins 1 D- Cosine D. Tang. D. Cotang. ; i 9-280599 1082 9-991947 41 9-288652 1123 10-711348 60 J I 1 281248 1081 991922 41 289326 1122 710674 59 P I 2 281897 1079 991897 41 289999 1120 710001 58) ) 3 282544 1077 991873 41 290671 1118 709329 57) ? 4 283190 1076 991848 41 291342 1117 708658 56? ) 5 283836 1074 991823 41 292013 1115 707987 55? ) 6 284480 1072 991799 41 292682 1114 707318 54/ ) 7 285124 1071 991774 42 293350 1112 706650 53) ) 8 285766 1069 991749 42 294017 1111 705983 52? / 9 286408 1067 991724 42 294684 1109 705316 51 ) ) 10 287048 1066 991699 42 295349 1107 704051 50, 11 9-287687 1064 9-991674 42 9-296013 1106 10-703987 49 < 12 288326 1063 991649 42 296677 1104 703323 48 < / 13 288904 1061 991624 42 297339 1103 702661 47 < < U 289600 1059 991599 42 298001 1101 701999 46 ( 15 290236 1058 991574 42 298662 1100 701338 45 ( 1(5 290870 1056 991549 42 299322 1098 700678 44 ( 17 291504 1054 991524 42 299980 1096 700020 43 ( 18 292137 1053 991498 42 300638 1095 699302 42 ( > 19 292768 1051 991473 42 301295 1093 698705 41 ( ) 20 293399 1050 991448 42 301951 1092 698049 40 ( i 21 9294029 1048 9-991422 42 9-302607 1090 10-697393 39 ) J 22 ( 23 f 24 294658 1046 991397 42 303261 1089 690739 38 S 295286 1045 991372 43 303914 1087 696086 37) 295913 1043 991346 43 304567 1086 695433 36 S '25 296539 1042 991321 43 305218 1084 694782 35 ) ( 2(j 297164 1040 991295 43 305869 1083 694131 34 ) 27 297788 1039 991270 43 306519 1081 693481 33 Hs 298412 1037 991244 43 307168 1080 092832 32 \ i 20 299034 1036 991218 43 307815 1078 692185 31 < | 30 299655 1034 991193 43 308463 1077 691537 30 J ! 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 689602 27) 34 302132 1028 991090 43 311042 1071 688958 26) 35 302748 1026 991064 43 311685 1070 688315 25) 3fi 303364 1025 991038 43 312327 1068 687673 24 ) 37 303979 1023 991012 43 312967 1067 687033 23 > 38 304593 1022 990986 43 313608 1065 686392 22) 39 305207 1020 990960 43 314247 1064 685753 21 ) 40 305819 1019 990934 44 314885 1062 685115 20) 41 9-306430 1017 9-990908 44 9 315523 1061 10684477 19 J 42 307041 1016 990882 44 316159 1060 683841 18 ) 43 307650 1014 990855 44 316795 1058 683205 17 ) 44 308259 1013 990829 44 317430 1057 682570 16) 45 308867 1011 990803 44 318064 1055 681936 15) 4(5 309474 1010 990777 44 318697 1054 681303 14 ) 47 310080 1008 990750 44 319329 1053 680671 13) 48 310685 1C07 990724 44 319961 1051 680039 12) 49 311289 1005 990697 44 320592 1050 679408 11 I i 50 311893 1004 990671 44 321222 1048 678778 10 ) 1 51 9312495 1003 9-990644 44 9-321851 1047 10-678149 A ) 52 313097 1001 990618 44 322479 1045 677521 8 53 313698 1000 990591 44 323106 1044 676894 7 54 314297 998 990565 44 323733 1043 676267 6/ 55 314897 997 990538 44 324358 1041 675642 5 < ) 56 315495 996 990511 45 324983 1040 675017 4\ ) 57 316092 994 990485 45 325007 1039 674393 3 ^ 58 316689 993 990458 45 326231 1037 673769 ! 1 59 317284 991 990431 45 326853 1036 673147 ) 60 317879 990 1 990404 ' 45 327475 1035 672525 On _L*b> 148 (12 Degrees.) A TABLE OF LOGARITHMIC 9-317879 990 9-990404 45 9-327474 1035 10672526 60 1 318473 988 990378 45 328095 1033 671905 59 2 319066 987 990351 45 328715 1032 671285 58 3 319G58 986 990324 45 329334 1030 670666 57 4 320249 984 990297 45 329953 1029 670047 56 5 320840 983 990270 45 330570 1028 669430 55 6 321430 982 990243 45 331187 1026 668813 54 7 322019 980 990215 45 331803 1025 668197 53 8 322607 979 990188 45 332418 1024 6G7582 52 9 323194 977 990161 45 333033 1023 666967 51 10 323780 976 990134 45 333646 1021 G66354 50 11 9324366 975 9-990107 46 9334259 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 989997 46 336702 1015 663298 45 16 327281 968 989970 46 337311 1013 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 339133 1010 660867 41 20 329599 962 989860 46 339739 1008 660261 40 21 9330176 961 9-989832 46 9-340344 1007 10659656 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 989665 47 343958 999 656042 33 28 334195 952 989637 47 344558 998 655442 32 29 334766 950 989609 47 345157 997 654843 31 30 335337 949 989582 47 345755 996 654245 30 31 9335906 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 989409 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 649486 22' 39 340434 937 989328 47 351106 985 648894 21 < 40 340996 936 989300 47 351697 983 648303 20 « 41 9341558 935 9-989271 47 9-352287 982 10 647713 19 ! 42 342119 934 989243 47 352876 981 647124 18 43 342679 932 989214 47 353465 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 644773 14 47 344912 927 989100 48 355813 975 644187 13) 48 345469 926 989071 48 356398 974 643602 12 S 49 346024 925 989042 48 356982 973 643018 " J 50 346579 924 989014 48 357566 971 642434 10 s 51 9-347134 922 9-988985 48 9-358149 970 10-641851 9 ? 52 347687 921 988956 48 358731 969 641269 6 f 53 348240 920 988927 48 359313 968 640687 ?? 54 348792 919 988898 48 359893 967 640107 6 ? 55 349343 917 988869 48 360474 966 639526 5 ) 5G 349893 916 988840 48 361053 965 638947 4/ 57 350443 915 988811 49 361632 963 638368 3 ? 58 350992 351540 914 988782 49 362210 962 637790 2> 59 913 988753 49 362787 961 637213 1 60 352088 911 988724 49 363364 960 636636 ° | M. 77 Degrees. SINES AND tangents. (13 Degrees.) D. I Cosine I Tang. | Cotang. | { o 9-352088 911 9-988724 49 9363364 960 10-636036 60) S i 352635 910 98S695 49 363940 959 636060 59 ) \ 2 353181 909 988666 49 364515 958 635485 58 ) S 3 353726 908 988636 49 365090 957 634910 57 ) 354271 907 988607 49 365664 955 634336 56) 354815 905 988578 49 366237 954 633763 55 ) 355358 904 988548 49 366810 953 633190 54 ) 1 7 355901 903 988519 49 367382 952 632618 53) I 8 356443 902 988489 49 367953 951 632047 52) I 9 356984 901 988460 49 368524 950 631476 51 ) MO 357524 899 988430 49 369094 949 630906 50 \ ) U 9358064 898 9-988401 49 9-369663 948 10-630337 49) ) 12 358603 897 988371 49 370232 946 629768 48/ ) I 3 359141 896 988342 49 370799 945 629201 47 > ( 14 359(578 895 988312 50 371367 944 628633 46 ) ( 15 360215 893 988282 50 371933 943 628067 45 ) < 16 360752 892 988252 50 372499 942 627501 44 ) ( 17 361287 891 988223 50 373064 941 626936 43 ) ( 18 361822 890 988193 50 373029 940 626371 42 ) < 19 362356 889 988163 50 374193 939 625807 41 ) (20 362889 888 988133 50 374756 938 625244 40 ) )21 9363422 887 9-988103 50 9-375319 937 10-624681 39 ( )22 363954 885 988073 50 375881 935 624119 38 ( S 23 364485 884 988043 50 376442 934 623558 37 < >24 (25 365016 883 988013 50 377003 933 622997 36 < 365546 882 987983 50 377563 932 622437 35 < >26 (27 ,28 J 29 \30 366075 881 987953 50 378122 931 621878 34 < 366604 880 987922 50 378681 930 621319 33 ( 367131 879 987892 50 379239 929 620761 32 ( 367659 877 987862 50 379797 928 620203 31 ( 368185 876 987832 51 380354 927 619646 30/ > 31 9-368711 875 9-987801 51 9-380910 926 10-619090 29( >32 369236 874 987771 51 381466 925 618534 28 < >33 369761 873 987740 51 382020 924 617980 27 < >34 370285 872 987710 51 382575 923 617425 26 < >35 370808 871 987679 51 383129 922 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 385337 918 614663 21 ( >40 373414 865 987526 51 385888 917 614112 20 < 41 9-373933 864 9-987496 51 9-386438 915 10-613562 19 / (42 374452 863 987465 51 386987 914 613013 18 ) ?43 374970 862 987434 51 387536 913 612464 17 ) <44 375487 861 987403 52 388084 912 611916 16 S ?45 376003 860 987372 52 388631 911 611369 15 > (46 376519 859 987341 52 389178 910 610822 14 ) ?47 377035 858 987310 52 389724 909 610276 13 ) ?48 377549 857 987279 52 390270 908 609730 12 '» ?49 378063 856 987248 52 390815 907 609185 11 ( i ?50 378577 854 987217 52 391360 906 608640 10 (51 9-379089 853 9-987186 52 9-391903 905 10-608097 9 i 1 (52 379601 852 987155 52 392447 904 607553 8 , (53 380113 851 987124 52 392989 903 607011 1 i 1 J54 380624 850 987092 52 393531 902 606469 6, 3, (55 381134 849 987061 52 394073 901 605927 56 381643 848 987030 52 394614 900 605386 ?57 382152 847 986998 52 395154 899 604846 (58 382661 846 986967 52 395694 898 604306 (59 383108 845 986936 52 396233 897 603767 (60 383675 844 986904 52 396771 896 603229 ° l~ Cosine Sine Cotan|. I 1 ^Tang^j ^j 76 Degreea 150 (14 Degree s.) A TABLE OP LOGARITHMIC ^ST Sine 1 »• | Cosme 1 D. 1 Tang. 1 D. Cotang. \ ( ° 9383675 844 9-986904 52 9396771 896 10-603229 60 , l 384182 843 986873 53 397309 896 602691 59 . 2 384687 842 986841 53 397846 895 602154 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 53 8 387709 836 986651 53 401058 889 598942 52 9 388210 835 986619 53 401591 888 598409 51 1 10 388711 834 986587 53 402124 887 597876 50 11 9389211 833 9986555 53 9-402656 886 10*597344 49 12 389711 832 986523 53 403187 885 596813 48 13 390210 831 986491 53 403718 884 596282 47 14 390708 830 986459 53 404249 883 595751 46 15 391206 828 986427 53 404778 882 595222 45 16 391703 827 986395 53 405308 881 594092 44 17 392199 826 986303 54 405836 880 594164 43 18 392695 825 986331 54 406364 879 593636 42 19 393191 824 986299 54 406892 878 593108 41 20 393685 823 986266 54 407419 877 592581 40 21 9394179 822 9-986234 54 9-407945 876 10-592055 39 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 390150 818 986104 54 410045 873 589955 35 26 396641 817 986072 54 410569 872 589431 34 27 397132 817 986039 54 411092 871 588908 33 28 397621 816 986007 54 411615 870 588385 32 29 398111 815 985974 54 412137 869 587863 31 30 398600 814 985942 54 412658 868 587342 30 31 9-399088 813 9-985909 55 9-413179 867 10-586821 29 32 399575 812 98.5876 55 413699 866 586301 28 33 400062 811 985843 55 414219 8C5 585781 27 34 400549 810 985811 55 414738 864 585262 26 35 401035 809 985778 55 415257 864 584743 25 36 401520 808 985745 55 415775 863 584225 24 37 402005 807 985712 55 416293 862 583707 23 38 402489 806 985679 55 416810 861 583190 22 < 39 402972 805 985646 55 417326 860 582674 21 40 403455 804 985613 55 417842 859 582158 20 41 9-403938 803 9-985580 55 9-418358 858 10581642 19 j 42 404420 802 985547 55 418873 857 581127 18 43 404901 801 985514 55 419387 856 580613 17 44 405382 800 985480 55 419901 855 580099 16 45 405862 799 985447 55 420415 855 579585 15 < 46 406341 798 985414 56 420927 854 579073 14 ) 47 406820 797 985380 56 421440 853 578560 13 48 407299 796 985347 56 421952 852 578048 12 49 407777 795 985314 56 422463 851 577537 11 50 408254 794 985280 56 422974 850 577026 10 51 9-408731 794 9-985247 56 9-423484 849 10-576516 »! 52 409207 793 985213 56 423993 848 576007 8 53 409682 792 985180 56 424503 848 575497 7 ) 54 410157 791 985146 56 425011 847 574989 6 J 55 410632 790 985113 56 425519 846 574481 5 J 56 411106 789 985079 56 426027 845 573973 4 5 57 411579 788 985045 56 426534 844 573466 3 ) 58 412052 787 985011 56 427041 843 572959 2 59 412524 786 984978 56 427547 843 572453 1 60 412996 785 984944 56 428052 842 571948 J L~^ Cosin* ^^^J Sine | I Cotang. ! ^Tang^J M. , 75 Degre es. AND TANGENTS. tl5 De rees.) 151 pr | Sine D. Cosine D. | Tane. i D. Cotang. 1 i ) o 9412996 785 9-984944 57 9-428052 842 10571948 1 60 < 413467 784 984910 57 428557 841 571443 59 I i 2 413938 783 984876 57 429002 840 570938 58 ( 57 56 ) 3 414408 783 984842 57 429506 839 570434 ) 4 414878 782 984808 57 430070 838 569930 i 5 415347 781 984774 57 4:'0573 838 569427 55 ( < 6 415815 780 984740 57 431075 837 568925 54 ( S 7 416283 779 98470(3 57 431577 836 568423 53 I ( 8 416751 778 984672 57 432079 835 567921 52 I ( 9 417217 777 984637 57 432580 834 567420 51 ( <10 417684 776 984603 57 433080 833 566920 50 j i n 9418150 775 9-984569 57 9-433580 832 10'5664SO 49 < ) 12 418015 774 984535 57 434080 832 5(55920 48 ) ) 13 419079 773 984500 57 434579 831 565421 47 ( ? u 419544 773 984466 57 435078 830 564922 46 ( ) 15 420007 772 984432 58 435576 829 564424 45 ( > 1G 420470 771 984297 58 436073 828 £63927 44 ( ) 17 420933 770 984363 58 436570 828 563430 43 ( S 18 421395 769 984328 58 437067 827 562933 42 ( \ 19 421857 768 984294 58 437563 826 562437 41 ( j 20 422318 767 984259 58 438059 825 561941 40 ( J 21 9422778 767 9-984224 58 9-438554 824 10-561446 39 J ) 22 423238 766 984190 58 439048 823 5G0952 38 ) ) 23 423697 765 984155 58 439543 823 560457 37 S / 24 424156 764 984120 58 440036 822 559964 36 ) ) 25 424615 763 984085 58 440529 821 559471 35 ) ) 26 425073 762 984050 58 441022 820 558978 34 S ) 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 426899 759 983911 58 442988 817 557012 30 < 1 31 9427354 758 9-983875 58 9-443479 816 10-556521 29 j ( 32 427809 757 983840 59 443968 816 556032 28 ) I 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 554565 25 ) ( 36 ) 37 429623 753 983700 59 445923 812 554077 24 ) 430075 752 983664 59 446411 812 553.589 23 ) I 38 ) 39 430527 752 983629 59 446898 811 553102 22 ) 430978 751 983594 59 447384 810 552616 21 ) MO 431429 750 983558 59 447870 809 552130 20 J (41 9-431879 749 9983523 59 9-448356 809 10-551644 19 \ <42 432329 749 983487 59 448841 808 551159 18 ) <43 432778 748 983452 59 449326 807 550674 I 7 ( ( 44 433226 747 983416 59 449810 806 550190 16 ) < 45 433675 746 983381 59 450294 806 549706 15 ) I 46 434122 745 983345 59 4.50777 805 549223 14 ) < 47 434569 744 983309 59 451260 804 548740 13 ; (4V 435016 744 983273 60 451743 803 548257 12 I 10 ) ?49 435462 743 983238 60 452225 802 547775 (5(. 435908 742 983202 60 452706 802 547294 ' 51 9-436353 741 9-983166 60 9-453187 801 10-546813 9 ( ( 52 436798 740 983130 60 453668 800 546332 8 ( (53 437242 740 983094 60 454148 799 545852 7 ( v 54 437686 739 983058 60 454628 799 545372 6 ( i 55 438129 738 983022 60 455107 798 544893 5 ( <■ 56 438572 737 982986 60 455586 797 544414 4 ( ( 57 439014 736 982950 60 456064 796 543936 3 ( I 58 . 439456 736 982914 60 456542 796 543458 2 ) (59 439897 735 982878 60 457019 795 542981 1 ) (60 440338 734 982842 60 457496 794 542504 > | Cosine | 74 Degrees. 152 (16 Degrees.) a table of logaritmic J ° Sine 9440338 " 1 D- 734 | Cosine | D. 1 Tang. | D. | Cotang. 1 9-982842 60 9-457496 794 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 541075 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 539651 54 ) 7 443410 728 982587 61 460823 789 539177 53 ) 8 443847 727 982551 61 461297 788 538703 52 ) 9 444284 727 982514 61 461770 788 538230 51 ) 10 444720 726 982477 61 462242 787 537758 50 i U 9445155 725 9-982441 61 9-462714 786 10537286 49 ) 12 445590 724 982404 61 463186 785 536814 48 ) 13 446025 723 982367 61 463658 785 536342 47 ) 14 446459 723 982331 61 464129 784 535871 46 ) 15 446893 722 982294 61 464599 783 535401 45 ) I 6 447326 721 982257 61 465069 783 534931 44 > I 7 447759 720 9*2220 62 465539 782 534461 43 ) 18 448191 720 9132183 62 466008 781 533992 12 ) I 9 448623 719 982146 62 466476 780 533524 41 J 20 449054 718 982109 62 466945 780 533055 40 J 21 9-449485 717 9-982072 62 9-467413 779 10-532587 39 ) 22 449915 716 982035 62 467880 778 532120 38 ) 23 450345 716 981998 62 468347 778 531653 37 ) 24 450775 715 981961 62 468814 777 531186 36 ) 25 451204 714 981924 62 469280 776 530720 35 ) 26 451G32 713 981886 62 469746 775 530254 34 ) 27 452060 713 981849 62 470211 775 529789 33 > 28 452488 712 981812 62 470676 774 529324 32 ) 29 452915 711 981774 62 471141 773 528859 31 J 30 453342 710 981737 62 471605 773 528395 30 1 31 9-453768 710 9-981699 63 9-472068 772 10-527932 29 > 32 454194 709 981662 63 472532 771 527468 28 ) 33 454619 708 981625 63 472995 771 527005 27 ) 34 455044 707 981587 63 473457 770 526543 28 } 35 455469 707 981549 63 473919 769 526081 25 ) 36 455893 706 981512 63 474381 769 525619 24 )37 456316 705 981474 63 474842 768 525158 23 ) 38 ) 39 456739 704 981438 63 475303 767 524697 22 457162 704 981399 63 475763 767 524237 21 j 40 457584 703 981361 63 476223 766 523777 20 ) 41 9-458006 702 9-981323 63 9-476683 765 10-523317 19 - ) 42 458427 701 981285 63 477142 765 522858 18 { ) 43 458848 701 981247 63 477601 764 522399 17 ) 44 459268 700 981209 63 478059 763 521941 16 W5 459688 699 981171 63 478517 763 521483 15 ) 46 460108 698 981133 64 478975 762 521025 14 \ 47 460527 698 981095 64 479432 761 520568 13 > 48 460946 697 981057 64 479889 761 520111 12 < v 49 461364 696 981019 64 480345 760 519655 11 J 50 461782 695 980981 64 480801 759 519199 It ) 51 9-462199 695 9-980942 64 9-481257 759 10 518743 9 > 52 462616 694 980904 64 481712 758 518288 8< ; 53 54 463032 693 980866 64 482167 757 517833 7 I 463448 693 980827 64 482621 757 517379 6 . 55 56 { 57 463864 692 980789 64 483075 756 516925 5 464279 691 980750 64 483529 755 516471 4 ) 464694 690 980712 64 483982 755 516018 3 ) , 58 465108 690 980673 64 484435 754 515565 2 . 59 465522 689 980635 64 484887 753 515113 1 60 465935 688 980596 64 485339 753 514661 °! 3Z&-J 73 Degrees G 1NES AND TANGENTS. (17 Degrees.) L53 "m. 1 Sine | ^D^ Cosine | D. Tang. | D. | Cotang. | ! ° 9-465935 1 688 9-980596 64 9-485339 755 10-514661 60 ( l i 466348 688 980558 64 485791 752 514209 59 ( 2 466761 687 980519 65 486242 751 513758 58 ( 3 467173 686 986480 65 486693 751 513307 57 ( ( 4 467585 685 980442 65 487143 750 512857 56 I I 5 467996 685 980403 65 487593 749 512407 55 ( I 6 468407 684 980364 65 488043 749 511957 54 ( 7 468817 683 980325 65 488492 748 511508 53 / / 8 469227 683 980286 65 488941 747 511059 52 | I 9 1 10 469637 682 980247 65 489390 747 510610 51 ; 470046 681 980208 65 489838 746 510162 50 , I ll 9-470455 680 9-980169 65 9-490286 746 10-509714 49 ) I 12 470863 680 980130 65 490733 745 509207 48 ( 13 471271 679 980091 65 491180 744 508820 47 I 14 471679 678 980052 65 491627 744 508373 46 } ( 15 472086 678 980012 65 492073 743 507927 45 ( ( 16 472492 677 979973 65 492519 743 507481 44 / ( 17 472898 676 979934 66 492965 742 507035 43 ( 18 473304 676 979895 66 493410 741 5CC590 42 ( { 19 473710 675 979855 66 493854 740 500146 41 ( j 20 474115 674 979816 66 494299 740 505701 40 J ) 21 9-474519 674 9-979776 66 9494743 740 10-505257 39* S 22 474923 673 979737 66 495186 739 504814 38 ) 23 475327 672 979697 66 495G30 738 504370 37 ) 24 475730 672 979658 66 496073 737 503927 36* \ 25 476133 671 979618 66 496515 737 503485 35 S 26 47G536 670 979579 66 496957 736 503043 34 S 27 476938 669 979539 66 497399 736 502601 33 S 28 477340 669 979499 66 497841 735 502159 32 S 29 477741 668 979459 66 498282 734 501718 31 j 20 478142 667 979420 66 498722 734 501278 30 31 9-478542 667 9-979380 66 9-499163 733 10-500837 29 ) 32 478942 666 979340 66 491)603 733 500397 28 ) 33 479342 665 9793C0 67 500042 732 499958 27 ) 34 479741 665' 979260 67 500481 731 499519 26 ) 35 480140 664 979220 67 50C920 731 499080 25 ) 36 480539 663 979180 61 601359 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 497328 21 ) 40 482128 661 979019 67 503109 728 496891 20 ( 41 9-482525 660 9-978979 67 9-503546 727 10-496454 19 ( 42 482921 659 978939 67 503982 727 496018 18 ( 43 483316 659 978898 67 504418 726 495582 17 ( 44 483712 658 978858 67 504854 725 495146 16 ( 45 484107 657 978817 67 505289 725 494711 15 I 46 484501 657 978777 67 505724 724 494276 14 ( 47 484895 656 978736 67 506159 724 493841 13 ( 48 485289 655 978696 68 506593 723 493407 12 ? 49 485682 655 978655 68 507027 722 492973 11 I 50 486075 654 978615 68 507460 722 , 41/2340 10 I 51 9-486467 653 9-978574 68 9507893 721 10-492107 9 < 52 486860 653 978533 68 508326 721 491674 8 < 53 487251 652 978493 68 508759 720 491241 7 ( 54 487643 651 978452 68 509191 719 490809 6 < 55 488034 651 978411 68 509622 719 490378 5 *> 56 57 ( 58 488424 650 978370 68 510054 718 489946 4 488814 650 978329 68 510485 718 489515 3 489204 649 978288 68 510916 717 489084 2 < 59 489593 648 978247 68 511346 716 488654 1 } 60 489982 648 978-206 68 511776 716 488224 | Cosine | | Sine | Cotang. | Tang. 72 Degrees 154 (18 Degrees.; a tabu; OF logarithmic ( M. | Sine ! D. | Cosine | D. | Tang. | D. | Cotan». J ( 9-489982 648 9-978206 68 9-511776 716 10-488224 60 ( 1 490371 648 978165 68 512206 716 487794 59 ( ! 2 490759 647 978124 68 512635 715 487365 58 ( ; 3 491147 646 978083 69 513064 714 486936 57 ( > 4 491535 646 978042 69 513493 714 486507 56 ( ) 5 491922 645 978001 69 513921 713 486079 55 ( ) <> 492308 644 977959 69 514349 713 485651 54 I ) 7 492695 644 977918 69 514777 712 485223 53 I ) 8 493081 643 977877 69 515204 712 484796 52 ( ) 9 49:1466 642 977835 69 515631 711 484369 51 ( J 10 493851 642 977794 69 516057 710 483943 50 j f H 9494236 641 9-977752 69 9-516484 710 10-483516 49 S ) 12 494621 641 977711 69 516910 709 483090 48 S ) 13 495005 640 977669 69 517335 709 482665 47 S ) M 495388 639 977628 69 517761 708 482239 46 S ) 15 495772 639 977586 69 518185 708 481815 45 S > 10 496154 638 977544 70 518610 707 481390 44 S >.17 ) 18 496537 637 977503 70 519034 706 480906 43 S 496919 637 977461 70 519458 706 480542 42 S ) 19 497301 636 977419 70 519882 705 480118 41 ) J 20 497682 636 977377 70 520305 705 479695 40 < < 21 9-498064 635 9-977335 70 9-520728 704 10-479272 39) ) 22 498444 634 977293 70 521151 703 478849 38 / 37 J 36 > 23 498825 634 977251 70 521573 703 478427 > 24 499204 633 977209 70 521995 703 478005 ) 25 499584 632 977167 70 522417 702 477583 35 34 b ) 2(5 ) 27 499963 632 977125 70 522838 702 477162 500342 631 977083 70 523259 701 476741 33 I 32/ 31) ) 28 500721 631 977041 70 523680 701 476320 ) 29 501099 630 976999 70 524100 700 475900 ) 30 501476 629 970957 70 524520 699 475480 30 > ) 31 9-501854 629 9-976914 70 9-524939 699 10-475061 29| I 32 502231 628 976872 71 525359 698 474641 28? / 33 502607 628 976830 71 525778 698 474222 27/ > 34 502984 627 976737 71 526197 697 473803 26/ ' 35 503360 628 976745 71 526615 697 473385 25/ > 36 503735 628 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 505234 623 976532 71 528702 694 471298 20 ( < 4i 9-505608 623 9-976489 71 9-529119 693 10-470881 19 < \ 42 505981 622 976446 71 529535 693 470465 18 ( 1 43 506354 622 976404 71 529950 693 470050 17 ( < 44 506727 621 976361 71 530366 692 469634 16 ( I 45 507099 620 976318 71 530781 691 469219 15 < • 46 507471 620 976275 71 531196 691 468804 14S I 47 507843 619 976232 72 531611 690 468389 13 < I 1 48 508214 619 976189 72 532025 690 467975 12 < / 49 508585 618 976146 72 532439 689 467561 11 ( { 50 508956 618 617 976103 72 532853 689 467147 10. ! 51 9-509326 9-976060 72 9-533266 688 10-466734 9< ( 52 509696 616 976017 72 533679 688 466321 8 53 510065 616 975974 72 534092 687 405908 7 54 510434 615 975930 72 534504 687 465496 6< 55 510803 615 975887 72 534916 686 465084 5< ( 56 511172 614 975844 72 535328 686 464672 4 (57 511540 613 975800 72 535739 685 464261 3< ( 58 511907 613 975757 72 536150 685 463850 2 v 59 C 60 512275 612 975714 72 536561 684 463439 1 512642 612 975670 72 536972 684 463028 Cosine | Sine | | Cotang. | I Tang. J^l 71 Degrees. SINES AND TANGENTS. (19 degrees.) 155 r*^"r Sine | D. | Cosine D. | Tanf?. D. Cotang. | ~\ ) ° 9 r >12642 612 9-975670 73 9-536972 684 10-463028 60 I 59 ( > i 513009 611 975627 73 537382 683 462618 ) 2 513375 611 975583 73 537792 683 462208 58 ( ) 3 513741 610 975539 73 538202 682 461798 57 < ) 4 514107 609 975496 73 538611 682 461389 56 ( ) 5 514472 609 975452 73 539020 681 460980 55 ( ? 6 514837 608 975408 73 539429 681 460571 54 ( ? 7 515202 608 975365 73 539837 680 460163 53 ? ) 8 515566 607 975321 73 540245 680 459755 52 ( / 9 515930 607 975277 73 540653 679 459347 51 ( 10 516294 606 975233 73 541061 679 458939 50 < ill" 9516657 605 9-975189 73 9-541468 678 10-458532 49 { ( 12 517020 605 975145 73 541875 678 458125 48 47 46 ( ( 13 517382 604 975101 73 542281 677 457719 < 14 517745 604 975057 73 542688 677 457312 < 15 518107 603 975013 73 543094 676 456906 45 ( I 16 518468 603 974069 74 543499 676 456501 44 ( < 17 518829 002 974925 74 543905 675 456095 43 ( ( 18 519190 601 974880 74 544310 675 455690 42 ( \ 19 519551 601 974836 74 544715 674 455285 41 ( (20 519911 600 974792 74 545119 674 454881 40 ( S 21 9-530271 600 9-974748 74 9-545524 673 10-454476 39 J S22 520631 599 974703 74 545928 673 454072 38 ) S23 520990 599 974659 74 546331 672 453669 37 ) S24 521349 598 974614 74 546735 672 453265 36 ) S25 521707 598 974570 74 547138 671 452862 35 ) >26 522066 597 974525 74 547540 671 452460 34 ) S27 522424 596 974481 74 547943 670 452057 33 ) S28 522781 596 974436 74 548345 670 451055 32 S S29 523138 595 974391 74 . 548747 669 451253 31 ) J 30 523495 595 974347 75 549149 669 450851 30 J ?31 9523852 594 9-974302 75 9-549550 668 10-450450 29 } /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 )37 525630 591 974077 75 551552 666 448448 24 ) 525984 591 974032 75 551952 665 448048 23 ) /38 526339 590 973987 75 552351 665 447649 22 ) )39 526693 590 973942 75 552750 665 447250 21 ) )40 527046 589 973897 75 553149 664 446851 20 ) <41 9527400 589 9-973852 75 9-553548 664 10-446452 19 \ (42 527753 588 973807 75 553946 663 446054 18 ? (43 528105 588 973761 75 554344 663 445656 17 / (44 528458 587 973716 76 554741 662 445259 16 / (45 528810 587 973671 76 555139 662 444861 15 / (46 529161 586 973625 76 555536 661 444464 14 I (47 529513 586 973580 76 555933 661 444067 13 / (48 529864 585 973535 76 556329 600 443671 12 ( (49 530215 585 973489 76 556725 660 443275 11 ( (50 530565 584 973444 76 557121 659 442879 10 ( 5i 9-530915 584 9-973398 76 9-557517 659 10-442483 9 (52 531265 583 973352 76 557913 659 442087 8 S )53 531614 582 973307 76 558308 658 441692 7 S )54 531963 582 973261 76 558702 658 441298 6 S )55 532312 581 973215 76 559097 657 440903 5 i )56 532661 581 973169 76 559491 657 440509 4 S )57 533009 580 973124 76 559885 656 440115 3 S ;58 533357 580 973078 76 560279 656 439721 2 S )59 533704 579 973032 77 560673 655 439327 1 ( J69 534052 578 972986 77 561066 655 438934 ( L. | Cosine | Sine J | Cotang. I ^Jang^^ ^M.j 70 Dejn-ees. 156 (20 Degrees.) A TABLE OP LOGARITHMIC M. | 1 2 3 4 5 G 7 2ii 24 25 <, 2(5 27 23 ( 29 J 30 31 ) 32 * 33 34 ) 35 ) 30 ) 37 S 38 S 39 j 40 j 41 > 42 ) 4 3 ) 44 > 45 ; 46 47 1 48 ) 49 50 51 52 53 54 55 56 57 58 59 60 I D. | D. Tang. | D. | Cotang. | 9-534052 578 9-972986 77 534399 577 972l)40 77 $34745 577 972894 77 535092 577 972848 77 535438 576 972802 77 535783 576 972755 77 530129 575 972709 77 530474 574 972663 77 53G818 574 972617 77 5371G3 573 972570 77 537507 573 972524 77 9-537851 572 9-972478 77 538194 572 972431 78 538538 571 972385 78 538880 571 972338 78 539223 570 972291 78 539505 570 . 972245 78 539907 5G9 972198 78 540249 569 972151 78 540590 5C8 972105 78 540931 508 972058 78 9541272 567 9-972011 78 541G13 507 971904 78 541953 506 971917 78 542293 566 971870 78 542G32 565 971823 78 542971 565 971770 78 543310 564 971729 79 543G49 564 971682 79 543987 5G3 971635 79 544325 5G3 971588 79 9544663 562 9-971540 79 545000 562 971493 79 545338 561 971446 79 545674 561 971398 79 546011 560 971351 79 54G347 560 971303 79 54GG83 559 971256 79 547019 559 971208 79 547354 558 971 1G1 79 547089 558 971113 79 9548024 557 9-971066 80 548359 557 971018 80 543693 556 970970 80 549027 556 970922 80 549360 555 970874 80 549693 555 970827 80 550020 554 970779 80 550359 554 97073K 80 550G92 553 970683 80 551024 553 970635 80 9-551356 552 9-970580 80 551087 552 970538 80 552018 5o2 970490 80 552349 551 970442 80 552080 551 970394 80 553010 550 970345 81 553341 550 970297 81 553070 549 970249 81 554000 549 970200 81 554329 548 970152 81 9-561066 501459 561851 562244 562636 563028 5G3419 563811 5642G2 564592 564983 9565373 5G5763 566153 566542 566932 567320 567709 568098 5G8486 568873 9-569261 569648 570035 570422 570809 571195 571581 571967 572352 572738 9-573123 573507 573892 574276 574660 575044 575427 575810 576193 576576 9-576958 577341 577723 578104 578480 578867 579248 579629 580009 580389 9-580769 581149 581528 581907 582286 582G65 583043 583422 583800 584177 655 654 654 653 653 653 C52 652 651 651 650 650 649 649 649 648 648 G47 647 646 646 645 645 645 644 644 643 643 642 642 642 641 641 640 640 639 639 639 638 638 637 637 636 636 636 635 635 634 634 634 633 633 632 632 632 631 631 630 630 629 10-438934 438541 438149 437756 437364 436972 436581 436189 435798 435408 435017 10-434627 434237 433847 433458 433068 432680 432291 431902 431514 431127 10-430739 430352 429965 429578 429191 428805 428419 428033 427648 427262 10-426877 426493 426108 425724 425340 424956 424573 424190 423807 423424 10-423041 422659- 422277 421896 421514 421133 420752 420371 410991 419611 10-419231 418851 418472 418093 417714 417335 416957 416578 41G200 415823 59 58 57 50, 55 > 54/ 53) 52) 51) 50 ) 49 48 47 46? 45 44 43 42 41 40 39 38 37 86 35 34 33 32 31 30 2<) 88 27 36 25 '24 23 2-2 21 20 19 18 17 1G 15 14 13 12, 11 10 9 8 7 G 5 4< fi Degrees. ai Tanjj^jJjLA SINES AND tangents. (21 Degrees.) 157 n. Sine D. Cosine D. Tan?. D. Cotang. 9-554329 548 9970152 81 9-584177 629 10-415823 60 l 554658 548 970103 81 584555 629 415445 59 2 554987 547 960055 81 584932 628 415068 58 3 555315 547 970006 81 585309 628 414691 57 4 555643 546 969957 81 585686 627 414314 56 5 555971 546 969909 81 586062 627 413938 55 6 556299 545 969860 81 586439 627 413561 54 7 556626 545 969811 81 586815 026 413185 53 8 556953 544 969762 81 587190 626 412810 52 9 557280 544 969714 81 587566 625 412434 51 10 557606 543 969665 81 587941 625 412059 50 11 2 557932 543 9-969616 82 9588316 625 10-411684 49 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 559234 541 969420 82 589814 623 410186 45 10 559558 541 969370 82 590188 623 409812 44 17 559883 540 969321 82 590562 622 409438 43 18 560207 540 969272 82 590935 622 409065 49 19 5C0531 539 969223 82 591308 622 408692 41 20 560855 539 969173 82 591681 621 408319 40 21 9561178 538 9-969124 82 9-592054 621 10-407946 39 22 561501 538 969075 82 592426 620 407574 38 23 561824 537 969025 82 592798 620 407202 37 24 562146 537 968976 82 593170 619 406829 30 25 562468 536 968926 83 593542 619 406458 35 20 562790 536 968877 83 593914 618 406086 34 27 563112 536 968827 83 594285 618 405715 33 2? 563433 535 968777 83 594656 618 405344 33 29 563755 535 968728 83 595027 617 404973 31 30 564075 534 968678 83 595398 617 404602 30 31 9-564396 534 9-968628 83 9-595768 617 10-404232 29 32 564716 533 968578 83 596138 616 403862 88 33 565036 533 968528 83 596508 616 403492 27 34 565356 532 968479 83 596878 616 403122 26 35 565676 532 968429 83 597247 615 402753 25 36 565995 531 968379 83 597616 615 402384 24 37 566314 531 968329 83 597985 615 402015 23 38 566632 531 968278 83 598354 614 401646 83 39 566951 530 968228 84 598722 614 401278 21 40 567269 530 968178 84 599091 613 400909 20 41 9-567587 529 9-968128 84 9-599459 613 10-400541 19 42 567904 529 968078 84 599827 613 400173 18 43 568222 528 968027 84 600194 612 399806 17 44 568539 528 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 569488 527 967826 84 601662 611 398338 13 48 569804 526 967775 84 602029 610 397971 12 49 570120 526 967725 84 602395 610 397605 11 50 570435 525 967674 84 602761 610 397239 10 51 9570751 525 9-967624 84 9603127 609 10396873 9 52 571066 524 967573 84 603493 609 396507 8 53 571380 524 967522 85 603858 609 396142 7 54 571695 523 967471 85 604223 608 395777 6 55 572009 523 967421 85 604588 608 395412 5 56 572323 523 967370 85 604953 607 395047 4 57 572636 522 967319 85 605317 607 394683 3 58 572950 522 967268 85 605682 607 394318 2 59 573263 521 967217 85 606046 606 393954 1 60 573575 521 967166 85 606410 606 393590 (22 Degrees.) A TABLE OF LOGARITHMIC M. Sine D. Cosine 1 D- Tan?. a Cotan 55 ) 4 574824 519 966961 85 607863 604 392137 5 575136 519 966910 85 608225 604 391775 6 575447 518 966859 85 608588 604 391412 54 ) 7 575758 518 966808 85 608950 603 391050 53 ) 8 576069 517 966756 86 609312 603 390688 52 > 9 576379 517 966705 86 609674 603 390326 51 > 10 576689 516 966053 86 610036 602 389964 50 ) > 11 9576999 516 9-966602 86 9-610397 602 10-389603 49 ) ) 12 577309 516 966550 86 610759 602 389241 48 ) 13 577618 515 966499 86 611120 601 388880 47 / 14 577927 515 966447 86 61 1480 601 388520 46 ) 15 578236 514 966395 86 611841 601 388159 45/ 16 578545 514 966344 86 612201 600 387799 44/ ► 17 578853 513 966292 86 612561 600 387439 43 > | 18 579162 513 966240 86 612921 600 387079 42 > 19 579470 513 966188 86 613281 599 386719 41 > > 20 579777 512 966136 86 613641 599 386359 40 > > 21 9-580085 512 9 966085 87 9-614000 598 10-386000 39 ( 22 580392 511 966033 87 614359 598 385641 38 ( > 23 580699 511 965981 87 614718 598 385282 37 ( 24 581005 511 965928 87 615077 597 384923 36 ( 25 581312 . r 510 965876 87 615435 597 384565 35 ( 26 581618 510 965824 87 615793 597 384207 34 I 27 581924 509 965772 87 616151 596 383849 33 ( 28 582229 509 965720 87 616509 596 383491 32 I 29 582535 509 965668 87 616867 5% 383133 31 / 30 582840 508 965615 87 617224 595 382776 30 ( 31 9583145 508 9-965563 87 9-617582 595 10-382418 29 < 32 583449 507 965511 87 617939 595 382061 28 S 33 583754 507 965458 87 618295 594 381705 27 \ , 34 584058 506 965406 87 618652 594 381348 26 S 35 584361 506 965353 88 619008 594 380992 25 < 36 584665 506 965301 88 619364 593 380636 24 \ 37 584968 505 965248 88 619721 593 380279 23 ( 38 585272 505 965195 88 620076 593 379924 22 ( 39 585574 504 965143 88 620432 592 379568 21 < ! 40 585877 504 965090 88 620787 592 379213 20 ( , 41 9-586179 503 9-965037 88 9-621142 592 10-378858 19 j 42 586482 503 964984 88 621497 591 378503 18 ) > 43 586783 503 964931 88 621852 591 378148 17 ) > 44 587085 502 964879 88 622207 590 377793 16 J > 45 587386 502 964826 88 622561 590 377439 15 ) 46 587688 501 964773 88 622915 590 377085 14 ) 47 587989 502 964719 88 623269 589 376731 13 ) 48 588289 501 964666 89 623623 589 376377 12 \ 49 588590 500 964613 89 623976 589 376024 11 > 50 588890 500 964560 89 624330 588 375670 10 \ 51 9-589190 499 9964507 89 9-624083 588 10-375317 9 I > 52 589489 499 9644.54 89 625036 588 374904 8 / ) 53 589789 499 964400 89 625388 587 374612 7 / ) 54 590088 498 964347 89 625741 587 374259 6 / ) 55 590387 498 964294 89 620093 587 373907 5/ ) 56 590686 497 964240 89 626445 586 373555 4 I ) 57 590984 497 964187 89 626797 586 3732(13 3/ ) 58 591282 497 964133 89 627149 586 372851 2 ) > 59 591580 496 964080 89 627501 585 372499 1 I ) 60 591878 496 964026 89 627852 585 372148 ) | Coiine 67 Degrees. sines and tangents. (23 Degrees.) S M. I Sine I 1). I Cosine 1 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 18 17 18 lit 20 SI 22 S3 24 SS S6 27 S8 S9 30 3] 33 33 34 35 36 37 38 3iJ 40 41 42 43 44 45 46 47 48 49 50 51 < 52 ; 53 j 54 r 55 ? 56 < 57 ( 58 59 60 9-591878 592176 592473 592770 593067 593363 593955 594251 594547 504842 9595137 595432 595727 590021 596315 596009 59(3903 597196 597490 597783 9-598075 598368 598660 598952 599244 599536 599827 600118 600409 600700 9-600990 601280 601570 601860 602150 602439 602728 603017 603305 603594 604170 604457 604745 6J05032 605319 605606 605892 606179 606465 9-606751 607036 607322 607607 607892 608177 60F4H1 608745 609029 609313 495 495 495 494 494 493 493 493 492 492 491 491 491 490 490 489 489 489 488 488 487 487 487 486 486 485 485 485 484 484 484 483 483 482 482 482 481 481 481 480 480 479 479 479 478 478 478 477 477 476 476 476 475 475 474 474 474 473 473 473 9-964026 963972 963919 963865 963811 963757 963704 963650 963596 963542 963488 9-963434 963379 963325 963271 903217 903163 963108 963054 9(52999 902945 9-962890 902836 902781 962727 962072 962617 962562 963508 962453 962398 9-962343 902288 902233 902178 962123 902067 962012 961957 961902 961846 9-961791 961735 961680 961624 961569 961513 961458 961402 961340 961290 9-961235 961179 961123 961067 961011 960955 9M899 960843 960786 960730 90 90 90 90 91 91 91 91 91 91 91 91 91 91 91 91 91 92 92 92 92 92 92 92 92 92 92 92 92 92 92 93 93 93 93 93 93 93 93 93 93 93 93 93 93 94 94 Tan?. D. | 9-627852 1 585 628203 585 628554 585 628905 584 629255 584 629606 583 629956 583 630306 583 630656 583 631O05 582 631355 582 9-631704 582 632053 581 632401 581 632750 581 633098 580 633447 580 633795 580 634143 579 634490 579 634838 579 9-635185 578 635532 578 635879 578 636226 577 636572 577 636919 577 637265 577 637611 576 637956 576 638302 576 9-638647 575 638992 575 639337 575 639682 574 6401,27 574 640371 574 640716 573 641060 573 641404 573 641747 572 9-642091 572 642434 572 642777 572 643120 571 643463 571 643806 571 644148 570 644490 570 -644832 570. 645174 569 9-645516 569 645857 569 646199 569 646540 508 64H881 568 647222 568 647562 567 647903 567 648243 567 648583 566 10-368296 307947 307599 367250 366902 366553 366205 365857 365510 365162 10-364815 364468 364121 363774 363428 363081 362735 302389 362044 361698 66 Degrees. 160 (24 Degrees.) A TABLE OP LOGARITHMIC JML Sine D. Cosine D. , Tansr. D. Cotang. "H / o 9-609313 473 9-960730 94 9-648583 566 10-351417 60) ) 1 609597 472 960674 94 648923 566 351077 59 2 609880 472 960618 94 649263 566 350737 58 ; 57 56) l 3 610164 472 960561 94 649602 566 350398 ) 4 610447 471 960505 94 649942 565 350058 ) 5 610729 471 960448 94 65(1281 565 349719 55 S > 6 611012 470 960392 94 650620 565 349380 54 ) 7 611294 470 960335 91 650959 564 349041 53 S ) 8 611576 470 960279 94 651297 564 348703 52 51 ( 9 611858 469 960222 94 651636 564 348364 >10 612140 469 960165 94 651974 563 348026 50 ( ; n 9-612421 469 9960109 95 9-652312 563 10-347688 49 j ) 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 346337 45) , 16 613825 467 959825 95 654000 562 346000 44) ) 17 614105 466 959768 95 654337 561 345663 43) ) 18 614385 466 959711 95 654674 561 345326 42) } 19 614665 466 959654 95 655011 561 344989 41) ) 20 614944 465 959596 95 655348 561 344652 40 J ! 21 9-615223 465 9959539 95 9655684 560 10-344316 39 j ? 22 615502 465 959482 95 656020 560 343980 38) / 23 615781 464 959425 95 656356 560 343644 37) I 24 616060 464 959368 95 656692 559 343308 36) ) 25 616338 464 959310 96 657028 559 342972 35) ) 26 616616 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 617727 462 959023 96 658704 558 341296 30 J ? 31 9-618004 461 9-958965 96 9-659039 558 10-340961 29) ( 32 618281 461 958908 96 659373 557 340627 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) I 36 I 37 619386 460 958677 96 660710 556 339290 24) 619662 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 620488 458 958445 97 662043 555 337957 20) S 41 9620763 458 9-958387 97 9662376 555 10-337024 19? ( 42 621038 457 958329 97 662709 554 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 J < 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) I 49 622956 455 957951 97 665035 553 334965 11 ) j 50 623229 455 957863 97 665366 552 334634 10) i 51 9-623502 454 9-957804 97 9-665697 652 10-334303 9 ( < 52 623774 454 957746 98 666029 552 3331>71 8 ( < 53 624047 454 957687 98 666360 551 333640 7) < 54 624319 453 957628 98 666691 551 333309 6 ) ( 55 624591 453 957570 98 667021 551 332979 5 ) ( 56 624863 453 957511 98 667352 551 332648 4 < 57 625135 452 957452 98 667683 550 332318 3) ( 58 625406 452 957393 98 668013 550 331987 2 J ( 59 625677 452 957335 98 668343 550 331657 1 ) ( 60 625948 451 957276 98 668672 550 331328 1 | Cosine Sine ^~- Cotang. ' i Tang. 65 Degrees. SINES AND TANGENTS. (25 Degrees.) 1G1 Sine D. Cosine ■_~dT" Tang. D. Cotang. J 9-625948 451 9-957276 98 9-668673 550 10-331327 60 ( 620219 451 957217 98 669002 549 330998 59 ( 626490 451 957158 98 669332 549 330668 58 ( 626760 450 957099 98 669661 549 330339 57 ( 627030 450 957040 98 669991 548 330009 56 ( 627300 450 956981 98 670320 548 329680 55 ( 627570 449 956921 99 670649 548 329351 54 ( 627840 449 956862 99 670977 548 329023 53 ( 628109 449 956803 99 671306 547 328694 52 ( 628378 448 956744 99 671634 547 328366 51 ( 628647 448 956684 99 671963 547 328037 50 , 9-628916 447 9956625 99 9-672291 547 10-327709 49 { 629185 447 956566 99 672619 546 327381 48 ) 629453 447 956506 99 672947 546 327053 47 ( 629721 446 956447 99 673274 546 388726 46 < 629989 446 956387 99 673602 546 326398 45 ( 630257 446 956327 99 673929 545 326071 44 ( 630524 446 956208 99 674257 545 325743 43 ( 630792 445 956208 100 674584 545 325416 42 ( 631059 445 9.56148 100 674910 544 325090 41 ( 631326 445 956089 100 675237 544 324763 40 j 9*631593 444 9-956029 100 9-675564 544 10324436 39 ) 631859 444 955969 100 675890 544 324110 38 ) 632125 444 955909 ► 100 676216 543 323784 37 ) 632392 443 955849 100 676543 543 323457 36 ) 632658 443 955789 100 676809 543 323131 35 ) 632923 443 955729 100 677194 543 322806 34 ) 633189 442 955669 100 677520 542 322480 33 ) 633454 442 955009 100 677846 542 322154 32 \ 633719 442 955548 100 678171 542 321829 31 ) 633984 441 955488 100 678496 542 321504 30 S 9'634249 441 9955428 101 9*678821 541 10-321179 29 ) 634514 440 955308 101 679146 541 320854 28 ) 634778 440 955307 101 679471 541 320529 27 ) 635042 440 955247 101 679795 541 320205 26 ) 635306 439 955186 101 680120 540 319880 25 / 635570 439 955126 101 680444 540 319556 24 ) 635834 439 955005 101 680768 540 319232 23 636097 438 955005 101 681092 540 318908 22 ) 636360 438 954944 101 681416 539 318584 21 ) 636623 438 954883 101 681740 539 318260 20 9'636886 437 9-954823 101 9682063 539 10-317937 19 ) 637148 437 954702 101 682387 539 317613 18 ) 637411 437 954701 101 682710 538 317290 17 16 ) 637673 437 954040 101 683033 538 310907 637935 436 954579 101 683356 538 310644 15 638197 436 954518 102 683679 538 316321 14 ) 6384.58 436 954457 102 684001 537 315999 13 ) 638720 435 954396 102 684324 537 315676 12 ) 638981 435 954335 102 684646 537 315354 11 / 639242 435 954274 102 684968 537 315032 io ) 9-639503 434 9-954213 102 9-685290 536 10-314710 9\ 639764 434 954152 102 685612 536 314388 8 640024 434 954090 102 685934 536 314066 7 ( 640284 433 954029 102 686255 536 313745 G( 640544 433 953968 102 686577 535 313423 5 640804 433 953906 102 686898 535 313102 4 641064 432 953845 102 687219 535 312781 3 641324 432 953783 102 687540 535 312460 2 641584 432 953722 103 687861 534 312139 1 ( 641842 431 953600 103 688182 534 311818 ( 64 Degrees. / x 162 (26 Degrees.) A TABLE OF LOGAR1TMIC Sine 1 D- Cosine D. Tang. D. Cotang. 9-641842 431 9-953660 103 9-688182 534 10'311818 60 642101 431 953599 103 688502 534 311498 59 642360 421 953537 103 688823 534 311177 58 642618 430 953475 103 689143 533 310857 57 642877 430 953413 103 689403 533 310537 56 643135 430 953352 103 689783 533 310217 55 643393 430 953290 103 690103 533 309897 54 643650 429 953228 103 690423 533 309577 53 643908 429 953166 103 690742 532 309258 52 644165 429 953104 103 691062 532 308938 51 644423 428 953042 103 691381 532 308619 50 9644680 428 9-952980 104 9-691700 531 10-308300 49 644936 428 952918 104 692019 531 307981 48 645193 427 952855 104 692338 531 307662 47 645450 427 952793 104 692656 531 307344 46 645706 427 952731 104 692975 531 307025 45 045962 426 952669 104 693293 530 306707 44 646218 420 952606 104 693612 530 306388 43 646474 426 952544 104 693930 530 306070 42 646729 425 952481 104 694248 530 305752 41 646984 425 952419 104 694566 529 305434 40 9-647240 425 9-952356 104 9-694883 529 10-305117 38 647494 424 952294 104 695201 529 304799 38 647749 424 952231 104 695518 529 304482 37 648004 424 952168 105 695836 529 304164 36 648258 424 952106 105 696153 528 303847 35 648512 423 952043 105 696470 528 303530 34 648766 423 951980 105 696787 528 303213 33 649G20 423 951917 105 697103 528 302897 32 649274 422 951854 105 697420 527 302580 31 649527 422 951791 105 697736 527 302264 30 9649781 422 9-951728 105 9-698053 527 10-301947 2!) 650034 422 951665 105 6983C9 527 301631 28 650287 421 951602 105 698C85 526 301315 27 650539 421 951539 105 699001 526 300999 26 650792 421 951476 1C5 699316 526 300684 25 651044 420 951412 105 699632 526 300368 24 651297 420 951349 106 699947 526 300053 23 651549 420 951286 106 700203 525 299737 22 651800 419 951222 106 700578 525 299422 Si 652052 419 951159 106 700893 525 299107 81 9-652304 419 9-951096 1C6 9-701208 524 10-298792 18 652555 418 951032 106 701523 524 298477 18 652806 418 950968 1.06 701837 524 298163 17 653057 418 950905 106 702152 524 297848 Hi 653308 418 . 950841 K-6 702466 524 297534 15 653558 417 950778 106 702780 523 297220 14 653808 417. 950714 106 703095 523 296905 13 654059 417 950650 106 703409 523 296591 IS 654309 416 950586 106 703723 523 296277 11 654558 416 950522 107 704036 522 295964 10 654808 416 9-950458 107 9704350 522 10-295650 9 655058 416 950394 107 704663 522 295337 8 655307 415 950330 107 704977 522 295023 7 655556 415 950266 107 705290 522 294710 6 655805 415 950202 107 705603 521 294397 5 656054 414 950138 107 705916 521 294084 4 656302 414 950074 107 706228 521 293772 3 65655] 414 950010 107 706541 521 293459 2 656709 413 949945 107 706854 521 293146 1 657047 413 949881 107 707166 520 292834 Cosine | 63 Degrees. SINES AND TANGENTS. (27 Degrees.) 1G3 M. Sine D. Cosine D. | Tang-. D. Cotangv 9657047 413 9-949881 107 9-707166 520 10-292834 (50 1 657295 413 949816 107 707478 520 292522 59 2 657542 412 949752 '107 707790 520 292210 58 S 657790 412 949688 108 708102 520 291898 57 4 658037 412 949623 108 708414 519 291586 98 5 658284 412 949558 108 708726 519 291274 55 6 658531 411 949494 108 709037 519 290963 54 7 658778 411 949429 108 709349 519 290651 53 8 659025 411 9493(54 108 709660 519 290340 52 9 659271 410 949300 108 709971 318 290029 51 10 659517 410 949235 108 710282 518 289718 50 11 9-659763 410 9-949170 108 9-710593 518 10-289407 49 12 660009 409 949105 108 710904 518 289096 48 13 660255 409 949040 108 711215 518 288785 47 14 660501 409 948975 108 711525 517 288475 4(5 15 6(50746 409 948910 108 711836 517 288164 45 16 660991 408 948845 108 712146 517 287854 44 17 661236 408 948780 109 712456 517 287544 1:1 13 661481 408 948715 109 712766 516 287234 1-2 19 661726 407 948650 109 713076 516 286924 41 20 661970 407 948584 109 713386 516 286614 40 21 9662214 407 9-948519 109 9-713696 516 10-286304 39 22 662459 407 948454 948388~ 109 714005 516 285995 39 23 662703 406 109 714314 515 285686 37 24 662946 406 948323 109 714624 515 285376 3(5 25 663190 406 948257 109 714933 515 285067 35 20 663433 405 948192 109 715242 515 284758 34 27 663677 405 948126 109 715551 514 284449 33 28 663920 405 948060 109 715860 514 284140 3-2 29 664162 405 947995 110 716168 514 283832 31 30 664406 404 947929 110 716477 514 283523 3Q 31 9664648 404 9-947863 110 9-716785 514 10-283215 29 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 282291 • 95 35 665617 403 947600 110 718017 513 281983 35 30 665859 402 917533 110 718325 513 281(575 24 37 666100 402 947467 110 718633 512 281367 ■Zi 38 666342 402 947401 110 718940 512 281060 22 39 666583 402 947335 110 719248 512 280752 SI 40 666824 401 947269 110 719555 512 280445 80 41 9 667065 401 9-947203 110 8-719862 512 10-280138 19 42 667305 401 847136 111 720169 511 279831 18 43 667546 401 947070 111 720476 511 279524 17 44 667786 400 947004 111 720783 511 279217 1(5 45 668027 400 940937 111 721089 511 278911 15 46 668267 400 946871 111 721396 511 278604 14 47 668506 309 940804 111 721702 510 278298 13 48 668746 309 946738 111 722009 510 277991 1-2 49 668986 399 946671 111 722315 510 277685 11 50 669225 399 946604 111 722621 510 277379 10 51 9-669464 398 9-946538 111 9-722927 510 10-277073 9 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 276156 6 55 670419 397 946270 112 724149 509 275851 5 56 6706.58 397 946203 112 724454 509 275546 4 57 670896 397 946136 112 724759 508 275241 3 58 671134 396 946069 112 725065 508 274935 2 59 671372 396 940002 112 725369 508 274631 1 60 671609 396 945935 112 725674 508 274326 r I Cosine | Sine | | Cotang. 62 Degrees. 164 (28 Degrees.) a i'able of logarithmic Cm. Sine D. Cosine D. Tang. D. Cotan^. ~1 ) o 9*671609 396 9-945935 112 9-725674 508 10-274326 60) 1 671847 395 945868 112 725979 508 274021 59 S ! 2 672084 395 945800 112 726284 507 273716 58 ) ) 3 672321 395 945733 112 726588 507 273412 57) ? 4 672558 395 945666 112 726892 507 273108 56) ) 5 672795 394 945598 112 727197 507 272803 55) ? 6 673032 394 945531 112 727501 507 272499 54 ) ) 7 673268 394 945464 113 727805 506 272195 53 ) J 8 673505 394 945396 113 728109 506 271891 52) ) 9 673741 393 945328 113 728412 506 271588 51 ) i 10 673977 393 945261 113 728716 506 271284 50 S 1 U 9674213 393 9-945193 113 9-729020 506 10-270980 49) / 12 674448 392 945125 113 729323 505 270677 48 I ) 13 674684 392 945058 113 729626 505 270374 47 I ) 14 674919 392 944990 113 729929 505 270071 46/ ) 15 675155 392 944922 113 730233 505 269767 45/ ) 16 675390 391 944854 113 730535 505 269465 44/ ) i7 675624 391 944786 113 730838 504 269162 43/ ) 18 675859 391 944718 113 731141 504 268859 42/ ) 19 676094 391 944650 113 731444 504 268556 41 / ) 20 676328 390 944582 114 731746 504 268254 40/ ) 21 9676562 390 9-944514 114 9-732048 504 10-267952 39 ( 1 1 22 676796 390 944446 114 732351 503 267649 38 ( I 1 23 677030 390 944377 114 732653 503 267347 37 ( I 1 24 677264 389 944309 114 732955 503 267045 36 ( / 25 677498 389 944241 114 733257 503 266743 35 C / 26 677731 389 944172 114 733558 503 266442 34 S / 27 677964 388 944104 114 733860 502 266140 33 ( ) 28 678197 388 944036 114 734162 502 265838 32 ( ) 29 678430 388 943967 114 734463 502 265537 31 ( ) 30 678663 388 943899 114 734764 502 265236 30 ' S 31 < 32 9678895 387 9-943830 114 9-735066 402 10-264934 29) 679128 387 943761 114 735367 502 264633 28 ) ( 33 679360 387 943693 115 735668 501 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 501 263430 24 ) ( 37 680288 386 943417 115 736871 501 263129 23 ) < 38 680519 385 943348 115 737171 500 262829 22 ) < 30 680750 385 943279 115 737471 500 262529 21 ) I 40 680982 385 943210 115 737771 500 262229 20 S ) 41 9681213 385 9-943141 115 9-738071 500 10-261929 19/ i 42 681443 384 943072 115 438371 500 261629 18/ 5 43 681674 384 943003 115 738671 499 261329 17? < 44 681905 384 942934 115 738971 409 261029 16 / < 45 682135 384 942864 115 739271 499 260729 15/ ( 46 682365 383 942795 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 11 ( j 50 683284 382 942517 116 740767 498 259233 10/ ) 51 9683514 382 9-942448 116 9-741066 498 10-258934 »i ) 52 683743 382 942378 116 741365 498 258635 8 ( 53 J 54 683972 382 942308 116 741664 498 258336 7{ 684201 381 942239 116 741962 497 258038 6( ) 55 684430 381 942169 116 742261 497 257739 5( ) 56 684658 381 942099 116 742559 497 257441 4 ' ) 57 684887 380 942029 116 742858 497 257142 3 < S 58 685115 380 941959 116 743156 497 256844 2( ) 59 685343 380 941889 117 743454 497 256546 1 ( > 60 685571 380 941819 117 743752 496 256248 J { Cosine Sine | Cotang. }^^~> _^ran|^ XT) 61 Degrees. BINES AND TANGENTS. (29 Degrees.) 165 9685571 380 9-941819 117 9-743752 496 10-256248 60? 1 685799 379 941749 117 744050 496 255950 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 56) ) 5 686709 378 941469 117 745240 496 254760 55) > 6 686936 378 941398 117 745538 495 254462 54) ) ' 687163 378 941328 117 745835 495 254165 53) 8 687389 378 941258 117 746132 495 253868 52) ) 9 687616 377 941187 117 746429 495 253571 51) >10 687843 377 941117 117 746726 495 253274 50) Ui 9-688069 377 9-941046 118 9747023 494 10-252977 49 ( '12 688295 377 940975 118 747319 494 252681 48? >13 688521 376 940905 118 747616 494 252384 47? 14 688747 376 940834 118 747913 494 252087 46? >15 688972 376 940763 118 748209 494 251791 45? >16 689198 376 940093 118 748505 493 251495 44? >17 689423 375 940622 118 748801 493 251199 43? )l8 689648 375 940551 118 749097 493 250903 42? '19 689873 375 940480 118 749393 493 250607 41 ? )20 680098 375 940409 118 749689 493 250311 40? 21 9-690323 374 9-940338 118 9-749985 493 10-250015 39 22 690548 374 940267 118 750281 492 249719 38 ( 23 690772 374 940196 118 750576 492 249424 37 < 24 690996 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 691G68 373 939911 119 751757 492 248243 33 ( 28 691892 373 939840 119 752052 491 247948 32 ( >29 692115 372 939768 119 752347 491 247653 31 ( 30 692339 372 939697 119 752642 491 247358 30 31 9-692562 372 9-939625 119 9-752937 491 10247063 29 32 692785 371 939554 119 753231 491 246769 28 ) 33 693008 371 939482 119 753526 491 246474 27) 34 693231 371 939410 119 753820 490 246180 26) 35 693453 371 939339 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 939123 120 754997 490 245003 22 ) 39 694342 370 939052 120 755291 490 244709 21 ) 40 694564 369 938980 120 755585 489 244415 20 ; 41 9-694786 369 9-938908 120 9-755878 489 10-244122 19 42 695007 369 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 696775 367 938258 121 758517 488 241483 10 ) > 51 9-696995 367 9-938185 121 9-758810 488 10-241190 9 )52 697215 366 938113 121 759102 487 240898 8 \ )53 697435 366 938040 121 759395 487 240605 7\ )54 697654 366 937967 121 759687 487 240313 6S j 55 697874 366 937895 121 759979 487 240021 5 ( )56 698094 365 937822 121 760272 487 239728 4 < )57 698313 365 937749 121 760564 487 239436 3 ( )58 698532 365 937676 121 760856 486 239144 2 ( )59 698751 365 937604 . 121 761148 486 238852 1( ^60 698970 364 937531 121 761439 486 238561 °) Cosine Sine Cotang. ^Tang^l JL) C Degre BS. 166 (30 Degrees.) A TABLE OP LOGARITHMIC (m. 1 Sine 1 D. Cosine 1 D. 1 Tang. 1 D. Cotang. 1 I 1 ° 9-698970 364 9937531 121 9-761439 486 10-238561 60 ) > 1 699189 364 937458 122 761731. 486 238269 59 ) ? 2 699407 364 937385 122 762023 486 237977 58 ; ? 3 699626 364 937312 122 762314 486 237686 57 ) f 4 699844 363 937238 122 762606 485 237394 56 ) ) 5 700062 363 937165 122 762897 485 237103 55 ) ) 6 700280 363 937092 122 763188 485 236812 54 > ) 7 700498 363 937019 122 763479 485 236521 53 ) ) 8 700716 363 936946 122 763770 485 236230 o2 ) ? 9 700933 362 936872 122 764061 485 235939 51 \ s 10 701151 362 936799 122 764352 484 235648 50 j \ 11 9701368 362 9-936725 122 9764643 484 10-235357 49 / < 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 ) s 1 5 702236 361 936431 123 765805 484 234195 45 I ( 16 702452 361 936357 123 766095 484 233905 44 / S 17 702669 360 936284 123 766385 483 233615 43 ) S 18 702885 360 936210 123 766675 483 233325 42 ( ( 19 703101 360 936136 123 766965 483 233035 41 / J 20 703317 360 936062 123 767255 483 232745 40 ) S 21 9-703533 359 9-935988 123 9-767545 483 10-232455 39 ( S 22 703749 359 935914 123 767834 483 232166 38 ( S 23 703964 359 935840 123 768124' 482 231876 37 ( > 24 734179 359 935766 124 768413 482 231587 36 < s 25 704395 359 935692 124 768703 482 231297 35 ( ) 26 704610 358 935618 124 768992 482 231008 34 < S 27 704825 358 935543 124 769281 482 230719 33 < S 28 705040 358 935469 124 769570 482 230430 32 ( S 29 705254 358 935395 124 769860 481 230140 31 ( J 30 705469 357 935320 124 770148 481 229852 30 ( ? 31 9-705683 357 9935246 124 9770437 481 10-229563 29 ) > 32 705898 357 935171 124 770726 481 229274 28 ) > 33 706112 357 935097 124 771015 481 228985 27 ) 34 706326 356 935022 124 771303 481 228697 26 ) ) 35 706539 356 934948 124 771592 481 228408 25 24 1 23 22 ) ) 36 706753 356 934873 124 771880 480 228120 I 37 706967 356 934798 125 772168 480 227832 ) 38 707180 355 934723 125 772457 480 227543 > 39 707393 355 934649 125 772745 480 227255 21 ) > 40 707606 355 934574 125 773033 480 226967 20 ) ( 41 9707819 355 9934499 125 9773321 480 10-226679 19 ( < 42 708032 354 934424 125 773608 479 226392 18 / S 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 ( S 47 709094 353 934048 125 775046 479 224954 13 I I 48 709306 353 933973 125 775333 479 224667 12 ? ( 49 709518 353 933898 126 775621 478 221379 11 \ J 50 709730 353 933822 126 775908 478 224092 10 ) 51 9-709941 352 9-933747 126 9-776195 478 10223805 9 ! S 52 710153 352 933671 126 776482 478 223518 8 ) ) 53 ) 54 710364 352 933596 126 776769 478 223231 7 { 710575 352 933520 126 777055 478 222945 6 S S 55 710786 351 933445 126 777342 478 222658 5 S \ 56 710997 351 933369 126 777628 477 222372 4 s ) 57 711208 351 933293 126 777915 477 222085 3 S S 58 711419 351 933217 126 778201 477 221799 2 S S 59 711629 350 933141 126 778487 477 221512 1 < 60 711839 350 933066 126 778774 477 221226 ( L~ Cosine ~— -, Sine Cotang. | Tang. ^MJ Degrees. SINES an/) TiNuENTS. (31 Degrees.) 167 CmT Sine D. Cosine D. | T. ng. | D. | Cotang. f ° 9-711839 350 9-933066 126 9-778774 477 1U-221226 00 } I 1 712050 350 932990 127 779000 477 220940 60 > 2 712200 350 932914 127 779346 476 220054 Rj) ) 3 712409 349 9328:18 127 779032 476 220368 57) ) 4 712079 349 932702 127 779918 476 220082 5'J ) 5 712889 349 932085 127 780203 476 219797 55/ > 6 713098 349 932809 127 780489 476 219511 54^ ) 7 ' 713303 349 932533 127 78J775 470 219225 53< ) 8 713517 348 932457 127 781000 470 213940 52? > 9 713726 343 932380 127 781346 475 218654 51 ; ( I0 713935 348 9323)4 127 781631 475 218369 50/ ) 11 9714144 343 9-932228 127 9-781916 475 10-218084 49 I ( 12 714352 347 932151 127 7822U1 475 217799 48 ( ( 13 714561 347 932075 128 782486 475 217514 47 ( ( 14 714769 347 931938 128 782771 475 217229 40 < ( 15 714978 347 931921 128 783056 475 216944 43 ( ( 16 715186 347 931345 128 783341 475 210659 41 ( ( 17 715394 340 931768 128 783026 474 21 6374 43 ( ( 18 715602 346 931691 128 783910 474 216090 42 ( / 19 715809 346 931014 128 784195 474 215805 41 ( >20 716017 346 931537 128 784479 474 215521 40 ( < 21 9-718224 345 9-931460 128 9-784704 474 10-215230 39 > (22 716432 345 931383 128 785048 474 214952 38 ) (23 716039 345 931306 128 785332 473 214668 37 S (24 716846 345 931229 129 785010 473 214384 30 ) (25 717053 345 931152 129 785900 473 214100 35 ( (26 717259 344 931075 129 78G184 473 213816 34 S (27 717466 344 930998 129 780408 473 213532 33 \ (28 717673 344 930921 129 78G752 473 213248 32 S (29 717879 344 930843 129 787036 473 212964 31 S (30 7180a 1 * 343 930760 129 787319 472 212081 30 \ 31 9718291 343 9-930083 129 9-787603 472 10-212397 29) ,32 718497 343 930611 129 787886 472 212114 28) S 33 718703 343 933533 129 788170 ' 472 211830 27) 34 718909 343 930450 129 788453 472 211547 20) ( 35 719114 342 930378 129 788730 472 211264 25) (30 719320 342 930300 130 789019 472 210981 24 ) (37 719525 342 930223 130 789302 471 210098 23) (38 719730 342 930145 130 789585 471 210415 22) (39 719935 341 930007 130 789868 471 210132 21 ) (40 720140 341 929989 130 790151 471 209849 20) >41 9-720345 341 9-929911 130 9-790433 471 10-209567 19? )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 7915G3 470 208437 15/ ) 40 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 792092 470 207308 11 ( [50 722181 339 929207 131 792974 470 207026 10) }51 9-722385 339 9-929129 131 9-793250 470 10-200744 9S ) 52 722588 339 929050 131 793538 409 206462 8\ ) 53 722791 338 928972 131 793819 469 200181 7 S ) 54 722994 338 928893 131 794101 469 205899 6 ( >55 723197 338 928815 131 794383 409 205017 5 ( ) 56 723400 338 928736 131 794664 469 205336 4 \ )57 723603 337 928657 131 794945 469 205055 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 ( 58 Debtees. 168 (32 Degrees.) A TABLE OF LOGARITHMIC [mT| S.ne | D. Cosine | D. Tan-. D. Cotang. i I ° 9724210 337 9928420 132 9795789 468 10-204211 60 ? ) 1 724412 337 928342 132 796070 468 203930 59 ( I 2 724614 336 928263 132 796351 468 203649 58 ( ? 3 724816 336 928183 132 796632 468 203368 57 < ) 4 725017 336 928104 132 796913 468 203087 56 ( ) 5 725219 336 928025 132 797194 468 202806 55 / ) 6 725420 335 927946 132 797475 468 202525 54 I ; 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 ( j 10 726225 335 927629 132 798596 467 201404 50 j ( 11 9726426 334 9-927549 132 9-798877 467 10201123 49 S / 12 726626 334 927470 133 799157 467 200843 48 S ) 13 726827 334 927390 133 799437 467 200563 47 < ) 14 727027 334 927310 133 799717 467 200283 46 \ > 15 727228 334 927231 133 799997 466 200003 45 S ? 16 727423 333 927151 133 800277 466 199723 44 S ) I 7 727628 333 927071 133 800557 466 199443 43 \ ) i8 727828 333 926991 133 800836 466 199164 42 S ) 19 728027 333 926911 133 801116 466 198884 41 S J 20 728227 333 926831 133 801396 466 198604 40 S ( 21 9728427 332 9-926751 133 9801675 466 10-198325 39) / 22 728026 332 926671 133 801955 466 198045 38 > ? 23 728825 332 926591 133 802234 465 197766 37 / / 24 729024 332 926511 134 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 ( 32) ) 28 729820 331 926190 134 803630 465 196370 > 29 730018 330 926110 134 803908 465 196092 31 / ) 30 730216 330 926029 134 804187 465 195813 30 < 31 9 730415 330 9-925949 134 9804466 464 10195534 29 { ( 32 730613 330 925868 134 804745 464 195255 28 ( ( 33 730811 330 925788 134 805023 464 194977 27 < < 34 731009 329 925707 134 805302 464 194698 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 135 806693 463 193307 21 < ( 40 732193 328 925222 135 806971 463 193029 20 ( < 41 9732390 328 9-925141 135 9-807249 463 10-192751 19 J < 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 327 924735 136 808638 462 191362 14 > < 47 733569 327 924654 136 808916 462 191084 13) ( 48 733765 327 924572 136 809193 462 190807 12) ( 49 733961 326 924491 136 809471 462 190529 11 I ( 50 734157 326 924409 136 809748 462 190252 10) / 51 9 734353 326 9-924328 136 9-810025 462 10189975 9< / 52 734549 326 924246 136 810302 462 189698 8^ ) 53 734744 325 924164 136 810580 462 189420 "M >54 734939 325 924083 136 810857 462 189143 6 \ ) 55 735135 325 924001 136 811134 461 188866 5 l ) 56 735330 325 923919 136 811410 461 188590 4 ( ; 57 735525 325 923837 136 811687 461 188313 3 \ ) 58 735719 324 023755 137 811964 461 188036 2 ( 59 735914 324 933673 137 812241 461 187759 1 ) 60 736109 1 324 923501 137 812517 461 187483 ( U 1 Cosine i^-~^> 1 Km L^^ Cotang. ^>~~ Tang. l_JLJ 57 Degrees. sines AND tangents. (33 Degrees.) 169 > M. Sine . D. Cosine D. Tang. 1 D- 1 Cotang. | \ 1 ° 9736109 324 J9-923591 137 9-812517 461 10-187482 60 { ( 1 736303 324 923509 137 812794 461 187206 59 ( 58 ( S 2 736498 324 923427 137 813070 461 186930 \ 3 736692 323 923345 137 813347 460 186653 57 ( 4 736886 323 923263 137 813623 460 186377 56? S 5 737080 323 923181 137 813899 460 186101 55/ ' 6 737274 323 923098 137 814175 460 185825 54 ( \» 737467 323 923016 137 814452 460 185548 53 I 737661 322 923983 137 814728 460 185272 52 (' S 9 737855 322 922851 137 815004 460 184996 51 ( > 10 738048 322 922768 138 815279 460 184721 50/ i 1J 9-738241 322 9-922686 138 9-815555 459 10184445 49 ( ) 12 738434 322 922603 138 815831 459 184169 48 < ) 13 738627 321 922520 138 816107 459 183893 47 ( ) 14 738820 321 922438 138 816382 459 183618 46 ( ) 15 739013 321 922355 138 81G658 459 183342 45 f ) 16 739206 321 922272 138 816933 459 183067 44 ( i n 739398 321 922189 138 817209 459 182791 43 ( > 18 739590 320 922106 138 817484 459 182516 42( ) 19 739783 320 922023 138 817759 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 S / 22 740359 320 921774 139 818585 458 181415 38 S > 23 740550 319 921691 139 818860 458 181140 37) I 24 740742 319 921607 139 819135 458 180865 36 S / 25 740934 319 921524 139 819410 458 180590 35) I 26 741125 319 921441 139 819684 458 180316 34 ( 33 ( . I 27 741316 319 921357 139 819959 458 180041 I 28 741508 318 921274 139 820234 458 179766 32 S I 29 741099 318 921190 139 820508 457 179492 31$ / 30 741889 318 921107 139 820783 457 179217 30 j I 31 9-742080 318 9-921023 139 9-821057 457 10178943 29) < 32 742271 318 920939 140 821332 457 178668 28 ) 27 S ( 33 7424G2 317 920856 140 821606 457 178394 ( 34 742652 317 920772 140 821880 457 178120 26) 25) < 35 742842 317 920688 140 822154 457 177846 ( 36 743033 317 920604 140 822429 457 177571 24) < 37 743223 317 920520 140 822703 457 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 S \ 41 9-743982 316 9-920184 140 9-823798 456 10176202 19) ) 42 744171 316 920099 140 824072 456 175928 18) ) 43 744361 315 92C015 140 824345 456 175655 17) ) 44 744550 315 919931 141 824619 456 175381 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 174561 13) 7 48 745306 314 919593 141 825713 455 174287 12) ) 49 J 50 745494 314 919508 141 825986 455 174014 11 ) 745683 314 919424 141 826259 455 173741 10 \ /51 > 52 9745871 314 9-919339 141 9-826532 455 10-173468 9 ( 746059 314 919254 141 826805 455 173195 8 ) 53 746248 313 919109 141 827078 455 172922 7 / < S 4 746436 313 919085 141 827351 455 172649 6) / 55 746624 313 919000 141 827624 455 172376 5 / ) 56 746812 313 918915 142 827897 454 172103 *i ) 57 746999 313 918830 142 828170 454 171830 3) ) 53 747187 312 918745 142 828442 454 171558 2 ) ) 59 747374 312 918659 142 828715 454 171285 1 ) 60 747562 312 918574 142 828987 454 J 71013 ( Cosine 1 Sine 1 ^ Cotang. 1 i~Jfe^ ISiJ 66 Degrees. 170 (34 Degrees.) a table OF LOGARITHMIC M. | Sine 1 D. | Cosine 1 D. Tansr. 1 D. I Cotang 1 . / 9-747562 312 9-918574 142 9-828987 454 10171013 60 ( 59 C 58 1 747749 312 918489 142 829260 454 170740 2 747036 312 918404 142 829532 454 170468 i 748123 311 918318 142 829805 454 170195 57 ( 4 748310 311 918233 142 830077 454 169923 56 ( 5 748497 311 918147 142 830349 453 169651 55 C (3 748683 311 918062 142 830621 453 169379 54 I 7 748870 311 917976 143 830893 453 169107 53 ( 8 749056 310 917891 143 831165 453 168835 52 ( 749243 310 917805 143 831437 453 168563 51 ( 10 749429 310 917719 143 831709 453 168291 50 ' 11 9749615 310 9-917634 143 9-831981 453 10-168019 49 J M 749801 310 917548 143 832253 453 167747 48 S 13 749987 309 917462 143 832525 453 167475 47 ) 14 750172 309 917376 143 832796 453 167204 46) 15 750358 309 917290 143 833068 452 166932 45 ) 1(3 750543 309 917204 143 833339 452 166661 44 S 17 750729 309 917118 144 833611 452 166389 43 S 18 750914 308 917032 144 833882 452 166118 42 S 19 751099 308 916946 144 834154 452 165846 41 S 90 751284 308 916859 144 834425 452 165575 40 S 21 9751469 308 9-916773 144 9-834696 452 10165304 39 ; 22 751654 308 916687 144 834967 452 165033 38 > 33 751839 308 916600 144 835238 452 164762 37 ) •24 752023 307 916514 144 835509 452 164491 36 > 25 752208 307 916427 144 835780 451 164220 35 ) 2(3 752392 307 916341 144 836051 451 163949 34 > 27 752576 307 910254 144 836322 451 163678 33 ) 28 752760 307 916167 145 836593 451 163407 32 ) 99 752944 306 916081 145 836864 451 163136 3i ; 30 753128 306 915994 145 837134 451 162866 30 > 31 9753312 306 9-915907 145 9-837405 451 10162595 29 \ 32 753495 306 915820 145 837675 451 162325 28 ( 33 753679 306 915733 145 837946 451 162054 27 < 34 753862 305 915646 145 838216 451 161784 26 ? 35 754046 305 915559 145 838487 450 161513 25? 36 754229 305 915472 145 838757 450 161243 24 ? 37 754412 305 915385 145 839027 450 160973 23? 38 754595 305 915297 145 839297 450 160703 22? 3!) 754778 304 915210 145 839568 450 160432 21 ? 40 754960 304 915123 146 839838 450 160162 20? 41 9755143 304 9-915035 146 9-840108 450 10159892 19 ( 42 755326 304 914948 146 840378 450 159622 18 ( 43 755508 304 914860 146 840647 450 159353 17 ( 44 755690 304 914773 146 840917 449 159083 16 ( 45 755872 303 914685 146 841187 449 158813 15 < 46 756054 303 914598 146 841457 449 158543 14 ( 47 756236 303 914510 146 841726 449 158274 13 ( 48 756418 303 914422 146 841996 449 158004 12 ( 49 756600 303 914334 146 842266 449 157734 u S 50 756782 302 914246 147 842535 449 157465 10 ( 51 9-756963 302 9-914158 147 9-842805 449 10157195 9 ) 52 757144 302 914070 147 843074 449 156926 8 ) 53 757326 302 913982 147 843343 449 156657 7 ) 54 757507 302 913894 147 843612 449 156388 c > 55 757688 301 913806 147 843882 448 156118 5S 96 757869 301 913718 147 844151 448 155849 4) 57 758050 301 913630 147 844420 448 155580 3 ) 58 758230 301 913541 147 844689 448 155311 i sy 758411 301 913453 147 844958 448 155042 60 758591 301 913365 147 845227 448 154773 ! Cosine 1 Sine l^~> Cotaiig. 1 Tang. ^2 5 j Degree S. SINKS AND TANGENTS. (35 Degrees.) 171 M. S^ D. Cosine D. Tariff. D. | Cotanff. 9-758591 301 9-913365 147 9.845227 448 10-154773 60 j 1 758772 300 913276 147 845496 448 154504 59) o 758952 300 913187 148 845764 448 154236 58) 3 759132 300 913099 148 84(5033 448 153967 57) 4 759312 300 913010 148 846302 448 153698 56) 5 759492 300 912922 148 846570 447 153430 55) 6 759672 299 912833 148 846839 447 153161 54) 7 759852 299 912744 148 847107 447 152893 53) 52> e 760031 299 912655 148 847376 447 152624 9 760211 299 912566 148 847644 447 152356 51) 10 760390 299 912477 148 847913 447 152087 50) n 9-760569 298 9-912388 148 9.848181 447 10151819 49? 12 760748 298 912299 149 848449 447 151551 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 150746 45 ? 16 761464 298 911942 149 849522 447 150478 44? 17 761642 297 911853 149 849790 446 150210 43? 18 761821 297 911763 149 850058 446 149942 42? 19 761999 297 911674 149 850325 446 149675 41? 20 762177 297 911584 149 850593 446 149407 40? 21 9762356 297 9-911495 149 9850861 446 10-149139 39 ( 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 ( 25 763067 296 911136 150 851931 446 148069 35( 26 763245 296 911046 150 852199 446 147801 34 ( 27 763422 296 910956 150 852466 446 147534 33 ( 28 763600 295 910866 150 852733 445 147267 32 ( 29 763777 295 910776 150 853001 445 146999 31 ( 30 763954 295 910686 150 853268 445 146732 30 ( 31 9-7641 31 295 9-910596 150 9-853535 445 10-146465 29 J 32 764308 295 910506 150 853802 445 146198 28 S 33 764485 294 910415 150 854069 445 145931 27 ) 34 764662 294 910325 151 854336 445 145664 26 S 35 764838 294 910235 151 854603 445 145397 25 S 36 765015 294 910144 151 854870 445 145130 24 \ 37 765191 294 910054 151 855137 445 144863 23$ 38 765367 294 909963 151 855404 445 144596 22 \ 21 ' 39 765544 293 909873 151 855671 444 144329 40 765720 293 909782 151 855938 444 144062 20 <, 41 9-765896 293 9-909691 151 9-856204 444 10143796 29 S 42 766072 2<>3 909601 151 856471 444 143529 18) 43 766247 293 909510 151 856737 444 143263 17) 44 766423 293 909419 151 857004 444 142996 16) 45 766598 292 909328 152 857270 444 142730 15) 46 766774 292 909237 152 857537 444 142463 14) 47 766949 292 909146 152 857803 444 142197 13) 48 767124 292 909055 152 858069 444 141931 12) 49 767300 292 908964 152 858336 444 141664 11 ) 50 767475 291 908873 152 858602 443 141398 10 j 51 9-767649 291 9-908781 152 9-858863 443 10-1411 r 2 9? 52 767824 291 908690 152 859134 443 140866 8 ? 53 767999 291 908599 152 859400 443 140600 7) 54 768173 291 908507 152 859666 443 140334 6) 55 768348 290 908416 153 859932 443 140068 5 ? 56 768522 290 908324 153 860198 443 139802 4 57 768697 290 908233 153 860464 443 139536 3) 58 768871 290 908141 153 860730 443 139270 S( 59 769045 290 908049 153 860995 443 139005 ) ^ 769219 290 907958 153 861261 443 138739 0) i 1 Cosine 1 | Sine 1 | Cotang. 1 I Tan?. M.? 54 Degrees. 172 (36 Degrees.) A TABLE OP LOGARITHMIC [m/ Sine 1 D- Cosine D. Tang. D. Cotang. ( ° 9-769219 290 9-907958 153 9-861261 443 10-138739 60 \ ? 1 769393 289 907866 153 861527 443 138473 59 ( ( 2 769566 289 907774 153 861792 442 138208 58 ( ( 3 769740 289 907682 353 862058 442 137942 57 < ( 4 '09913 289 907590 153 862323 442 137677 56 ( 5 770087 289 907498 153 862589 442 137411 55 ( ( 6 770260 288 907406 153 862854 442 137146 54 ( 7 770433 288 907314 154 863119 442 136881 53 ( 8 770606 288 907222 154 863385 442 136615 52 ( C 9 770779 288 907129 154 863650 442 136350 51 < ! 10 770952 288 907037 154 863915 442 136085 50 j \ n 9-771125 288 9-906945 154 9-864180 442 10135820 49 I 12 771298 287 906852 154 864445 442 135555 48 ( 13 771470 287 906760 154 864710 442 135290 47 < 14 771643 287 906667 154 864975 441 135025 46 ( 15 771815 287 906575 154 865240 441 134760 45 ( ( 10 771987 287 906482 154 865505 441 134495 44 ( ( 17 772159 287 906389 155 865770 441 134230 43 ( ) 18 772331 286 906296 155 866035 441 133965 42 ( ) 19 772503 286 90G204 155 866300 441 133700 41 ( J 20 772675 286 906111 155 866564 441 133436 40 } 21 9-772847 286 9-906018 155 9-866829 441 10133171 39 { ) 22 773018 286 905925 155 867094 441 132906 38 S ) 23 773190 286 905832 155 867358 441 232642 37 > ) 24 773361 285 905739 155 867623 441 132377 36 } 35 < s 32 ( ) 25 773533 285 905645 155 867887 441 132113 } 28 773704 285 905552 155 868152 440 131848 ; 27 773875 285 905459 155 868416 440 131584 ) 28 774046 285 905366 156 868680 440 131320 ) 29 774217 285 905272 156 868945 440 131055 31 ( ) 30 774388 284 905179 156 869209 440 130791 30 < ( 31 9-774558 284 9-9050135 156 9-869473 440 10130527 29 ) ( 32 774729 284 904992 156 869737 440 130263 28 S ( 33 774899 284 904898 156 870001 440 129999 27 ) ( 34 775070 284 904804 156 870265 440 129735 26 S ( 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 S ( 38 775750 283 904429 157 871321 440 128679 22 S ( 39 775920 283 904335 157 871585 440 128415 21 ( j 40 776090 283 904241 157 871849 439 128151 20 j J 41 9-776259 283 9-904147 157 9-872112 439 10-127888 19 ) ) 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 127097 16 ) \ 45 776937 282 903770 157 873167 439 126833 15 ) \ 46 777106 282 903676 157 873430 439 126570 14 > ) 47 777275 281 903581 157 873694 439 126306 13 ) S 48 777444 281 903487 157 873957 439 126043 12) S 49 777613 281 903392 158 874220 439 125780 11 10 < < 50 777781 281 903298 158 874484 439 125516 } 51 9777950 281 9-903203 158 9-874747 439 10-125253 9) ) 52 778119 281 903108 158 875010 439 124990 8 ( ) 53 778287 280 903014 158 875273 438 124727 7 I ) 54 778455 280 902919 158 875536 438 124464 6 ( ) 55 778624 280 902824 158 875800 438 124200 5 ) 56 778792 280 902729 158 870063 438 123937 4' ) 57 778960 280 902634 158 876326 438 123674 3 < ) 58 779128 280 902539 159 876589 438 123411 2) ) 59 779295 279 902444 159 876851 438 123149 1 ) j 60 779463 279 902349 159 877114 438 122886 j Cosine Smj Cotang. Tang. M. j 53 Degrees. INES AND tangents. (37 Degrees.) 173 p5^ 1 o sine D. Cosine D. Tan* D. Cotang. | ) 9-779463 279 9902349 159 9-877114 438 10-122886 60 ( ) i 779631 279 902253 159 877377 438 122623 59 ( l 2 779798 279 902158 159 877640 438 122360 58 ) f 3 779966 279 902063 159 877903 438 122097 57 / ) 4 780133 279 901967 159 878105 438 121835 56 ; 55 > / 5 780300 278 901872 159 878428 438 121572 > G 780467 278 90J"76 159 878691 438 121309 54 > / 7 780634 278 901681 159 878953 437 121047 53 ) ) 8 780801 278 901585 159 879216 437 120784 52 ) ) 9 780968 278 901490 159 879478 437 120522 51 ) 1 10 781134 278 901394 160 879741 437 120259 50 J c u 9-781301 277 9-901298 160 9-880003 437 10119997 49 ) ( 12 781468 277 901202 160 880265 437 119735 48 } I 13 781634 277 901106 160 880528 437 119472 47 > ( I 4 781800 277 901010 160 880790 437 119210 46 ) ( I 5 781966 277 900914 160 881052 437 118948 45 ) ( 16 ^32132 277 900818 160 881314 437 118686 44 c n 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 ) > 2a 782796 276 900433 161 882363 436 117637 40 ) ( 21 9-782961 276 9-900337 161 9882625 436 10117375 39 j ' 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 ) I 26 783788 275 899854 161 883934 436 116066 34 ) < 27 783953 275 899757 161 884196 •436 115804 33 ) ? 28 784118 275 899660 161 884457 436 115543 32 > < 29 784282 274 899564 161 884719 436 115281 31 S ? 30 784447 274 899467 162 884980 436 115020 30 S S 31 9-784612 274 9-899370 162 9-885242 436 10114758 29 ) C 32 784776 274 899273 162 885503 436 114497 28 ) < 33 784941 274 899176 162 885765 436 114235 27 ) S 34 785105 274 899078 162 886026 436 113974 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 1 J 2667 21 ) j 40 786089 273 898494 163 887594 435 112406 20 j ) 41 9-786252 272 9-898397 163 9-887855 435 10112145 19 ) ( 42 786416 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 ) < 46 787069 272 897908 163 889160 435 110840 14 ) ( 47 787232 271 897810 163 889421 435 110579 13 > < 48 787395 271 897712 163 889682 435 110318 12 ) < 49 787557 271 897614 163 889943 435 110057 11 ? ( 50 787720 271 897516 163 890204 434 119796 10 ) > 51 9-787883 271 9-897418 164 9-890465 434 10109535 9 ( S 52 788045 271 897320 164 890725 434 109275 8 ( J 53 788208 271 897222 164 890986 434 109014 7 I ; 54 788370 270 897123 164 891247 434 108753 6 ? ) 55 788532 270 897025 164 891507 434 108493 5 ( ) 56 788694 270 896926 164 891768 434 108232 4 ( ) 57 788856 270 896828 164 892028 434 107972 3 { 2 / ) 58 789018 270 896729 164 892289 434 107711 ) 59 789180 270 896631 164 892549 434 107451 1 ) J 60 789342 269 896532 164 892810 434 107190 ) Ly- Cosine Sine -^~^~ Cotang. •N.O^- -^>_ Tang. ^mJ 52 Degrees. 174 (38 Degrees.) a table of logarithmic CiC Sine D. | Cosine D. Tang. | ^dT^ Cotp.ng ( ( o 9789342 269 9896532 164 9 892810 434 10107190 60 ( ) i 789504 269 896433 165 893070 434 106930 59 S ; 2 789665 269 896335 165 893331 434 106669 58 S ; 3 789827 269 896236 165 893591 434 106409 57 S ) 4 789988 269 896137 165 893851 434 106149 56 ( ) 5 790149 269 896038 165 894111 434 105889 55 , \ 6 790310 268 895939 165 894371 434 105629 54 53 \ J 7 790471 268 895840 165 894632 433 105368 / 8 \ 9 790632 268 895741 165 894892 433 105108 52 \ 790793 268 895641 165 895152 433 104848 51 < ,10 790954 268 895542 165 895412 433 104588 50 j ) 11 9 791115 268 9-895443 166 9-895672 433 10104328 49 J ) 12 791275 267 895343 166 895932 433 104068 48 ) ) 13 791436 267 895244 166 896192 433 103808 47 ) ) 14 791596 267 895145 1(56 896452 433 103548 46 ) ) J 5 791757 267 895045 166 896712 433 103288 45 ) i 16 791917 267 894945 166 896971 433 103029 44 > ) 17 792077 267 894846 166 897231 433 102769 43) I 18 792237 266 894746 166 897491 433 102509 42) ) 19 792397 266 894646 166 897751 433 102249 41 ) ) 20 792557 266 894546 166 898010 433 101990 40 j 39/ ( 21 9792716 266 9-894446 167 9-898270 433 10101730 ( 22 792876 2(56 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 J00173 33) ( 28 793832 265 893745 167 900086 432 099914 32) f 29 793991 265 893645 167 900346 432 099654 31 ) ( 30 794150 264 893544 167 900605 432 099395 30 ) I 31 9794308 264 9-893444 168 9-900864 432 10099136 29( 33 794467 264 893343 168 901124 432 098876 28( 33 794626 264 893243 168 901383 432 098617 27 C ) 34 794784 264 893142 168 901642 432 098358 26 I ) 35 794942 264 893041 168 901901 432 098099 25 < S 36 795101 264 892940 168 902160 432 097840 24 ( ( 37 795259 264 892839 168 902419 432 097581 23 I S 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 j 1 41 9 795891 263 9-892435 169 9-903455 431 10 096545 19 ) 42 796049 263 892334 169 903714 431 096286 18 S ) 43 796206 263 892233 169 903973 431 096027 17 < ) 44 796364 262 892132 169 904232 431 095768 16 ( ) 45 796521 262 892030 169 904491 431 095509 15 < ) 46 796679 262 891929 169 904750 431 095250 14 ( ) 47 796836 262 891827 169 905008 431 094992 13 < ) 48 796993 262 891726 169 905267 431 094733 12 ) 49 797150 261 891624 169 905526 431 094474 11 ! J 50 797307 261 891523 170 905784 431 094216 10 ( ( 51 9797464 261 9-891421 170 9-906043 431 10093957 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 ) / 55 798091 261 891013 170 907077 431 092923 5 ; ( 56 798247 261 890911 170 907336 431 092664 4 ; < 57 798403 260 890809 170 907594 431 092406 3) / 58 798360 260 890707 170 907852 431 092148 2) / 59 798716 260 890605 170 908111 430 091889 15 ? 60 798872 260 890503 170 908369 430 091631 I I Cosine 51 Degrees. bines AND tang K NTS, (30 Degrees.) 175 > o 9-798872 260 9-890503 170 9-908309 430 10-091631 60? ) 1 799028 260 890400 171 908628 430 091372 59? 2 799184 260 890298 171 908886 430 091114 58/ 3 799339 259 890195 171 909144 430 090856 57 ) 4 799495 259 890093 171 909402 430 090598 56 ) ) 5 799051 259 889990 171 909600 430 090340 55 > ) <> 799806 259 889888 171 909918 430 090082 51? 7 799962 259 889785 171 910177 430 089823 53 ) > 8 800117 259 889682 171 910435 430 089565 52? > 9 800272 258 889579 171 910693 430 089307 51 1 10 800427 258 889477 171 910951 430 089049 50) 49 I > 11 9800582 258 9-889374 172 9911209 430 10-088791 12 800737 258 889271 172 9114G7 430 088533 48? 13 800892 258 889168 172 911724 430 088276 47 ( 14 801047 258 889064 172 911982 430 088018 46 < 45 I 15 801201 258 888961 172 912240 430 087760 > 10 801356 257 888858 172 912498 430 087502 44 ( > 17 801511 257 888755 172 912756 430 087244 43 < i 18 801665 257 888651 172 913014 429 C8C986 42? i 19 801819 257 888548 172 913271 429 086729 41? 20 801973 257 888444 173 913529 429 086471 40 J 21 9-802128 257 9-888341 173 9-913787 429 10-086213 39 j 22 802282 256 888237 173 914044 429 085956 38j 37 < 3«< 1 23 802436 256 888134 173 914302 429 085698 24 802589 256 888030 173 914560 429 085440 25 802743 256 887926 173 914817 429 085183 35$ 26 802897 256 887822 173 915075 429 084925 34 S '27 803050 256 887718 173 915332 4:29 084608 33 S 28 803204 256 887614 173 915590 429 084410 32 ( 29 803357 255 887510 173 915847 429 084153 31 ( 30 803511 255 887406 174 91G104 429 083896 30) 31 9-803664 255 9-887302 174 9916362 429 10-083638 29) 32 803817 255 887198 174 91C619 429 083381 28) 33 803970 255 887093 174 916877 429 083123 27) 34 804123 255 886989 174 917134 429 082866 26) 35 804276 254 886885 174 917391 429 082609 25) I 3# 804428 254 886780 174 917648 - 429 082352 24) 37 804581 254 886676 174 917905 429 082095 23) 38 804734 254 * 886571 174 918163 428 081837 22 ) 39 804886 254 886466 174 918420 428 081580 21) ! 40 805039 254 886362 175 918677 428 081323 20 j J 41 9-805191 254 9-886257 175 9-918934 428 10-081066 19? \ 42 805343 253 886152 175 919191 428 080809 18? ( 43 805495 253 886047 175 919448 428 080552 17 ( ( 44 805647 253 885942 175 919705 428 080295 16) ( 45 805799 253 885837 175 919962 428 080038 15) ( 46 805951 253 885732 175 920219 428 079781 14 » ( 47 806103 253 885627 175 920476 428 079524 13) (48 ( 49 806254 253 885522 175 920733 428 079267 12) 806406 252 885416 175 920990 428 079010 ] U j 50 806557 252 885311 176 921247 428 078753 10 ) 9806709 252 9-885205 176 9921503 428 10*078497 of ) 52 806860 252 885100 176 921760 428 078240 8 S ) 53 807011 252 884994 176 922017 428 077983 7 S 54 807163 252 884889 176 922274 428 077726 6 S 55 S 58 <57 807314 252 884783 176 922530 428 077470 5 ! 807465 251 884677 176 922787 428 077213 4 807615 251 884572 176 923044 428 076956 3 ) 58 807766 251 884466 176 923300 428 076700 2 S ) 59 807917 251 884360 176 923557 427 076443 l\ }60 808067 251 884254 177 923813 427 076187 °i> I ; Coeine J^ Sine I | Cotang\ l^~^ 1~-32T£~- ikj V-^. 1 Decree S. ^-^~/ 176 (40 Degrees.) a table op logarithmic < M. | Sine I D. Cosine J I Tang. | I Cotanjf. | 9-808067 251 9-884254 177 9-923813 427 10-076187 60 / 1 808218 251 884148 177 924070 427 075930 59 I ) 2 808368 251 884042 177 924327 427 075673 58 / ) 3 808519 250 883936 177 924583 427 075417 57 ) ) 4 808669 250 883829 177 924840 427 075160 56 I ) 5 808819 250 883723 177 925096 427 074904 55 / ) 6 808969 250 883617 177 925352 427 074648 54 ; > 7 809119 250 883510 177 925609 427 074391 53 / ) 8 809269 250 883404 177 925865 427 074135 52 / ) 9 809419 249 883297 178 926122 427 073878 51 / S 10 809569 249 883191 178 926378 427 073622 50 / > 11 9809718 249 9-883084 178 9-926634 427 10073366 49 ( ) J2 809868 249 882977 178 926890 427 073110 48 <, ) 13 810017 249 882871 178 927147 427 072853 47 ( ) 14 810167 249 882764 178 927403 427 072597 46 < > 15 810316 248 882657 178 927659 427 072341 45 < I 16 810465 248 882550 178 927915 427 072085 44 < I 17 810614 248 882443 178 928171 427 071829 43 ( ) 18 810763 248 882336 179 928427 928683 427 071573 42 ( ) 19 810912 248 882229 179 427 071317 41 ( ? 20 811061 248 882121 179 928940 427 071060 40 ( ( 21 9-811210 248 9-882014 179 9-929196 427 10070804 39 ) ( 22 811358 247 881907 179 929452 427 070548 38 ) ( 23 811507 247 881799 179 929708 427 070292 37 ) ( 24 811655 247 881692 179 929964 426 070036 36 ) ( 25 811804 247 881584 179 930220 426 069780 35 ) ( 26 811952 247 881477 179 930475 426 069525 34 S (27 812100 247 881369 179 930731 426 069269 33 S ( 28 8i2248 247 881261 180 930987 426 069013 32 S < 29 812396 246 881153 180 931243 426 068757 31 S ( 30 812544 246 881046 180 931499 426 068501 30 j ( 31 9812692 246 9-880938 180 9 931755 426 10068245 29 J ) 32 812840 246 880830 180 932010 426 067990 28 ) ) 33 812988 246 880722 180 932266 426 067734 27 ) )34 813135 246 880613 180 932522 426 067478 26 > > 35 813283 246 880505 180 932778 426 067222 25 ) ) 36 813430 245 880397 180 933033 426 066907 24) ) 37 813578 245 880289 181 933289 426 066711 23 ) > 38 813725 245 880180 181 933545 426 066455 22 ) ) 39 813872 245 880072 181 933800 426 066200 21 j ) 40 814019 245 879963 181 934056 426 065944 20 S J 41 9-814166 245 9-879855 181 9934311 426 10065689 19 ) (42 814313 245 879746 181 934567 426 065433 18 ) 43 814460 244 879637 181 934823 426 065177 17 ) 44 814607 244 879529 181 935078 426 064922 16 ) ( 45 814753 244 879420 181 935333 426 064667 15 ) (46 814900 244 879311 181 935589 426 064411 14 ) < 47 815046 244 879202 182 935844 426 064156 13 ) (48 815193 244 879093 182 936100 426 063900 12 ) (49 815339 244 878984 182 936355 426 063645 11 / (50 815485 243 878875 182 936610 426 063390 10 J )51 9-815631 243 9-878766 182 9-936866 425 10-063134 9 ( ) 52 815778 243 878656 182 937121 425 062879 8 ( ) 53 815924 243 878547 182 937376 425 062624 7 ( ) 5 4 816069 243 878438 182 937632 425 062368 6 / S 55 816215 243 878328 182 937887 425 062113 5 ) ) 56 816361 243 878219 183 938142 425 061858 4 ( > 57 816507 242 878109 183 938398 425 061002 3 ) ) 58 816652 242 877999 183 938653 425 061347 2 ) )59 816798 242 877890 183 938908 425 061092 1 ) S60 816943 242 877780 183 939163 425 060837 j r | Cosine I | Sine 1 | Cotan^. 1 I Tan*. 1 M. 1 49 Degrees. SINES 4ND TANGENTS. (41 Degrees.) 177 | D. | Cosine | Ttog | D. | Cotang. | ) 9-816943 242 9877780 183 9939163 425 10060837 60 ( 1 817088 242 877670 183 939418 425 060582 59 ) 2 817233 242 877560 183 939673 425 060327 58 ) 3 817379 242 877450 183 939928 425 060072 57 ', 4 817524 241 877340 - 183 940183 425 059817 56 ) 5 817068 241 877230 184 940438 425 059562 55 S 6 817813 241 877120 184 940694 425 059306 54 S 7 817958 241 877010 184 940949 425 059051 53 4 8 818103 241 876899 184 941204 425 058796 52 S 9 818247 241 876789 184 941458 425 058542 51 < 10 818392 241 876678 184 941714 425 058286 50 5 n 9-818536 240 9 876508 184 9-941968 425 10-058032 49 >13 < 13 818681 240 876457 184 942223 425 057777 48 818825 240 876347 184 942478 425 057522 47 \ 14 818969 240 876236 185 942733 425 057267 46 SJ5 819113 240 876125 185 942988 425 057012 45 (l6 819257 240 876014 185 943243 425 056757 44 S 17 819401 240 875904 185 943498 425 056502 43' ) 18 819545 239 875793 185 943752 425 056248 42 S j9 819689 239 875682 185 944007 425 055993 41 ^20 819832 239 875571 185 944262 425 055738 40' )21 9-819976 239 9-875459 185 9-944517 425 10055483 39 ;22 820120 239 875348 185 944771 424 055229 38 >23 820263 239 875237 185 945026 424 054974 37 )24 820406 239 875126 186 945281 424 054719 36 25 820550 238 875014 186 945535 424 054465 35) )26 820693 238 874903 186 945790 424 054210 34) }27 820836 238 874791 186 946045 424 053955 33) )28 820979 238 874680 186 946299 424 053701 32) S 29 821122 238 874568 186 946554 424 053446 31) J 30 821265 238 874456 186 946808 424 053192 30 ) 1 31 9821407 238 9-874344 186 9947063 424 10052937 29 J >32 821550 238 874232 187 947318 424 052682 28 ( X33 821693 237 874121 187 947572 424 052428 27 ( )34 821835 237 874009 187 947826 424 052174 26 ( )35 821977 237 873896 187 948081 424 051919 25 l )36 822120 237 873784 187 948336 424 051664 24 I )37 822262 237 873672 187 948590 424 051410 23 ( )38 822404 237 873560 187 948844 424 051156 22 ( )39 822546 237 873448 187 949099 424 050901 21 ) )40 822688 236 873335 187 949353 424 050647 20 J 41 9-822830 236 9-873223 187 9-949607 424 10.050393 19 (42 822972 236 873110 188 949862 424 050138 18 S ?43 823114 236 872998 188 950116 424 049884 17 ( /44 823255 236 872885 188 950370 424 049630 16 > ?45 823397 236 872772 188 950625 424 049375 15) ?46 823539 236 872659 188 950879 424 049121 14 \ )47 823680 235 872547 188 951133 424 048867 13 ( >48 823821 235 872434 188 951388 424 048612 12 > ■49 823963 235 872321 188 951642 424 048358 U ) >50 824104 235 872208 188 951896 424 048104 10 J 51 9-824245 235 9-872095 189 9-952150 424 10047850 9 ; 52 824386 235 871981 189 952405 424 047595 ? ) 53 824527 235 871868 189 952659 424 047341 7 ? 54 824668 234 871755 189 952913 424 047087 6) ( 55 824808 234 871641 189 953167 423 046833 Si (50 824949 234 871528 189 953421 423 046579 57 825090 234 871414 189 953675 423 046325 3) 58 825230 234 871301 189 953929 423 046071 2 ) (59 825371 234 871187 189 954183 423 0458 J 7 1 ) (60 825511 234 871073 190 954437 423 045563 w I I Cotaug. 48 Degrees. i M.r 178 (-12 Degrees ) a TABLE OF LOGARITHMIC smT Sine 1 D. | Cosine 1 I>- 1 Tan,?. 1 D- | Cotsng. ]~i J 9-825511 234 9-871073 | 190 9-954437 423 10-045563 60 { 1 825651 233 870960 ( 190 954691 423 045309 59 { 58 ( ) 2 825791 233 870846 1 190 954945 423 045055 3 ( * ( 5 825931 233 870732 190 955200 423 044800 57 ( 826071 233 870618 190 955454 423 044546 56 ( 826211 233 870504 190 955707 423 044293 55 ( 6 826351 233 870390 190 955961 423 044039 54 I 7 826491 233 870276 190 956215 423 043785 53 I 8 9 826631 233 870161 190 956469 423 043531 52 ( 826770 232 870047 191 956723 423 043277 51 ( 826910 232 869933 191 956977 423 043023 50 j ^ n 9-827049 232 9-869818 191 9-957231 423 10-042769 49 j 12 827189 232 869704 191 957485 423 042515 48 \ I 13 827328 232 869589 191 957739 423 042261 47 S ' 14 827467 232 869474 191 957993 423 042007 46 J t IS < 16 17 .18 19 20 1 21 827606 232 869360 191 958246 423 041754 45 \ 827745 232 869245 191 958500 423 041500 44 ) 827884 231 869130 191 958754 423 041246 43 S 828023 231 869015 192 959008 423 040992 42 \ 828162 231 868900 192 959262 423 040738 41 ; 828301 231 868785 192 959516 423 040484 40 • 0828439 231 9-868670 192 9-959769 423 10040231 39 ) '.22 828578 231 868555 192 900023 423 039977 38 > 23 828716 231 868440 192 9G0277 423 039723 37 ) 1 24 828855 230 868324 192 960531 423 039469 36 ) ) 25 828993 230 868209 192 960784 423 039216 35 ) 26 829131 230 868093 192 961038 423 038962 34 ) ',27 829269 230 867978 193 961291 423 038709 33 ; 32 / ' 28 829407 230 867862 193 961545 423 038455 29 i 30 829545 230 867747 193 961799 423 038201 31 > 829683 230 867631 193 962052 423 037948 30 ) ) 31 9829821 229 9-867515 193 9-962306 423 10037694 29 j > 32 829959 229 867399 193 962560 423 037440 28 ( ) 33 830097 229 867283 193 962813 423 037187 27 ( ) 34 830234 229 867167 193 963067 423 036933 26 ( ) 35 830372 229 867051 193 963320 423 036680 25 ( > 36 830509 229 866935 194 963574 423 036426 24 ( ) 37 830646 229 866819 194 963827 423 036173 23 I / 38 830784 229 866703 194 964081 423 035919 22 ( ) 39 830921 228 866586 194 964335 423 035665 21 ( ( 40 831058 228 866470 194 964588 422 035412 20 I ( 41 9-831195 228 9-866353 194 9-964842 422 10035158 19 J < 42 831332 228 866237 194 965095 422 034905 18 S { 43 831469 228 866120 194 965349 422 034651 17 > < 44 831606 228 866004 195 965602 422 034398 16 ) ( 45 831742 , 228 865887 195 965855 422 034145 15 > { 46 831879 ■ 228 865770 195 966109 422 033891 14 ) ( 47 832015 227 865653 195 966362 422 033638 13) ( 48 832152 227 865536 195 966616 422 033384 12 ) ( 49 832288 227 865419 195 966869 422 033131 11 ) J 50 832425 227 865302 195 967123 422 032877 10 j j 51 9-832561 227 9'865185 195 9-967376 422 10032624 9> ) 52 832697 227 865068 195 967629 422 032371 8 ( ) 53 832833 227 864950 195 967883 422 032117 7 ^ / 54 ) 55 5 56 832969 226 864833 196 968136 422 031864 6> 833105 226 864716 196 968389 422 031611 5? 833241 226 864598 196 968643 422 031357 4 I ) 57 833377 226 864481 196 968896 422 031104 3) 58 833512 226 864363 196 969149 422 030851 2 ? ) 59 833648 226 864245 196 969403 422 030597 J! | 60 833783 226 864127 196 969656 422 030344 ' \\ Cosine! *~^J Sine ! Cotang. | ^N^M«! ^^Tang^l . M -J 47 Degrees. SINES AND tangents. (43 Degrees.) 179 M. I I Cosine Tang. | D. | Cotang. | 9-833783 226 9-864127 196 9-969656 422 10-030344 60^ 1 833919 225 864010 196 969909 422 030091 59? 2 834054 225 863892 197 970162 422 029838 58? 3 834189 225 863774 197 970416 422 029584 57? 4 834325 225 863656 197 970669 422 029331 56? 5 834400 225 863538 197 970922 422 029078 55 6 834595 225 863419 197 971175 422 028825 54? 7 834730 225 863301 197 971429 422 028571 53 8 834865 2-25 863183 197 971682 422 028318 52? 9 834999 224 863064 197 971935 422 028065 51 ( 10 835134 224 862946 198 972188 422 02*812 50? 11 9-835269 224 9-862827 198 9-972441 422 10027559 49 j 12 835403 224 862709 198 972694 422 027306 48 ( 13 835538 224 862590 198 972948 422 027052 47 ( 14 835672 224 862471 198 973201 422 026799 46 < 15 835807 224 862353 198 973454 422 026546 45 ( 16 835941 224 862234 198 973707 422 026293 44 ( 17 836075 223 862115 198 973960 422 026040 43 ( 18 836209 223 861996 198 974213 422 025787 42 ( 19 836343 223 861877 198 974466 422 025534 41 C 20 836477 223 861758 199 974719 422 025281 40 ( 21 9 836611 223 9-861638 199 9-974973 422 10025027 39) 22 836745 223 861519 199 975220 422 024774 38 ( 23 836878 223 861400 199 975479 422 024521 37 < 24 837012 222 861280 199 975732 422 024268 36 ) 25 837146 222 861161 199 975985 422 024015 35 S 26 837279 222 861041 199 976238 422 023762 34S 27 837412 222 860922 199 976491 422 023509 33 S 28 837546 222 800802 199 976744 422 023256 32< 29 837679 222 860682 200 976997 422 023003 31 ( 30 837812 222 860562 200 977250 422 022750 30 J 31 9-837945 222 9-860442 200 9*977503 422 10-022497 29) 32 838078 221 860322 200 977756 422 022244 28) 33 838211 221 860202 200 978009 422 021991 27) 34 838344 221 860082 200 978262 422 021738 26) 35 838477 ' 221 859962 200 978515 422 021485 25) 36 8386 JO 221 859842 200 978768 422 021232 24 ) 37 838742 221 859721 201 979021 422 020979 23) 38 838875 221 859601 201 979274 422 020726 22 ) 39 839007 221 859480 201 979527 422 020473 21 ) 40 839140 220 859360 201 979780 422 020220 20) 41 9-839272 220 9-859239 201 9-980033 422 10019967 19 ) 42 839404 220 859119 201 980286 422 019714 18) 43 839536 220 858998 201 980538 422 019462 17 ) 44 839668 220 858877 201 980791 421 019209 16) 45 839800 220 858756 202 981044 421 018956 15) 46 839932 220 858635 202 981297 421 018703 14 ) 47 840064 219 858514 202 981550 421 018450 13 ) 48 840196 219 858393 202 981803 421 018197 12 ) 49 840328 219 858272 202 982056 421 017944 11 ) 50 840459 219 858151 202 982309 421 017691 io S 51 9-840591 219 9-858029 202 9-982562 421 10-017438 9( 52 840722 219 857908 202 982814 421 017186 8? 53 840854 219 857786 202 988067 421 016933 7 > 54 840985 219 857665 203 983320 421 016680 6) 55 841116 218 857543 203 983573 421 016427 5 ) 56 841247 218 857422 203 983826 421 016174 4 ) 57 841378 218 857300 203 984079 421 015921 3 ? 58 841509 218 857178 203 984331 421 015669 2) 59 841640 218 857056 203 984584 421 . 015416 l) 60 841771 218 856934 203 984837 421 015163 o) .X^S^l^J, 46 Degrees. 180 (44 Degrees.) a table of logarithmic M. | I D. Tan?. | Cotang. | ( 9-841771 ai8 9-856934 203 1 9-984837 421 10015163 60 I 1 841902 218 856812 2U3 985090 421 014910 59 < 2 842033 218 856690 204 985343 421 014657 58 ( 3 842163 217 856568 204 985596 421 014404 57 ( 4 842294 217 856446 204 985848 421 014152 56 ( 5 842424 217 856323 204 986101 421 013899 55 ( 6 842555 217 856201 204 986354 421 013646 54 ( 7 842685 217 856078 204 986607 421 013393 53 ( 8 842815 217 855956 204 986860 421 013140 52 l 9 842946 217 855833 204 987112 421 012888 51 J 10 843076 217 855711 205 987365 421 012635 50 i 11 9843206 216 9-855583 205 9-987618 421 10-012382 49 ( 12 843336 216 855465 205 987871 421 012129 48 ( 13 843466 216 855342 205 988123 421 011877 47 s 1 4 843595 216 855219 205 988376 421 011624 46 S 15 843725 216 855096 205 988629 421 011371 45 ( 16 843855 216 854973 205 988882 421 011118 44 I I 7 843984 216 854850 205 989134 421 010866 43 ( 18 844114 215 854727 206 989387 421 010613 42 ( 19 844243 215 854603 206 989640 421 010360 41 (20 844372 215 854480 206 989893 421 010107 40 < (21 9-844502 215 9854356 206 9-990145 421 10009855 39 S 22 844631 215 854233 206 . 990398 421 009602 38 < 23 844760 215 854109 206 990651 421 009349 37 < 24 844889 215 853986 206 990903 421 009097 36 <25 845018 215 853862 206 991156 421 008844 35 < 26 845147 215 853738 206 991409 421 008591 34 ( 27 845276 214 853614 207 991662 421 008338 33 < 28 845405 214 853490 207 991914 421 008086 32 < 29 845533 214 853366 207 992167 421 007833 31 ( 30 845662 214 853242 207 992420 421 007580 30 ) 31 9-845790 214 9-853118 207 9-992672 421 10-007328 29 ) 32 845919 214 852994 207 992925 421 007075 28 ) 33 846047 214 852869 207 993178 421 006822 27 ) 34 846175 214 852745 207 993430 421 006570 26 ) 35 846304 214 852620 207 993683 421 006317 25 ) 36 846432 213 852496 208 993936 421 006064 24 ) 37 846560 213 852371 208 994189 421 005811 23 S 38 846688 213 852247 208 994441 421 005559 22 S 39 846816 213 852122 208 994694 421 005306 21 S 40 846944 213 851997 208 994947 421 005053 20) j 41 9-847071 213 9-851872 208 9-995199 421 10004801 lo) > 42 847199 213 851747 208 995452 421 004548 18 P ) 43 847327 213 851622 208 995705 421 004295 n) ) 44 847454 212 851497 209 995957 421 004043 16? ) 45 847582 212 851372 209 996210 421 003790 15 ( ) 46 847709 212 851246 209 996463 421 003537 14 ( ) 47 847836 212 851121 209 996715 421 003285 13/ ) 4 8 847964 212 850996 209 . 996968 421 003032 12/ ) 49 848091 212 850870 209 997221 421 002779 11 / ) 50 848218 212 850745 209 997473 421 002527 10 \ <51 9-848345 212 9-850619 209 9-997726 421 10002274 9< (52 848472 211 850493 210 997979 421 002021 8< (53 848599 211 850368 210 998231 421 001769 7 ( <54 848726 211 850242 210 998484 421 001516 6S ( 55 848852 211 850116 210 998737 421 001263 5 ( < 56 • 848979 211 849990 210 998989 421 001011 4S < 57 849106 211 849864 210 999242 421 000758 3 S ? 58 849232 211 849738 210 999495 421 000505 2( < 59 849359 211 849611 210 999748 421 000253 1 ( I 00 849485 211 849485 210 10000000 421 000000 ( I Tang. I M.' 45 Degrees THE La OW 1 RETURN CIRCULATION DEPARTMENT TO— ► 202 Main Library ; LOAN PERIOD 1 HOME USE 2 3 4 5 5 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Renewals and Recharges may be made 4 days prior to the due date. Books may be Renewed by calling 642-3405 DUE AS STAMPED BELOW RECEIVED JUN 2 9 1996 CIRCULATION DEP t UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DD6 BERKELEY, CA 94720 @$ U. C. BERKELEY L 180 .^.?.. CD571EES22 V? : ffZLW i Qasi B THE UNIVERSITY OF CALIFORtflX^BRARY - ^ * %f. ■'**•