F LIBRARY OF CONGRESS. «{{.....?- (fippuftt |n- Shelf -J1-2-L UNITED STATES OF AMERICA. ELEMENTS OF GEOMETRY TRIGONOMETRY. APPLICATIONS IN MENSURATION BY CHARLES DA VIES. LL. D. (lOTnOR OF FIRST LESSONS IN ARITHMETIC, ELEMENTARY ALGEBRA, PRACTICAL MATHEMATICS FOR PRACTICAL MEN, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOMETRY, SHADES, SHADOWS, AND PERSPECTIVE, ANA- LYTICAL GEOMKTRY, DIFFERENTIAL AND INTEGRAL CALCULUS. - Copyright, 1858, by Charles Davies. Copyright Renewed, 1S8G, by Mary Ann Davjes- ::"L. GBOM, PREFACE. Those who are conversant with the preparation cf ele- mentary text-books, have experienced the difficulty of adapting them to the various wants which they aie in- tended to supply. The institutions of education are of all grades, from the college to the district school, and although there is a wide difference between the extremes, the level, in passing from one grade to the other, is scarcely broken. Each of these classes of seminaries requires text-books adapted to its own peculiar wants ; and if each held its proper place in its own class, the task of supplying suit- able works would not. be difficult. An indifferent college is generally inferior, in the system and scope of its instruction,to the academy or high school; while the district school is often found to be superior to its neighboring academy. The Geometry of Legendre, embracing a complete course oT Geometrical science, is all that is desired in the colleges and higher seminaries; while the Practical Mathematics for Practical Men, recently published, is designed to meet the wants of those schools which are strictly elementary and practical in their systems of instruction. PREFACE But still a large class of seminaries remained unsup- plied with a suitable text-book on Elementary Geometry and Trigonometry : viz., those where the pupils are car- ried beyond the acquisition of facts and mere practical knowledge, but have not time to go through with a full course of mathematical studies. It is tor such, that the following work is designed. It has been the aim of the author to present the striking and important truths of Geometry in a form moie simple and concise than could be adopted in a complete treatise and yet to preserve the exactness of rigorous reasoning. In this system of Geometry nothing has been taken foi granted, and nothing passed over without being fully de- monstrated. The Trigonometry, including the applications to the measurements of heights and distances, has been writ- ten upon the same plan and for the same objects : it embraces all the important theorems and all the striking examples. In order, however, to render the applications of Ge- ometry to the mensuration of surfaces and solids complete in itself, a few rules have been given which are not de- monstrated. This forms an exception to the general plan of the work, but being added in the form of an appendix, it does not materially break its unity. That the work may be useful in advancing the* interests of education, is the hope and ardent wish of the author. Fishkill Landing, May, 1861 CONTENTS. BOOK I. Pagb. Definitions and Remarks, ..... 9 — ifl Axioms, 16 Properties of Polygons, - 17 — 37 BOOK II. Of the Circle, 38 Problems relating to the First and Second Books, - • 53 -68 BOOK III. Ratios and Proportions, ... - 69--S1 BOOK IV. Measurement of Areas and Proportions of Figures, - • 82 — 108 Problems relating to the Fourth Book, .... 109 — 113 Appendix — Regular Polygons, - - 113 — 115 BOOK V Of Planea and their Angles, ... 110 — 126 BOOK VI. 01 Solids, - 126—102 Appendix, ....... 163—164 8, CONTENTS. TRIGONOMETRY. Page. Of Logarithms. - - - - 165—170 Of Scales, 176—181 Definitions, an i Ex| Janation of Tables, - 181—189 Theorems, 189—192 Examples, 193—201 Application to Heights and Distances, - 202—210 APPLICATIONS OF GEOMETRY. Mensuration of Surfaces, 211 General Principles, 211—213 Contents of Figures, .... 218—289 Mensuration of Solids, - 239 General Principles, .... 289—240 Solidities of Figures, ... 240—241 Mensuration of the Round Bodies, 248 To find the Surface. of a Cylinder, 248—249 To find the Solidity of a Cylinder, 249—250 To find the Surface of a Cone, 250—251 To find the Solidity of a Cone, 251- -252 To find the Surface of the Frustum of a Cone, 263 To find the Solidity of the Frustum of a Cone, 254 To find the Surface of a Sphere, .... 255 To find the Surface of a Spherical Zone, 255—256 To find the Solidity of a Sphere, 266—257 To find the Solidity of a Spherical Segment, - 258 To find the Solidity of a Spheroid, - 269—260 T: find the Surface of a Cylindrical Ring, 260—261 To find the Solidity of a Cylindrical Riq B» 261—262 ELEMENTARY GEOMETRY BOOK I. DEFINITIONS AND REMARKS. 1. Extension has three dimensions, length, breadth, ami thickness. Geometry is the science which has for its object : 1st. The measurement of extension ; and 2diy, To discover, by means of such measurement, the properties and relations of geometrical figures. 2. A Point is that which has place, or position, but not magnitude. 3. A Line is length, without breadth or thickness. 4. A Straight Line is one which lies in the same direction between any two of its points. 5. A Curve Line is one which changes is direction at every point. The word line when used alone, will designate a straight line ; and the word curve, a curve line. 6. A Surface is that which has length and breadth, with- out height or thickness. 7. A Plane Surface is that which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide in its whole length. 8. A Curved Surface has length and breadth without thick- ness, and like a curve line is constantly changing its direction 9. A Solid or Body is that which has length, breadth, and thickness. Length, breadth, and thickness are called dimen- 10 GEOMETRY De f i ni t i o i s. sions. Hence, a solid has three dimensions, a surface two and a line one. A point has no dimensions, but position only 10. Geometry treats of lines, surfaces, and solids. 1 1 . A Demonstration is a course of reasoning which estab- lishes a truth. 13. An Hypothesis is a supposition on which a demonstra- tion may be founded. 13. A Theorem is something to be proved by demonstration, 14. A Problem is something proposed to be done. 15. A Proposition is something proposed either to be done or demonstrated — and may be either a problem or a theorem. 16. A Corollary is an obvious consequence, deduced from something that has gone before. 17. A Scholium is a remark on one or more preceding propo* 6itions. 18. An Axiom is a self evident proposition. OF ANGLES. 19. An Angle is the portion of a plane included between two straight lines which meet at a common point. The tvrD straight lines are called the sides of the angle, and the common point of intersection, the vertex. Thus, the part of the plane included C between AB and A C is called an angle : /^ AB and AC are its sides, and A its vertex, a ft An angle is generally read, by placing the lettei at the vei iox in the middle. Thus, we say, the angle CAB. We may however, say simply, the angle A. 20. One line is said to be perpendicular to another when it inclines no more to the one side than to the other BOOK U Definitions, The two angles formed are then equal to each other. Thus, if the line DB is per- pendicular to AC, the angle DBA will be equal to DBC. 21. When two lines are perpendicular to each other, the angles which they form are called right angles. Thus, DBA and DBC are called right angles. 22. An acute angle is less than a right angle. Thus, DBC is an acute angle. 23. An obtuse angle is greater than a right angle. Thus, DBC is an obtuse angle. 24. The circumference of a cirole is a curve line all the points of which are equally distant from a certain poinJb within called the centre. Thus, if all the points of the curve AEB are equally distant from the centre (7, this curve will be the circumference of a circle. 25. Any portion of the circumference, as A ED, is called an arc 26. The diameter of a circle is a straight line passing through the centre and terminating at the circumference. Thus, A CB is a diameter. 27. One half of the circumference, as ACB is called a semicircumference ; and one quarter of the circumference, as A C is called a quadrant B £ D B ~C b e 8 rf 12 GEOMETRY. Definitions. 28. The circumference of a circle is used for the measure- ment of angles. For this purpose it is divided into 360 equal parts called degrees, each degree into 60 equal parts .called minutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds are marked thus ° ' " ; and 9° 18' 16", are read, 9 degrees 18 minutes and 16 seconds. 29. Let us suppose the circumference of a circle to be divided into 360 degrees, beginning at the point B. If through the point of division marked 40, we draw- ee, then, the angle ECB will be equal to 40 degrees. If CF were drawn through the point of division marked 80, the angle BCF would be equal to 80 degrees. OF LINES. 30. Two straight lines are said to be parallel, when being produced either way, as far as we please, they will not meet each other. 31. Two curves are said to be parallel or concentric, when they are the same dis- tance from each other at every point. 32. Oblique lines are those which ap- proach each other, and meet if sufficiently produced. 33. Lines which are parallel to the horizon, or to the water tevel, are called hor'zontal lines. 34. Lines which are perpendicular to the horizon, or to the water level are called vertical lines BOOK 1 13 D e fin it io ns. OF PLANE FIGURES. 35. A Plane Figure is a portion of a plane terminated on all sides by lines, either straight or curved. 36. If the lines which bound a figure are straight, the space wliich they inclose is called a rectilineal figure, oi polygon The lines themselves, taken together, are called the perimeter of the polygon. Hence, the perimeter of a poly gen is the sum of all its sides. 37. A polygon of three sides is called a triangle. 38. A polygon of four quadrilateral. sides is called 39. A polygon of five sides is called a pentagon. 40. A polygon of six sides is called hexagon. 41. A polygon ot seven sides is called a heptagon 42 A polygon of eight sides is called an octagon. 2 14 GEOMETRY De f in it jo ns. 43. A polygon of nine sides is called a nonagon. 44. A polygon of ten sides is called a decagon. 45. A polygon of twelve sides is called a dodecagon. 46. There are several kinds of triangles. First. An equilateral triangle, which has its three sides all equal. Second. An isosceles triangle, which has two of its sides equal. Third. A scalene triangle, three sides all unequal. rhich has its Fourth. A right angled triangle, which lias one right angle. In the right angled triangle ABC, the side AC, opposite the right angle, is called the hypothenuse. 47. The base of a triangle is the side on which it stands. Thus, AB is the base of the triangle ACB. The altitude of a triangle is a line drawn from the angle opposite the base and per- ,4 pendicular to the base, angle ACB Thus, CD is the altitude of the tri BOOK I 15 D e f ini ti ons. 48. There are three kinds of quadrilaterals. 1. The trapezium, which has none of Its sides parallel. 2. The trapezoid, which has only two of its sides parallel. 8. The parallelogram, which has its opposite sides parallel. 7 4y. There are four kinds of parallelograms 1. The rhomboid, which has no right / angle. / 2. The rhombus, or lozenge, which i3 an equilateral rhomboid. 3. The rectangle, which is an equian- gular parallelogram. 4. The square, which is both equilat- eral and equiangular. Iti GEOMETRY Of Axioms. 50. A Diagonal of a figure is a line which joins the vertices of two angles not adjacent. 51. The base of a figure is the side on which it is supposed to stand ; and the altitude is a line drawn from the opposite side or angle, perpendicular to the base. AXIOMS. 1 . Things wliich are equal to the same thing are equal to each other. 2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the wholes will be un« equal. 5. If equals be taken from unequals, the remainders will be unequal. 6. Things which are double of equal things, are equal to each other. 7. Things which are halves of the same thing, are equal to each other. 8. The whole is greater than any of its parts 9. The whole is equal to the sum of all its parts. 10. All right angles are equal to each other. 11. A straight line is the shortest distance between two points. 12. Magnitudes, which being applied to each other, coin- cide throughout their whole extent, are equal. BOOK 1 . 17 Of Angles. PROPERTIES OF POLYGONS. THEOREM I. Every diameter of a circle divides the circumference into tvn> equal parts. Let ADBE be the circumference of a circle, and ACB a diameter: then will the part ADB be equal to the part AEB. For, suppose the part AEB to be turn- ed around AB, until it shall fall on the part ADB. The curve AEB will then exactly coincide with the curve ADB, or else there would be some point in the curve AEB or A DB, unequally distant from the centre C, which is contrary to the definition ol a circumference (Def. 24). Hence, the two curves will be equal (Ax. 12). Corollary 1. If two lines, AB, DE, be drawn through the centre C perpen- dicular to each other, each will divide the circumference into two equal parts ; and the entire circumference will be divided into the equal quadrants DB, DA, AE, and EB. Cor. 2. Hence, a right angle, as DCB, is measured by one quadrant, or 90 degrees; two right angles by a semicireumfer- ence, or ISO degrees ; and four right angles by the whole cir- cumference, or 360 degrees 18 tx E IYI ETRV, Of Angle THEOREM II. Ij one straight line meet another straight line, the sum of thz two adjacent angles will be equal to two right angles. Let the straight line CD meet the straight line AB, at the point C; then will the angle DCB plus the angle DC A be equal to two right angles. A C B About the centre C, with any radius as CB, suppose a semicircumference to be described. Then, the angle DCB will be measured by the arc BD, and the angle DC A by the arc AD. But the sum of the two arcs is equal to a semicir- enmference ■ hence, the sum of the two angles is equal to two right angles (Th. i, Cor. 2). n Coi. 1. If one of the angles, as DCB, is a right angle, the other angle, DC A will also be a right angle. Cor. 2. Hence, all the angles which can be formed at any point C\ by any number of lines, CD, CE, CF, &c, drawn on the same side of AB, are equal to two right angles : for, they will be measured by a semicircumference. Cor. 3. If DC meets two lines CB, CA, making DCB plus DC A equal to two right angles, ACB will form one straight line. Cor. 4. Hence, also, all the angles which can be formed round any point, as C, are equal to four right angles. For, the sum of all the arcs which measure them, is equal to the entire circumference, which is the measure of four right angles (Th. i. Cor. 2). BOOK I 19 Of Triangles THEOREM III. Ij two straight lines intersect each other, the opposite or ver- tical angles which they form, are equal. Let the two straight lines A.B and y CD intersect each other at the point E : then will the opposite angle A EC — be equal to DEB, and AED=CEB. For, since the line AE meets the line CD, the angle AEC+AED= two right angles. Bui since the line DE meets the line AB, we have DEB+AED= two right angles. Taking away from these equals the com- mon angle AED, and there will remain the angle AEC equal to the angle DEB (Ax. 3). In the same manner we may prove that the angle AED is erjual to the angle CEB. THEOREM IV.. If two triangles have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal. Let the triangles ABC and DEF have the side A C equal to DF, CB to FE, and the angle C equal to the angle F: then will the triangle A CB be equal to the triangle DEF. For, suppose the side A C, of the A & & triangle ACB, to be placed on DF, so that the extremity C shall fall on the extremity F: then, since the sides are equal, A will fall on D. But since the angle C is equal to the angle F, the line CB 20 GEOMETRY Of Tria n g I c 8 will fall on FE ; and since CB is equal to FE, the extremity i? will fall on E ; and consequently the side AB will fall on the side DE (Ax. 11). Hence, the two triangles will fill the same space, and consequently are equal (Ax. 12.). ^ # ^ Scholium. Two triangles are said to be equal, when being applied the one to the other they exactly coincide (Ax. 12). Hence, equal triangles have their like parts equal, each to each, since those parts coincide with each other. The converse of the proposition is also true, namely, that two triangles which have all the parts of the one equal to the corresponding parts of the other, each to each, are equal : for if applied the one to the other, the equal parts will coincide. THEOREM V. If two triangles have two angles and the included side of tnv one, equal to two angles and the included side of the other, each to each, the two triangles will be equal. Let the two triangles ABC and DEF have the angle A equal to the angle D, the angle B equal to the angle E, and the included side AB equal to the included side DE • then will the triangle ABC bt equal to the triangle DEF. For, let the side AB be placed on the side DE, the extrem ity A on the extremity D ; and since the sides are equal, the point B will fall on the point E. Then since *he angle A is equal to the angle D, the side BOOK 1. ?*! Of Triangles AC will take the direction DF: and since the angle B is equal to the angle E, the side BC will fall or the side EF : hence, the point C will be found at the same time on DF and EF, and therefore will fall at the intersection F: consequently, all the parts of the triangle ABC will coincide with the parts of the triangle DEF, and therefore, the two triangles are equal THE O HEM VI. In an isosceles triangle the angles opposite the equal sides are equal to each other. Let ABC be an isosceles triangle, hav- ing the side AC equal to the side CB : then will the angle A be equal to the an- gle B. a n b For, suppose the line CD to be drawn dividing the angle C into two equal parts. Then, the two triangles A CD and DCB, have two sides and l he included angle of the one equal to two sides and the in- cluded angle of the other, each to each : that is, the side AC equal to BC, the side CD common, and the included angle .4 CD equal to the included angle DCB : hence the two than gles are equal (Th. iv) ; and hence, the angle A is equal to the angle B. Cor. 1. Hence, the line which bisects the vertical angle oi an isosceles triangle, bisects the base. It is also perpendicu- lar to the base, since the angle CD A is equal to the angle CDB. Cor. 2. Hence, also, every equilateral triangle, must also be equiangular: that is, have all its angles equal, each to each 22 GEOMETRY Of Triangles THEOREM VII. Conversely. — If a triangle has two of its angles equal, ifu sides opposite those angles will also be equal In the triangle ABC, let the angle A be fiqual to the angle B : tnen will the side BC be equal to the side AC. For, if the two sides are not equal, one of them must be greater than the other. Suppose AC to be the greater side. Then take a part AD equal to BC Now, in the two triangles ADB and ABC, we have thj side ADz=BC, by hypothesis ; the side AB common, and the angle A equal to the angle B : hence, the two triangles have two sides and the included angle of the one equal to two sides and the included angle of the other, each to each : hence, the two triangles are equal (Th. iv), that is, a part ADB is equal to the whole ABC, which is impossible (Ax. 8) : conse- quently, the side A C cannot be greater than the side CB, and hence, the triangle is isosceles. Scholium 1. The method of reasoning pursued in the last theorem, is called the " reductio ad absurdum," or a proof thai leads to a known absurdity. Let us analyze this method of reasoning. We wished to prove that the two sides AC, CB were equal. We supposed them unequal, anl AC the greater — that was an hypothesis (See Def. 12). We then reasoned on the hypothesis and proved a part equal to the whole, which we know to be false (Ax. 8) Hence, we conclude that the hypothesis is untrue, because after a correct chain of reasoning it leads to a result which we know to be absurd BOOK 2B Of Triangles Scholium 2. Generally, — If the demonstration is based on known principles, previously proved, or admitted in the ax- ioms, the conclusion will always be true. But, if the demon- stration is based on an hypothesis, (as in the last theorem, thai AC was the greater side), and the conclusion is contrary to what has been previously proved, or admitted in the axioms then, it follows, that the hypothesis cannot be true. The former is called a direct, and the latter an indirect demonstration. THEOREM VIII. If two triangles have the three sides of the one cqval to t/tf. three sides of the other, each to each, the three angles will aim be equal, each to each. Let the two triangles ABC, ABD, have the side A B equal to the side AB, the side A C equal to AD, and the side CB equal to DB : then will the corres- ponding angles also be equal, viz : the angle A will be equal to the angle A, the angle B to the angle B, and the angle C to the angle D. For, suppose the triangles to be joined by their longest equal sides A B, and the line CD to be drawn. Then, since the side A C is equal to AD, by hypothesis, the triangle ADC will be isosceles; and therefore, the angle ACD will be equal to the angle ADC (Th. vi). In like maimer, in the triangle CBD, the side CB is equal to DB : hence, the angle BCD is equal to the angle BDC. Now, by the addition o! equals, we have \ 24 GEOMETRY. Of Triangles. ACD+BCD=ADC+BDC that is, the angle ACB=ADB. Now, the two triangles ACB and ADB have two sides and the included angle of the one equal to two sides and the in- cluded angle of the other, each to each: hence, the remaining angles will be equal (Th. iv) : consequently, the angle CAB is equal to BAD, and the angle CBA to the angle ABD. Sch. The angles of the two triangles which are equal to each other, are those which lie opposite the equal sides. THEOREM IX. If one side of a triangle is produced, the outward angle is greater than either of the inward opposite angles. Let ABC be a triangle, having the side AB produced to D : then will the outward angle CBD be greater than either of the inward opposite angles A or C. For, suppose the side CB to be bisected at the point E Draw AE, and produce it until EF is equal to AE, and then draw BF. Now, since the two triangles AEC and BEFhave AE— EF and EC—EB, and the included angle AEC equal to the included angle BEF (Th. iii), the two triangles will be equal in all respects (Th. iv) : hence, the angle EBF will be equal to the angle C. But the angle CBD is greater than the angle CBF, consequently it is greater than the angle C. In like manner, if CB be produced to G, and AB be bi sected, it may be proved that the outward angle ABG, or its equal CBD (Th. iii), is greater than the angle A BOOK l. 2b Of Triangles THEOREM X. The sum of any two rides of a triangle is greater than th* third ride. let ABC ha a triangle ■ then will the siim of two of its sides, as AC, CB, be g eater than tlte third side AB. For the straight line AB is the short- est distance between the two points A and B (Ax. xi): hence AC+CB is greater than AB. THEOREM XI. The greater side of every triangle is opposite the greater ang^e, and conversely, the greater angle is opposite the grca'er side First. In the triangle CAB, let the an- 6 A gle C be greater than the angle B : then, will the side AB be greater than the side A C. ( i T For, draw CD, making the angle BCD equal to the angle B. Then, the triangle CBD will be isosceles : hence, the side CD = DB (Th. vii.) Hut, by the last theorem AC is less than AD-\- CD ; thai is loss than AD + DB, and consequently less than AB. Secondly. Let us suppose the side AB to be greater than A C; then will the angle C be greater than the angle B. For if the angle C were equal to B, the triangle CA /> would re isosceles, and the side A C would be equal to A B v Th. vii) , which would be contrary to the hypothesis. Again if the angle C were less than B, then, bv the first part of the theorem, the side AB would be less than AC, which is also contrary to the hypothesis Hence, since C 2G GEO M E T R Y Of Parallel Lines. cannot be equal to B, nor less than B, it follows that it must be greater THEOREM XII. // a straight line intersect two parallel lines, the alternate anglis will be equal. If two parallel straight lines, AB CD, are intersected by a third line GH, the ]?/ angles AEF and EFD are called alternate " /..*"'" ^ angles. It is required to prove that these C ~/P /) angles are equal. If they are unequal one of them must be greater than the other. Suppose EFD to be the greater angle. Now conceive FB to be drawn, making the angle EFB equal to the angle AEF, and meeting AE in B Then, in the triangle FEB the outward angle FEA is greater than either of the inward angles B or EFB (Th. ix.) ; and therefore, EFB can never be equalto AEFso long as FB meets EB. But since we have supposed EFD to be greater than AEF, it follows that EFB could not be equal to AEF, if FB fell be- low FD. Therefore, if the angle EFB is equal to the angle AEF, FB cannot meet AB, nor fall below FD, and conse- quently must coincide with the parallel CD ( Def. 30) : and once, the alternate angles AEF and EFD are equal. Cor. If a line be perpendicular to one of two parallel lines, it will also be per- pendicular to the other BOOK!. 27 Of P a i a I 1 e 1 L i i, c THEOREM XIII. Conversely, — If a line intersect two straight lines, making ifu alternate angles equal, those straight lines will be parallel. Let the line EF meet the lines AB, • CD making the angle AEF equal to the e/ angle EFD: then will the lines AB and CD be parallel. C For, if they are not parallel, suppose through the point F the line FG to be drawn parallel to AB. Then, because of the parallels AB, FG, the alternate angles, AEF and EFG will be equal (Th. xii). But, by Hypothesis, the angle AEF is equal to EFD : hence, the angle EFD is equal to the angle EFG (Ax. 1) ; that is, a part is equal to the whole, which is absurd (Ax. 8) : therefore, no line but CD can be parallel to AB. Cor. If two lines are perpendicular to the same line, they will be parallel to each other. ! THEOREM XIV. If a line cut two parallel lines, the outward angle is equal to the inward opposite angle on the same side; aid the two inward angles, on the same side, are equal to two right angles. Let the line EF cut the two -parallels A 5 CD . then will the outward angle s EGB be equa A to the inward opposite an- gle EHD ; and the two inward angles, BGH and GIID, will be equal to two right angles. 28 GEOMETRY. Of Parall-.l L First. Since the lines AB, CD, are parallel, the angle AG H is equal to the alternate angle GHD E (Tli. xii) ; but the angle AGH is equal ^ —ft — ti to the opposite angle EGB : hcmce, the _ -^- angle EGB is equal to the angle EHD p (Ax 1). Secondly. Since the two adjacent angles EGB anil BGH are equal to two right angles (Th. ii) ; and since the angle EGB has been proved equal to EHD, it follows that the sum of BGH plus GHD, is also equal to two right angles. Cor. 1. Conversely, if one straight line meets two other straight lines, making the angles on the same side equal to each other, those lines will be parallel. Cor. 2. if a line intersect two other lines, making the sum of the two inward angles equal to two right angles, those two lines will be parallel Cor. 3. If a line intersect two other lines, making the sum of the two inward angles less than two right angles, those lines will not be parallel, but will meet if sufficiently produced. THEOREM XV. All straight lines which are parallel to the same line, are parallel to each other. Let the lines AB and CD be each par- q ail el to EF: then will they be parallel to each other. For. let the line Gl be drawn perpen- dicular to EF : then will it also be per- pendicular to the parallels AB CD (Th. uii Cor.). A" ~~B c D E I F BOOK I . 29 Of Triangles Then, since the lines AB and CD are perpendicular to the line GI, they will be parallel to each other (I'll. xiii. Cor). THEOREM XVI. If cue side of a triangle be produced, the outward angle will U equal to the sum of the inward opposite angles. In the triangle ABC, let the side AB be produced to D : then will the outward angle CBD be equal to the sum of the in- ward opposite angles A and C. For, conceive the line BE to be drawn parallel to the side AC. Then, since BC meets the two pa- rallels AC y BE, the alternate angles A CB and CBE will be equal (Th. xii). And since the line AD cuts the two parallels BE and AC the angles EBD and CAB are equal to each other (Th. xiv) Therefore, the inward angles C and A, of the triangle ABC are equal to the angles CBE and EBD ; and consequently the sum of the two angles, A and C, is equal to the outward angle CBD (Ax. 1). THEOREM XVII. In any triangle the sum of the three anghs is equal to two righ angles. Let ABC be any triangle: then will tne sum of the three angles A -\-B-\- C = two right angles. For, let the side AB be produced to D Then, the outward angle CBD --A+C (T\i. xvi) 3* 30 GEOMETRY' Of Triangles, To each of these equals add the angle CBA, and we shall have CBD^ CBA=A-\-C+B. But the sum of the two angles CBD and CBA, is equal to two right angles A (Th.ii): hence A \- Z»4-C=two right angles (Ax. 1). Cor. 1. If two angles of one triangle be equal to two angles of another triangle, the third angles will also be equal (Ax. 3). Cor. 2. If one angle of one triangle be equal to one angle of another triangle, the sum of the two remaining angles in each triangle, will also be equal (Ax. 3). Cor. 3. If one angle of a triangle be a right angle, the sum of the other two angles will be equal to a right angle ; and each angle singly, will be acute. Cor. 4. No triangle can have more than one right angle, nor more than one obtuse angle ; otherwise, the sum of the three angles would exceed two right angles : hence, at least two angles of every triangle must be acute. THEOREM XV] II. I. A perpendicular is the shortest line that can be drawn from a given point to a given line. II. If any number of lines be drawn c rom the same point, those which arc nearest the perpendicular are less than those which are more remote. Let A be a given point, and DE a straight line. Suppose AB to be drawn perpendiculai to DE, and suppose the oblique lines AC and AD also to be BOOK J . 31 Of Triangles drawn : Then, AB will be shorter than either of the oblique lines, and AC will be less than AD First. Since the angle B, in the triangle ACB, is a righ angle, the angle C will be acute (Th. xvii. Cor. 3) : and since the greater side of every triangle is opposite the greater angle (Th. xi), the side AC will be greater than AB. Secondly. Since the angle ACB is acute, the adjacent angle ACD will be obtuse (Th. ii) : consequently, the angle D is acute (Th. xvii. Cor. 3), and therefore less than the an^rle AC J). And since the greater side of every triangle is oppo- site the greater angle, it follows that AD is greater than AC. Cor. A perpendicular is the shortest distance from a point to a line. THEOREM XIX. G C E if two right angled triangles have the hypothenuse and a stiL of the one equal to the hypothenuse and a side of the other, the. remaining parts will also be equal, each to each. Let the two right angled triangles A [) ABC and DEF, have the hypothe- nuse AC equal to DF, and the side AB equal to DE : then will the re- maining parts be equal, each to each. " For, if the side BC is equal to EF, the corresponding an- gles of the two triangles will be equal (Th. viii). If the sides are unequal, suppose BC to be the greater, and take apart, BG, equal to EF, and draw AG. Then, in the two triangles ABG and DEF. the angle B is equal to the angle E, the side AB to the side DE, and the side TIG to the side EF : hence, the two triangles are equal in all respects (Th. iv) and consequently, the side AG is equal to 32 GEOMETRY Of Polygons DF. But DF is equal to AC, by hypothesis; therefore. AG is equal to AC (Ax. 1). But this is impossible (TL xviii) ; hence, the sides BC and EF cannot be unequal ; con- sequently, the triangles are equal (Th. viii). THEOREM XX The sum of the four angles of every quadrilateral is equal to four right angles. Let A CBD be a quadrilateral : then will .^ A + B+C + D=z four right angles. /'[\ l^et the diagonal DC be drawn dividing 4/ I : fi the quadrilateral AB, into two triangles, ^\l/ BDC, ADC. C Then, because the sum of the three angles of each triangle is equal to two right angles (Th. xvii), it follows that the sum of the angles of both triangles is equal to four right angles. But the sum of the angles of both triangles, make up the angles of the quadrilateral. Hence, the sum of the four angles of the quadrilateral is equal to four right angles Cor. 1. If then three of the angles be right angles, the fourth angle will also be a right angle. Cor. 2. If the sum of two of the tour angles be equal to two right angles, the sum of the remaining two will also be equal lo two right angles. Cor. 3. Since all the angles of a square or rectangle, are equal to each other (Def. 48), and their sum equal to four right angles, it follows that each angle is equal to one right angle. THEOREM XXI. The .mm of all the interior angles of any polygon is equal to twice as many right angles, wanting four, as the figure has side • BOOK I . 33 O f P olygons Let ABCDE be any polygon: then will the sum of its inward angles A+B+C+D+ E be equil to twice as many right angles, wanting four, as the figure has sides. For, from an} point P, within the poly- A B gon, draw the lines PA, PB, PC, PD, PE, to each cf the angles, dividing the polygon into as many triangles ^s the figure has sides. Now, the sum of the three angles of each of these triangles is equal to two right angles (Th. xvii) : hence, the sum of the angles of all the triangles is equal to twice as many right an- gles as the figure has sides. But the sum of all the angles about the point P is equal to four right angles (Th. ii. Cor. 4) ; and since this sum makes no part of the inward angles of the polygon, it must bfc sub- tracted from the sum of all the angles of the triangles, before found. Hence, the sum of the interior angles of the polygon is equal to twice as many right angles, wanting four, as the figure has sides. Sch. This proposition is not applicable to polygons which have re-entrant angles. The reasoning is limited to polygons with salient angles, which may properly be named convex polygons. THEOREM XXII. [f every side of a polygon be produced out, the sum of all the oui ward angles thereby formed, will be equal to four righi angles (jEOMETRY. Of Polygons Let A, B, C, D, and E, be the outward angles of a polygon formed by producing all the sides. Then will A 4-B+ C+D + E=four right angles. For, each interior angle, plus its exte- rior angle, as A + a, is equal to two right angles (Th. ii). But there are as many exterior as interior angles, and as many of each as there are sides of the polygon : hence, the sum of all the interior and exterior angles will be equal to twice as many right angles as the polygon has sides. But the sum of all the interior angles together with four right angles, is equal to twice as many right angles as the polygon has sides (Th. xxi) : that is, equal to the sum of all the in- ward and outward angles taken together. From each of these equal sums take away the inward angles, and there will remain, the outward angles equal to four right angles (Ax. 3). THEOREM XXIII The opposite sides and angles of every parallelogram are equal, each to each : and a diagonal divides the parallelogram into two equal triangles. Let ABCD be any parallelogram, and DB a diagonal : then will the opposite sides and angles be equal to each other, each to each, and the diagonal DB will divide the parallelogram into two equal triangles. For, since the figure is a parallelogram, the sides AB, DC are parallel, as also the sides AD, BC. Now, since the BOOK 1. 30 Of Parallelograms. parallels are cut by the diagonal DB, the alternate angles will be equal (Th. xii) : that is the angle ADB^DBC and BDC=ABD. Hence the two triangles ADB BDC, having two angles in the one equal to two angles in the other, will have their third angles equal (Th. xvii. Cor. 1), viz. the angle A equal to the angle C, and these are two of the opposite angles of tho parallelogram. Also, if to the equal angles ADB, DBC, we add the equals BDC, ABD, the sums will be equal (Ax. 2) : viz. the wholo angle ADC to the whole angle ABC, and these are the other two opposite angles of the parallelogram. Again, since the two triangles ADB, DBC, have the side DB common, and the two adjacent angles in the one equal to the two adjacent angles in the other, each to each, the two triangles will be equal (Th. v) : hence, the diagonal divides the parallelogram into two equal triangles. Cor. 1. If one angle of a parallelogram be a right angle, each of the angles will also be a right angle, and the parallelo- gram will be a rectangle. Cor. 2. Hence, also, the sum of either two adjacent angles of a parallelogram, will be equal to two right angles. THEOREM XXIV. If the opposite sides of a quadrilateral, are equal, each to each ike equal sides will be parallel, and the figure will be a pa? rallelogram. 36 GEOMETRY O 1 Parallelograms Let ABCD be a quadrilateral, having its opposite sides respectively equal, viz. AB=CD and AD=BC then will these sides be parallel, and the figure will be a parallelogram. For, draw the diagonal BD. Then, the two triangles A BI) BDC, have all the sides of the one equal to all the sides of the other, each to each : therefore, the two triangles are equal (Th. viii) ; hence, the angle ADB, opposite ^the side AB, is equal to the angle DBC opposite the side DC ; therefore, the sides AD, BC, are parallel (Th. xiii). For a like reason DC is parallel to AB, and the figure ABCD is a parallelogram. THEOREM XXV. If two opposite sides of a quadrilateral are equal and parallel t 'lw remaining sides will also be equal and parallel, and the figure will be a varallelogram. Let ABCD be a quadrilateral, having the sides A B, CD, equal and parallel : then will the figure be a parallelogram. For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, since AB is parallel to DC, the alternate angles, ABD and BDC are equal (Th. xii) : moreover, the side BD is common ; hence the two triangles have two sides and the included angle of" the one, equal to two sides and the included angle of the O her : the triangles are therefore equal, and consequently AD is equal to BC, and the angle ADB to the angle DBC and consequently, AD is also parallel to BC (Th xii;) Therefore, the figure ABCD is a parallelogram. BOOK I 37 Of Parallelograms THEOREM XXVI. Th two diagonals of a parallelogram divide each other into equa. parts, or mutually bisect each other Let ABCD be a parallelogram, ai>d A C, BD its two diagonals intersecting at E. Then will AE = EC and BE = ED. D A B Comparing the two triangles AED and BEC, we rind the side AD = BC (Th. xxiii), the angle ADE=EBC and EAD — ECB: hence, the two triangles are equal (Th. v) : therefore, AE, the side opposite ADE, u equal to EC, the side opposite EBC; and ED is equal to EB Sch. In the case of a rhombus (Def. 48), the sides AB, BC being equal, the trian- gles AEB and BEC have all the sides of the one equal to the corresponding sides of the other, and are therefore equal. A Whence it follows that the angles AEB and BEC are equal. Therefore, the diagonals o( a rhomb n bisect each other at right angles. GEOMETRY BOOK II, OF THE CIRCLE DEFINITIONS. 1. The ciicumference of a circle is a curve line, all the points of which are equally distant from a certain point within called the centre. 2. The circle is the space bounded by this curve line. 3. Every straight line, CA, CD, CE, drawn from the centre to the circumference, is called a radius or semidiamster. Every line which, like AB, passes through the centre and terminates in the circumfe- rence, is called a diameter. 4. Any portion of the circumference, as EFG, is called an arc. 5. A straight line, as EG, joining tho^ extremities of an arc, is called a chord. 6 A segment is the surface or portion of a circle included between an arc and its chord. Thus EFG is a segment. BOOK II. 3^ Definitions 7. A sector is tue part of the circle in- cluded between an arc and the two radii diawn through its extremities. Thus. CAB is a sector 8. A straight line is said to be in- scribed in a circle, when its extremities are in the circumference. Thus, the line AB is inscribed in a circle. 9. An inscribed angle is one which is formed by two chords that intersect each other in the circumference. Thus, BAC is an inscribed angle. 1U. An inscribed triangle is one which has its three angular points in the circumference. Thus, ABC is an inscribed triangle. J) 11. Any polygon is said to be in- scribed in a circle when the vertices of all the angles are in the circumference. The ciicle is then said to circumscribe the polygon. 40 GEOMETRY Definitions 12 A secant is a line which meets the circumference in two points, and lies partly within and partly without the Thus A B is a secant. circle. 13. A tangent is a line which has but one point in common with the cir- cumference. Thus, CMB is a tangent. \ 14. Two circles are said to touch each other internally, when one lies within the other, and their circumfe- rences have but one point in common. 15. Two circles are said to touch each other externally, when one lies without the other, and their circumfe- rences have but one point in common BOOK II. 41 Of the Circle, THEOREM I. A diameter is greater than any other chord. Let AD be any chord. Draw the radii CA, CD to its extremities. We shall then have A C-\- CD greater than AD (Book I. Th. X*). But A AC+CD is equal to the diameter AB : hence, the diameter AB is creator than AD. THEOREM [I. If from the centre of a circle a line be drawn to the middle of a chord, I . It will be perpendicu \ar to the chord ; II. And it will bisect the arc of the chord. Let C be the centre of a circle, and AB any chord. Draw CD through D, the middle point of the chord, and produce it to E : then will CD be perpendicular to the chord, and the arc AE equal to EB. First. Draw the two radii CA, CB. Then the two triangles A CD, D CB, have the three sides of the one equal to the three sides of the *Note. When reference is made from one theorem to unother, in the same Book, the number of tne theorem referred to is alone given • but when the theorem referre 1 to is found ir a preceding Book, the number of the Book is also g ; ven. 4* 42 G E M £ T R V Of the Circle. other, each to each: viz. AC equal to CB, being radii, AD equal to DB, by hypothesis, and CD common: hence, the corresponding angles are equal (Hook I. Th, viii) : that is, the angle CD A equal to CDB, and the angle ACD equal to the angle DCB. But, since the angle CD A is equal to the angle CDB, the radius CE is perpendicular to tbe chord AB (Bk. I. Def. 20). Secondly. Since the angle ACE is equal to BCE, the arc A E will be equal to the arc EB, for equal angles must have equal measures (Bk. I. Def. 29). Hence, the radius drawn through the middle point of a chord, is perpendicular to the chord, and bisects the arc of the chord. Cor. Hence, a line which bisects a chord at right angles, bisects the arc of the chord, and passes through the centre of the circle. Also, a line drawn through the centre of the cir- cle and perpendicular to the chord, bisects it. THEOREM III. If more than two equal lines can be drawn from any point witfun a circle to the circumference, that point will be the centre. Let D be any point within the circle ABC. Then, if the three lines DA, DB, and DC, drawn from the point D to the circumference, are equal, the point D will be the centre. For, draw the chords AB, BC, bi- sect them at the points E and F, and ioin DE and DF. BOOK II 43 Of the Circle Then, since the two triangles DAE and DEB have the side AE equal to EB, AD equal to DB, and DE common, they will be equal in all respects ; and consequently, the angle DEA is equal to the angle DEB (Bk. I. Th. viii) ; and therefore, DE is perpendicular to AB (Bk. I. De[. 20) But if DE bisects AB at right angles, it wiL pass through the centre of the circle (Th. ii. Cor). In like manner, it may be shown that DF passes through the centre of the circle, and since the centre is found in the two lines ED, DF, it will be found at their common inter- section D. THEOREM IV. Any chords which are equally distant from the centre of a circle, are equal. Let AB and ED be two chords equally distant from the centre C : then will the two chords AB, ED be equal to each other Draw CF perpendicular to AB, and CG perpendicular to ED, and since these perpendiculars measure the distances from the centre, they will be equal. Also draw CB and CE. Then, the two right angled triangles CFB and CEG hav ing the hypothenuse CB equal to the hypothenuse CE, and the side CF equal to CG, will have the third side BF equal U: EG (Bk. I Th. xix) But, BF is the half of BA and EG the half oi^ DE (Th. ii. Cor); hence BA is equal to DE (Ax 6). 44 G E O M E T RY. Of the Cir THEOREM V. A line which is perpendicular to a radius at its extremity, is tangent to the circle. Lot the line ABD be perpendicular to the radius CB at the extremity B : then will it be tangent to the circle at the point B. For, from any other point of the line, as D, draw DFC to the centre, cutting the circumference in F. Then, because the angle B, of the triangle CDB, is a right angle, the angle at D is acute (Bk. 1. Th. xvii. Cor. 3), and consequently less than the angle B But the greater side of every triangle is opposite to the greater angle (Bk. I. Th. xi) ; therefore, the side CD is greater than CB, or its equal CF. Hence, the point D is without the cir- cle, and the same may be shown for every other point of the line AD. Consequently, the line ABD has but one point in common with the circumference of the circle, and therefore is tangent to it at the point B (Def. 13) Cor. Hence, if a line is tangent to a circle, and a radius be drawn through the point of contact, the radius will be perpen dicular to the tangent. THEOREM VI. if the distance betvieen the centres of two circles is equal to the sum of their radii, the two circles will touch each other externally. BOOK II. 45 Of the C irele Let C and D be the two centres, and suppose the distance between them to be equal to the sum of the radii, that is, to CA-\ AD The circumferences of the circles — "" will evidently have the points common, and they will have n » other. Because, if they had two points common, that, is if they cut each other in two points, G and H, the distance CD be- tween their centres would be less than the sum of their radii CH, HD (Bk. I. Th. x) ; but this would be contrary to the supposition. THEOREM VII. If the distance between the centres of two circles ts equal i the difference of their radii, the two circles will touch each otb internally. Let C and D be the centres of two circles at a distance from each other equal to AD— AC— CD. Fr Now, it is evident, as in the last theo- \ V rem, that the circumferences will have the point A common ; and they can have no other. For, if they had two points common, the difference be- tween the radii AD and FC would not be equal to CD, thv distance between their centres : therefore, they cannot have two points in common when the difference of their radii is equal to the distance between their centres : hence, they are tangent to each other. Sch If two circles touch each other, either externally 01 internally, their centres and the point of contact will be in the same straight line 46 GEO M E T K V Of the Circle THEOREM VIII An angle at the circumference of a circle is measured by half the arc that subtends it Lt-\ BAD he an inscribed angle : then will it be measured by half the arc BED, whuh subtends it. For, through the centre C draw the diameter ACE, and draw the radii BC, CD. Then, in the triangie ABC, the exte- rior angle BCE is equal to the sum of the interior angles B and A (Bk. I. Th. xvi). But since the triangle BAC is isosceles, the angles A and B are equal (Bk. I. Th. vi) ; therefore, the exterior angle BCE is equal to double the angle BA C. But, the angle BCE is measured by the arc BE, which subtends it ; and consequently, the angle BAE, which is hali of BCE, is measured by half the arc BE. It may be shown, in like manner, that the angle EAD is measured by half the arc ED: and hence, by the addition of equals, it would follow that, the angle BAD is measured by half the arc BED, which subtends it. Cor. 1. Hence, if an angle at the centre, and an angle it the circumference, both stand on the same arc, the angle at the ccntie will be double the angle at the circumference. Cor. 2. If two angles at the circumference stand on equal arcH thev will be equal to each other. BOOK II. 47 Of the Circle • THEOREM II. A 11 angles at the circumference, which, stand upon the same arc are equal to each other. Let the angles BAC, BDC, BFC, have their vertices in the circumference, and stand on the same arc BEC : then will they be equal to each other. For, each angle is measured by half ^ the arc BEC (Th. viii) ; hence, the an- gles are all equal. E B THEOREM X. An angle in a semicircle, is a right angle. Let ABBC be a semicircle : then will every angle, as B, B, inscribed in it, be a right angle. For, each angle is measured by half j the semic^rcumference ADC, that is, by a quadiant, which measures a right angle fBk I. Th. i. Cor. 2). D B THEOREM XI. If a quadrilateral be inscribed in a circle, the sum of either tiro of its opposite angles is equal to two right angles. Let A BCD be any quadrilateral in- scribed in a circle ; then will the sum of the two opposite angles, A and C, or B and D, be equal to two right angles. u For, the angle A is measured by half the arc DCB, which subtends it (Th. viii) ; 48 G E O M E T R Y . Of the Ci re! and the angle C is measured by half the arc DAB, which subtends it. Hence, the sum of the two angles, A and C is measured by half the entire circumference. I\ But half the entire circumference is the n measure of two % right angles; therefore, th? sum of the opposite angles A and C is equal angles. In like manner, it may be shown, that the wo angles B and D is equal to two right angles sum THEOREM XII. If the side of a quadrilateral, inscribed m a circle, be pro- duced out, the exterior angle will be equal to the inward opposite angle Let the side BA, of the quadrilateral A BCD be produced to E, then will the outward angle DAE be equal to the in- ward opposite angle C. For, the angle DAB plus the angle C, ts equal to two right angles (Th. xi). But DAB plus DA E is also equal to two right angles (Bk. I. Th. ii). Taking from each the common angle DAB, and we shall have the angle DAE equal to the interior opposite angle C. THEOREM XIII. Two parallel chords intercept equal arcs. BOOK II. 49 Of the Circle Let the chords AB and CD be parallel: then will the arcs AC and BD be equal. For, draw the line AD. Then, because the lines AB and CD are parallel, the alternate angles ADC and DAB will be equal (Bk. I. Th. xii). But the angle ADC is measured by half the arc AC, and the angle DAB by half the arc BD (Th. viii) : the two arcs A C and BD are themselves equal. henct THEOREM XIV. The angle formed by a tangent and a chord, is measured by half the arc of the chord. Let BAE be tangent to the circle at the point A, and AC any chord. From A, the point of contact, draw the diameter AD. Then, the angle BAD will be a right angle (Th. v. Cor), and therefore will be measured by half the semicircle AMD B (Bk. I, Th. i. Cor. 2). But the angle DA C being at the circumference, is measure 1 by half the arc DC: hence, by the addition of equals, the two angles BAD and DAC, or the entire angle BAC will be meas- ured by half the arc AMDC. It may be shown, by taking the difference between the two angles DAE and DAC, that the angle CAE is measured by half the arc AC included between its sides. 5 60 GEOMETRY Of the Circl THEOREM XV. If a tangent and a chord are parallel to each other , they will intercept equal arcs. Let the tangent ABC be parallel to the chord DF : then will the intercepted arcs DB, BF, be equal to each other. For, draw the chord DB. Then, since AC and DF are parallel, the angle ABD will be equal to the angle BDF. But ABD being formed by a tangent and a chord, will be measured by half the arc DB ; and BDF being an angle at the circumference will be measured by half the arc BF (Th. viii). But since the angles ?ire equal, the arcs will be equal : hence DB is equal to BF. THEOREM XVI The angle formed within a circle by the intersection of two chords, is measured by half the sum of the intercepted arcs. Let the two chords AB and CD inter- sect each other at the point E : then will the angle AEC, or its equal DEB, be measured by half the sum of the inter- cepted arcs AC, DB. For, draw the chord AF parallel to CD. Then because of the parallels, the angle DEB will be equal to the angle FAB (Bk I. Th. xiv), and the arc FD to the arc AC. But the angle FAB is meas- ured by half the arc FDB, that is, by half the sum of the arcs FD, DB. Now, since FD is equal to AC, it follows that the angle DEB, or its equal AEC, will be measured by half the sum of the arcs DB arifl A C BOOK II. 51 Of the Circle. THEOREM XVII. The angle formed without a circle by the intersection cf two secants is measured by half the difference of the intercepted arcs. Let the two secants DE and EB inter- sect each other at E : then will the angle DEB be measured by half the intercepted arcs CA and DB. Draw the chord AF parallel to ED. D> Then, because AF and ED are parallel, and EB cuts them, the angles FAB and and DEB are equal (Bk. I. Th. xiv). But the angle FAB. at the circumference, is measured by half the arc FB (Th. viii), which is the difference of the arcs DFB and CA : hence, the equal angle E is also measured by half the difference of the intercepted arcs DFB and CA THEOREM XVIII. An angle formed by two tangents is measured by half the difference of the intercepted arcs. Let CD and DA be two tangents to the circle at the points C and A : then will the angle CD A be measured by half the difference of the intercepted arcs CEA and CFA. For, draw the chord AF parallel to the tangent CD. Then, because the lines CD and AF are parallel, the angle BAF will be equal to the angle BDC (Bk. I. Th. xiv). But the ang-le BAF, formed by a tangent and a chord, is measured by 52 G E O ;f E T R V Of the Circle half the aic AF, that is, by half the difference of CFA and CF. But since the tangent DC and the chord A F are parallel, the arc CF is equal to the arc CA : hence the angle BAF, or its equal BDC, which is meas-/ ured by half the difference of CFA and CF, is also measured by half the differ- ence of the intercepted arcs CFA and CA. Ccr. In like manner it may be proved that the angle E, formed by a tangent and secant, is measured by half the difference of the intercepted arcs AC and DBA. THEOREM XIX The cJiOrd of an arc of sixty degrees is equal to the radius of the circle. " et AEB be an arc of sixty degrees and AB its chord: then will AB be equal to the radius of the circle. For, draw the radii CB and CA. Then, since the angle ACB is at the centre, it will be measured by the arc AEB: that is, it will be equal to sixty degrees (Bk. I. Def. 29). Again, since the sum of the three angles of a triangle is equal to one hundred and eighty degrees (Bk. I. Th. xvii), it. BOOK II. 53 Of the Circle. follows that the sum of the two angles A and B will be equal to one hundred and twenty degrees. But the triangle CA B is isosceles: hence, the angles at the base are equal (Bk. I. 1 h. vi) : hence, each angle is equal to sixty degrees, and consequently, the side AB is equal to AC or CB (Bk. I. Th vi) PROBLEMS RELATING TO THE FIRST AND SECOND BOOKS. The Problems of Geometry explain the methods of con structing or describing the geometrical figures. For these constructions, a straight ruler and the common compasses or dividers, are all the instruments that are ab- solutely necessary. DIVIDERS OR COMPASSES. The dividers consist of the two legs ha, be, which turn easily about a common joint at b. The legs of the dividers 64 GEOMETRY Problems are extended or brought together fry placing the forefinger on the joint at b, and pressing the thumb and fingers against the legs PROBLEM 1. On my line, as CD, to lay off a distance equal to A R. Take up the dividers with the thumb and second finger, and place the forefinger on the joint at b. A B Then, set one foot of the dividers ~ at A, and extend the legs with the ' thumb and fingers, until the other foot reaches B. Then, raise the dividers, place one foot at C, and mark with the other the distance CE : and this distance will evi- dently be equal to AB. E D PROBLEM II. To describe from a given centre the circumference of a circle having a given radius. Let C be the given centre, and CB the given radius. Place one foot of the dividers at C and extend the other hg until it reaches to B. Then, turn the di- viders around the leg at C, and the othei leg will describe the required circumference BOOK II. 53 Problems. OF THE RULER. A ruler of a convenient size, is about twenty inches in length, two inches wide, and one fifth of an inch in thickness. It should be made of a hard material, and perfectly straighl and smooth. PROBLEM III. To draw a straight line through two given points A and B. Place one edge of the ruler on A and slide the ruler around until he same edge falls on B. Then, with a pen, or pencil, draw the ine AB. B TROBLEM IV. To bisect a given line : that is, to divide it into two equal parts. Let A B be the given line to be divided. With iasa centre, and radius greater than half of AB, describe an arc IFE. Then, with Basa centre, and an equal radius BI describe the arc IHE. Join the points / and E by the line IE . the point D, where it intersects AB, will be the middle point of the line AB. £N. 56 G E O M E T R Problems. For, draw the radii AI, AE BI, and BE. Then, since these radii are equal, the triangles A IE pnd B1E have all the sides of the one equal to the corresponding sides of the other ; hence, iheir corres ponding angles are equal (Bk I. Th. viii) ; that is, the angle A IE is equal to ihe angle Bl F. Therefore, the two triangles AID and BID, have the sidt AI—IB, the angle AID = BID, and ID common: lionet thev are equal (Bk. I. Th. iv), and AD is equal to DB. PROBLEM V. To bisect a given angle or a given aic. Let A CB be the given angle, and AEB the given arc. From the points A and B, as centres, describe with the same radius two arcs cutting each other in D. Through D and the centre C, draw CED, and it will divide the angle ACB into two equal parts, and also bisect the arc AEB bx£. For, draw the radii AD and BD. Then, in the two triangles A CD, CBD, we have AC=CB, AD = BD and CD common : hence, the two triangles have their corres- ponding angles equal (Bk I. Th. viii), and consequently, A CD is equal to BCD. But since A CD is equal to BCD, it fol lows that the arc AE, which measures the former, is equal tc the arc BE. which measures the latter BOOK 11. 57 Problems. PROBLEM VI. At a given point in a straight line tc erect a perpendicular to ttu line. Let A be the given point, and BC the given line. From, A lay off any two distances, A B and A C, equal to each other Then, from the points B and C, as centres, with a radius greater than JTC--'" AB, describe two arcs intersecting each other at. D ; diaw DA, and it will be the perpendicular required. For, draw the equal radii BD, DC. Then, the two trian- gles, BDA, and CD A, will have AB—AC BD — DC and AD common : hence, the angle DAB is equal to the angle DAC (Bk. I. Th. viii), and consequently, DA is perpendicu- lar to BC. (Bk. I Def. 21). SECOND METHOD. WAen the point A is near the extremity of the line. Assume any centre, as P, out of the given line. Then with P as a centre, and radius from P to A, de- scribe the circumference of a circle Through C, where the circumference cuts BA, draw CPD. Then, through D, where CP produced meets the circumference, draw DA : then will DA be perpendicular tu BA, since CAD is an angle in a jsemicirclc (Bk. II. Th. x). 58 GEOMETRY Problems 'J5T • PROBLEM VII. From a given point without a straight line tc let fall a perpen dicular on the line. Let A be the given point, and BD the given line From the point A as a centre, with a radius greater than the shortest distance to BD, describe an arc cut- ting BD in the points B and D. Then, with B and D as centres, and the same radius, describe two arcs intersecting each other at E. Draw AFE, and it will be the perpendicular required. For, draw the equal radii AB, AD, BE and DE Then, the two triangles EAB and EAD will have the sides of the one equal to the sides of the other, each to each ; hence, their corresponding angles will be equal (Bk. I. Th. viii), viz. the angle BAE to the angle DAE. Hence, the two triangles BAF and DA F will have two sides and the included angle of the one, equal to two sides and the included angle of the other, and therefore, the angle AFB will be equal to the angle AFD (Bk. 1. Th. iv) : hence, AFE will be perpendiculat to BD. SECOND METHOD When the given point A is nearly opposite the extremity of the line. Draw A C, to any point C of the line BD. Bisect AC at P. Then, with P as a centre and PC as a ra- dius, describe the semiciicle CD A ; draw A D, and it will be perpendicular to CD since CD A is an angle in a semicircle (Bk. II. Th. x). BOOK II. 59 Problems. PROBLEM VIII. At a given point in a given line, to make an angle equal to a given angle Let A be the given point, AE , n the given line, and IKL the given y // \ y^\ angle. / V / \ From the vertex K, as a centre, *- ' J A -& with any radius, describe the arc IL, terminating in the two sides of the angle : and draw the chord IL. From the point A, as a centre, with a distance AE, equaJ to KI, describe the arc DE ; then with E, as a centre, and a radius equal to the chord IL, describe an arc cutting DE at D; draw AD, and the angle EAD will be equal to the angle K. For, draw the chord DE. Then the two triangles IKL and EAD, having the three sides of the one equal to the three sides of the other, each to each, the angle EAD will be equal to the angle K (Bk. I. Th. viii). PROBLEM IX. Through a given point to draw a line that shall be parallel to a given line. Let A be the given point and p p BC the given line. With A as a centre, and any ra- A^- l D dius greater than the shortest dis- tance from A to BC, describe the indefinite arc DE. From the point E, as a centre, with the same radius, describe the arc AF : then, make ED equa to A F and draw A D, and it wil] be the required parallel. bO GEOMETRY. Problem B~ For, since the arcs AF and ED are equal, the angles EAD and AEF, which they measure, are equal : hence, the line AD is parallel to BC (Bk 1. Th xiii). i^ E PROBLEM X. Two angles of a triangle being given or known, to find the. thirl Draw the indefinite line DEF. At any point, as E, make the angle DEC equal to one E of the given angles, and then CEFI equal to a second, by Prob. VIII ; then will the angle HEF be equal to the third angle of the triangle. For, the sum of the three angles of a triangle is equal to two right angles (Bk. I. Th. xvii) ; and the sum of the three angles on the same side of the line DE is equal to two right angles (Bk. I. Th. ii. Cor. 2) ; hence, if DEC and CEH are equal to two of the angles, the angle HEF will be equal to the remaining angle of the triangle PROBLEM XI. Three sides of a triangle being given, to describe the triangle Let A, B, and (7, be the given sides. Draw DE, and make it equal to the side A. From the point D, as a centre, with a radius equal to the -£" B\- 8P:cond side B. describe an arc O- BO K I 1. 61 \ Problems. ' from £asa centre, with the third side C, describe another arc intersecting the former in F: draw DF and FE: then will DEF be the required triangle. • For, the three sides are respectively equal to the three lines L B, and C. PROBLEM XII. The adjacent sides of a parallelogram, until Ctie angle uihtrh 'h/n f contain, being given, to describe the jmrallelogiain Let A and B be the given sides ,, and .C the given angle. / Draw the line DE and make it ^L equal to A. At the point D make A 1 ' the angle EDF equal to the angle C. Make the side DF equal to B. Then describe two ares, one from F as a centre, with a radius FG equal to DE, tht other from E, as a centre, with a radius EG equal to DF. Through the point 67, the point of intersection, draw the lines EG and FG, and D EGF will be the required parallelogram. For, in the quadrilateral DFGE, the opposite sides DE and FG are each equal to A : the opposite sides DF and EG are each equal to B, and the angle EDF is equa. lo C. But, since the opposite sides are equal, the}'- arc also parallel (Bk. I. Th. xxiv), and therefore the figure is a arallelogram PROBLEM XIII. To describe a square on a given line. 6 02 GEOMETRY Problems, Let AB be the given line. At the point B draw B C perpendicu- lar to AB, by Problem VI, and then make it equal to AB. Then, with A as a centre, and ra- dius equal to AB, describe an arc ; and with C as a centre, and the same radius AB, describe another arc; and through D, their point of intersection, draw AD and CD : then will ABCD be the required square. For, since the opposite sides are equal, the figure will be a parallelogram (Bk. I. Th. xxiv) : and since one of the angleo is a right angle, the others will also be right angles (Bk- I. Th. xxiii. Cor. 1 ) ; and since the sides are all equal, the figure will be a square. PROBLEM XIV. To construct a rhombus, having given the length of one of the equal sides, and one of the angles. Let AB be equal to the given side, and E the given angle. At B lay off an angle, ABC, equal to E, by Prob. VIII. and make BC equal to AB. Then, with A and C as centres, and a radius equal to AB, ^ & describe two arcs. Through D, their point of intersection, draw the lines AD, CD: then will ABCD be the required rhombus, For, since the opposite sides are equal, they will be parallel (Bk. 1. Th. xxiv). But they are each equal to AB. and the BOOK II fi3 Problems. angle B is equal to the angle E : hence, ABCD is the re- quired rhombus. PROBLEM xv. To find the centre of a circle Draw any chord, as AB, and bisect it by Problem IV. Then, through F, the middle point, draw DCE, perpendicular to AB, by Problem VI. Then DCE will be a diameter of the circle (Bk. II. Th. ii. Cor.). Then bisect DE at C, and C will be the centre of the circle. PROBLEM XVI. To describe the circumference of a circle through three given points not in the same straight line. Let A y B, C, be the given points. Join these points by the straight lines AC AB, BC. Then, bisect any two of these straight lines, as AB, BC, by the perpendiculars OD, OP (Prob. iv) ; and the point O, where these per- pendiculars intersect each other, will be the centre of the circle. Then with O as a centre, and a radius equal to OA, de« scribe the circumference of a circle, and it will pass through the points A, B, and C. For, the two right angled triangles OAP and OBP have the side AP equal to the side BP, OP common, and the included 64 GEOMETRY P ro b 1 e m* . angles OP A and OPB equal, being right angles; hence, the side OB is equal to OA (Bk. I. Th. iv). In like manner it may be shown that OC is equal to OB. IJence, a circumference described with the radius OA, will pass points B and C. through the Sch. This problem enables us to describe the circumference of a circle about a given triangle. For, we may consider the vertices of the three angles as the three points through which the circumference is to pass. PROBLEM XVII. 7krough a given point in the circumference of a circle, to drau a tangent line to the circle. Let A be the given point Through A, draw the radius AC to the centre, and then draw DAE perpendicu- lar to AC, by Problem VI. Then will DAE be tangent to the circle at the point A (Bk. II. fh. \) PROBLEM XVIII. Thrwgh a given point without the circumference, to draw o tangent line to the 'circle. BOOK 11. 65 Problems Let C be the centre of the circle, and A the given point without the circle. Join A and the centre C, and on A C as a d'.ameter, describe a circumference. 'I'h rough ihe points B and D where the two circumferences intersect each other, draw the lines AB and AD: these lines will be tangent to the circle »vhose centre is C. For, since the angles ABC and ADC are each inscribed in a semicircle, they will be right angles (Bk. II. Th. x). Again, since the tines AB, AD. are each perpendicular to a radius at its extremity, they will be tangent to the circle (Bk. II. Th. v). PROBLEM XIX To inscribe a circle in n given triangle. Let ABC be the given tri- angle. Bisect the angles A and B by the lines AO and BO, meet- ing at the point 0. From O, let fall the perpendiculars OD, OE, OF, on the three sides of the triangle — these perpendiculars will be equal to each other. For, in the two right angled triangles DAO and FAO, we ha/e the right angle D equal the right angle F, the angle FAO equal to DAO, and consequently, the third angles AOD and AOF are equal (Bk. I. Th xvii. Cor I) But the two triangles have a common side AO, hence, they are equal (Bk. I. Th v), and consequently, OD is equal to OF 6* 66 GEOMETRY Problems In a similar manner, it may be proved that OE and OD arc equal . hence, the three per- pendiculars, OD, OF, and OE, are all equal. Now, if with O as a centre,^ and OF as a radius, we describe the circumference of a circle, it will pass through the points D and E. and since the sides of the triangle are perpendiculai to the radii OF, OD, OE, they will be tangent to the circum- ference (Bk. II. Th. v). Hence, the circle will be inscribed in the triangle. PROBLEM XX. To inscribe an equilateral triangle in a circle. Through the centre C draw any diam- eter, as ACB. From 5asa centre, with a radius equal to BC, describe the arc DCE. Then, draw AD, AE, and DE, and DAE will be the required triangle. For, since the chords BD, BE, are ^ach equal to the radius CB, the arcs BD, BE, are each equal to sixty degrees (Bk. II. Th. xix), and the arc DBE to one hundred and twenty degrees ; hence, the angle DAE is equal to sixty degrees (Bk. II. Th. viii). Again, since the arc BD is equal to sixty degrees, and the arc BDA equal to one hundred and eighty degrees, it follows that DA will be equal to one hundred and twenty degrees : hence, the angle DEA is equal to sixty degrees, and conse- quently, the third an^le ADE, is equal to sixty degrees BOOK II 67 Problems. Therefore, the triangle ADE is equilateral (Bk. I. Th. vi Cor. 2). PROBLEM XXI. To inscribe a regular hexagon in a circle. Draw any radius, as AC. Then ap- ply the radius AC around the circum- ference, and it will give the chords AD, DE, EF, FG, GH, and HA, which will be the sides of the regular hexagon. For, the side of a hexagon is equal to the radius (Bk. II. Th. xix), PROBLEM XXII. To inscribe a square in a given circle. Let ABCD be the given circle. Draw the two diameters A C, BD, at right angles to each other, and through the points A, B, C and D draw the lines AB, BC, CD, and DA: then will ABCD be the required square. For, the four right angled triangles, AOB, BOC, COD, and DOA are equal, since the sides AO, OB, OC, and OD are equal, beinjj radii of the circle ; and the angles at O are equal in each being right angles: hence, the sides AB, BC, CD, and DA are equal (Bk. I. Th. iv). But each of the angles ABC, BCD, CD A, DA B, is a right angle, being an angle in a semicircle (Bk. II. Th x) : hence, the figure ABCD is a square (Bk. I. Def 48) 68 GEOMETRY Problems Sch. If we bisect the arcs AB, BC, CD, DA. and join the points, we shall have a regular octagon in- scribed in the circle. If we again bisect the arcs, and join the points of bisection, we shall have a regular polygon -of sixteen sides. ^ PROBLEM XXIII. To describe a square about a given liiclc. Praw the diameters AB, DE, at right angles to each other. Through the extremities A and B draw FA G and HBI parallel to DE, and through E and D, draw FEH and GDI par- allel to AB: then will FGIH be the required square. For,. since ACDG is a parallelogram, the opposite sides art-: equal (Bk. I. Th. xxiii): and since the angle at C is a right angle all the other angles are right angles (Bk. I. Th. xxiii. Cor. 1): and as the same may be proved of each of the figures CI, CH and CF, it follows lhat all the angles, F, G, I, and //, are right angles, and that the sides GI, IH, HF, and FG, are equal, each being equal to the diameter of the circle. Henc? the figure GIB F is a square (Bk I. Def. 48). GEOMETRY. BOOK III. OF RATIOS \ N D PROPORTIONS. DEFINITIONS. 1. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if the numbers 3 and 6 have the same unit, the ratio of 3 to 6 will he. expressed by 3 And in general, if A and B represent quantities of the same kind, the ratio of A to B will be expressed by B A 2 If there be four numbers, 2, 4, 8, 16, having such values ♦.hat the second divided by the first is t-qual to the fourth di- vided by the third, the numbers are said to be in proportion. And in general, if there be four quantities A, B, C, and D having such values that B D A~C' then, A is said to have the same ratio to B, that C has to D , or. the ratio of A to B is equal to the ratio of C to D When 70 GEOMETRY Of Ratios and Proportions. four quantities have this relation to each other, they are said to be in proportion. Hence, the proportion of four quantities results from an equality of their ratios taken two and two . To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus : A : B :: C : D ; and read, A is to B, as G to D. The quantities which are compared together are called tne terms of the proportion. The first and last terms are called the extremes, and the second and third terms, the means. Thus, A and D are the extremes, and B and G the means. 3. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the conse- quents ; and the last is said to be a fourth proportional to the other three taken in order. Thus, in the last proportion, A and C are the antecedents, and B and D the consequents. 4. Three quantities are in proportion when the first has the same ratio to the second, that the second has to the third ; and then the middle term is said to be a mean proportional between the two other. For example, 3 : 6 :: 6 : 12 ; and 6 is a mean proportional between 3 and 12. 5. Quantities are said to be in proportion by inversion, or inversely, when the consequents are made the antecedents and the antecedents the consequents. Thus, if we have the proportion 3 : 6 :: 8 : 16. the inverse proportion would be 6 : 3 :: 16 : 8. B*OJK III. 7i Of Ratios and Proportion' 6. Quantities are said to be in proportion by alternation, oi alternately, when antecedent is compared with antecedent and consequent with consequent. Thus, if we have the proportion 3 : 6 : : 3 : 16, the alternate proportion vrould be 3 : 8 : : 6 : 16. 7. Quantities are said to be in proportion by composition, when the sum of the antecedent and consequent is compared cither with antecedent or consequent. Thus, if we have the proportion 2 : 4 : : 8 : 16, the proportion by composition would be 2 + 4 : 4 :: 8+16 : 16; that is, 6 : 4 : : 24 : 16. 8. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with the antecedent or consequent. Thus, if we have the proportion 3 : 9 : : 12 : 36, the proportion by division will be 9-3 : 9 :: 36-12 : 36; that is, 6 : 9 : : 24 : 36. 9. Equimultiples of two or more quantities are the products wrhich arise from multiplying the quantities by the same number. Thus, if we have any two numbers, as 6 ami 5 and multiply 72 GEOMETRY. Of Ratios and Proportions them both by any number, as 9, the equimultiples will be 54 and 45 ; for 6x9 = 54 and 5x9 = 45. A J 30, mxA and mxB are equimultiples of A and 5, the common multiplier being m. 10. Two variable quantities, A and B, are said to be re- ciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. When this relation exists, either of them is equal to a constant quantity divided by the other. Thus, if we had any two numbers, as 2 and 4, so related to each other that if we divided one by any number we must multiply the other by the same number, one would increase in the same ratio as the other would dimmish, and their product would not be changed. THEOREM I. If four quantities are in proportion, the product of the two ea tremes will be equal to the product of the two means If we have the proportion A : B : : C : D we have, by Def. 2, B_D A~ C and by clearing the equation of fractions, we have BC = AD Sch The general principle is verified in the proportion between the numbers 2 : 10 : : 12 : 60 which gives 2 * 60=10 y 12 = 120 BOOK III. 73 Of Ratios and Proportions. THEOREM II. If four quantities are so related to each other, that the product of two of them is equal to the product of the other two ; then two of them may he made the means, and the other two the extremes of a proportion. Let A, B, C, and D y have such values that BxC=AxD Divide both sides of the equation by A and we have **C=D A Then divide both sides of the last equation by C, and we have B_D A~C hence, by Def. 2, we have A : B : : C : D. Sch. The general truth may be verified by the numbers 2x18 = 9x4 which give 2 : 4 : : 9 . 18 THEOREM III. fthiee quantities are in proportion, the product of the two extremes will be equal to the square cf the middle term. Let us suppose that we have A : B . : B : C Then, by Def. 2, we have B_C A~ B and by clearing the equation of its fractions, we have ~1 74 GEOMETRY. Of Ratios and Proportion Sch. The proposition may be verified by the numbers 3 : 6 : : 6 : 12 flitch givt 3x12-6x6 = 36 THEOREM IV. If four quantities are in proportion, they will be in proportion when taken alternately. Let A : B : : C : D Then, by Def. 2, we have B_D A~~C Q Multiplying both members of this equation by — , we have B CD A~B and consequently, A : C : : B : D. Sch. The theorem may be verified by the proportion . 10 : 15 : : 20 : 30 for, we have, by alternation, 10 : 20 : : 15 : 30. THEOREM V. If thei e be two sets of proportions, having an antecedent and a consequent in the one, equal to an antecedent and a consequent in the other; then, the remaining terms will be proportional If we have A : B : . C . D, and A : B : R : F ; then we shall have BOOK III . y 6 Of Ratios and Propor ions. B D B F 2 = C and A=£ Hence, by Ax. 1, we have D_F C^E and consequently, C : D : : E : F Sch. The proposition may be verified by the following proportions, 2 : 6 : : 8 : 24 and 2 : 6 : : 10 : 30 which give 8 : 24 : : 10 : 30. THEOREM VI. If four quantities are in proportion, they will bo in proportion when taken inversely. If we have the proportion A : B : : C : D we have, by Th. I, AxD=BxC, or BxC=AxD. Hence, we have, by Th. II, B : A : : D : C. Sch. The proposition may be verified by the proportion 7 : 14 : : 8 : 16; which, when taken inversely, gives 14 : 7 : : 16 : 8. THEOREM VII. 4./ four quantities are in proportion, they will be in proportion by composition. 76 GEOMETRY". Of R,atios and Propoi ions Let us suppose that we hue A : B : : C : D we shaL then have AxD=.-BxC. To each of these equals, add BxD, and we have (A+B)xD=z(C+D)xB; and by separating the factors by Th. II, we have A + B : B : : C+D : D. Sch. The proposition may be verified by the following proportion, 9 : 27 : : 16 : 48. We shall have, by composition, 9+27 : 27 : : 16 + 48 : 48, that is, 36 : 27 : : 64 : 48 in which the ratio is three fourths. THEOREM VIII. If four quantities are in p? oportion, they urill be in proportion by division. Let us suppose that we have A : B : : C : D , we shall then have AxD=BxC. From each of these equals let us subtract BxD, and we have (A-B)xD={C-D)xB; and by separating the factors by Th. II, we have, A-B : B : : C-D : D. Sch The proposition may be verified by the proportion, 24 • 8 : : 48 : 16 BOOK III. 77 Of Ratios and Pro port ions We have, by division, 24-8 : 8 : : 48-16 : 16; that is, 16 : 8 : : 32 : 16; in which the ratio is one -half. THEOREM IX. Equal multiples of two quantities have the same ratio as thf quantities themselves. It wo havo the proportion A : B ■ : C : D we shall have B_D A~C Now, let M be any number, and by it multiply the nu« merator and denominator of the first member of the equation which will not change its value : we shall then hn\ e MxB D MxA C and hence we have MxA : MxB :: C : D, that is, the equal multiples Mx A and MxB, have the same ratio as A to B. Sch The proposition may be verified by the proportion, 5 : 10 : : 12 ■ 24; for, by multiplying the first antecedent and consequent by any number, as 6, we have 30 : 60 : : 12 : 24, v Arhich the ratio is still 2. 7* 78 GEOMETRY Of Ratios and Proportion THEOREM X. If four quantities arc proportional, and one antecedent and lis consequent be augmented by quantities which have the same ratio as the antecedent and consequent, the four quantities will still bs in proportion Let us take the proportions A : B : : C : D, and A : B : : E : F, which give AxD=zBxC and AxF=BxE; adding these equals we have Ax{D + F) = Bx{C + E); and by Th. II, we have A : B : : C+£ : D+F in which the antecedent C and its consequent D, are augment- ed by the quantities E and F, which have the same ratio. Sch. The proposition may be verified by the proportion, 9 : 18 : : 20 : 40, in which the ratio is 2. If we augment the antecedent and its consequent bv 1 5 and 30, which have the same ratio, we have 9 : 18 : : 20+15 : 404 30 that is, 9 : 18 : : 35 • 70, in which the ratio is still 2. THEOREM XI. If four quantities are proportional, and one antecedent and its consequent be diminished by quantities which have the same ratio as the antecedent and consequent, the four quantities will still be ■in pi'oportion BOOK III. 79 Of Ratios and Proportions. Let us take the proportions A : B : : G : D, and A : B : : E : F. which give AxD=BxC and AxF=BxE. By subtracting these equalities, we have Ax{D-F)=Bx(C-E); and by Th. II, we obtain A : B : : C-E : D-F, m which the antecedent and consequent, C and D, are dimin- ished by E and F, which have the same ratio Sch. The proposition may be verified by the proportion, 9 : 18 : : 20 : 40, for, by diminishing the antecedent and consequent by 15 and 30, we have 9 : 18 :: 20 — 15 : 40 — 30; that is 9 : 18 : : 5 : 10 in which the ratio is still 2. THEOREM XII. If we have several sets of proportions, having the same ratio, any antecedent will be to its consequent, as the sum of the anle< cedents to the sum of the consequents. If we have the several proportions, A : B : : C : D which gives A x D= Bx C A : B : : E : F which gives AxF-BxE A : B :: G : H which gives AxH=BxG We shall then have, by addition, Ax{D+F-[-H) = Bx{C-rE+G); and consequently, by Th II. A : B : : C+E + G : D4-F+H. 80 GEOMETRY. Of Ratios and Proportion Sch. The proposition may be verified by the following proportions : viz. 2 • 4 : : G : 12 and 1 : 2 : : 3 : 6 Then, 2 : 4 : : 6 + 3 : 12 + 6; that is, 2 : 4 : • 9 : 18, in which the ratio is still 2. THEOREM XIII. If four quantities are in proportion, their squares or cubes will also be proportional. If we have the proportion A : B : : C : />, it gives B_D A~C Then, if we square both members, we have and if we cube both members, we havo B 3 D 3 T = C 3 and then, changing these equalities into a proportion, we have for the first, A 2 : £ 2 : : C 2 : D . and foi the second A 2 B* : C 3 : 1) Soh. We may verify the proposition by the proportion, 2 : 4 : : 6 : 12, and by squaring each lerm we have, 4 : 10 : . 36 ■ 144 BOOK III. 81 Of Ratios and Proportions numbers which are still proportional, and in which the ratio is 4. If we cube the numbers we have, 2 3 : 4 3 :• 3 12 3 that is, 8 : f>4 : ■ 2.6 • 172 in which the ratio is 8. THEOREM XIV. If we have two sets of proportional quantities, the products of the corresponding terms will be proportional. Let us take the proportions, A : B : : C : D which gives E : F : : G : H which gives Multiplying the equalities together, we have BxF DxN B_D A~C F_H E~G AxE CxG md this by Th. II, gives AxE : BxF :: CxG : DxH. Sch. The proposition may be verified by the followm. proportions : 8 : 12 : : 10 : 15, ami 3 : 4 : : 6 : 8 ; we sh nil then have 24 : 48 : : 60 : 120 whi^h are proportional, the ratio being 9. GEO M E T II Y BOOK IV O * THE MEASUREMENT OF AREAS, AND THD PROPORTIONS OF FIGURES. DEFINITIONS. 1 Similar figures, are those which have the angles of tne one equal to the angles of the other, each to each, and the sides about the equal angles proportional. 2. Any two sides, or any two angles, which are like placed in the two similar figures, are called homologous sides or angles. 3. A polygon which has all its angles equal, each to each, and all its sides equal, each to each, is called a regular polygon. A regular polygon is both equiangular and equilateral. 4. If the length of a line be computed in feet, one foot is the unit of the line, and is called the linear unit. If the length of a line be computed in yards, one yard is the linear unit 5. If we describe a square on the unit of length, such square is called the unit of surface. Thus, if the linear unit is one foot, one square foot will be the unit of surface, or superficial unit. BOOK IV. S3 Of Parallelograms 6. If the linear unit is one yard, one square yard will be the unit of surface ; and this square yard contains nine square feet. 1 yd. ==3 feet. 7. The area of a figure is the measure of its surface. The unit of the number which expresses the area, is a square, the side of which is the unit of length. 8. Figures have equal areas, when they contain the same measuring unit an equal number of times. 9. Figuies which have equal areas are called equivalent. The term equal, when applied to figures, implies an equality in all respects. The term equivalent, implies an equality in one respect only : viz. an equality in their areas. The sign =0=, denotes equivalency, and is read, is equivalent to. THEOREM i. Parallclog? ams which, have equal bases and equal altitudes, are equivalent. Place the base of one parallel- ogram on that of the other, so that AB shall be the common base of the two parallelograms ABCD and ABEF. Now, since the par- allelograms have the same altitude, their upper bases, DC and FE, will fall on the same line FEDC, parallel to AB. Since the opposite sides of a parallelogram are equal to each other (Bk. I Th. xxiii),.4D is equal to BC. Also, DC and FE are each equal to AB : and consequently, they are equal to each 34 GEOMETRY Of Triangles and Parallelogram j other (Ax. 1 ). To each, add ED : then will CE be equal to DF. But since the line FC cuts the two parallels CB and DA, the angle BCE will be equal to the angle ADF (Bk. I. Th. xiv) : hence, the two triangles ADF and BCE have two sides and the included angle of the one equal to two sides and the included angle of the other, each to each ; consequently, they are equal (Bk. I. Th. iv). If then, from the whole space ABCF we take away the tri- angle ADF, there will remain the parallellogram ABCD ; but if we take away the equal triangle BEC, there will remain the parallelogram ABEF : hence, the parallelogram ABEF is equivalent to the parallelogram ABCD (Ax. 3). Cor. A parallelogram and a rectangle, having equal bases and equal altitudes, are equivalent. THEOREM II Triangles which have equal bases and °.qual altitude equivalent. Place the base of one triangle F D_____E_ on that of the other, so that ABC and ABD shall be two trian- gles, having a common base AB, and for their altitude, the distance between the two parallels AB, FC : then will the triangle ABC be equivalent to the triangle ADB. For, through A draw AE parallel to BC, and AF parallel to W/). forming the two parallelograms BE and BF Thea BOOK IV. 85 Of Triangles and Parallelograms since these parallelograms have a common base and equal altitudes, they will be equivalent (Th. i). But the triangle ABC is half the parallelogram BE (Bk. L Th. xxiii) ; and A BD is half the equal parallelogram BF . hence, the triangle A BC Is equivalent :o the triangle AIM). THEOREM 111. Lj a triangle and a -parallelogram have equal basts and equal altitudes, the triangle will be half the parallelogram. Place the base of the triangle on the base of the parallelogram, so that AB shall be the common base of the tri- angle and parallelogram : then will the triangle ABE be half the parallelogram BD. For, draw the diagonal AC. Then, since the altitude of the triangle AEB is equal to that of the parallelogram, the vertex will be found some where in CD, or in CD produced. Now the two triangles ABC and ABE, having the same base A D, and equal altitudes, are equivalent (Th. ii). But the tri- angle ABC is half the parallelogram BD (Bk. I. Th. xxiii) : hence, the triangle ABE is half the parallelogram BD (Ax. IV Cor. Hence, if a triangle and a rect- angle have equal bases and equal alti- tudes, the triangle will be half the rectangle. For the rectangle would be equiva- lent to a parallelogram of the same base and altitude (Th. i. Cor.), and since the triangle is half the parallelogram, it is also equivalent to half 'he rectangle 8G GEOMETRY. Of Rectanglco D C 11 A B E 71 F THEOREM IV. Rectangles which are described on equal lines are equivalent Let BD and FHbe two rectangles, having the sides AB, BC, equal to the two sides EF, FG, each to each: then will the rectangle ABCD, described on the lines AB, BC, be equivalent to the rectangle EFGH, described on the lines EF, FG. For, draw the diagonals AC, EG, dividing each parallel- ogram into two equal parts. Then the two triangles, ABC, EFG, having two sides and the included angle of the one equal to two sides and the in- cluded angle of the other, each to each, are equal (Bk. 1. Th. iv). But these equal triangles are halves of the respective rectangles (Th. iii. Cor.) : hence, the rectangles are equal (Ax. 7) ; and consequently equivalent. Cor. The squares on equal lines are equal. For a square is but a rectangle having its sides equal. THEOREM V. Twc rectangles having equal altitudes are tc each other as their bases. Let AEFD and EBCF be two rectangles having the common alti- tude AD ; then will they be to each other as the bases AE and EB. For, suppose the base A E to be to the base EB, as any two numbers, say the numbers 4 and 3 Let AE be then divided D _ F V i J s 1*1 1 M i I i . i ■ i i" 1 \ A E B BOOK IV. 87 Of Rectangle into four equal parts, and EB into three equal parts, and through the points of division draw parallels to AD We Bliall thus form seven rectangles, all equivalent to each other since they have equal bases and equal altitudes (Th. iv). But the rectangle AEFD will contain four of these partial rectangles, while the rectangle EBCF will contain three ; hence, the rectangle AEFD wil) be to the rectangle EBCF as 4 to 3 ; that is, as the base AE to the base EB. The same reasoning may be applied to any other rect- angles whose bases are whole numbers : hence, AEFD : EBCF :: AE : EB. THEOREM VI. Any two rectangles are. to each other as the products of their bases and altitudes. Let A BCD and AEGF be // I) two rectangles : then will ABCD : AEGF ■ : ABxAD : AFxAE For, having placed the two rectangles so that BAE and G F DAF shall form straight lines, produce the sides CD and GE until they meet in H. Then, the two rectangles ABCD, AEHD, having the com- mon altitude AD, are to each other as their bases AB and AE (Th. v)* In like manner, the two rectangles AEHD AEGF, having the same altitude AE, are to each other as their bases AD and AF. Thus, we have the proportions A BCD : AEHD : : AB : AE, AEHD : AEGF : : AD : AF. 8* GEOMETRY Of Rectangles. If, now, we multiply the cor- responding terms together, the products will be proportional (Bk. III. Th. xiv.) ; and the common multiplier AEHD may be omitted (Bk. III. Th. ix.) : hence, we shall have ABCD : AEGF : : ABxAD H d q F \ i A B G AJSXAF. Sch. Hence, the product of the base by the altitude may be assumed as the measure of a rectangle. This product will give the number of superficial units in the surface : because, for one unit in weight, there are as many superficial units as there are linear units in the base ; for two units in height, twice as many; for three units in height, three times as many, &c. THEOREM VII. The sum of the rectangles contained by one line, arid tht several parts of another line any way divided, is equivalent to tlie rectangle contained by the two whole lines. Let AD be o e line, and AB the other, divided into the parts AE, EF, FB : then will the rectangles contained by AD and AE, AD and EF, AD and FB, be equiv- alent to the rectangle A C which is con- tained by the lines AD and AB. For, through the points E and F draw the lines EG and FH, parallel to the line AD : then will the rectangle AG D G H C t A I : I ' b BOOK I V . 8!) Of Areas of Parallelograms. be equal to the rectangle of AD x AE ; EH will be equal to EGx EF, or to AD x EF; and FC will be equal to FHx FB, or to AD x FJ5. But the rectangle AC is equal to the sum of the partial rectangles : hence, ADxAB=G=ADxAE+ADxEF+AD> FB THEOREM VIII. The area of any parallelogram is equal to the produn of its basr, by its altitude. Let ABCD be any parallelogram, and BE its altitude : then will its area be \~P ~ J equal to AB x BE. / For, draw AF perpendicular to the !/ base AB, and produce CD to F. Then, the parallelogram BD and the rectangle EF, having the saint' base and altitude are equivalent (Th. i. Cor.). But the arei of the rectangle BF is equal to the product of its base AB by the altitude AF (Th. vi. Sch.) : hence, the area of the paral lelogram is equal to AB x BE. Cor. Parallelograms of equal bases are to each other as then altitudes ; and if their altitudes are equal, they are to each other as their bases. For, let B be the common base, and C and D the altitudes 01' two parallelograms. Then, by the theorem, theii areas are to each other, as B x C : BxD, that is (Bk. III. Th ix), as C : D If A and B be their bases, and C their common altitude, then they w' 1 ] be to each other as A x C : BxC: that is, as A : F 90 GEOMETRY Areas cf Triang les and Trapezoids. THEOREM IX The area of a triangle is equal to half &•,„ product of its base by its altitude. Let ABC be any triangle and CD its altitude : then will its area be equal to half the product of ABx CD. For, through B draw BE parallel to AC, and through C draw CE parallel to AB : we shall then form the parallelogram AE, having the same base and altitude as the triangle ABC. But the area of the parallelogram is equal to the product of the base AB by its altitude DC ; and since the parallelogram is double the triangle (Th. iii), it follows that the area of the tri angle is equal to half this product : that is, to half the product of ABx CD. Cor. Two triangles of the same altitude are to each other as their bases ; and two triangles of the same base are to each other as their altitudes. And generally, triangles are to each other as the products of their bases and altitudes. THEOREM X. The area of a trapezoid is equal to half the product of its altitud* multiplied by the sum of its parallel sides. Let ABCD be a trapezoid, CG its altitude, and AB, DC its par- allel sides : then will its area be equal to half the product of CGx{AB + DC) I) C H ¥ \ \ A C ? / ? E BOOK IV. 91 Of II e c t an £ I For, produce AB until BE is equal to DC, and complete the rectangle AF ; also, draw BH perpendicular tc AB. Then, the rectangle A C will be equivalent to BF, since thev have equal bases and equal altitudes (Th iv). The diagonal BC will divide the rectangle GH into two equal triangles; and hence, the trapezoid A BCD will be equivalent to the trapezoid BEFC ; and consequently, the rectangle AF, is double the trapezoid ABCD. But the rectangle AF is equivalent to the product of ADxAE; that is, to CG X [AB-^- DC) ; and consequently the trapezoid ABCD is equal to half that product THEOREM XI. If a line be divided into two parts, the square described on the whole line is equivalent to the sum of the squares described an tlie two parts, together with twice the rectangle contained by the parti Let the line AB be divided into two parts at the point E : then wnl the square described on AB be equivalent to the two squares described on AE and EB, to- gether with twice the rectangle contained by AE and EB : that is D H F AR ■AE*-{- EB~ + '2AExEB. Foi let AC be a square on AB, and A F a sqi are on A E and produce the sides EF and GF to H and /. Then since EH is equal to AD, being the opposite side of a rectangle, it is also equal to AB ; and GI is likewise equaJ to AB If thrrefore, from these t-quals we take away EF and 92 GEOMETRY. Of Rectangle D H G GF, there will remain FH equal to FI, and each will be equal to HC or IC ; and since the angle at F is a right angle, it follows that FC is equal to a square de- scribed on EB. It also follows, that DF and FB are each equal to the rectangle of AE into EB. But the square ABCD is made up of four parts, viz., the square on AE ; the square on EB ; the rectangle DF , and the rectangle FB. Hence, the square on AB is equivalent to the square on AE plus the square on EB, plus twice the rectangle contained by AE and EB. B Cor. If the line AB be divided into two equal parts, the rectangles DF and FB would become squares, and the square described on the whole line would be equivalent to four times the square de- scribed on half the line. Sch. The property may be expressed in the language uf algebra, thus, {a + hf=.a+2ab + b? THEOREM XII. The square described on the hypothenuse of i right angled triangle, is equivalent to the sum of the squares described on tfu o*her two sides. BOOK IV. 9b Of Right Angled Triangles. Let BAC be a right an- gled triangle, right angled at A\ then will the square de- scribed on the hypotbenuse BC, bo equivalent to tbe two squares described on J] A and AC. Having described the squares BC, BL, and AI, let fall from A, on the hy- potbenuse, the perpendicular p ]g* Q AD, and produce it to E\ then draw tbe diagonals AF, CH, Now, the angle AJBF is made up of the right angle FBC and the angle CBA ; and tbe angle CBIl is made up of the right angle ABU and the same angle CBA: hence, tbe angle ABF is equal to CJBH. But FB is equal to BC, being sicles of the same square; and for a like reason, BA is equal to HB. Therefore, the two triangles ABF and CBH, having two sides and the included angle of the one equal to two sides and tbe included angle of the other, each to eacb, are equal (Bk. I. Tb. iv). Since tbe angles BAC and BAL are right angles, as jlso the angle ABH, it follows that CAL is a straight line parallel to BH. (Bk. I. Th. ii. Cor. 3). Hence, the square HA and the triangle HBC stand on the same base and be- tween tbe same parallels; therefore the triangle is ball* the square (Th. iii. Cor.). For a like reason, the triangle A-BF is half the rectangle BE. But it has already been proved that tbe triangle ABF is equal to tbe triangle CJBH : hence, the rectangle BE, which is double the former, is equivalent to the square BL, which is double the latter (Ax. 6). 94 GEOMETRY Of Right Angled T r ; a n g 1 e s . In the same manner it may be proved, that the rect- angle DG is equivalent to the square CK But the two rectangles BE, DG, make up the square BG : therefore, the square BG, described on the hypothenuse, is equiva- lent to the squares BL and CK, described on the other two sides. Cor. Hence, the square of either side of a right angled triangle is equivalent to the square of the hypothenuse diminished by the square of the other side. That is, in the light angled triangle ABC AB 4 o or £C 2 Sch. The last theorem may be illustrated by de- scribing a square on the hy- pothenuse BC, equal to 5, also on the sides BA, A C, respectively equal to 4 and 3 ; and observing that the num- ber of small squares in the large square is equal to the number in the two small squares ■■ACT-BCf AC'-AB 4 BOOK IV. 95 Of Triangle Sides cut Proportionally. THEOREM XIII. If a line be drawn parallel to the base of a triangle, it will divide the other two sides proportionally. Let ABC be any triangle, and DE a straight line drawn parallel to the base BC: then will AD : DB : : AE : EC. For, draw BE and DC. Then, the two triangles BDE and DCE have the same base DE, and the same altitude, B C since their vertices B and C, lie in the lin< BC parallel to DE : hence, they are equivalent (Th. ii). Again, the triangles ADE and BDE, ha\e a common ver- tex E, and the same altitude ; consequently, they are to each other as their bases (Th. ix. Cor.) ; hence, we have ADE : BDE : : AD : DB. But the triangles ADE and CDE, having a common vertex D, are to each other as their bases AE and EC : hence, we have ADE : CDE : : AE : EC. But the triangles BDE and CDE have been proved equiva- lent : hence, in the two proportions, the first antecedent and consequent in each are equal: therefore, by (Bk. 111. Th- v) we have AD : BD : : AE : EC. . Cor. The sides AB, AC, are also proportional to the parts AD, AE, or to BD, CE. For, by composition (Bk. III. Th. vii), we have AD+BD : BD :: AE+EC : EC. Then, by alternation (Bk. 111. Th. iv). AB : AC : : BD : EC, hence, also, AB : AC : : AD : AE 96 G EOMETRY. Proportions of T riant: lei THEOREM XIV. A line which bisects the vertical angle of a triangle divided ihe base into two segments which are proportional to the adjacent side. Let ACB be a triangk, hav- ing the angle C bisected by the line CD : then, will AD : DB : : AG : CB. For, draw BE parallel to CD and produce A C to E. Then, since CB cuts the two B D A parallels CD, EB, the alternate angles BCD and CBE are equal (Bk. I. Th. xii) : hence, CBE is equal to angle A CD. But, since AE cuts the two parallels CD, BE, the angle ACD is equal to GEB (Bk. I. Th. xiv) : consequently, the angle CBE is equal to the angle CEB (Ax. I) : hence, the «ide CB is equal to CE (Bk. I. Th. vii.) Now, in the triangle ABE the line CD is drawn parallel to BE: hence, by the last theorem, we have AD : DB : : AC i CE, and by placing for CE, its equal CB, we have AD : DB : : AC : CB. THEOREM XV. Equiangular triangles have their sides proportional, and are Let ABC and DEFbe two equi- angular triangles, having the angle A equal to the angle D, the angle C to the angle F, and the angle B to the angle E : then will AB : AC : : DE : Z>^ BOOK IV. 97 Proportions of Triangles. For, on the sides of the larger triangle DEF, make Dl equal to -AC and DG equal to AB, and join IG. Then thr> two triangles ABC and DIG, having two sides and the in- cluded angle of the one equal to two sides and the included angle of the other, each to each, will be equal (Bk. I Th. iv) Hence, the angles / and G are equal to C and B, and conse* quently, to the angles F and E : therefore, IG is parallel to EF (Bk. I. Th. xiv, Cor. 1 ). Now, in the triangle DEF, since IG is parallel to the base, we have (Th. xiii). DG : Dl : : DE : DF t that is, AB AC DL DF THEOREM XVI. Two triangles which have their sides proportional are equian- gular and similar. Let BAG and EDF be two triangles Laving BC . EF :: AB : ED, and BC : EF : : AC : DF; then will they have the corres- ponding angles equal, viz.. the angle B^E, A = D and C=F. For. at the point E make FEG equal to the angle /*, and at F make the angle EFG equal to the angle C. Then will the angle at G be equal to A, and the two triangles BAC and EGF will be equiangukr (Bk. I Th xvii. Cor 1). Therefore, by the last theorem, we shall have BC : EF : : AB : EG: 9 98 G E0METH1' Proportions of T r.i angles. AC AC DF; but by hypothesis, BC : EF : : AB : DE : hence, EG is equal to £D. By the last theorem we also have BC : EF : : and by hypothesis, BC : EF : : hence, FG is equal to DF. Therefore, the triangles DEF and EGF, having their three sides equal, each to each, are equiangular (Bk. I. Th. viii). But, by construction, the triangle EFG is equiangular with BAC : hence, the triangles BAC and EDF are equiangular, and consequently they are similar. Sch. By Theorem XV, it appears that if the corresponding angles of two triangles are equal, each to each, the correspond- ing sides will be proportional ; and in the last theorem it was proved that if the sides are proportional, the corresponding angles will be equal. Now, these proportions do not hold good in the quadrilate- rals. For, in the square and rectangle, the corresponding angles are equal, but the sides are not proportional ; and the angles of a parallelogram or quadrilateral, may be varied at pleasure, without altering the lengths of the sides. THEOREM XVII. // two triangles have an angle in the one equal to an angle in the oihdr, and the sides containing these angles proportional y tke two triarchies will he equiangular and similar. BOOK V . 99 Proportion 9 of Tiianglee. Let ABC and DEF be two tri- angles having the angle A equal to the angle D, and AS DE : : AC 1 DF ; .hen will the two triangles be similar. For, lay off AG equal to DE, and through G draw GI par allel to BC. Then the angle AG I will be equal to the angle ABC (Bk. I. Th. xiv) ; and the triangles AGI and ABC will be equiangular. Hence, we shall havo AB : AG : : AC : AI. But, by hypothesis, we have AB : DE : : AC : DF, and by construction, AG is equal to DE; therefore, AI is equal to DF, and consequently, the two triangles AGI and DEF are equal in all their parts (Bk. I. Th. it). But the tri- angle ABC is similar to AGI, consequently it is similar to DEF THEOREM XVIII. ij from tlie right angle of a right angled triangle, a perpen- dicular be let fall on the hypothcnuse, then I. The two partial triangles thus formed will be similar to each other and to the whole triangle. lx. Either side including the right angle will be a mean pro- portional between the hypothenuse and the adjacent segment. III. The perpendicular will be a mean proportional between the segments of the hypothenuse 100 GEO M E T H Y Proportions of Triangles Let ABC be a right angled triangle, and AD perpendicular to the hypothenuse. The two triangles BAC and BAD having the common angle 5, and the right angle BAC equal to the right angle at D, will be equiangular (lik. I. Th. xvii Cor. 1); and, consequently, similar (Th. xv). For a like reason the triangles BAC and CAD are similar. Now. from the triangles BA C and BAD, we have BC : BA : : BA : BD. From the triangles BA C and CAD, we have PC : CA : : CA : CD: and from the triangles BAD and DAC, we have BD : AD :: AD -. DC. Cor. If from a point A, in the circumference of a circle, AD be drawn perpendicular to any diam- eter as BC, and the chords AB A C be also drawn, then the an- gle BA C will be a right angle (Bk. II. Th. x): and by the theorem we shall have, 1st The perpendicular AD a mean proportional between the segments BD and DC. 2d Fach chord will be a mean proportional between the diameter and the adjacent segment. That is, AD*=BDxDC AB~=BCxBV A~c 2 =r>cxCD COOK IV 11. Proportions o f Triangles ?B THEOREM XIX. Similar triangles are to each other as the squares described on their homologous sides Let ABC and DEF be two siinilai triangles, and A L and DN t\ie squares de- scribed on the homologous "TO sides AB, DE: then will the triangle ABC : DEF : : AL : DN. M N For, draw CG and FH perpendicular to the bases AB, DE. and draw the diagonals B K and EM. Then, the similar triangles ABC and DJEF, having their homologous sides proportional, we have AC : DF : : AB and the two ACG, DFB, give DE : FH; AC : DF :: CG hence, (Bk. III. Th. v), we have AB : DE :: CG : FH, or (Bk. III. Th. iv), AB : CG : . DE : FH. Now, the two triangles ABC and AKB have the common base AB ; and the triangles DEF and DEM have the common case DE ; and since triangles on equal bases are to each othei as their altitudes (Th. ix, Cor.), we have he triangle ABC : ABK :: CG : AK or AB and the triangle, DEF : DME : : F!J : DM or DE. 102 GEOMETRY. Proportions of Triangles. But we have proved CG : AB : : FH : DE ; hence, ABC : ABK : : . DEF : DME, or, alternately, ABC : DEF : ABK : DUE. But the squares AL and DN being each double of the triangles A KB and DME liave the sajne ratio ; hence, ABC : DEF : : AL : DN THEOREM XX. Two similar polygons may be divided into an equal number of triangles, similar each to each, and similarly placed. Let ABCDE and FGHIK be two similar polygons. Fiom the angle A draw the diagonals AC, AD : J) and from the homologous angle F, draw FH, FL Now, since the poly- gons are similar, the ho- mologous angles B and G will be equal, and the sides about the equal angles propor tional (Def. 1): that is, AB : BC : : FG : GH. Hence, the triangles ABC and FGH have an angle in each equal, and the sides about the equal angles proportional . there- fore, they are similar (Th. xvii), and consequently, the angle ACB is equal to FHG. Taking these from the equal angles BCD and GUI, there will remain ACD equal to FHL The BOOK IV. 103 Proportions of Polygons. two triangles .4 CD and FHI will then have an angle in each equal, and the sides about the equal angles proportional : hence, they will be similar. In the same manner it may be shown that the triangles AED and FKI are similar: and, hence, whatever be the number of sides of the polygons, they may be divided into an equal number of similar triangles. THEOREM XXI. Similar polygons are to each other as the squares described on their homologous sides. Let ABODE and FGNIK, be two similar polygons ; then will they be to each other as the squares described on AB, FG, or any other two homologous sides. For, let the polygons be divided, as in the last the- orem, into an equal num- F G ber of similar triangles. Then, by Theorem XIX, we have triangles ABC : FGN : : AJ? . FG 2 ADC : FIN : : DC? : m 2 ADE FIK : : DE 2 : IK 2 But since the polygons are similar, the ratio of the last ante- cedent to its consequent, in each of the proportions, is the same : hence, we have (Bk. III. Th. xii). ABC+ADC+ADE : FGN+FIN+-FIK : : AB 2 : FG\ that is, ABCDE : FGNIK : : AB 2 : FG 2 ; Hence, the areas of similar polygons are to each other as the squares described on their homologous sides 104 GEO M P TRY Proportions of Polygons. THEOREM XXII. If similar polygons are inscribed in circles, tlieir hmnologom SvJes, and also their perimeters, will have the s ime t i f ic to each Other as the diameters of the circles in which they are inscribed Let ABODE, FGNIK, be two similar figures, in- scribed in the circles whose diameters are A L and FM : then will each side, AB, BC, &c, of the one, be to the homologous side FG, GN. &c, of the other, as the diameter AL to the diameter ¥M. Also, the perimeter AB-\-BC-{- CD &c, will be to the perimeter FG+GN+N1 &c, as the diameter AL to the dianeter FM For, draw the two corresponding diagonals A C, FN, as also the lines BL and GM. Then, the two triangles ACB and FNG will be similar (Th. xx) ; and therefore, the angle A CB is equal to ;he angle FNG. But, the angle ACB is equal to the angle ALB, and the angle FNG to the angle FMG (Bk. II. Th. ix) : hence, the angle ALB is equal to the angle FMG (Ax. J ) ; and since ABL and FGM are right angles (Bk. II. Th. x), the two tri- angles ALB and FMG will be equiangular (Bk. I. Th. xvii ' L'or. 1), and consequently similar (Th. xv). Therefore, AB : FG : : AL : FM. Again, since any two homologous sides are to each other in the name ratio as AL to FM, we have (Bk. III. Th xii), AB + BC+CD Ac : FG + GN±Nl &c. : : AL : FM. BOOK I V . 105 Proportions of Polygons. THEOREM XXIII. Similai polygons inscribed in circles are to each other as the squares of the diameters of the circles. Let ABODE, FGNIK, *ji> Uso polygons inscribed in the circles whose diam- eters are AL and FM: then will the polygon ABODE, be to the poly- gon FGNIK as the square of AL to the square of FM. For, the polygons being similar, are to each other as the squares of their like sides (Th. xxi) ; that is, as AB 2 to FG 2 But, by the last theorem, AB : FG therefore (Bk III. Th. xiii), FG 1 AL AL* FM FM' FGNIK AL' FM\ AB' consequently, ABODE Sch. If any regular polygon, ABDEFG,be inscribed in a ciicle, and then the arcs AB, BE, &c, be bisected, and lines be drawn through these points of bisection, a new poly- gon will be formed having double the number of sides. It is plain that this ^ ~^B new polygon Avill differ less from the circle than the first polygon, and its sides will lie nearer the circumference than the sides ~f the first polygon. If now, we suppose the number of sides to be continually increased, the length of each side will constantly diminish too GEOMETRY. Proportions of Circles. until finally the polygon will become equal to the circle, and the perimeter will coincide with the circumference. When this takes place, the line OH drawn perpendicular to one of the sides, will become e^ual to the radius of the circle. THEOREM XXIV. The circumferences of circles arc to each other as their diameters Let there be two circles whose diameters are AL and FM: then will their circumferences be to each other as AL to FM For, suppose two similar polygons to be inscribed in the circles : their perimeters will be to each other as AL to FM (Th. xxii). Let us now suppose the arcs which subtend the sides of tho polygons to be bisected, and new polygons of double the num- ber of sides to be formed : their perimeters will still be to each other as AL to FM, and if the number of sides be in- creased until the perimeters coincide with the circumference, we shall have the circumferences to each other as the diam- eters AL and FM. THEOREM XXV. 77*? areas of circles are to each other as the squares of ifair diameters. BOOK IV 107 Area of the Circle. Let there be two circles whose diameters are AL and FM: then will their areas be to each other as the square of AL to the square of FM. For, suppose two similar polygons to be inscribed in the circles : then will they be to each other as AL 2 to FM* (Th xxiii). Let us now suppose the number of sides of the polygons to be increased, by bisecting the arcs, until their perimeters shall coincide with the circumferences of the circles. The polygons will then become equal to the circles, and hence, the areas of the circles will be to each other as the squares of theii diameters. Cor. Since the circumferences of circles are to each other as their diameters (Th. xxiv), it follows, that the areas which are proportional to the squares of the diameters, will also be proportional to the squares of the circumferences THEOREM XXVI. The area of a regular polygon inscribed in a circle, is equal to Iwlf the product of the perimeter and the perpendicular let fall frwn the centre on one of the sides. Let C be the centre of a circle cir- cumscribing the regular polygon, and CD a perpendicular to one of its sides : then will its area be equal to half the product of CD by the perimeter. For, from C draw radii to the ver- tices of the angles, forming as many 103 GEOMETRY. Area of Circle. equal triangles as the polygon has sides, in each of which the perpen- dicular on the base will be equal to CD. Now, the area of one of them, as A CB, will be equal to half the pro- duct of CD by the base AB ; and the same will be true for each of the other triangles : hence, the area of the poly- gon will be equal to half the product of CD by the perimeter THEOREM XXVII. Tlie area of a circle is equal to half the product of the radius by the circumference. Let C be the centre of a circle : then will its area be equal to half the product of the radius A C by the cir- cumference ABE. For, inscribe within the circle a regular hexagon, and draw CD perpen- dicular to one of its sides. Then, the area of the polygon will be equal to half the product oi 3D multiplied by the perimeter (Th. xxvi). Let us now suppose the number of sides of the polygon to oe increased, until the perimeter shall coincide with the cir- cumference ; the polygon will then become equal to the chela and the perpendicular CD to the radius CA. Hence, the area of the circle will be equal to half the product of the radius by the circumference. BOOK IV. 109 Pr o d 1 e m s PROBLEMS RELATIXG TO THE FOURTH BOOK. PROBLEM I. To divide a line into any proposed number of equal parti Let AB be the line, and let it be required to divide it into four equal parts. Draw any other line, A C, forming an angle with AB, and take any dis- tance, as AD, and lay it off four times on A C. Join C and B and through the points D, E, and F, draw parallels to CB These parallels to BC will divide the line AB into parts pro portional to the divisions on A C (Th. xiii) : that is, into equal parts. PROBLEM II. To find a third proportional to two given lines. Let A and B be the given lines. Make AB equal to A, and draw p ~~_ C A C, making an angle with it. On "^"V A C lay oflf AC equal to B, and join \j\ HC . then lay off AD, also equal to ]) and through D draw DE parallel to BC : ihen will AE be the third proportional sought For, since DE is parallel to BC, we have (Th. xiii) AB : AC : : AD or AC : AE; ♦herefore, AE is the third proportional sought 10 B I J GEOMETRY Problems PROBLEM III. To find a fourth proportional to the lines A, B, and C. Place two of the lines forming an A angle with each other at A ; that is, # make AB equal to A, and AC equal C K^\ B ; also, lay off AD equal to C. ^s^ \ Then join BC, and through D draw A V B DE parallel to BC, and AE will be the fourth proportional sought. For, since DE is parallel to BC, we have AB : AC n AD : AE; therefore, AE is the fourth proportional sought. PROBLEM IV. To find a mean proportional between two given lines, A and b Make AB equal to A, and BC equal to B: on AC de- scribe a semicircle. Through B draw BE perpendicular to A C, and it will be the mean proportional sought (Th. xviii. Cor) PROBLEM V. To make a square which shall be equivalent to the sum of twe given squares. Let A and B be the sides of the given squares. Draw an indefinite line AB, and make AB equal to A. At B draw BC perpendicular to AB, and make BC equal to B : then draw A C and the square described on AC will be equivalent to the squares on A and B (Th. xii). BOOK IV. Ill Problems PROBLEM VI. To make a square vuhich shall be equivalent to the difference be tween two given squares. Let A and B be the sides of , a 'B »lie given squares. Draw an indefinite line, and make CB equal to A, and CD ' q — ~jy equal to B. At D draw DE perpendicular to CB, and with C as a centre, and CB as a radius, describe a semicircle meeting DE in E, and join CE: then will the square described on ED be equal to the differ- ence between the given squares. For, CE is equal to CB, that is, equal to A, and CD is equal to B : and by (Th. xii. Cor.), ED l =CE 2 -CD l . PROBLEM VII. To make a triangle which shall be equivalent to a given quad- rilateral. Let A BCD be the given quadri- ateral. Draw the diagonal A C, and through D draw DE parallel to AC, meeting BA produced at E. Join EC: then will the triangle CEB be equivalent to the quadrilateral BD. For, the two triangles ACE and ADC, having the same base A C, and the vertices of the angles D and E in the same line DE parallel to AC, are equivalent (Th. ix). If to each, we add ACB, we shall then have the triangle ECB equivalent to the quadrilateral BD (Ax. 2). E A B 112 CEO I\l E T K Y Problems PROBLEM VIII. To make a triangle which shall be equivalent to a given polygon Let ABODE be the polygon. Draw the diagonals AD, BD. Produce AB in both directions, and through C and E draw CG and EF, respectively parallel to AD and BD : then join FD and DG, and the triangle FDG will be equivalent to the polygor, ABODE. For, the triangle AED is equivalent to the triangle AFD and DBC to DBG (Th. ii); and by adding ADB to the equals, we shall have the triangle FDG equivalent to the polygon ABODE. PROBLEM IX. To make a rectangle that shall be equivalent to a given triangle. Let ABC be the given triangle. Bisect the base AB at D, and draw DH perpendicular to AB. Through C, the vertex of the triangle, draw CHG parallel to AB, and draw BG perpen- dicular to it : then will the rectangle DG be equivalent to the triangle ABC. For, the triangle would be half a rectangle having the same base and altitude : hence, it is equivalent to DG, whose base ia the half of AB, and altitude equal to that of the triangle. BOOK IV 113 Appendix PROBLEM X. To inscribe a circle in a regular polygon. Bisect any two sides of the polygon by the perpendiculars GO, FO, and with their point of intersection O, as a centre, and OGas a radius describe the circumference of a circle — this circle will touch all the sides of the polygon. For, draw OA. Then in the two right angled triangles OA G and OAF, the side AO is common, and A G is equal to if, since each is half of one of the equal sides of the polygon : hence, OG is equal to OF(Bk. I.Th. xix). In the same man- ner it may be shown that OH, OK and OL are all equal to each other : hence, a circle described with the centre O and radius OF will be inscribed in the polygon. C.r. Hence, also the lines OA, ON &c, drawn to the angles of the polygon are equal. APPENDIX OF THE REGULAR POLYGONS. 1. In a regular polygon the angles are all equal to each other (Def. 3). If then, the sum of the inward angles of a regular polygon be divided by the number of angles, the quo- tient will be the value of one of the angles. But the sum of the inward angles is equal to twice as many right angles, wanting four, as the polygon has sides, and we shall find the value in degrees by simply placing 90° for the right angle. 114 GEOMETRY. Appendix. ^^ 2. Thus, for the sum of all the angles of an equilateral 'riangle, we have 6x90°-4x90 o = 540 o -360 o = 180° and foi each angle 180°-f-3 = 60°: Hence, each angle of an equilateral triangle, is equal to GO degrees. 3. For the sum of all the angles of a square, we have 8x90°-4x90 o z=720 o -360 o = 360°, nnd for each of the angles 360° ^4 = 90° 4. For the sum of all the angles of a regular pentagon, wc have 10 x 90° -4 X 90° = 900° -360° = 540°, and for each angle 540° ^-5= 108°. 5. For the sum of all the angles of a regular hexagon, we have 12 x 90° -4 x 90° = 1080° -360° = 720°, and of each angle 720°^ 6 = 120°. 6. For the sum of the angles of a regular heptagon, we have 14x90°-4x90° = 1260° — 360° = 900°: and for one of the angles 900° h- 7 = 128° 34'+. 7. For the sum of the angles of a regular octagon, we ha\o 16x90°-4x90 o = 1440 o -360°=rl080 o : and for each angle l080°--r8=13S p BOOK IV 115 Regular Polygons. 8. Since the sum of the angles about any point is equal tc four right angles (Bk. I. TL ii. Cor. 3), it may be observed thai there are only three kinds of regular polygons, which can be ai ranged around any point, as C, so as exactly to fill up the space. These are, First. — Six equilateral triangles, in which each angle about C is equal to ■60°, and their sum to 60° x 6 = 360. Second.- Four squares, in which each angle is equal to 90°, and their si un to 90° x 4 = 360° n Third. — Three hexagons, in -.vhich each angle is equai to 120, and the sum of the three to 120° X 3 = 360°. GEOMETRY, BOOK V. OF PLANES AND .THEIR ANGLES. DEFINITIONS. 1 . A straight line is perpendicular to a plane, when it is per pendicular to every straight line of the plane which it meets The point at which the perpendicular meets the plane, is called the foot of the perpendicular. 2. If a straight line is perpendicular to a plane, the piano is also said to be perpendicular to the line. 3. A line is parallel to a plane when it will not meet that plane, to whatever distance both may be produced. Con- versely, the plane is then parallel to the line. 4. Two planes are parallel to each other, when they will not meet, to whatever distance both are produced. 5. If two planes are not parallel, they intersect each other in a line that is common to both planes : such line is called their common intersection. 6. The space included between two planes is called a diedral angle : the planes are the faces of the angle, and their intersection the edge. A diedral angle is measured by two lines, one in each plane, and both perpendicular to the common intersection at the same point. This angle may be acute, obtuse, or a right angle. When it is a right angle, the planes are said to be perpendicular tc each other. BOOK 117 Of Planes. Z> B E J Let AB be a plane coinciding with H the j lane of the paper, and ECF a plane intersecting it in the line FH. Now, if from any point of the common intersection as C, we draw CD in the plane AB, and CE in the plane ECF, and both perpendicular to CF at C, then will the angle DCE measure the inclination between the two planes. It should be remembered that the line £C is direct Iv ov«i the line CD. 7. A polyedral angle is the angular space included between several planes meeting at the same point. Thus, the polyedral angle S is formed by the meeting of the planes ASB, BSC, CSD, DSA. 8. The angle formed by three planes is called a triedral angle. THEOREM I. Two straight lines which intersect each other, lie ni tliv. saw. plane, and determine its position. Let A B and AC be two straight lines which intersect each other at A. Through AB conceive a plane to be pft3sed, and iet this plane be turned around AB until it embraces the point C : the plane will then contain the two lines AB, AC, and if it be turned either way it will depart from the point C, and consequently from the line A C. Hence, 118 GEOMETRY. Of Planes. the position of the plane is determined by the single condition of containing the two straight lines AB, A C. Cor. 1. A triangle ABC, or three points A, B, C, not in a straight line, determine the position of a plane. Cor. 2. Hence, also, two parallels AB, CD determine the position of a plane. For drawing EF, we see that the plane of the two straight lines AE, EF is that of the parallels AB, CD. K C /F THEOREM II. .1 •perpendicular ts the shortest line which can be drawn from a point to a plane. Let A be a point above the plane DE, and AB a. line drawn perpen- dicular to the plane : then will A B be shorter than any oblique line A C. For, tlirough B, the foot of the per- pendicular, draw BC to the point where the oblique line A C meets the plane. Now, since A B is perpendicular to the plane, the angle ABC will be a right angle (Def. 1.), and consequently less than the angle C: therefore, AB, opposite the angle C, will be less than AC opposite the angle B (Bk. I. Th. xi). BOOK V 11G Of Planes. Cor It is evident that if several lines be drawn from the point A to the plane, that those which are nearest the perpen- dicular AB, will be less than those more remote. Sch. The distance from a point to a plane is measured on die perpendicular : hence, when the distance only is named, the shortest distance is always understood. THEOREM III. The common intersection of two planes is a straight line Let the two planes AB, CD, cut each other. Join any two points E and F, in the common intersection, i.^ by the straight line EF. This line will lie wholly in the plane AB, and also wholly in the plane CD (Bk. I. Def. 7) ; therefore, it will be in both planes at once, and consequently, is their common intersection. .JB THEOREM IV. A straight line which is perpendicular to two straight lines at their point of intersection, will be perpendicular to the plane of those lines. Let the line PA be perpen- dicular to the two lines AD, AB : then will it be perpendic- ular to the plane BC which con- tains them. For, if AP is not perpendicular to the plane BC, suppose a plane 120 GEOMETRY Of Plan to be drawn through A, that shall be peqiendicular to AP Now. every line drawn through ,4, and perpendicular to A P. n'ill be a line of this last plane (l)ef. 1): hence, this last plane will contain the lines AB, AD, and consequently, a line winch is perpendicular to two lines at the point of intersection, will be perpendicular to the plane of those lines. D C p A B THEOREM V. if two straight lines are perpendicular to the same plane they will be parallel to each other. Let the two lines AB, CD, be perpendicular to the plane EF : then will they be parallel to each other For, join B and D, the points in which the lines meet the plane EF Then, because the lines AB, CD, are perpendicular to the plane EF, they will be perpendicular to the line BD (Def. 1). Now, if BA and DC are not parallel, they will meet at some point as : then, the triangle OBD would have two right angles, which ie impossible (Bk. I. Th. xvii. Cor. 4). Cor. If two lines are parallel, and one of them is perpen- dicular to a plane, the other will also be perpendicular to the same plane. A C A 1 » l\ I >' / > KOOK V 121 Of Planea. THEOREM VI. If two planes intersect each other at right angles, and a line bo drawn in one plane perpendicular to the common intersection t this line will be perpendicular to the other plane. Let the plane FE be perpen- dicular to MN, and AP be drawn m the plane FE, and perpen- dicular to the common intersec- tion DE: then will AP be per- pendicular to the plane MN. For, in the plane MN draw CP perpendicular to the common intersection DE. Then, because the planes MN and FE arc- perpendicular to each other, the angle APC, which measures their inclination, will be a right angle (Def. 6). Therefore, the line AP is perpendicular to the two straight lines PC and PD ; hence, it is perpendicular to their plane MN (Th. iv). THEOREM VII. If one p'anc intersect another plane, the sum of the angles on he same side will be equal to two right angles. Let the plane GEF intersect the plane A B in the line FE : then will the sum of the two angles on the SMne side be equal to two right angles. ! or, from any point, as E, in llio common intersection, draw the lines EG and DEC, one in each plane, and botn perpen- dicular to the common intersection at E. Then, the line G U makes, with the line DEC, two angles, which together are 11 12'J GEOMETRY. Of Planca equal to two right angles (Bk I. Th. ii): but these angles measure the inclination of the planes ; there- fure, the sum of the angles on the same side, which two planes make with each other, is equal to two right angles. Cor. In like manner it may be demonstrated, that planes which intersect each other have their vertical or opposite angles equal. THEOREM VIII. Two planes whch are perpendicular to the same straight line we parallel to each other. Let the planes MN and PQ be perpendicular to the line AB: q then will they be parallel. \. For, if they can meet any where, let O be one of their their common points, and draw OB, in the plane PQ, and OA, \ __\ in the plane MN. Now, since AB is perpendicular to both planes, it will be perpendicular to OB and OA (Def. 1) : hence, the triangle OAB will have two right angles, which is impossible (Bk. I. Th. xvii. Cor. 4) ; therefore, the pianes can have no point, ds 0, in common, and consequently, they are parallel (Def 4). M D N THEOREM IX. If a plane cuts two parallel planes, the lines of intersection wilJ be parallel 13 O O K V 123 Of Plane Let the parallel planes MIS and PA be intersected by the plane EH: then will the lines of inter- section EF, GH, be parallel. For, if the lines EF, GH, were not parallel, they would meet each other if sufficiently produced, since they lie in the same plane. If this were so, the planes MN, PA , would meet each other, and, consequently, could not be parallel; which would be contrary to the supposition. THEOREM X. Ij two lines are parallel to a third line, though not in the sama plane with it, they will be parallel to each other. Let the lines AB and CD be each parallel to the third line EF, though not in the same plane with it : then will they be parallel to each other. Foi since EF and CD are parallel, they will lie in the same plane FC (Th. i. Cor. 2), and AD, EF will also lie in the plane EB. At any point, G, in the line EF, let GI ami GH be drawn in the planes FC, BE, and each perpendicular to FE at G Then, since the line EF is perpendicular to the lines GH GI, it will be perpendicular to the plane HGI (Th. iv). And since FE is perpendicular to the plane HGI, its parallels AB and DC will also be perpendicular to the same plane (Th. v). Hence, since the two lines AB, CD, are both per* pellicular to the plane HGI, the} will be parallel to each other 12 1 G E () M E T R Y Of Planes THEOREM XJ. If two angles, not situated in the same plane, have their side j parallel and lying in the same direction, the angles will br> equal. Let the angles ACE and BDF have the sides AC parallel to BD, and CE to DF: then will the angle ACE be equal to the angle BDF. For, make AC equal to BD, and CE equal to DF, and join A B, CD, iwA EF ; also, draw AE, BF. Now since AC is equal and par- allel to BD, the figure AD will be a parallelogram (Bk. 1. Th. xxv); there- fore, AB is equal and parallel to CD. Again, since CE is equal and parallel to DF, CF will be a parallelogram, and EF will be equal and parallel to CD. Then, since AB and EF are both parallel to CD, they will be parallel to each other (Th. x) ; and since they are each equal to CD, they will be equal to each other. Hence, the figure BAEF is a parallelogram (Bk. 1. Th. xxv), and conse- quently, AE is equal to BF. Hence, the two triangles ACE and BDF have the th^ee sides of the one equal to the three sides of the other, each to each, and therefore the angle AC'E is equal to the angle BDF (Bk. I. Th. viii). THEOREM XII. If tu?o planes are parallel, a straight line vihich is perpendicular to the one will also be perpendicular to the oilier. BOOK v 123 Of Plane M Let MN and PQ be two par- allel planes, and let AB be per- pendicular to MN : then will it be perpendicular to PQ. For, draw any line, BC, in the plane PQ, and through the lines AB, BC, suppose the plane ABC to be drawn, intersecting the plane MN m the line AD : then, the intersection AD will be parallel to BC (Th. ix). But since AB is perpendicular to the plane NM, it. will be perpendicular to the straight line AD, and consequently, to its parallel BC (Bk. I. Th. xii. Cor.) In like manner, AB might be proved perpendicular to any other line of the plane PQ, which should pass through B ; \ *\ A r> N \ \ hence, it is perpendicular to the plane (\)v\\ 1). Cor. It from any point as H, any oblique linos, as HEF, HDC, be drawn, the parallel planes will cut these lines proportionally. For, draw II AB perpendicular to the plane MN : then, by the theorem, it will also be perpendi- cular to PQ. Then draw AD, AE, BC, BE. Now, since AE, BE, are the intersections of the plane FIIB, with the two parallel planes MN, PQ, they are paral- lel (Th ix.); and so also are AD, BC. Then, HA ITB : HE HE, and HA IIP HD HC, hence, HE : HE : : HD HC 11* GEOMETRY. BOOK VI OF SOLIDS. DEFINITIONS 1. Ever}' solid bounded by planes is called a polyadnm. 2. The planes which bound a polyedron are called facsa. The straight lines in which the faces intersect each other, are called the edges of the polyedron, and the points at which the edges intersect, are called the vertices of the angles, or vertices of the polyedron. 3. Two polyedrons are similar, when they are contained by the same number of similar planes, and have their polyedral angles equal, each to each. 4. A prism is a solid, whose ends arc equal polygons, and whose side faces are parallelograms. Thus, the prism whose lower base is the pentagon ABCDE, terminates in an equal and parallel pentagon FGHIK, which is called the upper base. The side faces of the prism are the parallelograms DH, DK, EF, A G, and BH. These are called the convex, or lateral surface of the nrism ROOK V .27 Of the Prism 5. The altitude of a prism is the distance between its upper and lower bases : that is, it is a line drawn from a point of the upper base, perpendicular, to the lower base 6, A right prism is one in which the edges AF, BC, EK, HC, and DI, are perpendicular to the bases. In the right prism, either of the per- pendicular edges is equal to the altitude. In the oblique prism the altitude is less than the edge. K '£'--. H / D B C 7'. A prism whose base is a triangle, is called a triangular prism ; if the base is a quadrangle, it is called a quadrangular prism ; if a pentagon, a pentagonal prism ; if a hexagon a hexagonal prism ; <&c. 8. A prism whose base is a parallelo- gram, and all of whose faces are also parallelograms, is called a parallelopipe- don. If all the faces are rectangles, it is called a rectangular parallelopipedon. 9. If the faces of the rectangular par- allelopipedon are squares, the solid is called a cube: hence, the cube is a prism bounded by six equal squares 128 GEOMETRY. Of the Pyramid 10. A pyramid is a solid, formed by several triangles united at the same point S, and terminating in the differ- ent sides of a polygon ABCDE. The polygon ABCDE, is called the lise of the pyramid ; the point *S, is called the vertex, and the triangles ASB, BSC, CSD, DSE, and ESA. form its lateral, or convex surface. 1 1 . A pyramid whose base is a triangle, is palled a titan- gular pyramid ; if the base is a quadrangle, it is called a quadrangular pyramid ; if a pentagon, it is called a petagonal pyramid; if the base is a hexagon, it is called a hexagonal pyramid; &c. 12. The altitude of a pyramid, is the perpendicular let fall from the vertex, upon the plane of the base. Thus, SO is the altitude of the pyramid * —ABCDE. 13. When the base of a pyramid is a. regular polygon, and the perpendicular SO passes through the middle point of the base, the pyramid is called a right pyramid, and the line SO is called the axis BOOK VI. J2U Pyramid and Cylinder. 14. The slant height of a right pyramid, is a line drawn from the ver- tex, perpendicular to one of the sides of the polygon which forms its base. Thus. SF is the slant height of the pyramid S— ABODE. 15. If from the pyramid S—ABCDE the pyramid S — abede be cut off by a plane parallel to the base, the remain- ing solid, below the plane, is called the frustum of a pyramid. The altitude of a frustum is the per- pendicular distance between the upper and lower planes. 16. A Cylinder is a solid, described by the revolution of a rectangle, AEFD, about a fixed side, EF. As the rectangle AEFD, turns around the side EF, like a door upon its hinges, the lines AE and FD describe circles, and the line AD describes the convex sur- face of the cylinder. The circle described by the line AE, is called the base of the cylinder, and the circle described by DF, is \he upper bas-e.. lower callod 130 GEOMETRY Of the Cy Under The immovable line EF is called the axis of the cylinder A cylinder, therefore, is a round body with circular ends 17. If a plane be passed through the axis of a cylinder, it will intersect the cylin- der in a rectangle, P#, which is double the revolving rectangle DE. ^F 18. ff a cylinder be cut by a plane par- allel to the base, the section will be a cir- cle equal to the base. For, while the side FC y of the rectangle MC, describes the lower base, the equal side MP, will describe the circle MLKN, equal to the lower base. 19 If a polygon be inscribed in the lower base of a cylinder, and a corres- ponding polygon be inscribed in the upper base, and their vertices be joined by straight lines, the prism thus formed is said to be inscribed in the cylinder. BOOK VI. L31 Of the Cone 20. A cone is a solid, described by the revolution of a right angled triangle, ABC, about one of its sides, CB. The circle described by the revolving side, AB, is called the base of the cone. The hypothenuse, AC, is called the slant height of the cone, and the surface described by it, is called the convex surface of the cone. The side of the triangle, CB, which remains fixed, is called the axis, or altitude of the cone, and the point C, the vertex of the cone. 21. If a cone be cut by a plane par- allel to the base, the section will be a circle. For, while in the revolution of the right angled triangle SAC, the line CA describes the base of the cone, its parallel FG will describe a circle FKHI, parallel to the base. If from the cone S— CDB,the cone S—FKH be taken away, the remaining part is called the frustum of the cone 22. If a polygon be inscribed in the base of a cone, and straight lines be drawn from its vertices to the vertex of the cone, the pyra- mid thus formed is said to be in- scribed in the cone. Thus, the pyramid S — ABCD is inscribed in the cone 132 GEOMETRY Of the Sphere, 23. Two cylinders are similar, when the diameters of their bases are proportional to their altitudes. 24. Two cones are also similar, when the diameters of theii bases are proportional to their altitudes. 25. A sphere is a solid terminated by a curved surface, ali the points of which are equally distant from a certain point within called the centre. 26. The sphere may be described by revolving a semicircle, ABD, about the diameter AD. The plane will describe the solid sphere, and the semicircumference ABD will describe the surface. 27. The radius of a sphere is a line drawn from the centre to any point of the circumference. Thus, CA is a radius. 28. The diameter of a sphere is a line passing through the centre, and terminated by the circumfer once. Thus. AD is a diameter BOOK VI. L33 Of the Sphere 29. All diameters of a sphere are equal to each other; and each is double a radius. 30. The axis of a sphere is any line about which it re- volves ; and the points at which the axis meets the surface, ire called the poles. 31. A plane is tangent to a sphere when it has but one point in com- mon with it. Thus, AB is a tan- gent plane, touching the sphere at B< 32. A zone is a portion of the sur- face of a sphere, included between two parallel planes which form its bases. Thus, the part of the surface included between the planes AE and DF is a zone. The bases of this zone are the two circles whose diameters are AE and DF. 33. One of the planes which bound a zone may become tangent to the sphere ; in which case the zone will have but one base. Thus, if one plane be tangent to the sphere at A, and another plane cut it in the circle DF, the zone included be- tween them, will have but one base. 12 134 GEOMETRY Of the P i ; s m 34. A spherical segment is a portion of the solid sphere in- cluded between two parallel planes. These parallel planes are its bases. If one of the planes is tangent to the sphere, the segment will have but one base. 35. The altitude of a zone or segment, is the distance be Kveen the parallel planes which form its bases t THEOREM I. The convex surface of a right prism is equal to the perimeter of its base multiplied by its altitude. Let ABODE— K be a right A prism : then will its convex surface be equal to {AB-) BC+CD + DE-{-EA)xAF. For, the convex surface is equal to the sum of the rectangles AG> BH, CI, DK, and EF, which com- pose it ; and the area of each rectan- gle is equal to the product of its base B U by its altitude. But the altitude of each rectangle is equal to the altitude of the prism : hence, their areas, that is, the con- vex surface of the prism, is equal to (AB + BC + CD+DE + EA) x A F; that is, equal to the perimeter of the base of the prism multi plied by its altitude. E'"< D THEOREM II. The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude BOOK VI. 135 Of the Pr Let DB be a cylinder, and AB the diameter of its base : the convex sur- face will then be equal to the altitude 4 /) multi] lied by the circumference of the base. Tor, suppose a regular prism to be inscribed within the cylinder. Then, the convex surface of the prism will be equal to the perimeter of the base mul- tiplied by the altitude (Th. i). But the altitude of the prism is the same as that of the cylinder ; and if we suppose the sides of the polygon, which forms the base of the prism, to be indefinitely increased, the polygon will become the circle (Bk. IV.Th.xxiii. Sch.), in which case, its perimeter will become the circumference, and the prism will coincide with the cylinder. But its convex surface is still equal to the perimeter of its base multiplied by its altitude: hence, the convex surface of a cylin- der is equal to the circumference of its base multiplied by its al- titude. THEOREM III. In every prism the sections formed by planes pirallcl to the bast are equal polygons. Let A G be any prism, and IL a sec- tion made by a plane parallel to the base AC: then will the polygon IL be equal to A C. For, the two planes A C, IL, being parallel, the lines AB, IK, in which they intersect the plane AF, will also be parallel (Bk. V. Th. ix). For a like reason, BC and KL will be par- K \ G / \ \ A \ \ K I) i , I 3 ( > 136 GEOMETRY Of the Pyramid allcl; also, CD will be parallel to LM, and AD to IM. But, since AI and BK are parallel, the figure AK is a parallelogram : hence AB is equal to IK (Bk. 1. Th. xxiii). In the same way it may be shown that BC is equal to KL, CD to LM, and AD to IM. But, since the sides of the polygon AC are respectively parallel to the sides of the polygon IL, it follows that their corresponding angles are equal (Bk. V. Th. xi), viz., the angle A to the angle /, the angle B to K, the angle C to L, and the angle M to D ; hence, the polygon IL is equal to AC. Sch. It was shown in Definition 18, that the section oi a cylinder, by a plane parallel to the base, is a circle equal to the base. \ \ \ \ M A' \ D \ B THEOREM IV. If a pyramid be cut by a "plane parallel to the base, I. The edges and altitude will be divided proportionally. II. The section ioill be a. polygon similar to the base. Let the pyramid S—ABCDE, of which SO is the altitude, be cut by the plane abede parallel to the base : then will, Sa : SA : : Sb : SB, and the same for the other edges ; and the polygon abede will be similar to the base ABCDE. First, Since the planes A BC and ah*- B O O R V 1 . 137 Of the Pyramid are parallel, their intersections, AB, ab, by the plane SA B, will also be parallel (Bk. V. Th. ix) ; hence, the triangles SAB. sab, are similar, and we have SA : Sa : : SB : Sb ; fax a similar reason, we have SB : Sb '. : SC : Sc ; end the same for the other edges ■ hence, the edges SA, SB % SC, &c, are cut proportionally at the points a, b, c, &c. The altitude SO is likewise cut proportionally at the point The altitude SO is likewise cut in the same proportion at the point o ; for, since BO is parallel to bo, we have SO : So : : SB : Sb. Secondly. Since ab is parallel to AB, be to BC, cd to CD Sic. ; the angle abc is equal to ABC, the angle bed to BCD and so on (Bk. V. Th. xi). Also, by reason of the similar triangles, SAB, Sab, we have AB : ab : SB : Sb, and by reason of the similar triangles SBC, Sbc, we have SB : Sb : : BC : be; hence (Bk III. Th. v), AB : ab : : BC : be; and for a similar reason, we also have BC : be : : CD : cd, Sic. Hence, tne polygons ABCDE, abede, having their angles respectively equal, and their homologous sides proportional are similar. 12* L3S GEOMETRY. Of the Pyramid. THEOREM V. If two pyramids, having equal altitudes and their bases in the, same plane, be intersected by planes parallel to the plane of lh& bases, the sections in each pyramid v:ill fi proportional to the bases Let S—ABCDE, and S — XYZ, be two pyra- mids, having a common vertex, and their bases sit- uated in the same plane. If these pyramids are cut by a plane parallel to the plane of their bases, giv- ing the sections abede, A ^^~~^1 xyz, then will the sections Y abede, xyz, be to each other as the bases ABCDE, XYZ. For, the polygons ABCDE, abede, being similar, their sur* faces are as the squares of the homologous sides AB, ab ; but AB : ab : : ■ SA : Sa: abede SA' Sa' hence, ABCDE For the same reason, XYZ : xyz : : SX* : «?. But since abc and xyz are in one plane, the lines SA, Sa, SX, Sx, are proportional to SO, So : (Bk. V.Th. xii. Cor.), therefore, SA : Sa : : SX : Sx : hence, ABCDE : abede : : XYZ : xyz. consequently, the sections abede, xyz, are to each other as the bases ABCDE, XYZ. Cor. If the bases ABCDE, XYZ, are equivalent, any sec- tions abede, xyz, made at equal distances from the bases, will be also equivalent BOOK. VI 139 Of the Pyramid THEOREM VI. The convex surface of a right pyramid is equal to halfth pro- duct cf the perimeter of its base multiplied by the slant height. Let S— ABODE he a right pyra- mid, SF its slant height : then will its convex surface be equal to half the product 8Fx(AB+BC+CD+DE+EA). For, since the pyramid is right, the point O, in which the axis meets the base, is the centre of the polygon ABODE; hence, the lines 0.4, OB, dec drawn to the vertices of the base, are equal (Bk. IV. prob. x.Cor). Now, in the right angled triangles SAO, SBO, the bases and perpendiculars are equal : hence, the hypothenuses are equal ; and in the same way it may be proved that all the edges of the pyramid are equal. The triangles, therefore,, which form the convex surface of the prism, are all equal tn ftach other. But the area of either of these triangles, as SAB, is equal to half the product of the base AB, by the slant height of the pyramid SF: hence, the area of all the triangles, which form the convex surface of the pyramid, is equal to half the product of the perimeter of the base by the slant height. THEOREM VII. The convex surface of the frustum of a regular pyramid is equal to half the sum of tl/e perimeters of the upper and lower bases multwlied by the slant height. 140 GEOMETRY. Of the Cone. Let a — ABCDE be the frustum of a regular pyramid : then will its convex surface be equal to half the product of the perimeter of its two bases multi- plied by the slant height Ff For, since the upper base abcde, is similar to the lower base ABCDE (Th. iv), and since ABCDE is a regular polygon, it follows that the sides ab, be, cd, de, and ea, are all equal to each other. Hence, the trapezoids EAae, ABba, &c, which form the convex surface of the frustum are equal. But the perpen- dicular distance between the parallel sides of these trapezoids is equal to Ef the slant height of the frustum. Now, the area of either of the trapezoids, as AEea, is equal to half the product of Ffx{EA + ea) (Bk. IV. Th. x): hence, the area of all of them, that is, the convex surface of the frustum, is equal to half the sum of the perimeters of the upper and lower bases, multiplied by the slant height. THEOREM VIII. The convex surface of a cone is equal to half the product of th<> circumference of the base multiplied by the slant height. In the circle which forms the base of the cone, inscribe a regular poly- gon, and join the vertices with the vertex S, of the cone We shall then have a right pyramid in- scribed in the cone. The convex surface of this pyra- mid will be eoual to half the product BOOK VI. 14) Of the Cone. of the perimeter of the base by the slant height (Th. vi). Let us now suppose the number of sides of the polygon to be indefi- nitely increased : the polygon will then coincide with the base of the cone, the pyramid will become the cone, and the line Sf which meas- ures the slant height of the pyramid, will then measure the slant height of the cone. Hence, the convex surface of the cone is equal to half the product of the slant height by the circumference of the base. THEOREM IX. The convex surface of the frustum of a cone is equal to half the sum of the circumferences of its two bases multiplied by tin slant height. For, if we suppose the frustum of a right pyramid to be inscribed in the frustum of a cone, its convex surface will be equal to half the pro- duct of its slant height by the perim- eters of its two bases. But if we increase the number of sides of the polvgon indefinitely, the frustum of the pyramid will become \te frustum of the cone : hence, the area of the frustum of the cone is equal to half the sum of the circumferences of its twe beses multiplied by the slant height <42 GEOMETRY Of Parallelopipedon THEOREM X. Two rectangular parallelopipedons, having equal altitudes and equal bases, are equal. Let E — ADCD, and F — KG HI, be two rectangular [jar ullelopipedons having equal % p bases, AC and KH, and equal altitudes, AE and KF : then will they be equal. For, apply the base of the one parallelopipedon to that B C G R of the other, and since the bases are equal, they will coincide Again, since the edges are perpendicular to the bases, the edges of the one parallelopipedon will coincide with those of the other; and since the altitude AE is equal to KF, the planes of the upper bases will coincide. Hence, the paral- lelopipedons will coincide, and consequently they are equal D KS \ \ T \ THEOREM XI. Two rectangular parallelopipedons, which have the same base, are to each other as their altitudes. Let the parallelopipedons AG, AL, have the same base BD, then will they be to each other as their altitudes AE A I. Suppose the altitudes AE, A I, to be to each other as two whole num- bers, as 15 is to 8, for example. Di- vide AE into 15 equal parts, whereof Al will contain 8; and through a?, y, ,s, &c, the points of division, draw planes BOOK VI 143 Of Parallelo pipe dons. parallel to the base. These planes will cut the solid A G into 15 partial parailelopipedons, all equal to each Otlicr, because they have equal bases and equal altitudes — equal bases, since every section, IL, made parallel to the base BD, of a prism, is equal to that base ; equal altitudes, because the altitudes are the equal divisions Ax, xy,yz, &c. But of these 15 equal par- ailelopipedons, 8 are contained in AL; hence, solid AG : solid AL : : or generally, solid AG : solid AL : : ff x i 15 AE & D V? AL THEOREM XII. Two regular parailelopipedons, having the same altitude, are to each oilier as their bases. Let the parailelopipe- dons AG, AK, have the same altitude AE ; then will they be to each other as their bases AC, AN. Having placed the two solids by the side of each other, as the figure re- presents, produce the plane ONKL until it meets the plane DCGH in PQ ; you will thus i E \ "OV liio 144 GEOMETRY Of Parallolopipedona have a third parallelo- pipedon A Q, which may be compared with each of the parallelopipedons AG,AK. The two sol- ids AG, AQ, having the same base AEHD, are £o each other as their altitudes AB, A O ; in like manner, the two solids A Q AK, having the same base AOLE, are to each other as their altitudes A D, AM. Hence, we have the two proportions, sohd AG : solid AQ :: AB : AO, solid AQ : solid AK : : AD : AM. Multiplying together the corresponding terms of these pro- portions, and omitting the common multiplier solid A Q, we have solid AG : solid A K :: ABxAD : AO x AM. But AB x AD represents the base ABCD ; and AOxAM represents the base AMNO: hence, two rectangular parallel- opipedons of the same altitude are to each other as their bases. THEOREM XIII. Any two rectangular parallelopidedons are to each other as tlu products of their three dimensions. For, having placed the two solids AG, AZ, (see next figure) so that their surfaces have the common angle BAE, produce the planes necessary for completing the third parallelopipedon AK, having the same altitude with the parallelopipedon AG By the last proposition we shall have the proportion, BOOK V 145 Of Parallelopipedons solid AM solid AK : : AB CD : AMNO But the two paral- lelopipedons AK, AZ, having the same base AMNO, are to each other as their altitudes AE, AX ; hence, we have solid AK : solid AZ : : AE : AX Multiplying together the corresponding terms of these pro- portions, and omitting in the result the common mullipliei solid AK, we shall have solid AG solid AZ : : ABCDxAE : AMNOxAX. Instead of the bases A BCD and AMNO, put ABxAP and A O x AM, and we have solid A : solid AZ • : ABxADxAE : A X AM X A X. Hence, any two rectangular parallelopipedons are to each other as the product of their three dimensions. Si-k. We are consequently authorized to assume, as the measure of a rectangular parallelopipedon, the product, of its three dimensions. In order to comprehend the nature of this measurement, i: i? neccr-sarv to reflect, that the number o( linear units in one 13 146 GEOMETRY. Of Parallelopipedons dimention of the base multiplied by the number of linear unit? of the other dimension of the base, will give the number of superficial units in the base of the parallelopipedon (Bk. IV Th. vi. Sch). For each unit in height, there are evidently jw many solid units as there are superficial units in tho base. Therefore, the product of the number of superficial units in the base multiplied by the number of linear units in the altitude is the number of solid units in the parallelopipedon. If the three dimensions of another parallelopipedon are valued according to the same linear unit, and multiplied together in tho same manner, the two products will be to each other as the solids, and will serve to express their relative magnitude Let us illustrate this by an example. Let ABCD be the base of a parallelopipedon, and suppose AB = 4 feet, and BC = 3 feet. Then the number of square feet m the base ABCD will be equal to 3x4 = 12 square feet Therefore, 12 equal cubes of 1 foot each, may be placed by the side of each other on the base. If the parallelopipedon be J foot in height, it will contain 12 cubic feet ; were it 2 feet in height, it would contain two tiers of cubes, or 24 cubic feet ; were it 3 feet in height, it would contain three tiers of tubes, or 36 cubic feet. The magnitude of a solid, its volume or extent, forms what is called its solidity ; and this word is exclusively employed to designate the measure of a solid ; thus, we say the s )Iidity of a rectangular parallelopipedon is equal to the product of it? base by its altitude, or to the product of its three dimensions BOOK VI. 14' Of Parali e 1 op ip edons . As the cube has all its three dimensions equal, il the side is 1, the solidity will be 1x1x1 = 1; if the side is 2, the solidity will be 2x2x2 = 8; if the side is 3, the solidity will be 3x3 X 3 = 27; and so on: hence, if the sides of a series of cubes are to each other as the numbers 1, 2, 3 Sic. the cubes themselves, or their solidities, will be as the num- bers 1, 8, 27, &c. Hence it is, that in arithmetic, the cube of a number is the name given to a product which results from three factors, each equal to this number. THEOREM XIV. If a parallelopipedon, a prism, and a cylinder, have equivalent bases and equal altitudes, they will be equivalent. Let F—ABCD, be a parallelopipedon ; F— ABODE, a prism ; and D — ABC, a cylinder, having equivalent bases and equal altitudes : then will they be equivalent. D B C For, since their bases are equivalent they will contain the same number of units of surface (Bk. IV. Def. 9). Now, for each unit of height there will be one tier of equal cubes in each solid, and since the altitudes are equal, the number of tiers in each solid will be equal : hence, the solidities will be equal, and therefore the solids will be equivalent. Cjr Hence, we conclude, that the solidity of a prism or cylinder is equal tj the area of its base multiplied by its altitude. MS GEOMETRY. Of Triangular Pyramids THEOREM XV. Two triangular 'pyramids, having equivalent bums and equal altitudes, are equivalent, or equal in solidity. Let their equivalent bases, ABC, abc, be situated in the same plane, and let AT be their common altitude. If they are not equivalent, let S — abc be the smaller ; and suppose Aa to be the altitude of a prism, which, having ABC for its base, is equal to their difference. Divide the altitude AT into equal parts Ax, xy, yz, &c. each less than Aa, and let k be one of those parts : through the points of division pass planes parallel to the plane of the bases : the corresponding sections formed by these planes in the two pyramids will be respectively equivalent, namely DEF to def, GUI to ghi, &c (Th. v. Cor.) BOOK VI. 149 Of Triangular Pyra m ids. This being granted, upon the triangles ABC, DEF, G///, &c, taken as bases, construct exterior prisms having foi edges the parts AD, DG, GR, &c, of the edge SA ; in like manner, on bases def, ghi, klm, &c , in the second pyramid construct interior prisms, having for edges the corresponding parts of Sa. It is plain that the sum of the exterior prisms o\ the pyramid S — ABC will be greater than the pyramid; while the sum of the interior prisms of the pyramid *S — abc, will be less than the pyramid. Hence, the difference between these sums will be greater than the difference between the pyramids. Now, beginning with the bases ABC, abc, the second ex- terior prism EFD — G is equivalent to the first interior prism efd — a, because they have the same altitude k, and their bases DEF, def, are equivalent ; for like reasons, the third exterior prism HIG — K, and the second interior prism hig — d, are equivalent ; the fourth exterior and the third interior ; and so on, to the last of each series. Hence, all the exterior prisms of the pyramid *S — ABC, excepting the first prism BCA — D, have equivalent corresponding ones in the interior prisms 01 the pyramid 5 — abc: hence, the prism BCA — D is the differ- ence between the sum of all the exterior prisms of the pyramid S — ABC, and of the interior prisms of the pyramid S — abc But this difference has already been proved to be greater than that of the two pyramids : which, by supposition, differ by the prism a — ABC: hence, the prism BCA — D, must be greater than the prism a — ABC. But in reality it is less, foi they have the same base ABC, and the altitude Ax, of the first, is less than Aa, the altitude of the second. Hence, the supposed inequality between the two pyramids cannot exiat; hence, the two pyramids; f. ON=AFxcirf. ON. Cor. The surface described by any portion of the perim- eter, as EDO, is equal to the distance between the two per- pendiculars let fall from its extremities, on the axis, multiplied by the circumference of the inscribed circle. For, the sur- face described by DE is equal to HIx circumference ON and the surface described by DC is equal to /P X circumfe- BUOK VI 159 Of the Sphere nmce ON: hence, the surface described by ED+DC, is equal co {111+ IP)X circumference ON, or equal to HP xcu cum- Terence ON. THEOREM XXIII. The surface of a sphere is equal to the product of its diameter by the circumference of a great circle. Let ABODE be a semicircle. In- scribe in it any regular semi-polygon, and from the centre O draw OF per- pendicular to one of the sides. Let the semicircle and the semi- polygon be revolved about the axis AE: the semicircumference ABODE will describe the surface of a sphere (Def. 26) ; and the perimeter of the semi-polygon will describe a surface which has for its measure AEx cir- cumference OF (Th. xxii) ; and this will be true whatever be the number of sides of the polygon. But if the number of sides of the polygon be indefinitely increased, its perimeter will coincide with the circumference ABODE, the perpen- dicular OF will become equal to OE, and the surface do- Bcribed by the perimeter of the semi-polygon will then be the same as that described by the semicircumference ABODE Hence, the surface of the sphere is equal to AE x circum ference OE. Cot. Since the area of a great circle is equal to the product oi its circumference by half the radius, or by one-fourth of the diameter (Bk. IV. Th. xxvii), it follows that the surface of a sphere is equal to four of its great circles. 160 3 E O M E T R Y Of the Zone THEOREM XXIV. The surface of a zone is equal to its altitude multiplied by the circumference of a great circle. For, the surface described by any portion of the perimeter of the in- scribed polygon, as BC-\-CD is equal to EH x circumference OF (Th. xxii. Cor). But when the number of sides of the polygon is indefinitely increased, BC+CD, becomes the arc BCD, OF becomes equal to OA, and the surface described by BC+CD, becomes the surface of the zone described by the arc BCD: hence, the surface of the zone is equal to EHx circumference OA. Sch. 1. When the zone has but one base, as the zone dc scribed by the arc ABCD, its surface will still be equal to the altitude AE multiplied by the circumference of a greal circle. Sch. 2. Two zones taken in the same sphere, or in equal spheres, are to each other as their altitudes ; and any zone is to the surface of the sphere as the altitude of the zone is to the diameter of the sphere. THEOREM XXV. The solidity of a sphere is equal to one third of the product if the surface multiplied by the radius. For, conceive a polyedron to be inscribed in the sphere. BOOK VI. L6J Of the Sphere This polyedron may be considered as formed of pyramids, each having for its vertex the centre of the sphere, and for its base one of the faces of the polyedron. Now, the solidity of each pyramid, will be equal to one third of the product of its base i>y its altitude (Th. xvii). But if we suppose the faces of the polyedron to be continu- ally diminished, and consequently, the number of the pyra- mids to be constantly increased, the polyedron will finally become tlie sphere, and the bases of all the pyramids will become the surface of the sphere. When this takes place, the solidities of the pyramids will still be equal to one third tha product of the bases by the common altitude, which will then be equal to the radius oi^ the sphere. Hence, the solidity of a sphere is equal to one third of the product of the surface bv the radius. THEOREM XXVI. The surface of a sphere is equal to the convex surface of the circumscribing cylinder ; and the solidity of the sphere is two thirds the solidity of the circumscribing cylinder. Let MPNQ be a great circle of the sphere ; ABCD the circum- scribing square : if the semicircle PMQ, and the half square PADQ, are at the same time made to re- volve about the diameter PQ, the semicircle will describe the sphere, vhile the half square will describe the cylinder circumscribed about that sphere. The altitude AD, of the cylinder, is equal to the diameter 14* 162 GEOMETRY Of the Sphere. PQ; the base of the cylinder is equal to the great circle, since its diameter A B is equal MN; hence, the convex surface of the cylin- der is equal to the circumference of the great circle multiplied by its diameter (Th. ii). This meas- ure is the same as that of the sur- face of the sphere (Th. xxiii) : hence the surface of the sphere is equal to the convex sur- face of the circumscribing cylinder. In the next place, since the base of the circumscribing cylinder is equal to a great circle, and its altitude to the di- ameter, the solidity of the cylinder will be equal to a great circle multiplied by a diameter (Th. xiv. Cor). But the so- lidity of the sphere is equal to its surface multiplied by a third of its radius ; and since the surface is equal to four great circles (Th. xxiii. Cor.), the solidity is equal to four great cir- cles multiplied by a third of the radius ; in other words, to one great circle multiplied by four-thirds of the radius, 01 by two-thirds of the diameter, hence, the sphere is two-thirds of the circumscribing cylinder. BOOK VI 163 Appendix APPENDIX OF THE FIVE REGULAR PCLYEDRONS. A regular polyedron, is one whose faces are all equal poly- gons, and whose polyedral angles are equal. There are fiVe such solids. 1. The Tetraedron, or equilateral pyramid, is a solid bounded by four equal triangles. 2. The hexacdron or cube, is a solid, bourded by six equal squares. 3. The oclaedrrm, is a soiid, bounded by eight equal e.ftu lateral triangles. 104 GEOMETRY. Appendix 4. The dodeecudron, is a solid bounded by twelve oquaJ pentagons 5. The icosacdron, is a solid, bounded by twenty equa equilateral triangles. G. The regular solids may easily be made of pasteboard. Draw the figures of the regular solids accurately on paste board, and then cut through the bounding lines : this will give figures of pasteboard similar to the diagrams. Then, cut the other lines half through the pasteboard, after which, turn up the parts, and glue them together, and you will form the bodies which have been described. ELEMENTS OF TRIGONOMETRY. INTRODUCTION. SECTION I. OF LOGAniTOMS. 1. The logarithm of a number is the exponent of the \iowcr to which it is necessary to raise a fixed number, in order to produce the first number. This fixed number is called the base of the system, and maj beany number except 1 : in the common system 10 is assumed as the base. 2. Tf we form those powers of 10, which are denoted by entire exponents, we shall have 10° = 1 10 1 = 10 , 10 3 =r 1000 10 2 = 100 , 10 4 = 10000, &c. &c. From the above table, it is plain, that 0, 1, 2, 3, 4, . 9.584 908 : sin B' 45° 13' 55", or ABC 134° 4G' 05" 9.851236 The ambiguity in this, and similar examples, arises in con sequence of the first proportion being true for either of the angles ABC, or AB'C, which are supplements of each other, and therefore have the same sine (Art. 30). As long as the two triangles exist, the ambiguity will continue. But if the side CB, opposite the given angle, is greater than AC, the arc BB' will cut the line ABB', on the same side of the point A, in but one point, and then there will be only one triangle an- swering the conditions. If the side CB is equal to the perpendicular Cd, the art BB' will be tangent to ABB', and in this case also there will be but one triangle. When CB is less than the perpen- dicular Cd, the arc BB' will not intersect the base ABB', and in that case, no triangle can be formed, or it will be impossible to fulfil the conditions of the problem. 2. Given two sides of a triangle 50 and 40 respectively, and the angle opposite the latter equal to 32° : required the re- maining parts of the triangle. Aks. If the angle opposite the side 50 is acute, it is equal to 41° 28' 59" ; the third angle is then equal to 106° 31' 01", und the third side to 72.368. If the angle opposite the sidp TRIGONOMETRY. 19T Applications. 50 is obtuse, it is equal to 138° 31' 01", the third- angle to 9° 28' 59", and the remaining side to 12.436. case in. When the two sides and their included angle are given. B X, Let ABC be a triangle; AB } BC, the given sides, and B the given angle. Since B is known, we can find the sum of the two other angles : for A- A + C == 180° - B and l(A + C) = 1(180° - B) We next rind half tho difference of the angles A and C by Theorem ii., viz. BC + BA: BC- BA : : tan {(A + C) : tan \(A - C): in which we consider BC greater than BA, and therefore A ia greater than C\ since the greater angle must be opposite the greater side. Having found half the difference of A and C, by adding it to the half sum, ^(A -\- C), we obtain the greater angle, and by subtracting it from half the sum, we obtain the less. That b \{A + C) + \{A - C) = A, and \{A+ C)-i(A - C)= a Having found the angles ^1 and C, the third side AC may be found by the proportion. sin A : sin B : : BC : AC. EXAMPLE8. 1. In the triangle ABC, Jet BC = 540, AB = 450, and the included angle B = 80° : required the remaining parts. 17* 198 TRIGONOMETRY. Application a. GEOMETRICALLY. Draw an indefinite right line BC and from any point, as B t lay off a distance BC = 540. At B make the angle CBA — 80°: draw BA and make the distance Bi -- 450; tlraw AC; then will ABO be the required triangle. TRIGONOMETRICALLY. BC + BA = 540 4- 450 = 990; and BC — BA = 540 — 450 = 90. A + C 7 = 180° — 5 = 180° —80° = 100°, and therefore, J(,l + C) = 3(100°) = 50° To find \{A— C). As BC -\- BA 990 . ar. comp. . 7.004365 BC — BA 90 ... 1.954243 : tan ](A + C) 50° ... 10.076187 tan 1(^1— 6') 6° 11' . . . 9.03 4795 Hence, 50° 4- 0° 11' = 5G° 11' == A; and 50° — 0° 11' == 43° 49' = C. To find the third side AC. As sin C 43° 49' . ar. comp. . . 0.159672 : sin B 80° 9.993351 :: vlS 450 2.653213 AC 640X82 2.806230 2o Given two sides of a plane triangle, 1686 and 960, and their included angle 128° 04': required the other parts. Ans. Angles, 33° 34' 39" ; 18° 21' 21" ; side 2400. TRIGONOMETRY. 199 Applications. CASE IV. Having given the three sides of a plane triangle, to find the angles. Let fall a perpendicular from the angle opposite the greater tide, dividing the given triangle into two right-angled triangles : then find the difference of the segments of the base by Theo- rem iii. Half this difference being added to half the base gives the greater segment ; and, being subtracted from half the base, gives the less segment. Then, since the greater segment belongs to the right-angled triangle having the greatest hypo- thenuse, we have the sides and right angle of two right-angled triangles, to find the acute angles. EXAMPLES. 1. The sides of a plane trian- gle being given; viz. BC = 40, AC = 34 and AB — 25 : required the angles. H GEOMETRICALLY. With the three given lines as sides construct a triangle as in Bk. II. Prob. xi. Then measure the angles of the triangle either with the protractor or scale of chords. TRIGONOMETRIC ALLY. As BC :AC + AB : : AC - AB : CD - BD That is, 40 : 59 : : 9 : 59 X 9 = 13.275 40 40 -f- 13.275 Then, 40 = 26.6375 = CD . , 40 - 13.275 AJid = 13.3625 = BD. 200 TRIGONOMETRY Applications. In the triangle DAC, to find the angle DAC. As AC 34 . . ar. corap. . 8.468521 DC 26.6375 .... 1.425493 sin D 90° 10.000000 sin DAC 51° 34' 40" . . . 9.8940U In the triangle BAD. to find the angle BAD. As AB 25 ar. cornp. . 8.602060 BD 13.3625 . . . 1.125887 sin D 90° ... 10.000000 sin BAD 32° 18' 35" . . . 9.727947 Hence 90° — D A C = 90° — 51° 34' 40" = 38° 25' 20" = C and 90° — BAD = 90° — 32° 18' 35" = 57° 41' 25" = B and BAD + DAC = 51° 34' 40" -f 32° 18' 35" = 83° 53' 15" = A. 2. In a triangle, in which the sides are 4, 5 and 6, what are the angles ? Ars. 41° 24' 35"; 55° 46' 16" ; and 82° 49' 09". SOLUTION OP RIGHT-ANGLED TRIANGLES. The unknown parts of a right-angled triangle may be found by either of the four last cases : or, if two of the sides are given, by means of the property that the square of the hypo- thenuee is equivalent to the sura of the squares of the two other sides. Or the parts may be found by Theorems iv. and v. EXAMPLES. 1. In a right-angled triangle BAC, there are given the hypothenuse BC — 250, and the base AC = 240: re- C quired the other parts. TRIGONOMETRY. 201 Applications. To find the angle B. As BC 250 . ar com p. 7.602060 : AC 240 ... 2.380211 : : sin A 90° ... 10.000000 : sin B 73° 44' 23" . 9.982271 But C = 00° — B = 90° — 73° 44' 23" = 16° 15' 37" : Or C may be found from the proportion. A& CB 250 ar. comp. . 7.602060 2.380211 10.000000 CB 250 AC 240 R . C 16° 15' 37' cos C 16° 15' 37" . . . 9.982271 To find side AB by Theorem Iv. As R ar. comp. . 0.000000 tan C 16° 15' 37" . . . 9.404889 AC 240 .... 2.3S0211 AB 70.0003 .... 1.845100 2. In a right- angled triangle BAC, there are given AC ~ 384, and B = 53° 08' : required the remaining parts. Ans. AB= 287.96; £(7= 479.979; (7= 36° 52'. DEFINITIONS. 1. A horizontal angle is one whose sides are horizontal ; its plane is also horizontal. 2. An angle of elevation or depression, has one horizontal side, said the other oblique, but lying directly above or below the first. 202 TRIGONOMETRY Applications. APPLICATION TO HEIGHTS AMD DISTANCES. PROBLEM I. To determine the horizontal distance to a point which is inac- cessible by reason of an intervening river. Let lane. Suppose C to be directly over the given object, and A the point through which the horizontal plane is supposed to pass. Measure a horizontal base line ^^/J^^^^^^^^^^g AB, and at the stations A and B SfeS^SP conceive the two horizon taF lines AC, BC\ to be drawn. The oblique lines from A and B to the object will be the hypothenusea of two right-angled triangles, of which AC, BC, are the bases. The perpendiculars of these triangles will be the dis- tances from the horizontal lines AC, BC, to the object. If we turn the triangles about their bases AC, BC, until they become horizontal, the object, in the first case, will fall at C", and in the second at C". Measure the horizontal angles CAB, CBA, and also thfl angles of depression C AC, C" BC. Let us suppose that we have AB = 672 yards BAC --= 72° 29' ABC = 39° 20' C'AC=21° 49' . C"BC = 19° 10' First: In the triangle ABC, the horizontal angle ACS =? 80° - (A 4- B) = 180° - 111° 49' = 68° 11'. found 208 TRIGONOMETRY Applications. As Aft To find tbe horizontal distance AC. sin C G8° 11' ar. comp. . 0.032275 sin B 39° 20' ... 9.801973 AB 672 ... 2.827369 AC 45S.79 . . . 2.661017 To find the horizontal distance BC. sin C 68° 11' . ar. comp. . ..0.032275 sin A 72° 29' .... 9.979380 AB 672 2.827369 BC 690.28 2.839024 As As In the triangle CAC' y to find CC. R . ar. comp. . . 0.000000 tan C'AC 27° 49' .... 9.722315 AC 458.79 .... 2.661617 CC 242.06 .... 2.383932 In the triangle CBC", to find CC" R . ar. comp. . . 0.000000 tan C'BC 19° 10' . . . 9.541061 BC 690.28 .... 2.839024 CC" 239.93 . , 2.380083 TRIGONOMETRY 209 Applications, Hence also, CC - CC" = 242.06 - 239.93 = 2.13 yard?, which is the height of the station A above station B. PROBLEMS. 1. Wanting to know the distance between two inaccessible objects, which lie in a direct line from the bottom of a tower of 120 feet in height, the angles of depression are measured, and are found to be, of the nearer 57°, of the more remote 25° 30' : required the distance between them. Ans. 113.656 feet. 2. In order to find the distance between two trees A and B, which could not be directly measured because of a pool which occupied the intermediate space, the dis- tances of a third point C from each of them wore measured, and also the included angle A CB : it was found that CB — 612 yards CA = 588 yards ACB = 55° 40'; required the distance AB. Ans. 592.907 yards. 3. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible bill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then mea- suring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45'; required tl.e height of the towei. Ans. 83.998 feet. 18* 210 TRIGONOMETRY, Application 4. Wanting to know the horizon- tal distance between two inaccessi- ble objects E and W y the following measurements were made, f AB = 536 yards BAW — 40° 16' WAE = 57° 40' ABE = 42° 22' EBW= 71° 07' required the distance EW. A Ans. 939.634 yards. 5. Wanting to know the horizontal distance between two inaccessible objects A and B, and not finding any station from which both of them could be seen, two points C and D, were chosen, at a distance from each other, equal to 200 yards ; from the former of these points A could be seen, and from the latter B, and at each of the points C and D a staff was set up. From C a distance CF was measured, not in the direction DC y equal to 200 yards, and from T> a distance DE equal to 200 yards, and the follow- ing angle* taken, r AFC = 83° 00' BDE = 54° 30' viz. } ACD = 53° 30' BBC = 156° 25' [ ACF= 54° 31' BED = 88° 30' Ans. AB = 345.467 yards. APPLICATIONS GEOMETRY. MENSURATION OF SURFACES. DEFINITIONS. 1 The area of any figure has already been defined to be the measure of its surface (Bk. IV. Def. 7). This measure is merely the number of squares which the figure contains. A square whose side is one inch, one foot, or one yard, &c, is called the measuring unitj and the area or contents of a figure is expressed by the number of such squares which the figure contains. 2. In the questions involving decimals, the decimals are generally carried to four places, and then taken to the nearest figure. That is, if the fifth decimal figure is 5, or greater than 5, the fourth figure is increased by one. 3. Surveyors, in measuring land, generally use a chain called Gunter's chain. This chain is four rods, or 66 feet in length, and is divided into 100 links. 4. An acre is a surface equal in extent to 10 square chains; that is, equal to a rectangle of which one side is ten chains and the other side one chain. One quarter of an acre, is called a ~ood. Since the chain is 4 rods in length, 1 square cha'n contains 16 square rods; and therefore, an acre, which is 10 square chains, contains 160 square rods, and a rood contains 40 square rods. The square rods are called perches. 212 APPLICATIONS Mensuration of Surfaces. 5. Land is generally computed in acres, roods, and perched which are respectively designated by the letters A, R, P. When the linear dimensions of a survey are chains or linke the area will be expressed in square chains or square links, and it is necessary to form a rule for reducing this area to acres, roods, and perches. For this purpose, let us form ibe following TABLE. 1 square chain = 1 00 x 1 00 = 1 0000 square links 1 acre = 10 square chains = 100000 square links 1 acre = 4 roods = 1 GO perches. I square mile = 6400 square chains = 640 acres. 6. Now, when the linear dimensions are links, tho area will be expressed in square links, and may be reduced to acres by dividing by 100000, the number of square links in an acre : that is, by pointing off five decimal places from the right hand. If the decimal part be then multiplied by 4, and five places of decimals pointed off from the right hand, the figures to the left hand will express the roods. If the decimal part of this result be now multiplied by 40, and five places for decimals pointed off, as before, the figures to the left will express the perches. If one of the dimensions be in links, and the other in chains, the chains may be reduced to links by annexing two ciphers, or, the multiplication may be made without annexing the ci- phers, and the product reduced to acres and decimals of an acre, by pointing off three decimal places at the right hand. When both dimensions are in chains, the product is re- OF GEOMETRY. 213 Mensuration of Surfaces. luced to acres by dividing by 1 0, or pointing off one decimal place. From which we conclude : that, I. If Imks be multiplied by links, the product is reduced to acres by pointing off five decimal places from the right hand. II. If chains be multiplied by links, the product is reduced to acres by pointing off three decimal places from the right hand. III. If chains be multiplied by chains, the product is reduced to acres by pointing off one decimal place from the right hand. 7. Since there are 16.5 feet in a rod, a square rod is equaj to 16.5 x 16.5=272.25 square feet. If the last number be multiplied by 160, we shall have 272.25 X 160 = 43560 the square feet in an acre. Since there are 9 square feet in a square yard, if the last number be divided by 9, we obtain 4 840 = the number of square yards in an acre problem I. To find the area of a square, a rectangle, a rhombus, or a parallelogram. RULE. Multiply the base by the perpendicular height and the produc- will be the area (Bk. IV. Th. viii). EXAMPLES. I Required the area of the square A BCD, each of whose sides is 36 feet D C 1 A B 214 APPLICATIONS Mensuration of Surfaces We multiply two sides of the square together, and the product is the area in square feet. Operation. 36x36=1296 sq. ft. 2. How many acres, roods, and perches, in a square whose side is 35.25 chains? Arts. 124 A. 1 R. 1 P 3. What is the area of a square whose side is 8 feet 4 inches? Ans. 69 ft 5' 4" 4. What is the contents of a square field whose side is 46 rods? Ans. 13 A. R. 36 P. 5. What is the area of a square whose side is 4769 yards 1 Ans. 22743361 sq. yds 6. What is the area of the parallelo- gram ABCD, of which the base AB is 64 feet, and altitude DE, 36 feet ? e i c 7 We multiply the base 64, by the perpendicular height 36, and the product is the re- quired area. Operation 64x36 = 2304 sq ft 7. What is the area of a parallelogram whose base is 12,25 yards, and altitude 8.5? Ans 104,125 sq. yds. 8. What is the area of a parallelogram whose base is 8.75 chains, and altitude 6 chains ? Ans. 5 A. 1 R OP. 9. What is the area of a parallelogram whose base is 7 r eot 9 inches, and altitude 3 feet 6 inches ? Ans. 27 sq.ft. 1 ' 6' OF GEOMETRY 215 Mensuration of Surfaceo 10. To find the area of a rectangle A BCD, of which the base AB = A5 yards, and the altitude AD = 15 yards. Here we simply multiply the base by the altitude, and the product is the area. B Operation 45xl5=z675 sq. yds. 1 1. What is the area of a rectangle whose base is 14 feot 6 inches, and breadth 4 feet 9 inches ? Ans. 68 sq.ft. 10' 6". 12. Find the area of a rectangular board whose length is 112 feet, and breadth 9 inches. Ans. 84 sq. ft. 13. Required the area of a rhombus whose base is 10.51 and breadth 4.28 chains. Ans. 4 A. 1 R. 39.7 P+. 14. Required the area of a rectangle whose base is 12 feot 6 inches, and altitude 9 feet 3 inches. Ans. 115 sq. ft. T 6" PROBLEM II. To find the area of a triangle, whea the base and altitude are known. RULE. I. Multiply the base by the altitude, and half the product will be the area. II. Multiply the base by half the altitude and the product inill be the area (Bk. IV. Th. ix). EXAMPLES. I Required the area of the trianglo ABC, whose base AB is 10,75 feet, and altitude 7,25 feet. 15 216 APPLICATIONS Mensuration of Su rfaces. Operation. We first multiply the base 10,75x7,25=77,9375 by the altitude, and then di- and vide the product by 2. 77,9375^2 = 38,96875 = area 2. What is the area of a triangle whose base is 18 feet 4 inches, and altitude 11 feet 10 inches? Ans. 108 sq. ft. 5' 8". 3. What is the area of a triangle whose base is 12.25 chains, and altitude 8.5 chains? Ans. 5 A. OR. 33 1\ 4. What is the area of a triangle whose base is 20 feet, and altitude 10.25 feet. Ans. 102.5 sq. ft. 5. Find the area of a triangle whose base is 625 and alti- tude 520 feet. Ans. 162500 sq. ft 6. Find the number of square yards in a triangle whose base is 40 and altitude 30 feet. Ans. 66^ sq. yds. 7. What is the area of a triangle whose base is 72.7 yards, and altitude 36.5 yards ? Ans. 1326,775 sq. yds problem ill. To find the area of a triangle when the three sides are known. RULE, I. Add the three sides together and take half their sum. II. From this half sum take each side separately. III. Multiply together the half sum and each of the three remainders, and then extract the square root of the product, which will be the required area. OF GEOMETRY. 217 Mensuration of Surface.?-. EXAMPLES. 1. Find the area of a triangle whose sides are 20, 30, and 10 rods. 20 45 45 45 30 20 30 40 _5_^ 25 1st rem. 15 2d rem. 5 3d rem 45 half sum, Then, to obtain the pruduct, we have 45x25x 15x5 = 84375; from which we find area^v 784375 ^' 290 ' 4737 perches. 2. How many square yards of plastering are there in a tri- angle, whose sides are 30, 40, and 50 feet ? Ans. 66§. 3. The sides of a triangular field are 49 chains, 50.25 chains, and 25.69 : what is its area ? Ans. 61 A. 1 R. 39,68 P 4. What is the area of an isosceles triangle, whose base is 20, and each of the equal sides 15 ? Ans. Ill 803. 5. How many acres are there in a triangle whose three sides are 380, 420 and 765 yards. Ans. 9 A. OR. 38 P. 6. How many square yards in a triangle whose sides are 13, 14, and 15 feet. Ans. 9j. 7 What is the area of an equilateral triangle whose side is 25 feet ? Ans. 270.6329 sq. ft. 8. What is the ares of a triangle whose sides are 24, 36, and 48 yards? Ans 418.282 sq. yds. 2i8 APPLICATIONS Mensuration of Surfaces. PROBLEM IV. To find the hypothenuse of a right angled triangle when the base and perpendicular are known RULE. I. Square each of the sides separately. II. Add the squares together. III. Extract the square root of the sum, which will be ifa Inj- pothenuse of the triangle (Bk. IV. Tli. xii). EXAMPLES. 1. In the right angled triangle ABC, we have, AB = 30 feet, BC = 40 feet, to find tIC. We first square each side, and then take the sum, of which we extract the square root, which gives 3(T 900 40 =1600 sum =2500 ^4 0=^/2500 = 50 feet. 2. The wall of a building, on the brink of a river, is 120 feet high, and the breadth of the river 70 yards : what is the length of a line which would reach from the top of the wall to the opposite edge of the river? Arts. 241.86 ft. 3. The side roofs of a house of which the eaves are of the same height, form a right angle at the top. Now, the length of the rafters on one side is 10 feet, and on the other 14 feet : what is the breadth of the house ? Arts. 17.204 ft. 4. Wh it would be the width of the house, in the last ex« ample, if the rafters on each side were 10 feet? Ans. 14.142 ft. OF GEOMETRY. 219 Mensuration of Surfaces 5. What would be the width, if the rafters on each side were 14 feet ? Ans. 19.7989 ft. PROBLEM V. When the hypothenuse and one side af a right angled tri- angle are know n, to find the other side RULE. Square the hypothenuse and also the other given side, and take their difference : extract the square root of this difference, and the result will be the required side (Bk. IV. Th. xii. Cor.). EXAMPLES. 1. In the right angled triangle ABC, there are given AC=50 feet, and AB — 40 feet, required the side BC. We first square the hypoth- enuse and the other side, after which we take the difference, and then extract the square root, which gives BC=*/900=z30 feet. 2 The height of a precipice on the brink of a river is 103 feet, and a line of 320 feet in length will just reach from the top of it to the opposite bank : required the breadth of the river. Ans. 302.9703 ft. 3. The hypothenuse of a triangle is 53 yards, and the per pendicular 45 yards : what is the base 1 Ans. 28 yds. 4 A ladder 60 feet in length, wil] reach to a window 40 Operation 50* = 2500 40 2 rrl600 Diff.= 900 220 APPLICATIONS Mensuration of Surfaee feet from the ground on one side of the street, and by turning it over tu the other side, it will reach a window 50 feet from the ground : required the breadth of the street. Ans. 77.8875 ft. PROBLEM VI. To find the area of a trapezoid. RULE. Multiply the sum of the parallel sides by the perpendicular distance between them, and then divide the product by two : the quotient will be the ana (Bk. IV. Th. x). EXAMPLES. 1 . Required the area of the trapezoid A BCD, having given E B ,45-321.51 feet, DC = 214.24 fret, and OE = 171.16 feei Operation. We first find the sum of the sides, and then multiply it by the perpendicular height, after which, we divide the product by 2, for the area. 321.51+214.24 = 535.75- sum of parallel sides. Then, 535.75x171.16 = 91698.97 91698.97 2 i =the area. 2 What is the area of a trapezoid, the parallel sides of which, are 12.41 and 8.22 chains and the perpendicular dis- tance between them 5.15 chains ? Ans. 5 A. 1 R. 9.956 P. 3. Required the area of a trapezoid whose parallel sides OF GEOMETRY. 221 Mensuration of Surfaces. are 25 feet 6 inches, and 18 feet 9 inches, and the perpen- dicular distance between them 10 feet and 5 inches. Ans. 230 sq. ft. 5' 7". 4. Required the area of a trapezoid whose parallel sides are 20.5 and 12.25, and the perpendicular distance between them 10.75 yards. Ans. 176.03125 sq. yds. 5. What is the area of a trapezoid whose parallel sides are 7.50 chains, and 12.25 chains, and the perpendicular height 15.40 chains? Arts. 15 A. R. 33.2 P PROBLEM VII. To find the area of a quadrilateral. Measure the four sides of the quadrilateral, and also one of the diagonals : the quadrilateral will thus be divided into two trian- gles, in both of which all the sides will be known. Then, find the areas of the triangles separately, and their sum will be the area of the quadrilateral. EXAMPLES. 1. Suppose that we have meas- ured the sides and diagonal A C, of the quadrilateral ABCD, and found A A 5 = 40.05 chains; CD =29.87 chains, 50=26.27 chains AD = 37.01 chains, and A C = 55 chains : required the area of the quadrilateral Ans. 101 A. 1 R 15 P 19* 122 a PPLICATIONS Mensi rat of Surfaces Remark. — Instead of measuring the four sides of the quadrilateral, we may let fall the perpendicu- lars Bb, Dg, on the diagonal AC. The area of the triangles may then be determined by measuring these perpendiculars and diagonal AC. The pendiculars arv,Dg — 18.95 chains, and BL=z]7.92 chains. 2. Required the area of a quadrilateral whose diagonal ifi HO. 5, and two perpendiculars 24.5, and 30.1 feet. Arts. 2197.65 sq.ft. 3. What is the area of a quadrilateral whose diagonal is 108 feet 6 inches, and the perpendiculars 56 feet 3 inches, and 60 feet 9 inches ? Ans. 6347 sq.ft. 3'. 4. How many square yards of paving in a quadrilateral whose diagonal is 65 feet, and the two perpendiculars 28, and 33| ^et ? Ans. 222 T V sq. yds. 5. Required the area of a quadrilateral whose diagonal is 42 feet, and the two perpendiculars 18, and 16 feet. Ans. 714 sq. ft. 6. What is the area of a quadrilateral in which the diago- nal is 320.75 chains, and the two perpendiculars 69.73 chains, and 130.27 chains ? Ans. 3207 A. 2 R. PROBLEM VIII. To find the area of a regular polygon. Multiply half the perimeter of the figure by the perpendicular Lei fall from the centre on one of the sides, and *lte product will he the area (Bk. IV. Th. xxvi) OP GEOMETRY. 223 Mensuration of Surfaces. EXAMPLES. 1. Required the area of the regular pentagon ABCDE, each of whose -® 1 sides AB, BC, &c, is 25 feet, and the perpendicular OP, 17.2 feet. We first multiply one side by the number of sides and divide the product by 2 : this gives half the perimeter which we multiply by the perpen- dicular for the area. 25x5 : 62.5 = half the perim- eter. Then, 625x17.2 = 1075 sq. /*.=the area. 2. The side of a regular pentagon is 20 yards, and the per- pendicular from the centre on one of the sides 13,76382 ; re- quired the area. A'ns. 688.191 sq. yds. 3. The side of a regular hexagon is 14, and the perpen- dicular from the centre on one of the sides 12.1243556: re- quired the area. Ans. 509.2229352 sq.ft. 4. Required the area of a regular hexagon whose side is 14.6, and perpendicular from the centre 12.64 feet. Ans. 553.632 sq ft. 5. Required the area of a heptagon whose side is 19,3" ar.d perpendicular 20 feet. Ans. 1356.6 sq. ft. The following table shows the areas of the ten regular ;24 A PPLICATiONS Mensuration of Surfaces polygons when the side of each is equal to 1 : it also shows the length of the radius of the inscribed circle. Number of 6ides. Names. Areas. Radius of inscribed] circle. 3 Triangle, 0.4330127 0.2886751 4 Square, 1.0000000 0.5000000 5 Pentagon, 1.7204774 0.6881910 6 Hexagon, 2.5980762 0.8660254 7 Heptagon, 3.6339124 1.0382617 8 Octagon, 4.8284271 1.2071068 9 Nonagon, 6.1818242 1.3737387 10 Decagon, 7.6942088 1.5388418 11 Undecagon, 9.3656404 1.2028437 12 Dodecagon, 11.1961524 1.8660254 Now, since the areas of similar polygons are to each othei as the squares described on their homologous sides (Bk. IV Th. xx), we have l 3 : tabular area : : any side squared : area. Hence, to find the area of a regular polygon, we have the following RULE. I Square the side of the polygon. II. Multiply the square so found, by the tabular area set oppo- site the polygon of the same number of sides, and the product will be the irea. EXAMPLES. 1 . What is the area of a regular hexagon whose side is 20 20 =400 and tabular area =2,5980762. Hence, 2.5980762 x 400 = 1039.23048 = the area. OFGEOMETRV. 225 Mensuration of Surfaces 2. What is the area of a pentagon whose side is 25 ? Ans. 1075.298375. 3. What is the area of a heptagon whose side is 30 feet Ans. 3270.52116 4. What is the area of an octagon whose side is 10 feet \ Ans. 482.84271 sq. ft b. The side of a nonagon is 50 : what is its area ? Ans. 15454.5605 6. The side of an undecagon is 20 : what is its area ? Ans. 3746.25616. 7. The side of a dodecagon is 40 : what is its area ? Ans. 17913.84384 PROBLEM IX. To find the area of a long and irregular figure, bounded on one side by a straight line. RULE. I. Divide the right line or base into any number of equal parts, and measure the breadth of the figure at the points of di vision, and also at the extremities of the base. II. Add together the intermediate breadths, and half the sum of the extreme ones- Ill. Multiply this sum by the base line, and divide the product bif ite number of equal parts of the base. EXAMPLES. 1. The breadths of an irregu- a lar figure, at five equidistant a r \^^\ \ \ places, A, B, C, D, and E, be- j_. i — ^ — -4 ^ ing 8.20 chains, 7.40 chains. 220 APPLICATIONS Mensuration of Surfaces. 9.20 chains, 10.20 chains, and 8.60 chains, and the whole length 40 chains : required the area. 8.20 35.20 8.60 40 2 )16\80 4)1408.00 8.40 mean of the extremes. 352.00 square chains. 7.40 9.20 1O.20 35.20 the sum. Ans. 35 A. 32 P. 2. The length of an irregular piece of land being 21 chains and the breadths, at six equidistant points, being 4.35 chains 5.15 chains, 3.55 chains, 4.12 chains, 5.02 chains, and 6.10 chains : required the area. Ans. 9 A. 2 R. 30 P. 3. The length of an irregular figure is 84 yards, and the breadths at six equidistant places are 17.4 ; 20.6 ; 14.2 ; 16.5; 20.1 ; and 24.4 : what is the area ? Ans. 1550.64 sq. yds. 4. The length of an irregular field is 39 rods, and its breadths at five equidistant places, are 4.8; 5.2; 4.1; 7.3, and 7.2 rods : what is its area ? Ans. 220.35 sq. rods. 5. The length of an irregular field is 50 yards, and its breadths at seven equidistant points, are 5.5 ; 6.2 ; 7.3 ; 6 ; 7.5 ; 7 ; and 8.8 yards : what is its area ? Ans. 342.916 sq. yds. 6. The length of an irregular figure being 37.6, and the breadths at nine equidistant places, 0; 4.4 ; 6.5 ; 7,6 ; 5.4 ; 8; 5.2 ; 6.5 ; and 6.1 : what is the area? Ans. 219.255. PROBLEM X. To find the circumference of a circle when the diameter is known. OF GEOMETRY. 227 Mensuration of Surfaces. __ RULE Multiply the diameter by 3.1416, and the product will be th& circumference. EXAMPLES. 1. What is the circumference of a circle whose diameter is 17? We simply multiply the number 3.141G by the diam- eter and the product is the circumference Operation. 3.1416x17 = 53.4072, which is the circumference. 2. What is the circumference of a circle whose diameter ie 40 feet? Ans. 125.664 ft. 3. What is the circumference of a circle whose diameter is 12 feet? Ans. 37.6992 ft. 4. What is the circumference of a circle whose diameter is 22 yards? Ans. 69.1152 yds. 5. What is the circumference of the earth — the mean diam- eter being about 7921 miles? Ans. 24884.6136 mi. PROBLEM XI. To find the diameter of a circle when the circumference is known. RULE. Divide the circumference by the number 3.1416 and the quo- tient unll be the diameter. EXAMPLES. 1. The circumference of a circle is 69.1152 yards: what is the diameter' 228 APPLICATIONS Mensuration of Surfaces. We simply divide the cir- cumference fey 3.1416, and the quotient 22 is the diam- eter sought. Operation. 3.1416)691152(22 62832 62832 62832 2. What is the diameter of a circle whose circumference is 11652.1944 feet ? Ans. 3709. 3. What is the diameter of a circle whose circumference is 6850? Ans. 2180.4176. 4. What is the diameter of a circle whose circumference is 50? Ans. 15.915. 5. If the circumference of a circle is 25000.8528, what is the diameter ? Ans. 7958. PROBLEM XII. To find the length of a circular arc, when the number ot degrees which it contains, and the radius of the circle are known. RULE. Multiply the number of degrees by the decimal .01745, and the product arising by the radius of the circle. EXAMPLES. 1 . What is the length of an arc of 30 degrees, in a circle whose radius is 9 feet. We merely multiply the Operation. given decimal by the number .01 745 x 30 x 9 = 4.71 1 5, of degrees, an 1 by the radius, which is the length of the arc Remark. — When the arc contains degrees and minutes, re- duce the minutes to the decimals of a degree, which is done by dividing them by 60. OF GEOMETRY. 229 Mensuration of Surfaces 2. What is the length of an arc containing 12° 10' oj \2i° the diameter of the circle being 20 yards ? A as. 2.1231 3. "What is the length of an arc of 10° 15' or 10j°, in a circle "whose diameter is 68? Ans. 6.0813. PROBLEM XIII. To find the length of the arc of a circle when the chord dad radius are given. RULE. 1. Find the chord of half the arc. 1 T From eight times the chord of half the arc, subtract the chord of the whole arc, and divide the remainder by 3, and the quotient will be the length of the arc, nearly. EXAMPLES. 1. The chord AB = 30 feet, and the radius AC =20 feet: what is the length of the arc ADB ? First draw CD perpendicular to the chord AB : it will bisect the chord at P, and the arc of the chord at D. Then AP =15 feet. Hence, AC 2 -AP*=CP l : that is, 400—225 = 175 and V 175=1 3.228 = CP Then OD-CP=20-13.228 = 6.772=Z>P. Again, hencej Then, ^D = yjlP 2 +PD 2 = y'225 + 45.859984 AD = 16.4578 = chord of the half arc. l6 -± 578 * 8 - 3 ° = 33.8874 = arc ADB. 20 230 APPLICATIONS Mensuration of Surfac 2 What is the length of an arc the chord of which is 24 feet, and the radius of the circle 20 feet ? Ans. 25.7309 jt. 3. The chord of an arc is 16 and the diameter of the circle 20 : what is the length of the arc ? Ans. 18.5178. 4. The chord of an arc is 50, and the chord of half the arc is 27 : what is the length of the arc 1 Ans. 55 $. PROBLEM XiV. To find the area of a circle when the diameter and circum- ference are both known. RULE. Multiply the circumference by half the radius and the product will be the area (Bk. IV. Th. xxvii). EXAMPLES. 1. What is the area of a circle whose diameter is 10, and circumference 31.416 ? If the diameter be 10, the radius is 5, and half the ra- dius is 2| : hence, the cir- cumference multiplied by 2^ gives the area. 2. Find the area of a circle whose diameter is 7; and cir- cumference 21.9912 yards. Ans. 38.4846 yds. 3. How many square yards in a circle whose diameter is 3£ feet, and circumference 10.9956. Ans. 1.069016. 4. What is the area of a circle whose diameter is 100, and circumference 314.16? Ans. 7854 Operation. 31.416 x 2^=78.54; which is the area. OF GEOMETRY. 23 1 Mensuration of Surfaces 5. What is the area of a circle whose diameter is I , and circumference 3.1416. Ans. 0.7854. 6. What is the area of a circle whose diameter is 40, and circumference 131.9472 ? Ans. 1319.472. PROBLEM XV. To find the area of a circle when the diameter only Is known. RULE. Square the diameter, and then multiply by the decimal .7854 EXAMPLES. What is the area of a circle whose diameter is 5 ? We square the diameter, which gives us 25, and we then multiply this number and the decimal .7854 to- gether. Operation. .7854 5j=_25 39270 15708 area= 19.6350' 2. What is the area of a circle whose diameter is 7 ? Ans. 38.4846. 3. What is the area of a circle whose diameter is 4,5 ? Ans. 15.90435. 4. What is the number of square yards in a circle whose diameter is 1 1 yards ? Ans. 1.069016. 5. What is the area of a circle whose diameter is 8.75 feet? Ans. 60.1322 sq.ft. PROBLEM XVI. To find the area of a circle when the circumference only is known. 232 APPLICATIONS Mensuration of Surface3 RULE. Multiply tlie square of the circumference by the decimal .07958, and the product will be the area very nearly EXAMPLES. 1. What is the area of a circle whose circumference ie 3.1416? We first square the cir- cumference, and then multi- ply by the decimal .07958. Operation. )65 ,07958 3.1416 2 =9,86965056 area =.7854 + 2. What is the area of a circle whose circumference is 9H Ans. 659.00198. 3. Suppose a wheel turns twice in tracking 16-2 feet, and that it turns just 200 times in going round a circular bowling- green : what is the area in acres, roods, and perches ? Ans. 4 A. 3 R. 35.8 T 4. How many square feet are there in a circle whose cir cumference is 10.9956 yards? Ans. 86.5933. 5. How many perches are there in a circle whose circuin ference is 7 miles ? Ans. 399300.608. PROBLEM XVII. Having given a circle, to find a square which shall have an squal area. RULE. I. The, diameter X. 8862 = side of an equivalent square II. TJw circumference X .2821= side of an equivalent square OF GEOMETRY. 233 Mensuration of Surfaces EXAMPLES. 1. The diameter of a circle is 100 : what is the side of a square of equal area ? Ans. 88.62. 2. The diameter of a circular fishpond is 20 feet, what would be the side of a square fishpond of an equal area? Ans. 17.724 ft. 3. A man has a circular meadow of which the diameter is 875 yards, and wishes to exchange it for a square one of equal size : what must be the side of the square ? Ans. 775.425. 4. The circumference of a circle is 200 : what is the side of a square of an equal area 1 Ans. 56.42. 5. The circumference of a round fishpond is 400 yards : what is the side of a square pond of equal area 1 Ans. 112.84. 6. The circumference of a circular bowling-green is 412 yards : what is the side of a square one of equal area ? Ans. 116.2252 yds. 7. The circumference of a circular walk is 625 : what is the side of a square containing the same area ? Ans. 176.3125. PROBLEM XVIII. Having given the diameter or circumference of a circle, to find the side of the inscribed square. RULE. I. The diameter X .7071 =zside of the inscribed square. II. The circumference X .2251 —side of the inscribed square. 20* 231 APPLICATIONS Mensuration of Surfaces, EXAMPLES. 1. Tlio diameter AB of a circle is 400 : what is the value of AC, the side of the inscribed square 1 Here, .7071 x 400=282.8400= AC. 2. The diameter of a circle is 412 feet: what is the side of the inscribed square? Ans. 291.3252 ft. 3. If the diameter of a circle be 600 what is the side of the inscribed square ? Ans 424.26. 4. The circumference of a circle is 312 feet: what is the side of the inscribed square ? Ans. 70.2312 ft. 5. The circumference of a circle is 819 yards : what is the side of the inscribed square 1 Ans. 184.3569 yds. 6. The circumference of a circle is 715 : what is the side of the inscribed square ? Ans. J 60.9465. 7. The circumference of a circular walk is 625 : what is ihe side of an inscribed square 1 Ans. 140.6875. PROBLEM XIX To find the area of a circular sector. RULE. I. Find the length of the arc by Problem XII. II. Multiply the arc by one half the radius, and the product will be the area OF GEOMETRY. 23 s Mensuration of Surfac EXAMPLES. 1. What is the area of the circular sector ACB, the arc AB containing 18°, and the radius CA being equal to 3 feet. First, .0T1745X IS x3 = . '94230 = length AB. Then, ,94230x1 J=l. 4 1345= area 2. What is the area of a sector of a circle in which the ra- dius is 20 and the arc one of 22 degrees ? Ans. 76.7800. 3. Required the area of a sector whose radius is 25 and the arc of 147° 29'. Ans, 804.2448. 4. Required the area of a semicircle in which the radius is 13. Ans. 265.4143. 5. What is the area of a circular sector when the length of the arc is 650 feet and the radius 325 ? Ans. 105625 sq. ft. PROBLEM XX. To find the area of a segment of a circle. RULE. I. Find the area of the sector having the same arc with ihs. segment, by the last Problem. II. Find the area of the triangle formed by the chord of the segment and the two radii through its extremities. Ill If the segment is greater than the semicircle, add the two areas together; but if it is less, subtract them, and the result in eithei case, will be the area required 236 A PPLICATIONS Mensuration of Surf EXAMPLES. 1. What is the area of the seg- ment ADB, the chord AB = 24 feet and CA =20 feet. First, CP=^/CA 9 - AP 4 = ^400 — 144 = li Then, PZ>=OD-OP = 20 — 16 = 4. And, AD=^/AP i +PD i =zy/T44-{- 16 = 12,04911 then, arc ^B = 12 ' 6491 ' X8 - 8 i =8S ,7309. Arc ADB =25,7309 half radius = 10 area sector AD£C = 257,3090 area 04£=192 AP= CP= area CAB = 1~92 65,309 = area of segment ADB 2. Find the area of the segment AFB\ knowing the following lines, viz: .45 = 20.5; FP= 17.17; AF —20; FG=ll-5; and 04 = 11.64. . .__ ^Gx8-i4F 11.5x8-20 o Arc AGF= = =24 : o o and sector AGFBC=24x 11.64=279.36 : but CP—FP— 40=17.17— 11.64=5.53: ,^„ .4£xCP 20.5x5.53 Then, area ACB= = = 56.6825 O * GEOMETRY 237 Mensuration of Surfaces. Then, area of sector ^4F5C = 279.36 do. of triangle ABC — 56.6825 gives area of segment AFB = 336.0425 3 What is the area of a segment; the radius of the circle being 10 and the chord of the arc 12 yards ? Ans. 16.324 sq. yds. 4. Required the area of the segment of a circle whose chord is 16, and the diameter oi the circle 20. Ans. 44.5903. 5. What is the area of a segment whose arc is a quadrant, the diameter of the circle being 18 ? Ans. 63.6174. 6. The diameter of a circle is 100, and the chord of the seoment 60 : what is the area of the segment ? A /is. 408, nearly. PROBLEM XXI. To find the area of an ellipse. Multiply the two axes together, and their product by the decimal ,7854, and the result will be the required area. EXAMPLES. 1. Required the area of an ellipse, whose transverse axis AB = !0 feet, and the conjugate axis DE = 50 feet. ^Bx£E = 70x50 = 3500: Then, ,7854x3500=2748.9 = area. 2. Required the area of an ellipse whose axes are 24 and 1 \ Ans. 339.2928. •238 APPLICATIONS Mensuration of Surfaces 3. What is the area of an ellipse whose axes are 80 and CO ? Ans. 3769.92. 4. What is the area of an ellipse whose axes are 50 and 45? Ans. 1767.15. PROBLEM XXII. To find tlie area of a circular ring : that is, the area in- cluded between the circumferences of two circles, having a common centre. RULE. I. Square the diameter of each ring, and subtract *he square of the less from that of the greater. II. Multiply the difference of the squares by thb d ximai 7854, and the product will be the area. EXAMPLES. 1. In the concentric circles having the common centre C, w T e have A! AS =10 yds., and DE = 6 yards : what is the area of the space in- cluded between them ? BA* = 10 2 = \00 DE*= ?= 36 Difference = 64 Then, 63 X. 7854 = 50.2656 =area. 2. What is the area of the ring when the diameters of the circle aie 20 and 10 1 Ans. 235.62. OF GEOMETRY. 2375: then, 9675- 3=3225 which is the solidity expressed in solid feet. 2. Required the solidity of a square pyramid, each side of its base being 30 and its altitude 25. Ans. 7500 solid ft. 3. How many solid yards are there in a triangular pyramid whose altitude is 90 feet, and each side of its base 3 yards? Ans. 38.97117. 4. How many solid feet in a triangular pyramid the altitude ;)f which is 14 feet 6 inches, and the three sides of its base 5, 6 and 7 feet? Ans. 71.0352. 5. What is the solidity of a regular pentagonal pyramid, its altitude being 12 feet, and each side of its base 2 feet ! Ans 27-527G solid ft. 21* 2iC) APPLICATIONS Mensuration of Solids 6 How many solid feet in a regular hexagonal pyramid whose altitude is G.4 feet, and each side of the base 6 inches' .4ns. 1.38504. 7. How many solid feet are contained in a hexagonal pyra- mid the height of which is 45 feet, and each side of the base 10 feet? . Ans. 3897.1113. 8. The spire of a church is an octagonal pyramid, each sido of the base being 5 feet 10 inches, and its perpendicular height 45 feet. Within is a cavity, or hollow part, each side of the base being 4 feet 11 inches, and its perpendiculai height 41 feet: how many yards of stone does the spire contain 9 Ans. 32.197353 PROBLEM VI. To imd the solidity of the frustum of a pyramid. RULE. Add together the areas of the tico bases of the frustum and a geometrical mean proportional between them ; and then multi- ply the sum by the altitude, and take one-third the product for the solidity. EXAMPLES. 1. What is the solidity of the frus- tum of a pentagonal pyramid the area of the lower base being 16 and of the upper base 9 square r eet, the altitude being 7 feet ? OF GEOMETRY. 24T Mensuration of Solida. First, 16x9 = 144: then, -/l 44= 12, the mean Then, area of lower base =16 area of upper base = 9 mean of bases* =12 _ 37 height 7 3) 209 solidity = 86^ solid ft. 2. What is the number of solid feet in a piece of timbei whose bases are squares, each side of the lower base being 15 inches, and each side of the upper base being 6 inches, the length being 24 feet ? Ans. 19.5. 3. Required the solidity of a regular pentagonal frustum, whose altitude is 5 feet, each side of the lower base 18 mchcs, and each side of the upper base 6 inches. Ans. 9.31925 solid ft. 4. What is the contents of a regular hexagonal frustum, whose height is 6 feet, the side of the greater end 18 inches, and of the less end 12 inches ? Ans. 24.681724 cubic ft. 5. How many cubic feet in a square piece of timber, the areas of the two ends being 504 and 372 inches, and its length 31| feet? Ans. 95.447. 6. What is the solidity of a squared piece of timber, its length being 18 feet, each side of the greater base 18 inches, and each side of the smaller 12 inches ? Ans. 28.5 cubic ft. 7. What is the solidity of the frustum of a regular hexago- nal pyramid, the side of the greater end being 3 feet, that of the less 2 feet, and the height 12 feet? Ans. 197.453776 solid ft !48 APPLICATIONS Mensuration of So lido. MEASURES OF THE THREE ROUND BODIES PROBLEM I To find the surface of a cylinder. Multiply the circumference of the base by the altitude, and the product will be the convex surface ; and to this, add the areas of the two bases, when the entire surface is required (Bk. VI. Th. ii) EXAMPLES. 1. What is the entire surface of the cylinder in which AB, the diameter of the base, is 12 feet, and the altitude EF 30 feet ? First, to find the circumference of the base, (Prob. X. page 180) : we have 3.1416 x 12 = 37.6992 = circumference of the base. Then, 37.6992 x 30= 1 130.9760 = convex surface. Also, 12 2 =144: and 144 x .7854= 1 13.0976 = area of the base. Then. convex surface =1130.9760 lower base 1 13.0976 upper base 1 13.0976 Entire area =1357.1712 2. What is the convex curface of a cylinder, the diameter of whose base is 20, and the altitude 50 feet ? Arts 3141.6*7./*. OF GEOMETRY 249 Mensuration of the Round Bodies. 3. Required the entire surface of a cylinder, whose altitude is 20 feel and the diameter of the base 2 feet. Ans. 131.9472 ft. 4. What is the convex surface of a cylinder, the diametei ol whose base is 30 inches, and altitude 5 feet? Ans. 5654.88 sq. in. 5. Required the convex surface of a cylinder, whose alti- tude is 14 feet, and the circumference of the base 8 feet 4 inches. Ans. 116.6666, &c, sq. ft. PROBLEM n. To find the solidity of a cylinder. RULE. Multiply the area of the base by the altitude, and the prodw will be the solidity. EXAMPLES. 1. What is the solidity of a cylinder, the diameter of whose base is 40 feet, and altitude EF, 25 feet ? First, to find the area of the base, we have (Prob. xv. page 231). 40 2 =1600: then, 1 600 x .7854=1256.64. =area of the base. Then, 1256.64 x 25=31416 solid feet, which is the solidity. 2. What is the solidity of a cylinder, the diameter of whose base is 30 feet, and altitude 50 feet ? Ans 35343 cubic ft. 250 APPLICATIONS Mensuration of the Round Bodie 6. What is the solidity ol a cylinder whose height is 5 feet, and the diameter of the end 2 feet? Ans. 15.708 solid ft. 4. What is the solidity of a cylinder whose height is 20 feet, and the circumference of the base 20 feet ? Ans. 636.64 cubic ft 5. The circumference of the base of a cylinder is 20 feet, and the altitude 19.318 feet: what is the solidity? Ans. 614.93 cubic ft. 6. What is the solidity of a cylinder whose altitude ie» 12 feet, and the diameter of its base 15 feet ? Ans. 2120.58 cubic ft. 7. Required the solidity of a cylinder whose altitude is 20 feet, and the circumference of whose base is 5 feet 6 inches ? Ans. 48.1459 cubic ft. 8. What is the solidity of a cylinder, the circumference of whose base is 38 feet, and altitude 25 feet ? Ans. 2872.838 cubic ft. 9. What is the solidity of a cylinder, the circumference of whose base is 40 feet, and altitude 30 feet ? 10. The diameter of the base of a cylinder is 84 yards, and the altitude 21 feet: how many solid or cubic yards does it contain ? Ans. 38792.4768. PROBLEM III. To find the surface of a cone. RULE. Multiply the circumference of the base by the slant height, and divide the product by 2 ; the quotient will be the convex surface, to which add the area of the base, when the entire surface is require i (Bk. VI. Th viii) OF GEOMETRY. 251 Mensuration of the Round Bodies. EXAMPLES. 1. What is the convex surface of the cone whose vertex is C, the diameter AD, of its base being 8^ feet, and the side CA, 50 feet. First, 3.1416 X 81=26.7036 = circumference of base Then 2 -^L 6 2^ = G«7. 5 9 = convex surface. 2 2. Required the entire surface of a cone whose side is 36 and the diameter of its base 18 feet. Ans. 1272.348 sq. ft. 3. The diameter of the base is 3 feet, and the slant height J 5 feet : what is the convex surface of the cone ? Ans. 70.686 sq. ft. 4. The diameter of the base of a cone is 4,5 feet, and* the slant height 20 feet : what is the entire surface ? Ans. 157.27635 sq. ft. 5. The circumference of the base of a cone is 10. " 7 5. and the slant height is 18.25 : what is the entire surface ? Ans. 107.29021 sq. ft PROBLEM IV. To find the solidity of a cone. RULE. Multiply the area of the base by the altitude; and divide the pro- duct by 3, the quotient will be the solidity (Bk. VI. Th. ^viii). 252 APPLICATIONS Mensuration of the Round Bodies. EXAMPLES. 1. What is the solidity of a cone, the area of whose base is 380' square feet, and altitude CB, 48 feet ? We simply multiply the area of the base by the alti- tude, and then divide the pro- duct by 3. Operation. 380 48 3040 1520 3)18240 area = 6080 2. Required the solidity of a cone whose altitude is 27 feet, and the diameter of the base 1 feet. Ans. 706.86 cubic ft. 3. Required the solidity of a cone whose altitude is 1 0^ fpet, and the circumference of its base 9 feet ? Ans. 22.5609 cubic fl. 4. What is the solidity of a cone, the diameter of whose base is 18 inches, and altitude 15 feet? Ans. 8.83575 cubic ft. 5 The circumference of the base of a cone is 40 feet, and the altitude 50 feet: what is the solidity? Ans. 2122.1333 solid /?. OF GEOMETRY. 253 Mensuration of the Round Bodies PROBLEM V. To Jind the surface of the frustum of a cone RULE. Add together the circumferences of the twe bases; and multi- ply the sum by half the slant height of the frustum ; the product will be the convex surface, to which add the areas of the bases- when the entire surface is required (Bk. VI. Th. ix). EXAMPLES. . I. What is the convex surface of the frustum of a cone, of which the slant height is 12| feet, and the circumfe- rences of the bases 8,4 and 6 feet. half side Operation. 8.4 6 14~4 6.25 We merely take the sum of the circumferences of the bases, and multiply by half the slant height, or side. area = 90 sq. ft. 2. What is the entire surface of the frustum of a cone, the side being 16 feet, and the radii of the bases 2 and 3 feet ? Ans. 292.1688 sq.ft. 3. What is the convex surface of the frustum of a cone, the circumference of the greater base being 30 feet, and of die less 10 feet; the slant height being 20 feet? Ans. 400 sq. ft. 4. Required the entire surface of the frustum of a cone whose slant height is 20 feet, and the diameters of the bases 8 and 4 feet Ans. 439.824 sq. ft 22 254 APPLICATIONS Mensuration of the Round B o d i e i PROBLEM VI. To find the solidity of the frustum of a cone RULE. I. Add together the areas of the two ends and a geometneal mean between them. II? Multiply this sum by one-third of the altitude and the product will be the solidity. EXAMPLES. 1. How many cubic feet in the frus- tum of a cone whose altitude is 26 feet, M and the diameters of the bases 22 and II l 1 8 feet 1 m//l/li First, 22 2 X.7854:=380.134 = area of Uk lower base : and 18 2 X .7854 = 254.47 = area of upper base. Then, ^380.134 X 254.47 = 31 1.0 18= mean. 26 Then, (380.134 + 254,47 + 31 1.018) x- =8195.39 which is life solidity. 2. How many cubic feet in a piece of round timber the di- ameter of the greater end being 18 inches, and that of the less 9 inches, and the length 14.25 feet ? Ans. 14.68943. 3. What is the solidity of a frustum, the altitude being 1 8. the diameter of the lower base 8, and of the upper 4 ? Ans. 527.7888. 4. If a cask, which is composed of two equal conic frus- tums joined together at their larger bases, have its bung di- ameter 28 inches, the head diameter 20 inches, and the length OF GEOMETRY. 255 Mensuration of the Round Bodies. 40 inches, how many gallons of wine will it contain, there being 231 cubic inches in a gallon ? Ans. 79.0613. PROBLEM Vil. To find the surface of a sphere. RULE. Multiply the circumference of a great circle by the diameter, and the product will be the surface (Bk. VI. Th. xxiii). EXAMPLES. 1. What is the surface of the sphere whose centre is C, the diameter being 7 feet ? Ans. 153.9384 sq. ft. 2. What is the surface of a sphere whose diameter is 24 ? Ans. 1809.5616. 3. Required the surface of a sphere whose diameter is 7921 miles. Ans. 19711 1024 sq. miles. 4. What is the surface of a sphere the circumference o( whose great circle is 78.54? Ans. 1963.5. 5. What is the surface of a sphere whose diameter is 1 3 feet ? Ans. 5.58506 sq. ft. PROBLEM VIII. To find the convex surface of a spherical zone. RULE. Multiply the height of the zone by the circumference of a great circle of the sphere, and the product will be the convex surface (Bk. VI. Th. xxiv) 256 APPLICATIONS Mensuration of the Round Bodie EXAMPLES. 1. What is the convex surface of the zone ABD, the height BE being 9 inches, and the diameter of the sphere 42 inches ? First, 42x3.1416=131 .9472 = circumference, height = 9 surface =1187.5248 square inches. 2. The diameter of a sphere is 12| feet : what will be »he surface of a zone whose altitude is 2 feet ? Ans. 78.54 sq. ft. 3. The diameter of a sphere is 21 inches : what is the sur- face of a zone whose height is \\ inches ? Ans. 296.8812 sq. in. 4. The diameter of a sphere is 25 feet and the height of the zone 4 feet : what is the surface of the zone ? Ans. 314.16 sq. ft. 5. The diameter of a sphere is 9, and the height of a zone 3 feet : what is the surface oi the zone ? Ans. 84.8232. PROBLEM IX. To find the solidity of a sphere. rule :. Multiply the surface by one-third of the radius and the product will be the solidity (Bk. VI. Th. xxv)- OF GEO AI ETRY. 251 Mensuration of the Round B o 1 i 3 s . EXAMFLES 1. What is the solidity of a sphere whose diameter is 12 feet? First, 3.1416x12—37.6992 = circumference of sphere. diameter = 12 surface =452.3904 one-third radius = 2 Solidity =904.7808 cubic feet. 2. The diameter of a sphere is 7957.8: what is its solidity 7 Ans. 263863122758.4778. 3. The diameter of a sphere is 24 yards : what is its solid contents ? Ans. 7238.2464 cubic yds. 4. The diameter of a sphere is 8 : what is its solidity? Ans. 268.0832. a The diameter of a sphere is 16 : what is its solidity ? Ans. 2144.6656 RULE II. Cube the diameter and multiply the number thus found, by t/u decimal .5236, and the product will be the solidity. EXAMPLES. f . What is the solidity of a sphere whose diameter is 20 ? Ans. 4188.8. 2 What is the solidity of a sphere whose diameter is 6? Ans. 113.0976. 3. What is the solidity of a sphere whose diameter is JO" 22* Ans 523.6 258 APPLICATIONS Mensuration of t Round 13 o (1 i o s . PROBLEM X. To find the solidity of a spherical segment with one base.. RULE. I . T. three times the square of the radius of the base, add the square of the height. II. Multiply this sum by the height, and the product by the decimal .5236, the result will be the solidity of the segment. EXAMPLES. [ . What is the solidity of the seg- ment ABD, the height BE being 4 feet, and the diameter AD of the base being 14 feet ? First, 163: 7 2 x-3 + 4 2 = 147+16 Then, 163 x 4 x. 5236 = 341.3872 solid feet, which is the solidity of the segment. 2. What is the solidity of the segment of a sphere whose neight is 4, and the radius of its base 8 ? Ans. 435.6352. 3. What is the solidity of a spherical segment, the diam- eter of its base being 17.23368, and its height 4.5 ? Ans. 572.5566. 4. What is the solidity of a spherical segment, the diam- eter of the sphere being 8, and the height of the segment 2 feet ? Ans. 41.888 cubic Jt. 5 What is the solidity of a segment, when the diame'er of the sphere is 20, and the altitude of the segment 9 feet ? Ans. 1781.2872 cubic ft OF GEOMETRY 259 Mensuration of the Spheioid OF THE SPHEROID. A spheroid is a solid described by the revolution of an ellipse about either of its axes. If an ellipse A CBD, be re- volved about the transverse or onger axis AB, the solid de- scribed is called a prolate spheroid : and if it be revolved about the shorter axis CD, the solid described is called an oblate spheroid. The earth is an oblate spheroid, the axis about which il revolves being about 34 miles shorter than the diameter per- pendicular to it. PROBLEM XI. To find the solidity of an ellipsoul RULE. Multiply the fixed axis by the square of the revolving axu,, and the product by the decimal .5236, the result will be the re- quired solidity. EXAMPLES. 1. Tn the prolate spheroid ACBD, the transverse axis AB — 90, and the revolving axis CD = 70 feet: what is die solidity ? D Here, AB- 90 feet: CD* = 70 2 = 4900 : hence ABx CD 2 X. 5236 = 90 x 4900 x. 5236 = 230907.6 cubic feet., which is the solidity. 2G0 A PPLICATIONS Mensuration of Cylindrical Kings 2. What is the solidity of a prolate spheriod, whoso fixod axis is 100, and revolving axis 6 feet ? Ans. 1 88 1.96. 3. What is the solidity of an oblate spheroid, whose fix*>d axis is 60, and revolving axis 100 ? Ans. 314160 4. What is the solidity of a prolate spheroid, whose d*e& are 40 and 50? Ans. 41888. 5. What is the solidity of an oblate spheroid, whose axes are 20 and 10? Ans. 2094.4. 6. What is the solidity of a prolate spheroid, whose axes are 55 and 33 ? Ans. 31361.022. 7. What is the solidity of an oblate spheroid, whose axes are 85 and 75 ? Ans. OF CYLINDRICAL RINGS A cylindrical ring is formed by bending a cylinder until the two ends meet each other. Thus, if a cylinder be bent round until the axis takes the position mow, a solid will be formed, which is called a cylin- drical ring. The line AB is called the outer, and cd the inner diameter. PROBLEM XII. To rind the convex surface of a cylindrical ring. RULE. I. To the thickness of the ring add the inner diameter. II. Multiply this sum by the thickness, and the p'oduel 9.8696, the result will be the area. OF GEO IYI E T R Y 26 Mensuration of Cylindrical Rings EXAMPLES. 1 . The thickness A c, of a cylindri- cal ring is 3 inches, and the inner diameter cd, is 1 2 inches : what is (lie convex surface ? Ac+cd — 3 (-12 = 15: 15x3x9.8696 = 444.132 square inches = the surface. 2. The thickness of a c* lindrical ring is 4 inches, and the inner diameter 18 inches : what is the convex surface ? Ans. 868.52 sq. in. 3. The thickness of a cylindrical ring is 2 inches, and the innnr diameter 18 inches ■ what is the convex surface ? Arts. 391.784 sq. tn. PROBLEM XIII. To find the solidity of a cylindrical ring. RULE. I To the thickness of a ring add the inner diameter II. Multiply this sum by the square of half the thickness, and [he product by 9.8696, the result will be the required solidity. EXAMPLES. i . What is the solidity of an anchor ring, whose inner di- ameter is 8 inches, and thickness in metal 3 inches ? 84-3 = 11: then, 11 x (|) 2 X 9.8696=244.2?2G ; which ex- presses the solidity in cubic inches. 2. The inner diameter of a cylindrical ring is 18 inches, and the thickness 4 inches : what is the solidity of the ring ? Ans. 868.5248 cubic inches 262 APPLICATIONS Mensuration of Cylindrical Rings. 3. Required the solidity of a cylindrical riiig whose thick* ness is 2 inches, and inner diameter 12 inches ? Ans. 138.1744 cubic in 4. What is the solidity of a cylindrical ring, whose thick- ness is 4 inches, and inner diameter 16 inches? Ans. 789.568 cubic in. 5. What is the solidity of a cylindrical ring, whose thick- ness is 8 inches, and inner diameter 20 inches ? Ans. 6. What is the solidity )f a cylindrical ring whose thick ness is 5 inches, and inner diameter 18 inches ? Am\ A TABLE LOGARITHMS OF NUMBERS From 1 to 10,000 H. Lo g- j N. Log. N. 1 Log. N. 76 Log. i o-oooooo 26 I-4U973 5i 1 -707570 1 -880814 2 o-3oio3o 3 i-43i364 52 i-7i6oo3 77 1-886491 3 o-477»2i j 1. 4471 58 53 1-724276 78 1-892085 4 0-602060 29 i-4'«2398 54 1-732394 79 1-897627 5 0-698970 3o 1-477121 55 1 -i4o363 80 1 -903090 6 0-778151 3i i-49i362 56 1-748188 81 1 -908485 I 0-843098 32 1 -5o5i5o 57 I-755875 82 i- 9 .38i4 0-903090 33 1 -5i85i4 58 1-763428 83 1 -919078 9 0-954243 34 1 -53 1 479 1 • 544068 5 9 1 • 770852 84 1-924279 10 1 • 000000 35 60 1 - 7781 5i 85 1 -929410 ii 1 041393 36 1 -5563o3 61 i-78533o 86 1.934498 12 l -079181 11 1-568202 62 1 -792392 87 1-939519 i3 1 -1 13943 1-579784 63 1 -799341 88 1-944483 ' 14 1-146128 39 1 -591065 64 1 -806181 89 1.949.390 i5 1 • 1 7609 1 40 1 -602060 65 1 -812913 90 1 -954243 16 1 -204120 41 1-612784 66 1 -819544 9 1 1 -959041 17 1 • 23o44o 1-250273 42 1 -623249 67 1-826075 92 1.963788 18 43 1-633468 68 i-8325o9 9 3 1.968483 '9 1-278734 44 1 .643453 69 1-838849 94 1-973128 20 1 -3oio3o 45 1 -6532i3 70 1-845098 95 1.977724 21 1 -322219 46 1-662758 71 1 -85i258 96 1 .982271 22 1-342423 47 1 -672098 72 1-857333 97 1-986772 23 1-361728 43 1-681241 73 1-863323 98 1-991226 24 1 -380211 49 1 -690196 74 1-869232 99 1 .995635 2D 1-397940 5o 1 -698970 75 1-875061 100 2 • 000000 Remark. — In the following table, in the nine right- hand columns of each page, where the first or lead- ing figures change from 9's to O's, points or dots are introduced instead of the O's, to catch the eye, and to indicate that from thence the two figures of the Log- arithm to be taken from the second column, stand in the next line below. z A T i.BLK OF L O (4 A KITH MS F R(J M 1 TO 10,00a N. 100 1 I 2J3l/ 4 |5j6l7|8 9 pq 432 000000 0434 0868 i3oi! 1734 1 2166 2598 3029 3461 38gi 101 432 1 475ii 5i8ij 56og 6o38 6466 6894 7J2 1 ; 7748 8.74 428 103 8600 9026J o45i; 9876. 8 3oo '724 1147 ^70 igg3 *4i5 424 io3 012837 3259 368o 4100, 4521 1 4940 536o 5779 6197 6616 419 104 7o32 745i 7868, 8284 8700 01 16 g532, 9947 •36i •775 416 io5 02 1 1 89 i6o3 2016: 2428 2841 3252 3f64 4075 4486 4896 413 106 53o6 5713 6i25 6533 6o42 735o 7-57; 8164 8571 8978 408 tci o384 o33424 9780' e ig5 # 6oo 38s il 4227 4628 1004 1408 l8l2 ; 2216 2619 3021 404 ro8 5029 543o 583o: 623o 662g 7028 4<>o i '°9 7426 7825! 8223, 8620 9017 0414 3362 981 1 *207 3 7 55i 4148 •602 •998 3 9 6 :j o a4i3g3 1787 2182 2576 68 > 8? 454o 49-32 3 9 3 38 9 in 5323 5714 6io5 6495 Q093 «38o 7275 7664 ! 8o53; 8442 883o m 0218 05J078 9606 3463 •766 n53 i538 1924J 2309 2694 386 n3 3846 423o 46i3 4gg6| 5378 5760 6142 6524 382 U4 6905 7286 7666' 8046 8426 88o5! 9185 c563 2582! 2g58 3333 9942 •320 379 i "J 060698 1075 i452| 1829 2206 3709 7443 4o83 376 Ii6 4438 4832 52o6' 558o 5 9 53 6326 66gg' 7071 78i5 3 7 2 "I ll3 8180 8557 8928, 9298 9668 3352 ••38 •4071 '776 1 1 45 i5i4 36g 071882 325o 2617 2985 3 7 i8 4085! 445 1 4816 5i82 366 ! "j 5547 5912 o543 3 1 44 6276; 6640 7004 7368 773 1 8og4 8457 88 1 g 363 j 12 3 079 181 9904 1 *266 3oo3 386i •626 •987 i347 1707 2067 2426 36o 121 082780 4219 4576 4g34 52gi 5647 6004 35 7 122 636o 6716 707ij 7426 7781 8 1 36 8490 8845 gig8 o552 J071 355 123 9 9 o5 093422 •258 •611 « 9 63 i3i5 1667 2018 2370 2721 35 1 124 3772 4i22 4471 4820 5i6g! 55i8| 5866 62i5 6562 34g 125 6910 10037 1 -2 5 7 7604! 795i 8298 8644 8ggo g335 0681 3119 ••26 346 126 0715 ioSg 1403 H47 2ogi 2434 2777 3462 343 ! 127 128 38o4 4146 4487J 4828 5169 8565 55io 585i 6191 653 1 6871 340 7210 7549 7888j 8227 8go3 9241 9 5 79 0916 3275 •253 338 129 1 10590 0926 1263 1 5 9 9 1934 2270 26o5 2g4o 36og 335 i3o 1 13943 4277 4611 4944 52 7 8 56u 5g43 6276 g586 6608 6940 333 1 3 1 7271 7603 7934 8265 85 9 5 1888 8g26 9256 ogi5 3 1 98 •245 33o l32 120574 0903 I 23 I i56o 2216 2544 2871 3525 328 1 ^ 3 3852 4178 45o4 483o 5i56 548i 58o6 6i3i 6456 6781 325 1 i34 7io5 7429 77 53 8076 83 99 8722 9045 9368 9690 ••12 323 i35 i3o334 o655 0977 1298 i6iq l 9 3g 5i33 2260 258o 2900 3219 640J 321 1 36 353q 3858 4H" 7 4496 4814 545 1 5769 6086 3i8 'M 6721 7037 7354 7671 7987 83o3 8618 8 9 34 924g 9564 3i5 9879 •t 9 4 •5o8 •822 n36 i45o 1763 4885 2076 238g 2702 3i4 139 u3oid 332 7 363 ? 395i 4263 4574 5ig6 55o7 58i8 3i 1 140 146128 6438 6748 9 835 7o58 736 7 7676 79 85 82 9 4 86o3 891 1 3og 1 141 9219 152288 9 5 27 •142 •449 • 7 56 io63i 1370 1676 1982 3o 7 142 25g4 2900 32o5 35io 38r5 4120 4424 4728 5o32 3o5 143 5336 5640 5943 6246 6549 6852 7154 7457 77 5 9 8061 3o3 144 8362 8664 8 9 65 9266 9 56 7 g868 •168 •46g •769 1068 3oi u5 i6i368 1667 1967 2266 2564 2863 3i6i 346o 3 7 58 4o55 299 146 4353 465o 4947 5244 554i 5838 6i34 i.43o 6726 7022 297 147 7317 76i3 7908 0848 8203 I4o4 8792 9086 g38o 9 6 74 9968 293 148 170262 o555 1141 1726 2oig 23n 26o3 2895 293 149 3i86 3478 3769 4060 435i 4641 4932! 5222 7825 8n3 55i2 58o2 291 r5o 1 7609 1 638i 6670 6959 7248 7536 8401 8689 i558 28g i5i 8?77 9264 9^52 9839 •126 •4i3 •6gg # g85 1272 287 1 52 l8l844: 2I2Q 241 5, 2700 2985 5825 3270 3555 383g 4i23 4407 285 1 53 4691 4975 1 5259] 5542 6108 6391 6674 6g56 723g 283 1 54 7521 7803 8084 8366 8647 8g28 92og| g4go 977 1 •05, 281 1 55 190332 06 [2 0892 1171 I45i 1730 2010J 228g 2567 2846 $ 1 56 3i25 34o3 368 11 395g 4237 45i4 4792I 5069 7 556; 7832 5346 5623 1 57 58 99 6176 6453 6729 7oo5j 7281 8107 8382 276 1 58 86d 7 8 9 32 9206 9481 1 Q755J »»2g •3o3; »577 •85o 1 1 24 274 159 IN. 201397 1670 1943 2216 2488 2761) 3o33 33o5 35 77 3848 272 i 1 1 2 . 3 | 4 j 5 6 | 7 8 9 I D. A TABLE OF LOGARITHMS FROM ] [ TO 10,000. 3 NT. 1 60 1. 2 3 | 4 i 5 6 1 1 8 ( 9 3." 204120 4391 4663 4934 52o4 5475 5746 6016 6286, 6556 27 » 161 6826 7096 7365 7634; 79°4 : 8173 8441 8710 8979' 9247 269 162 95i5 9783 ••5 1 •3ig ! «586 *853 1 121 1388 1 654 1921 206 1 63 212188 2454 2720 2986 3252 35i8 3783 4049 43i4 4579 164 4844 5109 53 7 3 5638 5go2 6166 643o 6694 6o5 7 9 585 7221 264 i65 7484 7747 8010 8273 8536 8798 9060 9323 9846 262 166 220108 0370 o63i 0892 n53 1414 1675 1936 4533 2106 2456 261 16a 2716 2976 3236 3496 3755 4oi5 4274 4792 5o5i 259 5309 5568 5826 60S4 6342 6600 6858 71 id 7372 763oj 258 l6«; 7887 8144 8/oo 865 7 8913 9170 9426 9682 99 38 •193 256 170 330449 0704 oq6o 1 2 r 5 1470 1724 1979 2 234 2488 2742 254 171 2996 325o 3304 3757 401 1 4264 45i7 4770 5o23 5276 253 172 5028 5 7 8i 6o33 6285 6537 6789 7041 7292 7544 1 779 5 l 25s I 7 3 8046 8297 8548 8799 9049 9299 955o 9800 ••5o *3oot 55o 174 240349 0799 1048 1297 i546 1795 2044 2293 254i 2790 249 52661 248 l 7 5 3o38 3286 3534 3782 4o3o 4277 4525 4772 5019 176 55i3 5759 6006 6252 6499 6745 6991 7237 7482, 77281 246 177 7973 8219 8464 8709 8 9 54 i 9198 9443 9687 9932: •176I 245 178 250420 0664 0908 u5i i3 9 5 i638 1881 2125 2368 2610 243 l l 9 2853 3096 3338 358o 3822 4064 43o6 4548 4790' 5o3i 242 180 255273 55i4 5755 5996 6237 6477 6718 6958 71981 7439 241 181 7679 7918 8i58 83 9 8 8637 8877 9116 g355 9594I 9833 23 9 182 260071 o3io o548 0787 1025 1263 i5oi 1739 1976 2214 238 1 83 245 1 2688 2925 3i62 3399 3636 38 7 3 4109 4346' 4582 23 7 18; 4818 5o54 5290 5525 5761 5 99 6 0232 6467 6702 1 6937 235 1 85 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 186 9 5i3 9746 9980 •2l3 •446 •679 •912 1 144 1377, 1609 233 187 271842 2074 23o6 2538 2770 3ooi 3233 3464 3696 3927 232 188 4i58 438 9 4620 485o 5o8i 53u 5542 5772 6002 6232J 23o 189 6462 6692 6921 71 5 1 7 38o 7609 7 838 8067 8296! 8525; 229 •578j •800! 228 I 190 278754 8982 9211 943o 171 5 9667 9895 •l23 •35i 191 2 8io33 1261 1488 1942 2169 23o6 4606 2622 2849J 3o75 227 192 33oi 3527 3 7 53 3979 42o5 443 1 4882 5107 5332 226 193 5557 5782 6007 6232 6456 6681 6905 7i3o 7354 7578 225 194 7802 8026 8249 8473 8696 8920 9143 9 366 95891 9812 223 195 290035 0257 0480 0702 0925 1 147 i36 9 1 5 9 i i8i3 2o34 222 196 2256 2478 2699 2920 3i4i 3363 3584 38o4 402DI 4246 221 197 4466 4687 4907 5127 5347 5567 5787 6007 8.98 6226 6446 220 i 9 8 6665 6884 7104 7 323 7542 7761 7979 8416 8635 2IQ 199 8853 9071 9289 9 5o 7 9725 9943 •161 •378 •5 9 5 •8i3! 218 200 3oio3o 1247 U64 1681 1898 21 14 233i 20471 2"?64j 2980I 217 201 3196 3412 3628 3844 4o5g 4275 4491 4706 4921 5i36| 216 202 535 1 5566 5 7 8i 5996 62 1 1 642J 6639 6854 7068! 7282 2t5 203 7496 7710 7924 8i3 7 ! 835i 8564 8778 8991 9204J 9417 2l3 204 963o 9843 ••56 •268 «48i •6g3 •906 1118 i33o i542 2i2 2o5 311754 1966 2177 2389 2600 2812 3o23 3234 3445 36561 211 206 3867 4078 4289 4499 4710 4920 5i3o 5340 555i 576c. 210 207 5970 6180 6390 8481 6599 6809 7018 8689 8898 9106 7227 74361 7646 7854! 200 9938 208 208 8o63 8272 93 1 4 9522 9730 209 320146 o354 o562 0769 0977, 1184 i3qf 1 5 9 8 i8o5 2012 207 210 322219 2426 2633 2839 3o46 3252 3458 3665 38 7 i 4077 2o6 211 4282 4488 4694 4899 5io5 53io 55i6 5721 5926 6i3i 205 2!2 6336 654i 6745 6950 7 1 55 7359 7 563 7767 7972 8176 204 213 838o 8583 878- 899 1' 9194 s 9398 9601 9805 i832 •••8 •21 li 203 214 33o4i4 0017 0819 I022i 1225 1427 i63o 2o34 2236; 202 2l5 2438 2640 2842 3o44J 3246 ; 3447 364q 385o! 4o5i 4253 202 216 4454 4655 4856 5o57 5257 5458 -5658 585 9 ! 6o5 9 6260 201 217 6460 6660 6860 7060 7260 7459 7 65 9 7 85Sj 8o58 8257; 200 1 218 8456 8656 8855 9054 1 9203 945 1 9600 9849 ••47! «246! 199 N. 340444 C642 0S41 1039 1237 1435 [632 i83o' 2028! 2225 _L?t I 2 3 i 4 1 5 6 n 1 8 I 9 D 4 A TABLE OP LOGARITHMS FROM 1 TO 10,000. N. | I | 2 3 4 5 | 6 ■> 8 9 220 342423| 2620 1 2817 3oi4 32i 2( 3409' 36o6 38o2 3999 4iq6 221 43o2 4589! 4785 6353 i 6549' 6744 4981 5178 5374! 5570 5766 5962; 6137 222 6939 7 135) 733o 7525i 7720 7915, 8110 9860 ••54 195 223 83o5; 85oo 8694 8889 9083; 9278 9472 1 9666 194 1 224 350248 0442 o636 0829 1023 12161 1410 i6o3 1796 1989 1-3 225 2i83 2375 2568 2761 2954I 3i47 3339 4876 5o68i 526o 3532 3724 3916 i 9 3 226 4108 43oi 4493 4685 5452 5643 5834 192 227 6026 6217 6408 6099 6790' 6981' 7172 7363 7554 7744 19? 228 7 o35 8125 83i6 85o6 8696; 8886 9076J 9266 9456 9646 189 188 22Q 9«35 ®*25 •2l5 •404 •593: •7S3! ®972| 1 161 2482 2671' 285g 3o48 i35o i539 23o 361728 1917 3 800 2105 2294 3236 3424 23l 36i2 3988 5862 4176 4363 455 1 j 4739 4926 5n3 53oi 188 232 5488 56 7 5 6049 6236 6423; 66jo' 6796 6983, 7169 187 233 7356 7542 7729 79i5i 81 01 i 8287 8473 865o •328, »5i3 8845. go3o 186 234 9216 9401 93871 9772 99 58 •i43 •698 •883 1 85 235 371068 1253 1437J 1622 1806 1991 383 1 2175 236o 2544 2/28 184 236 2912 3096 328ol 3464 3647 4oi5 4198 4382 4565 184 23 7 4748 4932 5:i5| 5298 5481 5664 5846 6029 6212 63 9 4 1 83 238 65 77 6759 6942! 7124 8761 8943 7306 7488 7670 9487 7852 8o34 8216 182 239 83 9 8 858o 9124 93o6 9668 9849 ••3o 181 240 38o2 11 0392 0573 0754 0934 1 1 1 5 1296 1476 i656 i83 7 181 241 2017 3996 5 7 85 2377 2557 2 7 3 7 2917 3o 97 3277 3456 3636 180 242 3Si5 4174! 4353 4533 4712 4891 5070 5249 5428 179 243 56o6 5964 6142 632i 6499 6677 6856 7o34 7212 '78 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 8989 ^8 245 9166 9343 9320 9698 9875 ••5i •228 •4o5 •382 •739 '77 246 390935 1 112 1288 1464 1 641 1817 1993 2169 2345 2521 176 247 2697 4452 2873 3048 3224 34oo 35 7 5 375i 55oi 3926 4101 4277 176 248 4627 4802 4977 5i52 5326 5676 585o 6o25 i 7 5 249 6199 6374 6548 6722 6896 8634 7071 8808 7245 74i9 7 5 9 2 7766 '74 230 397940 8114 8287 8461 8981 9i54 9328 95oi i 7 3 231 9674 9847 ••20 •192 •365 •538 •711 •883 io56 1228 i 7 3 252 401401 1573 •745| 1017 2089 2261 2433 26o5 2777 2949 172 253 3l2I 3292 3464 3635 3807 3 97 8 4149 4320 4492 4663 171 254 4834 5oo5 5176 5346 55i 7 5688 5858 6029 6199 6370 171 255 654o 6710 6881 7o5i 7221 73 9 i 7 56i 7 7 3i 7901 8070 170 256 8240 8410 8579 8749 8918 9087 9 25 7 9426 95 9 5 9764 169 25t 9933 •102 •271 e 44o •600 2 29 6 •777 •946 1114 1283 i45i 3 258 411620 1783 1956 2124 2461 2629 2796 2964 3i32 239 33oo 3467 3635 38o3 3970 4i3 7 43o5 4472 4639 63o8 4806 167 260 4U973 5i4o 53o7 6 973 5474 564i 58o8 5974 6141 6474 8i35 167 261 6641 6807 7i3o 73o6 7472 8964! 9129 7 638 7804 7970 166 262 83oi 8467 8633 8798 •45 1 9 2 9 5 9460! 9625 9791 i65 263 9956 *i 21 ©286 •616 »78i •945 mo 1275 1439 1 65 264 421604 1788 1933 2 °97 3737 2261 2426 2390 2754 2918 4392! 4555 3o82 164 265 3246 3410 3574 3901 4o65 4228 4718 164 ?66 4882 5o45 5208 53 7 i 5334 5697 586o 6o23* 6186 6349 7973 1 63 2O7 65 1 1 6674 6836 6999 7161 7324 7 486 7648 7811 162 268 8i35 8297 8459 •» 7 5 8621 8783 8944 9106 92681 9429 9591 162 269 7752 9914 •236 •398| e 559 •720 •88l I042| 1203 161 270 43 ,364 i525 i685 1846 2007! 2167 2328 2488 2649 2809 161 271 2969 4669 6i63 3i3o 3290 345oj 36io ! 3770 3g3o 4090 4249: 4409 160 272 4729 4888 5o48 5207; 5367 5326 5685 5844! 6004 1 5 9 2 7 3 6322 6481 6640 6798! 6957 7116 72-5 7433J 7592 \U 274 77 5i 7909 8067 8226! 8384 8542 8701 8859 1 9017 9175 275 9 333 9491 9648 9806 9964 »I22 1224! i38i 1338 1695 •279 ®437j ^594! ^52 1 58 276 440909 1066 1832 2009 2166 2323 1 5 7 277 2480 2637 2793J 2950 3io6 3263; 3419 43371 45 1 3 4669 4825 1 4981 3576 3732 3889 ! ?I 2-8 4045 4-01 5 1 37 5293 1 5449 i56 1 ?79_ N. 3004 5760 1 591 5 6071 1 ~T~i~r| 7~i~| 6226 6382| 6537 6092 6848, 7003 i55 4 5 6 7 | 8 \ 9 | D. j A TABLE OF LOGARITHMS FROM 1 TO 10,000. e N. 1 J • 2 1 3 j 4 1 5 1 6 ~8o~88 1 * 8 9 '""lv] 280 447158! t3i3 7468 7623, 777^1 7Q33 8242 8397 8552 1 55 281 870&J 8861 1 9015 9170 9324 9478 9633 9787 994i •• 9 5 1 54 282 450249! 04 ?3 o557 0711' o865| 1018 1172 i3 2 6 U79 1 633 i5 4 283 1 7861 1940 2093 2247 240c "J553 2706 285 9 3oi2, 3i65 1 53 284 33i8j 3471 3624 3777, 393c, 4082 4235 438 7 454o, 4692 1 53 285 4845 4997 ! 5i5o 53o2 : 5454 1 56o6 575S 5gio 6062 6214 l52 286 6366 1 65i8' 6670 6821 I 6 973 | 7125, 7276 7428 7 5 79 773i i5i 287 7882| 8o33| 8iS4 ! 8336' 8487! 8638 S789 8940 9091 9242 1 5 1 288 93921 g543 9694 9S45 9993 1 •146, # 2g6 •447 •5 97 1 # 748 i5i 289 460898 1048 1 198 1 348 1499 j 1649 (799 1948 2098 2 24& i5o 29c 462398 2548, 2697; 2847 2997 3i46 3296 3445 35 9 4 3 7 44 i5o 291 3893, 4042 41 9 1 434o 4490 463g 4788 4936 5o85 5234 149 292 5383 5532 568o 5829' 5977 6126 6274 6423 6571 6719 \% 293 6868 7016 7164 73 1 2 ; 7460 7608 7756 7904 8o52 8200 294 8347 8495 8643 8790: 8938: go85, 9233 9 38o 9 52 7 9675 148 29D 9822 9969 •116 # 263! *4io •557, '704 •85 1 •998 1 145 147 296 471292 1438 1 585, 1732; 1878 2025 2171 23i8 2464 2610! 146 297 2706 2903 3049 3i95 334i 3487, 3633 3779 4g44' 5090' 5235 3g25 4071! 146 298 4216 4362 45o8 4653 4799 538i 5526 146 299 5671 58i6 5962 6107 6232 6397 6542. 6687 6832 6976! 145 3oo 4771 2 1 7266 ! 741 1 7555 7700 7844; 7989' 8i33 8278 97'9 8422! 145 3oi 8566 87 uj 8855 8999 9U3 9287, 943 1 j 9575 9 863 j 144 302 480007 oi5n 02g4 0438 o582 0725, 0869 1012 n56 1299 144 3o3 1443 1 586 1729 1872 2016 2l59 2302 2445 2588 2 7 3l 143 3o4 2874 3oi6 3159' 33o2 3445 3587| 3730 3872 4oi5 4157 143 3o5 43 00 4442 4585 | 4727 4869 5oii| 5i53 5295 5437 5579 142 3o6 5721 5863 6oo5 6147 6289 643o 6572 6714 6855 6997, 142 307 7i38 7280 7421 7563 7704 7845j 7986 8127 8269 84.101 141 3o8 855i 8692 8833 8974 9»4 9255 9396 9537 9 6 77 9818 141 309 9958 491362 • # 99 •23 9 »38o a ;)2o •661 j •8oi| »94i 1081 1222 140 3io l502 1642 17S2 1922 2062: 2201 j 2J41 2481 262 1 140 3u 2760 2900 3o4o 3i 79 3319 3458 3597I 3737 3876 401 5 1 3 9 3l2 4i55 4294 4433 4572 4711 485o 4989' 5i28 5267 54o6 1 3 9 3i3 5544 5683 5822 5960 6099 62381 6376 65i5 6653 6791 1 3 9 3i4 6930 7068 7206 7344 8724 7483 7621 7759 1 7897 8o35 8n3 i33 3i5 83 1 1 8448 8586 8862 8999I 9137 9275 9412 955o 1 38 3i6 9687 9824 9962 ••99 »236 •374J •Si 1 1 '648 •785 •922 .37 3i7 5oio59 1 196 i333 1470 ! 1607 1744' 1880' 2017 2 1 54 2291 i3 7 3i8 2427 2564 2700 2837! 2973 4o63 4199 4335 3io 9 ! 3246! 3382 35x8 3655 1 36 319 3 79 i 3 9 2 7 447 1 1 4607 4743 4878 5oi4 1 36 320 5o5i5o 5286 542 1 ! 5557 '• 5693 5828 5 9 64i 6099 6234 6370 1 36 321 65o5 6640 6776 691 ij 7046 7181 7316 7451 853o| 8664 1 8799 7586 7721 i35 322 7856 799 1 8126; 8260; 83g5 8q34 9068 i35 3j3 9>o3 9337 9471 9606 9740 9874: •••9 1 «i43 •277 •411 i34 324 5io545 0679 081 3i 0947 1081 I2i5j 1349I 1482 1616 1750 i34 325 1 883 2017 2i5i j 2284 2418 255i 2684J 2818 2951 3o84 i33 3^6 32i8 335i 3484 36i7 375o 3883 4oi6 ; 4149 4282 44i4 1 33 32 7 4548 4681 48 1 3 4946 5079 521 1 ! 53441 5476 5509 5741 133 328 58 7 4 6006 6139 6271 64o3 6535 6668; 6800 6 9 3 2 7064 132 ^ 9 c l 1 - 96 7328 7460 7592 8777 | 8 909 7724 7855 7987! 81 19 825i 8382 132 33o 5i85i4 8646 9040 9171 93o3 9434 9566 9697 i3i 33i t 9 8 ?2 9959 ••goj # 22I •353 •48 * •6i5| «745| '876 1007 i3i 333 52ii38 1269 1 i4oo ; i53o 1661 1792 1922J 2o53 2i83 23i4 i3i 333 2444 25751 2705! 2835 2966 3o 9 6l 32261 3356| 3486 36i6 i3o 334 3746 3876| 4006! 4i36 4266 4396 4526 4656 4780 4qi5 i3o 335 5o45 5i74j 53o4 5434' 5563 56g3 6 9 85 5822J 595i 6081 O210, 129 336 633 9 6469' 65q3 6727 6856 7ii4j 7 2 43 7372 7001 129 337 763o 7739 7888! 8016 1 8i45 8274 8402 853 1 8660 8788' 12Q 338 8917 9045, 9174 9302 943o 9 55 9 9687 j g8i5 9943 ••72; J 28 339 530200 o328; 0456; o584J 0712 0840J O968: IO96 1 T223 i35i| 128 N. 1 1 | 2 ; 3 4 5 j 6 7 1 3 . 9 | D. 6 A TALSLK OF LOGARITHMS FROM 1 TO L 0,000. N. 1 | 2 3 4 5 6 1 7 8 9 D. 34o 53U79 1607 1734 1862 1990 2117 2245 ! 2372 2D00 2627 128 34i 2754 2882 3009 3i36 3264 3391 35 1 8 3645 3772 38 99 127 342 4026 4i53 4280 4407 4534 4661 4787 4914 5o4i 5167 127 343 5294 5421 5547 56 7 4 58oo 5927 6o53| 6180 63o6! 6432 126 344 6558 6685 681 1 6 9 3 7 7063 8322 11% 73i5 8574 7441 756 7 7693 8 9 5i 126 345 7819 7945 8071 8197 8699 8825 126 346 9076 9202 9327 945a 9 5 7 8 9 7 o3 9829 99^4 ••79 •204 125 347 34« 54o329 0455 o58o 0705 o83o 0955 1080 I2o5 i33o 14^4 125 1579 1704 1829 1953 2078 2205 2327 2452 25 7 6 271-11 125 549 282! 2950 3074 3i 99 3323 3447 3571 36 9 6 4936 3820 3c>44 124 3jo 544068 4192 43i6 4440 4564 4688 4812 5o6o 5i83 124 35i 53o 7 543 1 5555 5678 58o2 5925 6049 6172 6296 6419 124 35; 6543 6066 6789 6913 7036 7i5 9 7282 74o5 7 52 ? 8738 7652 123 3; 53 7775 7898 8021 8144 8267 838 9 85i2 8635 8881 123 354 9003 9126 9249 0473 93 7 i 9494 9616 97 3 9 9861 9984 •106 123 355 550228 o35i 0595 0717 0840 0962 1084 1206 i328 122 356 i45o 1572 1694 1816 1938 2060 2181 23o3 2425 2547 122 357 2668 2790 2911 3o33 3 1 55 3276 33 9 8 35i9 364o 3762 121 358 3883 4004 4126 4247 4368 4489 4610 473i 4852 4973 6182 121 35 9 5094 52i5 5336 5457 55 7 8 5699 5820 5 9 4o 6061 121 36o 5563o3 6423 6544 6664 6785 6905 7026 7146 7267 738 7 I20 36 1 7D07 7627 7748 8948 7868 7988 8108 8228 8349 8469 858 y 120 36a 8709 8829 9068 9188 93o8 •5o4 9428 9548 9667 97B7 120 363 9907 ••26 •146 •265 •385 •624 •743 •863 •982 119 364 56 1 1 01 1221 1 34o U59 1578 1698 1817 1936 2o55 2174 II9 365 2293 2412 253i 265o 2769 3955 2887 3oo6 3l2D 3244 3362 119 366 348i 36oo 3718 3837 4074 4192 43u 4429 4548 H9 36 7 4666 4784 4903 5021 5i39 5257 5376 5494 56i2 5730 Il8 368 5848 5966 6084 6202 6320 6437 6555 66 7 3 7849 6791 6909 Il8 369 7026 568202 7144 7262 7379 7497 8671 7614 8788 7732 7967 8084 Il8 370 83i 9 8436 8554 8 9 o5 9023 9140 9257 117 3 7 i 9374 9491 9608 9725 9842 99 5 9 ••76 •, 9 3 •3o 9 •426 117 3 7 2 570543 0660 0776 0893 IOIO 1 1 26 1243 i3^o 2523 1476 i5o2 2755 >'7 3 7 3 1709 i825 1942 2o58 2174 2291 2407 3568 2639 116 374 2872 2988 3 1 04 322o 3336 3452 3684 38oo 3 9 i5 116 3 7 5 4o3i 4i47 4263 4379 44Q4 565o 4610 4726 4841 4957 5072 116 3 7 6 5i88 53o3 5419 5534 5 7 65 588o 5996 6111 6226 u5 377 634i 6457 65 7 2 6687 6802 6917 8066 7032 7147 7262 7377 1 1 5 378 7492 7607 7722 7836 8 9 83 79 5i 8181 82 9 5 8410 85?5 1 1 5 379 8639 8 7 54 8868 9097 9212 9 326 9441 9555 9669 114 38o 5 79 7«4 9898 ••12 •126 •241 •355 •469 •583 •697 1 836 •811 114 38i 580925 1039 u53 1267 i38i I4g5 263 1 1608 1722 1950 114 382 2o63 2177 2291 2404 25i8 2745 2858 2972 3o85 114 383 3199 33i2 3426 353 9 3652 3765 38 79 3 99 2 4io5 4218 1 13 384 433 1 4444 4557 4670 4783 4896 5009 5l22 5235 5348 n3 385 5461 5574 5686 5799 5912 6024 6i3 7 6250 6362 6475 n3 386 6587 6700 6812 6925 7037 7i49 7262 8384 7374 7486 7599 U2 387 7711 7823 7935 8047 8160 8272 8496 8608 8720 112 388 8832 8944 9o56 9167 9279 9391 95o3 96l5 9726 9838 112 38 9 9950 ••61 •i 7 3 •284 •3 9 6 •5o 7 •619 •730 •842 • 9 53 112 390 591065 1176 1287 l3 99 i5io 1621 1732 i843 1955 2066 III 3 9 i 2177 2288 2399 25io 2621 2 7 32 2843 2954 3o64 3i 7 5 III 3 9 2 3286 33 9 7 35o8 36i8 3729 3840 3 9 5o 4o6l 4171 4282 III 3 9 3 4393 45o3 4614 4724 4834 4945 5o55 5i65 5a 7 6 5386 no ? 9 i 5496 56o6 5717 5827 5 9 3 7 6047 6i5 7 6267 63 77 6487 no 3 9 5 65 97 7695 8791 6707 6817 6927 7037 7U6 8243 7256 7366 7476 7586 no 3 9 6 7 8o5 7914 8024 8i34 8353 8462 8572 8681 no 1 l 9 l 8900 9009 91 19 9228 9337 9446 9556 9 665 9774 109 3 9 8 9 883 999 2 •iOIJ *2I0 •319 1408 •428 •53 7 •646 •755 •864 109 1 399 * N. 600973 I082 I I9I ! I2Q9 i5i7 i625 1734 i843! 1901 109 1 ! > ! 3 4 5 6 7 ~8~yv D. A TkBLE OF LOGARITHMS FR OM ] 6 TO 10,000. * N. 400 1 ~ T_ 3 2494 2603 7 8 JL D. 602060 2169 3253 2277 2386 2711 2819 2928 3o36 108 401 3i44 336 1 3469 3577 4658 3686 3794 3902 4010 4118 108 402 4226 4334 4442 455o 4766 4874 4982 3089 5197 108 4o3 53o5 54i3 552i 5628 5 7 36| 5844 5951 6059 6166 6274 108 404 638 1 6489 65 9 6 6704 681 1 6919 7026 7 i33| 7241 7348 8419 107 403 7455 7362 7669 8740 7777 8847 7884 7991 8098 8 2 o5 83i2 07 406 8526 8633 8934 9061 9167 9274 9 38i 9488 107 407 9394 9701 9808 9914 ••21 •128 •234 •341 •447 •554 107 408 610660 0767 0873 0979 1086 1192 1298 i4o5 1 5i 1 1617 106 409 1723 1829 1936 2042 2148 2234 236o 2466 2572 2678 106 410 612784 2890 2996 3l02 3207 33i3 3419 3525 363o 3736 106 411 3842 3947 4o33 4159 4264 4370 4475 458 1 4686 4792 106 412 4897 5oo3 5io8 5 2 i3 5319 5424 5329 5634 5740 5845 io5 4i3 5930 6o55 6160 6265 6370 6476 658i 6686 6790 68 9 5 io5 4U 7000 7io5 7210 825-1 73i5 7420 7325 7629 7734 7 83 9 7943 8989 io5 41 5 8048 8i53 8362 8466 85 7 i 8676 8780 8884 io5 416 9093 9198 9302 94*06 95i 1 9 6i5 9719 9824 9928 ••32 104 417 620106 0240 o344 0448 o552 o656 0760 0864 0968 1072 104 418 1 176 1280 1 384 1488 1592 i6o5 2732 '799 igo3 2007 21 10 104 419 2214 23i8 2421 2525 2628 2833 2g3o 3973 3o42 3i46 104 420 623249 3353 3456 3559 3663 3766 386 9 4076 4H9 io3 421 4282 4385 4488 4591 4695 479S 4901 5oo4 5i07 5210 io3 422 53i2 54i5 55i8 562i 6724 5827 5929 6o32 6i35 6238 io3 423 634o 6443 6546 6648 6 7 5 1 6853 6956 7038 7161 7263 io3 424 7366 7468 7^71 7673 7775 7878 7980 8082 8i85 8287 102 425 838 9 8491 85 9 3 86 9 5 «797 8900 9002 9104 9206 9 3o8 102 426 9410 9312 961 J 9713 98'7 o835 9919 ••21 •123 •224 •326 102 427 63o428 o53o o63i 0733 0936 io38 1 139 2i53 1241 i342 102 428 1444 1 545 1647 1748 1849 1961 2052 2235 2356 101 429 2457 2559 2660 2761 2862 2963 3o64 3i65 3266 336 7 101 43o 633468 3369 3670 3771 38 7 2 3973 4074 4n5 5i82 4276 43 7 6 ioo 43 1 4477 5484 4378 4679 3685 4779 5 7 85 4880 4981 5o8i 5283 5383 100 432 5584 5886 5 9 86 6087 6187 6287 6388 ioo 433 6488 6588 6688 6789 6889 6989 1089 8090 9088 7.89 7290 73oo 838 9 9 38 7 ICO 434 435 7490 8489 858 9 7690 8689 fiX 7890 8888 7990 8988 8190 9188 8290 9287 99 99 436 94S6 9586 9686 9783 9885 99B4 ••84 •i83 •283 •382 99 437 438 640481 o58i 0680 0779 0879 0978 1077 "77 1276 i375 99 1474 i573 1672 1771 1871 1970 2069 3o58 2168 2267 2366 99 43 9 2465 2563 2662 2761 2860 2939 3i56 3255 3354 $ 44o 643453 355i 365o 3 749 3847 3 9 46 4044 4i43 4242 434o 44 1 4439 4537 4636 4734 4832 493i 5029 5 1 27 5226 5324 98 442 5422 5521 5619 5 7 i-7 6698 58i5 5913 6894 601 1 6i 10 6208 63o6 98 443 6404 65o2 6600 6796 6992 7089 •7187 7285 98 444 7383 7481 8458 2 5l 9 7676 7774 7872 8848 7969 8945 8067 8i65 8262 98 445 836o 8553 8653 8 7 5o 9043 9140 9237 97. 446 9 335 9432 953o 9627 9724 9821 9919 0890 ••16 •1 13 •210 97 447 448 65o3o8 o4o5 0502 0599 0696 0793 0987 1084 1181 97 1278 1375 1472 1569 1666 1762 1839 1936 2o53 2i5o 97 449 2246 2343 2440 2536 2633 2730 2826 2923 3019 3i 16 97 45o 6532i3 3309 42 7 3 34o5 35o2 35 9 8 36 9 5 3791 4754 3888 3 9 84 4080 96 431 4177 436 9 4465 4562 4658 485o 4946 5o42 96 452 5i38 5 2 35 533 1 5427 5523 5619 5 7 i5 58io 5906 6864 6002 96 453 6098 6194 6290 6386 6482 65 77 6673 6769 7725 6960 96 454 7036 7i52 7247 7343! 7438 7 534 7629 8584 t: 7916 8870 9 6 455 801 1 8107 8202 8298I 83 9 3 92301 9346 8488 8679 9 5 456 8 9 65 9060 9i55 9441 9 536 9 63i 9726; 9821 ^ 457 458 9016 66oS65 ••11 °io6 •201 1 ^296 •3oi 1 33 9 •486 •58i •676 0771 9 5 0960 1033 iiSol 1245 1434 i52o 2475 1623 1718 9 5 459 i8i3 1907 2002 2096! 2191 2286 238o 256 9 2663 93 N. I 2 3 | 4 * 6 7 8 j 9 I>. a:^ tf A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 1 2 3 | 4 | 5 j 6 | 7 J 8 | 9 j-b7 460 662708 2852 2947 3o4ii 3i35 323o| 3324; 34i8 35I2 1 3607 94 461 3701 3795 3889 39831 4078! 4172] 4266 436o 4454j 4548 94 462 4642 4736 483 4924 ! *5oi8 5862 5956 5ii2 52o6! 5299 6o5o ! 6i43 6237 5393: 5487 94 463 558 1 5675 5769 633 1 6424 94 464 65i8 6612 6705 6799 68 9 2 6986 7079! 7 i 7 3 7266! 7360 94 465 7453 7546 7640 77331 7826 8665! 8759 7920 8oi3l 8106 8199! 8293 ! 93 466 8386 8479 8372 8852 8945 9 o38 9i3ij 9224I 93 467 9317 9410 g5o3 9596 9689 9782 9875 9967 0895 ••60! »i53 93 468 670246 o339 1265 043i o524 0617 07101 0802 0988 1080 93 469 1173 1358 145 1 1 543 1636 1728 1821 iqi3 2oo5 9 3 470 672098 2190 2283 23 7 5 2467 256o 2652 2744 2836 2929 92 47i 3021 3n3 32o5 3297 3390 3482 3574 3666 37D8 385o 92 47^ 3942 4861 4o34 4126 4218 43 10 4402 4494 4586 4677 4769 92 4 7 3 4953 58 7 o 5o45 5i3 7 5228 5320 5412 55o3 5595 568 7 92 474 6694 5962 Oo53 6145 6236 6328 6419 65n 6602 92 475 6785 6876 6968 7o5 9 71D1 7242 8i54 7 333 7424 8336 75i6 9' 476 760-7 7698 7789 7881 7972 8882 8o63 8245 8427 91 477 85i8 8609 8700 8791 8 97 3 9064 oi55 9246 9 33 7 9' 478 9428 9 5l 9 9610 9700 979 « 9882 9973 ««63 •i54 •245 9i 479 68o336 0426 o5i7 0607 0698 0780 0879 0970 1060 1 1 5 1 9 1 480 681241 i332 1422 i5i3 i6o3 1693 1784 1874 1964 2o55 90 481 2145 2235 2326 2416 25o6 2596 2686 2777 2867 2957 90 482 3o47 3i3 7 3227 33i 7 3407 3497 3587 36 7 7 3767 385 7 90 483 3947 4037 4127 4217 43o7 43q6 4486 4576 4666 4756 90 484 4845 4935 5o25 5ii4 52o4 5294 5383 5473 5563 565 2 si 485 5742 583 1 5921 6010 6100 6,69 6279 6368 6458 6547 486 6636 6726 68i5 6904 6994 7886 7o83 7172 8064 7261 735i 744o i 9 487 7 52 9 7618 7707 85 Q 8 7796 7975 8865 8i53 8242 833 1 89 488 8420 8509 8687 8776 8 9 53 0841 9042 9i3i 9220 89 489 9309 9 3n8 9486 9575 9664 97 53 99 3o ••10 090D •107 S o 9 490 690 1 96 1081 0285 o373 0462 o55o 0639 0728 0816 ^993 8 o2 491 1 170 1258 1 347 1435 i524 1612 1700 1789 1877 88 492 196D 2o53 2142 2230 23i8 2406 24Q4 2583 2671 2759 88 493 1847 2o35 38i5 3o23 3i 1 1 3 '99 3287 3375 3463 355i 363 9 88 494 3727 3903 3 99 i 4868 4078 4166 4254 4342 443o 4517 88 495 46o5 4693 4781 4 9 56 5o44 5i3i 5219 5307 53 9 4 88 496 5482 556q 5657 5744 5832 5 9 i 9 6007 6094 6182 6269 £ 7 497 6356 6444 653 1 6618 6706 6 79 3 6880 6968 7o55 7142 ? 7 498 7229 73i7 8188 7404 749i 7 5 7 8 8449 7665 77 52 7 83 9 7926 8014 S 7 499 8101 8275 8362 8535 8622 8709 8796 8883 ^ . 5oo 698970 9057 9144 923i 9317 94o4 9491 9 5 7 8 9664 975 1 87 5oi 9838 9924 ••11 ••98 •184 •271 •358 •444 •53 1 •617 s 502 700704 0790 0877 0963 io5o n36 1222 1 309 i3 9 5 1482 5o3 1 568 1 654 1741 1827 1913 1999 2086 2172 2258 2344 86 5o4 243 1 2517 26o3 2689 2 77 5 2861 2947 3o33 3i 19 32o5 86 £o5 32qi 4i5i 3377 3463 3549 3635 3721 3807 38 9 3 475i 3979 4o65 86 5o6 4236 4322 4408 4494 4579 4665 4837 4922 86 507 5oo8 5094 5179 6o3d 5265 535o 5436 5522 5607 56 9 3 5 77 8 86 5o8 5864 68o3 6120 6206 6291 63 7 6 6462! 6547 6632 85 509 6718 6888 6974 7059 7144 T229 7 3i5 74oo 7485 8336 85 5io 707570 7655 7740 85 9 i 7826 791 1 7996! 8081 8166 825i 85 5n 8421 85o6 8676 8761 8846 8 9 3i 9015 9100 9 i85 85 5l2 9270 9 355 9440 9524 9609 9694 9779 9 863 9948 ••33 85 5i3 710117 0202 0287 0371 0456 0040, 0625 0710 0794 0879 85 5i4 0963 1048 Il32 1217 i3oi 1 385 1470 1 554i 1639 1723 2566 84 5i5 1807 1892 1976 2818 2060 2144 2229 23i3 2397 2481 84 5i6 2o5o 2734 2902 2986 3826 3070 3i54 3238 3323 3407 84 5l I 5i8 349i 35i5 3659 3742 458i 3910 3994 4833 4078 4162 4246 84 433o 4414 4497 4665, 4749 4916 5ooo 0084 84 619 5167 525i 5335 54i8 55o2 5586 5669 5 7 53 5836 5 9 2o 84 D. jr. J 1 2 3 4 ! 5 6 7 8 J 9 j A TABLE OF LOGARITHMS FROM L TO 10,000. 9 N. 1 1 . 1 2 1 3 j 4 5 6 j 7 j 8 | 9 1 ' -xi 520 716003 6087; 6170) 6254; 6337 6421 65o4, 6588 667 il 6754 S3 521 6838 6921; 7004] 7088] 7171 7204 7338 7421 75o4 7^87 83 522 7671 7754] 7837 792c 8oo3 8086 8169 8253 8336 8419 83 523 85o2 8585! 8668 8 7 5i 8834 8917 9000 1 9083! 9 1 651 9248 83 524 933i 94i4i 9497 j 958oi 9663 9740J 9828 991 1 1 99941 ee 77 83 525 720159 0242, o325 04071 0490 o573 ; o655i 0738! 0821! ogo3 83 626 0986 1068 ii5i 1233 i3i6 i3 9 8 148 1 1 563. 1646 1728 *2 52 7 1811 1893 1975 2o58: 2140 2222 2«io5 2387 2469; 2552 i* 528 2634 2716 2798 2881 2 9 63 3702! 3784 3o45 3127 3209 3291 3374 82 529 3456 3538 3620 3866 3948 4o3o 41 12, 4iq4 82 53o 724276 4358 4440 452 2 i 4604 4685 4767 4849 493 1 5oi3 82 53 1 5095 5176 5258, 534o| 5422 55o3 5585 56 il 5748 583o 82 532 5912 5993 6075 6i56 6238 632o 6401 6483 6564 6646 82 533 6727 6809 7623 6890 697 2 1 7o53 7i34 7216 7297 8110 7379 746o 81 534 7^41 7704 77801 7866 8097 8678 7948 8029 8191 8273 81 5*5 8354 8435 85i6 8759 8841 8922 9003 9084 81 536 9165 9246 9327 9408 9489 9 5 7 o 965i 97 32 9 8i3 9893 81 53 7 9974 ••55 . •i36 •217 *298 •3 7 8 1 186 •459 •540 •621 •702 81 538 730782 o863 0944 1024' no5 1266 1 347 1428 i5o8 81 539 1589 1669 1750 i83o| 191 1 1991 2072 2l52 2233 23i3 81 54o 732394 2474 2555 2635 2715 2796 2876 2 9 56 3o37 3117 80 541 3i 97 3278 3358 3438 35i8 3598 3679 3759 383 9 3919 80 543 3999 4079 4160 4240] 4320 4400 4480 456o 4640 4720 80 543 4806 4880 4960 5o4oi 5 1 20 5200 5279 6078 5359 5439 55i9 80 544 6599 56 79 5 7 59 5838 5 9 i8 5998 6157 623 7 63i 7 80 540 6397 6476 6556 6635! 6715 6795 6874 6954 7o34 7 1 13 80 545 7193 7272 7352 743 i| 75 1 1 7390 8384 7670 7749 7829 8622 7908 79 54? 79^7 8781 9572 8067 8146 8220 83o5 8463 8543 8701 79 048 8860 8 9 39 9018 9097 9177 9256 9 335 94i4 9493 79 54 9 965 1 97 3i 9S10J 9889 0600! 0678 9968 ••47 •126 •205 •284 79 5do 74o363 0442 0521 0757 o836 0915 0994 1073 79 55i I 102 1230 i3o9 1 388 1467 1 546 1624 1703 1782 i860 79 552 1939 2018 2096 2175 2254 2332 241 1 2489 2568 2647 31 553 2725 2804 2882 2961 3o39 38 2 3 3n8 3 196 3275 3353 343 1 554 35io 3588 3667 3745 3902 3980 4o58 4i36 42i5 7» 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5i53 5 2 3i 53o9 5387 5465 5543 562i 0699 5777 78 55 7 5855 5 9 33 60 1 1 6089 6167 6245 6323 6401 6479 6556 78 558 6634 6712 6790 6868 6945 7023 7101 7H9 7256 7334 78 539 74i2 7489 8266 7567 7645 7722 7800 7878 79 55 8o33 8110 78 56o 748188 8343 8421 8498 85 7 6 8653 8731 8808 8885 77 56i 8 9 63 9040 9118 9 i 9 5 9272 935o 9427 95o4 9 582 9 65 9 77 562 9736 9814 989 1 9968 ••45 •123 •200 •277 •354 •43 1 77 563 75o5o8 o586 o663 0740 0817 0894 09-1 1 1048 1123 1202 77 564 1279 1 3 56 1433 i5io i5S7 1664 1 74 1 1818 1890 1972 77 565 2048 2125 2202 2279 2356 2433 2509 2 586 2663 2740 77 566 2816 28 9 3 297O 3o47 3i23 3200 3277 3353 3430 35o6 77 D67 3583 366o 3 7 36 38i3 388 9 39661 4042 4119 4195 4272 77 568 4348 4425 45oi 45 7 8 4654 4730! 4807 4883 4960 5o36 76 56 9 5ll2 0189 5 2 65 5341 5417 5494 5570 5646 5722 5799 76 570 755875 5951 6027 6io3 6180 6256 6332 6408 6484 656o 76 i 11 6636 6712 6788 6864 6940 70i6| 7092 7168 7244 8oo3 7320 76 572 7396 8i5d 7472 7548 7624 7700 ■7775, 7801 7927 8079 76 573 823o 83o6 8382 8458 8533! 8609 8685 8761 8836 -6 5n4 8912 8988 9063 9 i3 9 9214 0290I 9366 944i 9517 q5q2 16 575 068 9743 1 9819 9894! 997° ••45! «I2I •196 0920 •272 •347 75 57& 760422 0498 j o573 0649 0724 0799! 0870 1025 1:01 7 ? t\l 1176 I2DI 1326 1402 1477 IDD2I 1627 23o3l 2378 1702 1778; i853 75 Ig28 2oo3J 2078 2i53 2228 2453 25291 2604 32 7 8j 3353 75 i 579 2679 2754 2829 2904! 2978 3oo'3 3128 32o3 75 L*. I 2 | 3 _^J 5 1 1 6 _7__ 8 1 9 D. 10 A TABLE OF LOGARITHMS FROM 1 TO 10,000. - N : 1 2 "35^8 3 | 4 | 5 6 1 7 1 8 I 9 1). 58o 763428 35o3 3653! 3727 1 38o2 3877 3952I 4027 4101 75 58i 4176 425i 4326 44oo 44761 455o 4624 4699' 4774 4848 75 582 4923 4098 5072 5i47 522i| 5296! 5370 3445 552o| 5394 75 583 566 9 5743 58i8 58 9 2 i 5 9 66i 6041 6636 6710! 6785 6n5j 6190 6264 6839' 6933 7007 6338 74 584 64l3 6487 6562 7082 74 585 7i56 723o 73o4 7379 1 7433 7527 7601 7675 7749 7823 74 586 7898 7972 8046 8120; 8194! 8268 8342 8416 8490 8564 74 58 7 8638 8712 8786 886o| 8934! 9008 9082 9 1 56 9230' 9 3o3 74 588 9 3 77 945 1 9525 9399 9673, 9746 9820 9894 9968! »»42 74 58 9 7701 i5 0189 0263: o336 0410 0484 o557 o63i 07051 0778 1367 1440' i5i4 74 590 770802 0926' 0999 1073 1 146 I22o| I2g3 74 5 9 , 1 58 7 1661 1734 1808 1881 igSS 2028 2102' 21751 2248 73 592 2322 23 9 5 2468 2542 26i5 2688 2762 2835 2908 2981 73 5 9 3 3o55 3i28 3201 3274 3348 3421 3494 3567 364o; 37i3 73 5 9 4 3 7 86 386o 3 9 33 4006 4079 4i52 4225 4298 43 7 i 4444 73 5g5 45i7 4590 4663 4736 4809 4882 4955 5o2^ 5 loo 5. 7 3 73 5 9 6 5246 5319 5392 5465 5538 56 10 5683 5756 1 3829 5902 73 5q7 5 97 4 6047 6120 6193 6265 6338 641 1 64*83 6556 6629 73 5q8 6701 6774 6846 6919 6992 7064 7i3 7 7209 7282 7934 8006 8658 8730 7354 73 5 99 7427 7499 7 5 7 2 7644 8368 7717 7789 7862 8079 72 600 778131 8224 8296 8441 85i3 8585 8802 72 601 8874 8947 9019 9091 9i63 9236 9 3o8 938o 9432 9524 72 602 9596 9669 9741 9813 9 885 9957 ••29 •101 • 173 •245 72 6o3 780317 o38 9 0461 o533 o6o5 0677 0749 0821 o8 9 3 0965 72 604 1037 1 109 1 181 1253 i324 1396 1468 i54o 161 2 1684 72 6o5 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 606 2473 2544 2616 2688 2 7 5 9 283 1 2902 2974) 3046 3u 7 72 607 3i8 9 326o 3332 34o3 3475 3546 36i8 368o 3761 44o3' 4475 3832 7i 608 3904 3 97 5 4046 4118 4189 4261 4332 4546 7i 609 4617 4689 4760 483 1 4902 4974 5o45 5n6' 5187 5259 71 610 78533o 54oi 5472 5543 56i5 5686 5 7 5 7 5828! 58 9 9 5 97 o 71 611 604 1 6112 6i83 6254 6325 63 9 6 6467 6538 ! 6609 6680 7i 612 6 7 5 1 6822 68 9 3 6964 7o35 7106 7H7 , 7248 7319 7390 7« 6i3 746o 753i 7602 7673 7744 78i5 8522 7885 79561 8027 8663 8734 8098 7i 614 8168 823 9 83io 838i 845i 85 9 3 8804 7> 6i5 88 7 5 958i 8946 9016 9087 9157 9228 9299 9369! 9440 95io 7i 616 965i 9722 9792 9863 9933 •••4 ••74 # i44 •2l5 70 6,7 790285 o356 0426 0496 0567 o63 7 0707 0778 0848 0918 70 618 0988 1059 1 1 29 1199 1269 1 340 1410 1480 i55o 1620 70 619 1691 1761 i83i 1901 1971 2041 21 1 1 2181 2252 2322 70 620 792392 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 3092 3i62 323i 33oi 33 7 i 3441 35 1 1 358i 365i 3721 70 622 3790 3 860 3930 4000 4070 4i39 ! 4209 4279 4349 4976 5o45 44l8 70 623 4488 4558 4627 4697 4767 4836 ' 4906 5n5 70 624 5i85 5254 5324 53 9 3 5463 5532 §602 5672 5741 58n 70 625 588o 5g49 6019 6088 6i58 6227, 6297 6366 6436 65o5 69 626 65 7 4| 6644 6 7 i3 6782 1 6852 6921 6990 7060 7 1 29 7,98 69 627 7268J 7337 74o6 7475 7545 7614 1 7683 83o5j 8374 7752 7821 7890 69 628 79601 8029 8098 8467 8236 8443 85 1 3 8582 69 629 865i 8720 8780 947« 8858! 8927 8996 9065 9134 9203 9272 6 9 63o 799341 9400 8300291 0098 9547I 9°i° 9685i 9754 9823 9892 9961 69 63 1 0167 o236| o3o5 0373. 0442 o5n' o58o 0648 69 632 0717 0786 o854 o 9 23j 0992 1 06 1 1 1 29 1747! i8i5 1 198 1266 i335 69 633 1404 1472 i54i 1609I 1678 1884 1952 2021 69 634 2089 2 1 58 2226 229s 1 2363 2432; 25oo 2568 2637 2705 & 635 2774 2842 2910 2979: 3o47, 3»n6 ! 3 1 84 3252. 332i J 3389 636 3457 3523 3394 3662I 3730; 3798 3867 3 9 35 4oo3 4071 68 637 4i3g 4208 4276 1 4344! 44i2 4480 4548 .4&i6' 4685 4753 68 638 4821, 4889 4957; 5o25 5093 5i6r 5229 5297I 5365 5433 68 63q 55oi| 5569 563 7 i 5705I 5773 584i; 5 9 o8 6 5976; 6044 61 1 2 68 .jy .0 | 1 2 j 3 | 4 | 5 | 1). A TABLE OF LOGARITHMS FROM 1 TO 10,000. n N. 1 2 3 4 5 6 7 8 9 D. ~j 64o 806180 6248 63i6 6384 645 1 6519 6587 6655 6723 6790 68 641 6858 6926 6994 7061 7129 7197 7264 7332 74oo 7467 8i43 68 642 7535 7603 7670 7738 7806 7873 794i 8008 8076 68 643 821 1 1 8279 8346 8414 8481 8549 8616 8684 8751 8818 67 644 8886 8953 9021 9088 9i56 9223 9290 9358 9425 9492 67 645 956o| 9627 9694 9762 9829 9896 99 6 4 ••3 1 ••98 •i65 67 646 8i0233 o3oo o36 7 0434 o5oi 0569 o636 0703 0770 0837 67 647 0904' 0971 1039 1709 1106 1 173 1240 i3o7 1374 i44i i5o8 67 648 i575 1642 1776 i843 1910 1977 2044 21 1 1 2178 67 649 2245 23l2 2379 2445 25l2 25 79 2646 2713 2780 2847 67 65o 8l29l3 2980 3o47 3 1 14 3i8i 3247 33i4 338i 3448 35i4 67 65i 358i | 3648 3714 3 7 8i 3848 3914 3g8i 4048 4114 4181 67 65? 4248 43i4 438i 4447 45i4 458 1 4647 4714 4780 4847 67 653 49i3 4980 5o46 5n3 f/ 79 5246 53i2 53 7 8 5445 55n 66 654 5578 5644 5711 5 777 5843 5910 5976 6042 6109 6173 66 655 6241 63o8 63 7 4 644o 65o6i 65 7 3 663 9 6705 6771 6838 66 656 6904' 6970 7o36 7102 7169 ^ 3 ? 73oi 736 7 7433 7499 66 657 7565 7 63i 7698 7764 783o 7896 7962 8028 8094 8160 66 658 8226I 8292 8358 8424 8490 8556 8622 8688 8 7 54 8820 66 65 9 88851 8q5i 9017 9083 9149 92 1 5 9281 9346 9412 9478 66 660 8i9544 9610 9676 974i 9807 9873 99 3 9 •••4 ••70 •i36 66 661 820201 0267 o333 0399 0464 o53o o5 9 5 0661 0727 0792 66 662 o858 0924 0989 io55 1 1 20 1 186 I25l i3i7 i382 1448 66 663 i5i4 1D79 2233 1643 1710 i 77 5 1841 1906 1972 2037 2io3 65 664 2168 2299 2364 243o 2495 256o 2626 2691 2756 65 665 2822 2887 2932 3oi8 3o83 3i48 32i3 3279 3344 3409 65 666 3474 3539 36o5 3670 3 7 35 38oo 3865 3930 3996 4061 65 667 4126 4191 4256 432i 4386 445 1 4016 458 1 4646 471 1 65 668 4776 4841 4906 4971 5o36 5ioi 5i66j 523i 5296 536i 65 669 5426 5491 5556 562i 5686 575i 58 1 51 588o 5 9 45 6010 65 670 826075 6140 6204 6269 6334 6399 6464! 6528 6393 6658 65 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 73o5 65 m 7369 8oi3 7434 7499 7563 7628 7692 7757 7821 7886 7g5i 65 673 8080 8i44 8209 8273 8338 8402 J 8467 853 1 8595 64 674 8660 8724 8789 8853 8918 9361 8982 9046! 91 1 1 9175 9239 64 675 93o4 9 368 9432 9497 •i3 9 9625 9690 9754 9818 9882 64 676 9947 ••11 ••75 •204 •268 •332 •396 •460 •325 64 678 830D89 o653 0717 0781 o845 0909 1330 0973 1037 167& 1102 1 166 64 I23o 1294 i358 1422 i486 1614 1742 1806 64 679 1870 1934 1998 2637 2062 2126 2189 2253; 23T7 238i 2445 64 680 832D09 2D73 2700 2764 2828 2892 2956 3020 3o83 64 681 3i47 321 I 3275 3338 3402 3466 353o 3593 4ib6' 423o 365 7 3721 64 682 3 7 84 3848 3ol2 3975 4039 4io3 4294 435 7 64 683 4421 4484 4548 461 1 46 7 3- 4739 5373 4802 4866 4929 4993 64 684 5o56 5l20 5i83 5247 53ro 5437 55oo 5564 5627 63 685 5691 5754 5817 588 1 5 9 44 6577 6oot 607 1 J 6 1 34 6197 6261 63 68b 6324 6387 6401 65u 6641 6704 1 6767 683o 6894 63 687 8219 7020 7o83 7U6 7210 7273 7336 7399 7462 7525 8i56 63 688 7602 77i5 7778 8408 7841 mt 7967 8o3o 85 9 7 8660 8093 63 689 8282 8345 8471 8723 8786 63 690 638849 8912 8 97 5 9038 9101 9164 9227 9289 9 855 9918 9352 94i5 63 691 9478 9341 9604 9667 0294 9729 9792 9981 ••43 63 692 840106 0169 C232 o35 7 0420 0482 0343' 0608 0671 63 693 0733 0796 o85o 1485 0921 1547 0984 1046 1109J 1 172 1735, 1797 1234 "97 63 694 1359 1422 1610 1672 i860 IQ22 2547 63 695 19851 2047 2110 2172 2235 2297 236o 2422 2484 62 696 2609 26T2 2734 2796 2839 2921 2983 3o46 3io8 3170 62 697 698 3233 3290 3357 3420 3482 3544 36o6 366n 3 7 3i 3793 62 3855! 3oi8 4477 4539 3980 •4042 4104 4166 4229 1 4291 4353 44i5' 62 699 4601 4664 4726 4788 485o 4912 4974 5o36 62 N. J i 2 _J__L 4 __ 5 6 ! 7 8 9 I D. | 12 A TABLE OF LOGARITHMS FROM ] TO 10,000. N. 1 2 3 4 5 6 7 8 9 D. 62 700 845098 5i6o 5222 5284 5346 5408 5470 5532 55g4 5656 701 5718 5780 5842 5904 5 9 66 6028 6090 6i5i 6213 6273 62 702 633 7 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 -ro3 6955 7017 7079 7i4i 7202 7264 7326 7 388 7449 8066 761 1 62 704 7373 8189 7634 7696 7738 7819 788! 7943 8559 8004 8128 £1 7o5 82D1 83i2 83 7 4 8435 8497 8620 8682 8743 62 706 88o5 8866 8928 8989 905 1 9112 ,9 X 74 9235 9297 9 358 61 707 9419 948i 9542 9604 9665 9726 9788 9849 991 1 9972 o585 61 708 85oo33 0095 oi56 0217 0279 o34o 0401 0462 0324 61 l 709 0646 0707 0769 o83o 0891 0932 1014 1075 u36 1 197 61 710 85i258 1320 i38i U42 i5o3 1 564 i625 1 686 1747 1809 61 711 1870 193 1 1992 2o53 2114 2175 2236 2297 2358 2419 61 712 2480 2541 2602 2663 2724 2 7 85 2846 2907 2968 3029 61 7 i3 3090 3i5o 321 1 3272 3333 33 9 4 3455 35i6 3577 363 7 61 7i4 36 9 8 3759 3820 388i 3941 4002 4o63 4124 4i85 4245 61 7 i5 43o6 4367 4428 4488 4549 4610 4670 473i 4792 4852 61 ■ 716 4913 4974 5o34 5095 5i56 52i6 5277 5337 53 9 8 5459 61 717 5019 558o 564o 5701 5 7 6i 5822 5882 5943 6oo3 6064 61 718 6124 6i85 6245 63o6 6366 6427 6487 6548 6608 6668 60 719 6729 6789 685o 6910 6970 703 1 7091 7162 7212 7272 60 720 85 7 332 7 3 9 3 7453 7 3i3 7374 7634 7694 7755 78i5 7875 60 721 7 9 35 7 Q9 5 8o56 8116 8176 8236 8297 835 7 8417 8477 60 722 8537 85 97 865 7 8718 8778 8838 8898 8g58 9018 9078 60 7 23 9 i38 9,98 9258 9 3i8 9379 9978 9439 9499 9559 Q619 9679 60 724 97 3 9 9799 9 85o 99.8 o5i8 ••38 ••98 •i58 •218 •278 60 720 86o338 o3 9 8 0458 0378 0637 0697 0737 0817 0877 60 726 0937 0996 io56 1116 1 1 76 1236 1295 i355 i4i5 U75 60 727 1 534 1594 1 654 17U 1773 1 833 i8 9 3 2489 io52 2349 2012 2072 60 728 2l3l 2191 225l 23lO 2370 243o 2608 2668I 60 729 2728 2787 2847 2906 2966 3o25 3o85 3i44 3204 3263 60 73o 8o3323 3382 3442 35oi 356i 362o 368o 3739 3 799 3858 5 9 7 3i 3917 3977 4o36 4096 4i55 4214 4274 4333 4392 4g85 4452 5 9 732 431 1 4570 463o 4689 4748 4808 4867 4926 5o45 5 9 7 33 5io4 5i63 5222 5282 534i 5400 5459 55: 9 5578 563 7 59 7 34 56 9 6 6287 5 7 55 58i4 58 7 4 5 9 33 5o 9 2 6583 6o5i 6110 6169 6228 5 9 735 6346 64o5 6465 6524 6642 6701 6760 6819 5 9 736 6878 6937 6906 7o55 7114 7173 7232 7201 735o 7409 7998 5 9 7 3 7 7467 1626 7385 7644 8233 77°3 7762 83 5o 7821 7880 7939 5 9 738 8o56 8u5 8174 8292 8409 8468 8527 8586 5 9 7 3 9 8644 8703 8762 8821 8879 8 9 38 8997 9o56 9u4 1 9H3 59 740 869232 9290 9349 9408 9466 9325 9 584 9642 9701 9760 5 9 74i 9818 9877 9935 9994 ••53 •in • 170 •228 •287 •345 s 742 870404 0462 o52i 0379 o638 0696 1281 0755 o8i3 0872 0930 i5i5 743 0989 1047 1 106 1 164 1223 1 339 i3 9 8 1 456 5P 744 1073 i63i l6qo 1748 1806 i865 i 9 23 1981 2564 2040 2098 58 745 2i56 22l5 2273 233i 2389 2448 25o6 2622 2681 58 746 2 7 3 9 2797 2855 2913 2972 3o3o 3o88 3i46 3204 3262 58 747 332i 3379 3437 3 49 5 4076 3553 36u 3669 2727 3 7 85 3844 58 748 3902 3 9 6o 4018 4i34 4192 425o 43o8 4366 4424 58 749 4482 4540 45 9 8 4656 47U 4772 483o 4888 4Q45 5324 5oo3 58 700 875061 5i 19 56 9 8 5177 5235 5293 535i 5409 5466 5582 58 7 5i 564o 5756 58i3 58 7 i if 9 6307 5987 6o45 6102 6160 58 702 6218 6276 6333 6391 6449 7026 6564 6622 6680 6 7 3 7 58 ^ 3 6 79 5 6853 6910 6968 7083 7U1 7199 7256 73i4 58 7 C 5 4 7 3 7 i 7429 7487 7344 7602 765 9 8234 77n 7774 8349 7832 7889 58 7 55 8522 8004 8062 8119 8177 8292 8407 8464 ?7 7 56 8579 863 7 8694 8 7 52 8809 8866 8924 8981 9039 P 7 58 9096 9 i53! 921 1 9268 9325 9383 9440 9497 9555 9612 5 J 9609. 9726! 9784 9841 9898 q 9 56 0328 ••i3 ••70 •127 •i85 i 1 75 9 880242 j 0299 s °356 o4i3 0471 o585 0642 0699 o 7 56 57 N. | 1 j 2 3 4 5 J_ '- 7 8 _9__ D. A TABLE OF LOGARITHMS FROM 1 to 10,000 13 N. 760 1 1 > I'M 5 | 6 | 7 j 8 9 ! I>- 880814 0871 0928 0985, 1042 1099' n56 i2i3 1271 1328' 57 761 i3S5 1442 1499 i556 i6i3 1670; 1727 1784 1841 1898 57 762 1955 2012 2069 258i| 2638 2126: 2i83 2240 2297! 2354 241 1 2468 5 7 763 2523 269a. 2752: 2809 2866; 2923, 2980 3o3 7 57 764 3093 3i5o 3207 3264I 332i 1 3377 3434 34qi 354S 36o5 ^7 760 366 1 3718 3 77 5 3832; 3888 3^45J 4002! 40D9I 41 15 4172 57 766 4229 4285, 4342 4399' 4455 43i2i ^569 4625 4682 4739 57 768 4795 4852 4909 4965. 5o22 5078! 5i35. 5192 5248 53o3 57 536 1 5418 5474 553i. 5587 5644' 5700 5757J L8i3 58 7 o 57 769 5926 5q83 6039 6096, 6i32 6209 6265 632i! 6378 6434 56 77c 886491 6547 6604 6660! 6716; 6773: 6829 6885, 6942 6998 56 771 7004 71 1 1 7167 7223; 7280 7336 7392 7449 75o5 756i 56 772 7617 7674 7730 7786 7842I 7898: 7955 801 1 8067 8123 56 77 3 8179 8236 8292 8348 8404 8460 8016 8573 8629 8685 56 774 8741 8797 8853 8909 89651 9021 9077 9134 9!9° 9246 56 775 9J02 9358 9414 9470 9026: 9582J 9038 ••3o ••86 l »i4i # I97 9 6g4 ! 9730 9806 56 776 9862 9918 9974 •253 »3o9 •365 56 777 890421 0477 o533 o589j 0645 1 070OJ 0760 0812 0868 0924 56 778 0980 io35 1091 1147J i2o3' 1259 1 i3i4 1370 1426 1482 56 779 :53 7 1593 1649 I7o5 1760 1816 1872 10281 1983 2o3g 56 780 892095 2 1 DO 2206 2262 23i 7 23 7 3 24:9 2484I 254o! 25o5 56 781 265 1 2707 2762 2818 2S7I 2929 298:1 3o4oj 3096 3i5i 56 782 3207 3262 33i8 33 7 3 3429 1 3484 33^0, 35g5 365i 3 7 o6 56 783 3762 38i 7 38 7 3 3928 3984' 4o39 4094 4;5o 4203 4261 55 784 43i6 43 7 i 4427 4482 4338, 4593 4648 4704 4759 4814 55 785 4870 4925 49^0 5o36 5ooi 5i46 5201 5257 53 1 2 5367 55 786 5423 5478 5533 5588 5644 5699 5734 58o 9 ' 5864 5920 55 787 5 ?7 5 6o3o 6o85 6140 6iq5 62DI 63o6 636i| 6416 6471 55 788 6326 658 1 6636 6692 6747, 6802 6837 691 2 j 6967 7022 55 789 7077 7]32, ?I 8 7 7242 7297; 7332 7407 7462! 7317 8012 8067 7572 8122 55 790 897627 7682 ( 77 3 7 823i 8286 7792 7847; 79 02 7957 55 791 8176 834i 83o6 845i 85o6 856i! 86i5 8670 55 792 8^25 8780 8835 8890 8944; 8999 9054 9109 9164 9218 55 7 9 3 927,3 9328 9 383 9437 9492; 9 5 47 9602 9656 971 1 9766 55 794 9821 9873 9930 9985 ••3g »*94 •i4g •2o3 »258 •3l2 55 7 9 5 900367 0422 0476 033 1 o586 0640 0693 0749 0804 0859 1404 55 796 0913 0968 1022 1077 ii3i| u 86 1240 1295 1349 55 797 1458 i5i3 1567 1622 1676 i 7 3i 1783, 1840 1894 1948 54 798 2003 2057 2112 2166 2221 2275 232Q 2384 2438 2492 54 799 2547 2601 2655 2710 2764 1 2818 2873 2927, 2981 3o36 54 800 903090 3 144 3199 3253 33o 7 ; 336i 34i6 3470 3524 35 7 8 -54 801 3633 368 7 3741 3 79 5 4337 3849 3904 3958 4012 4066 4120 54 802 4i74 4229 4283 43gi i 4445 4499 4553 4607 4661 54 8o3 4716 477° 4824 4878 4932 4986 5o4o 5094 5 1 4 s 5202 54 804 5256 53 10 5364 5418 5472- 5526 558o 5634 5688 3742 54 8o5 5796 585o 5904 5g58 6012 6066 61 19 6173 6227 6281 54 806 6335 638 9 6443 6497 655 1 1 66o4 6658 6712 6766 6820 1 54 807 808 6874 6927 6981 7o35 70S9 7U3 7196 7250 73o4 73581 54 74i 1 7465 7319 7 5 7 3 8110 7626: 7680 7734 7787 7841! 7895, 54 8270 8324 83 7 8 843 11 54 809 7949 8002 8o56 8i63| 8217 810 908485 8539 85g2 8646 8699: 8 7 53 8807 | 8860 8c 14 8967! 54 8n 9021 9074 9128 9181 9235 9289 9342 9396 9449 93o3 34 812 9556 9610 9663 9716 9770 9823 9877 0411 993o 9984 ## 3 7 53 8i3 910091 0144 0197 025l o3o4' o338 0464 o5i8, o5 7 i 53 814 0624 0678 0731 0784 o838, 0891 0944 0998 io5i: 1 1 04 53 8i5 n58 1211 1264 i3i7 i3 7 i 1424 1477, i53o i584: 1637 53 8i5 1690 1743 I7Q7 i85o 1903 1936, 2009 2o63 21 16 2169 53 8l l 818 2222 2275 2328 238i 2435 2488 254i 2594; 2647 1 2700 2066 3oiqI 3o72 ; 3i25 3178 3a3i 53 2 7 53 2806 285g 2913 53 819 3284 333/ 3390 3443 34o6' 3549 36o2i 3655 3708 3761 53 N. 1 ~?~ 3 1 4 ! 5 | 6 _J.1. 8 ._ ~?~[}>Z 14 A TABLB OF LOGARITHMS FROM 1 TO 10,0OU. N. 1 « ! 3973 4 | 5 6 ! 7 8 | 9 | r>. 820 9i38i4 386 7 3920I 4026; 4079 4i32| 4184 4237 4290J 53 821 4343 4396 4449 45o2 45551 4608! 4660! 4713 4766! 4819! 53 822 4872 4925 4977I 5o3o 5o83! 5i36i 5i8g ! 524i 52 9 4 5347 53 823 5400 5453: 55o5 5558 56iil 5664! 5716 5769 5822 5875 53 824 5 9 2 7 5980 1 6o33 6o85 6i38 6191I 6243 6296 6349| 6401 53 823 6454 65o7i 655g 6612 6664' 6717! 6770 6822 6875J 6927 53 826 6980 7o33 7083 7i38 71901 7243 7295; 7348 7400 7453 53 827 75o6 7558! 7611 7663 77i6, 7768 7820' 7873 82 9 3 8345 8397 7925; 7978 5 2 828 8o3o 8o83: 8i35 8188 824o ; 845oj 85o2 52 82Q 8555 8607. 8659 8712 8 7 64 ! 8816 8869 1 8921 8973! 9026 52 83o 919078 9i3o 9183 9235 9287; 9340' 9392! 9444 9496; 9549 52 83 1 9601 9653 9706 97 58 9810 9862! 9914 9967 ••i 9 -.7! 52 832 920123 0176 0228 0280 o332 o384 o436 0489 0341 1 0593 52 833 o645 0697 0749 0801 o853 0906 0958' 1010 1062 1114 52 834 1166, 1218 1270 1322 1374 1426 1478 1 i53o 1 582 1 634 52 835 1686 1738 1790 1842 1894 1 946 1998; 2o5o 2102 2 1 54 52 836 2206 2258 23l0 2362 24U 2466 23l8' 2570 2622 2674 52 837 838 2725 2777 2829 2881 2933 2985 3o37 3089 3i4o 3io2 52 3244 3296 334« 3399 345i! 35o3 3555 36o7| 3658, 3710 52 83 9 3762 38i4 3865 3917 3969' 4021 4072 4124 4176 4228 52 840 924279 4 33 1 4383 4434 4486 4538 4589 4641 4693I 4744 52 841 4796 4848 4899 5364 54 1 5 49^1 5oo3 5o54 5io6 5i5~i 5209 1 5261 52 842 53i2 5467 55i8 55 7 o 5621 • 5673 5725 5776 52 8i3 5828 5S79 5 9 3i 5982 6o34 6o85 6i37 6188! 6240 6291 5i 844 6342 63g4 6445 6497 6548 6600 665 1 6702! 6754 68o5 5i 845 6857 1 6908 6959 701 1 7062 71 1 4 7 165 7216 7268 7319 5i 846 7 3 7°| 7422 7473 7524 7576 7627 7678 77 3o 778i 7832 5i 847 7883; 7Q 35 7986 8o3j 8088 8140 9191 8242 8293 8345 5i 848 83 9 6' 8447 8498 8549 8601 8652 8703 8754 88o5 885 7 5! 849 8908 j 8959 9010 9061 9112 9163 92 1 5 9266 9317! 9 368 5i 85o 929419; 9470 9 521 9572 9623 96-74 9725 9776 9827 9879 5i 85i 9930 1 g 9 8l «»32 ••83 •i34 # i85 •236 ^287 •338 •38 9 5i 852 930440 0491 o542 0592 0643 0694 0745 0796 0847 0898 5i 853 0949 1000 io5i 1 102 1 1 53 1 1204 :254 i3o5 i356 1407 5i 854 1458 1 509 i56o 1610 1661 17 1 2 1763 1814 i865j igi5 5i 855 1966 2017 2068 2118 2169' 2220 227I 2322 2372 2423 5i 856 2474 2524 2575 2626 2677 2727 2778! 2829 3285- 3335 2879 1 2g3o 5i 857 2981 3o3i 3o82 3i33 3i83 3234 3386 343 7 5i 858 3487 3538 3589 363g 3690 3740 3791] 384i 3892 3 9 43 5i 85 9 3993 I 4044 4094 4145 4195 4246 4296 4347 4397 4448 5i -860 934498 4549 4599 465o 4700 475i 4801 4852 4902 4953 5o 861 5oo3 ! 5o54 5 1 04 5i54 52o5 5255 53o6 5356 5406 5457 5o 862 5507 5558 56o8 5658 5709 5759 5809 586o 5gio 5g6o 5o 863 601 1 6061 61 ii 6162 6212 6262 63 1 3 63631 641 3' 6463 5o 864 65 14 1 6564 6614 6665 6715 6765 68 1 5 6865 6916 6966 5o 865 7016 7066 71 17 7167 7217: 7267 7317: 7367 7418 7468 5o 866 75i8 7568 7618 8069' 81 19 7668 7718, 7769 7819 7869 7919 7969 5o 867 8019 8169 8219 8269 8320 83701 8420 8470 5o , 868 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 5o 869 9020 9070 9120 9170 9220 9270 9320 9369 9 4iQ }46q 5o 870 939519 1 9569 9619 9669 9719 9769 9819J 9869 9918 5968 5o 871 940018 0068 01 18 0168 0218 02671 °3i] 0367: 0417I 0467 0716 0765I 0810J o865' 091 5 0964 5o 8,2 o5i6 o566 0616 0666 5o 873 1014 1064 1 1 1 4 n63 i2i3 1263! i3i3 ! i362; 1 4-i 2 1 1462 5o 874 1 5 1 1 1 56 1 161 1 1660 1710 1760 1809 1859 1909 1958 5o 875 200S 2o58 2107 2157 2207j 2256 23o6 2355 24o5j 2455 5o 876 25o4 2554 26o3 2653 2702I 2752 2801 ! 285i 2901 1 2950 5o ^ 3oooi 3049 3099 3i48 3i 9 8 3247 3297 3346 33g6J 3445 i 5 9 3495 3544 3590 3643! 36o2 ! 3742 3791! 384i 38901 3g3g 4285 4335 43841 4433 ^ 9 879 3989 4o38 4o88| 4i3-»' 4186 4236 Ls. . N ' | 1 1 2 1 3 | 4 j 5 b | 7 i 8 1 9 : !.?•__ A TABLE OF LOGARITHMS FROM 1 TO 10,000. IS N. j ! .! 2 3 ! 4 * 1 6 1 7 _!_[ 9 1 880 9444H3 4532 458i 463 1 468o : 4729' 4779 1 4828| 4877; 4927 1 881 4976, 5o2D| 3074 5 1 24 5l73 5222 5272 532 1 1 5370 5419 882 0469! 55i8 5567 56i6 5665 57i5 5764 ; 58i3| 5862 5912 883 0961 6010 6059 6to8 6157, 6207 6256 63o5' 6354 64o3 884 64D2 1 65oi 655 i 6600 6649' 0DQ 8 6747 6796: 6845, 6894 885 6943 6992 7041 7090 7140, 7189 7238 7287; 7336 7385 886 7434 ; 7483 7532 758i 763o 7679 8119' 8168 7728 7777' 7826 7875 887 888 7924 : 7Q73 8022 8070 8217 8266 83 1 5 8364 84i3, 8462I 85n 856o 8609 8657 8706 8 7 55, 8804 8853 889 8902 949390 895 1 8999 9048 9097 9146 9195 9244 9292 9341 890 9439! 9488! 9536 9585 9634 9683 9731, 9780 9829! 891 9878, 9926 9975! ••24 ••73 •I2t •170 '219 ^267 »3i6 892 95o365- 04 14I 0462 o5i 1 o56o 0608 0657 ' 0706 0754 o8o3. 693 o85i| 0900! 0949 0997 1046 1095 1 i43 1192 1240 1289! 894 i338 i386 1435 1483 i532 i58o 1629 16771 1726 1775I 895 1823I 1872 1 92o| 1969' 2017! 2066 2114 2i63 2211: 2260 896 23o8j 2356 24o5, 2 453 2502 255o 2599 2647' 2696 2744 3o83: 3i3i! 3i8o 3228 897 2792I 2841 2889 1 2933, 2986 3o34 898 1 3276I 3325 33 7 3j 342 1 3470 35i8 3566 3(Si 5: 3663 3711 899 3 7 6c 38o8 3856 3905: 3953 4001 4049' 4098, 4146 4194 900 954243 4291 4339 438 7 4435 4484' 4532! 458o, 4628 4677 901 4725 4773 4821 4869 4918 4966 5oi4l 5o62i 5no 5 1 58 902 6207 5255 53o3 535i 53 99 588o 5447 54 9 5' 5543 1 5592 5640 903 5688 5736 5 7 84 5S3 2 i 5928 5 97 6 6024 6072 6120 904 6168 6216 6265 63i3 636 1 64091 6888 6457 65o5! 6553j 6601 9o5 6649 1 6697 6745 6793 6840 6 9 36 6984! 7032 i 7080 906 71281 7176 7224 7272 7320 7368 74i6 7464', 75i2; 7559 907 7607 ! 7655 77 o3 775i 7799 7847 7894 7942 7990 8o38 908 8086 j 8 1 34 8181 8229 8277 8325 83 7 3 8421 8468-, 85 1 6 909 8564 8612 865g 8707 8 7 55 88o3 885o 8898 8946 1 8994 910 959041 1 9089 9137 9 i85 9232 9280 9328 93]5 9423 9471 911 95i8i 9566 9614 9661 9709 97D7 9804 9852! 9900' 9947 912 9995 ••42 ••90 •i38 •i85 •233 •280 •328 •376, »423 913 96047 1 o5i8 o566 o6i3 0661 0709 0756 0804 o85i 0899 914 0946 0994 1041 1089 i/36 1 184 I 23 1 1279 i326 ! 1374 gi5 1421 1469 i5i6 1 563 161 1 1658 1706 1753 1801 j 1848 916 i8 9 5 1943 1990 2038 2oS5 2l32 2180 2227 2275i 2322 91 I 2369 2417 2464 25ll 2559 2606 2653 2701 2748| 2 79 5 918 2843 2890 2937 2985 3o32 3079 3i26 3n4 322I 1 3268 919 33i6 3363 34io 3457 35o4 3552 35 9 9 3646 36 9 3; 3741 920 963788 3835 3882 3929 3977 4024 4071 4118 4i65 4212 921 4260 43o7 4354 1 4401 4448 4495 4542 45go 4637 4684 922 473i 4778 4825 4872 4919 ! 4966 5oi3 5o6i 5 1 08 5i55 923 5202 5249 5296 5343 53 9 o ! 5437 5484 553 1 5578 5625 924 5672 5719 5766 58 1 3 5S6o 5907 5 9 54 6001 6048 6og5 926 6l42 6189 6658 6236 6283 6329! 6J76 6423 6470 65i 7 6564 926 661 1 6700 6752 6799 ! 6845 6892 6939 6986 7 o33 927 7080 7127 7173 7220 7267| 73i4 736i 7408 7454 75oi 928 7548 7595 7642 7688 77 35| 7782- 7829 7 8 7 5 7922 8343 83go 7069 929 8016 8062 8109 8i56 82o3i 8249 8296 8436 93o 968483 853o 8576 8623 8670 8716 8763 8810 8856 8903 93i 8950 8996 904^ 9090 9i36 9183 9229 9 6 9 5 9276 g323 9369 932 94l6 9 463 95oc 9556 9602 9649 9742 9789 9 833 933 9882' 9918 997: ••21 ••681 •uA •161 •207 •254: - 3oo 934 970347; o3 9 3 044c 0486 o533J 0579 0626 0672 0719 0765 9 35 0812 o858| 090; k 0931 0997 i 1044 1090 u37 u83 1229 9 36 1276 i322i i36c > I4i5 1461 j i5o8 1 1 534 1601 1647 1 6^3 SIIOJ 2137 III 1740' 1786 i83s 1879 1925 1971 2383 2434 2018 2064 22o3 2249 2295, 2342 2481 2527J 2573 ; 2619 9 3g 2666! 2712! 2758) 280/ ! 285 1 i 2897 L294 3 2989, 3o35 3082 1 *. O 1 «■ 1 2 ! 3 ! 4 i s | 6 rr ! 8 i 9 D. 49 49 49 49 49 49 49 49 49 49 49 49 49 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 46 46 46 46 46 46 IX 10 A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 940 1 2 3 4 5 6 7 8 9 'DTI 46 973128 3i74 3220 3266 33i3 335 9 34o5 345 1 3497 3543 941 3D90 3636 3682 3728 3774 3820 3866 3913 3909 400 5 46 942 4o5i a 4143 4189 4235 4281 4327 4374 4420 4466 46 943 45i2 4604 465o 4696; 4742 4788 4834 4880 4926 46 y44 4972 5oi8 5o64 5no 5i56j 5202 5248 5294 5 7 53 534o 5386 46 945 5432 5478 5524 5570 56i6| 5662 5707 5799 6258 5845 46 946 5891 635o 5o3 7 63 9 6 6854 5 9 83 6029 6075 6121 6167 6625 6212 63o4 46 947 6442 6488 6533 65 79 6671 6717 6 7 63 46 948 6808 6900 6946 6992 7037 7083 7129 7175 7220 46 949 7266 7312 7 358 74o3 7449 7495 754i 7586 7632 8089 7678 46 95o 977724 7769 8226 7 8i5 8272 7861 7906 7952 8363! 8409 8454 8o43 8i35 46 95i 8181 83i 7 85oo 8546 85 9 i 46 952 8637 8683 8728 8774 8819 1 8865 9275, 9321 891 1 8 9 56 9002 9047 46 9 53 9093 9i38 9184I 9230 9366 9412 9457 95o3 46 954 9048 9594 9639' 9685 973o, 9776 9821 9867 &; 99 58 46 955 980003 0049 0094 0140 01 85] 023 1 0640' o685 0276 0322 0412 45 9D6 0458 o5o3 o549 o5g4 0730 O776 0821 0867 45 9D7 0912 0957 ioo3 1048 io 9 3 n3g 1 184 I229 I2ff5 l320 45 9 58 1 366 i4n 1 456 i5oi i547 ( 1592 1637 1 683 1728 1773 45 9 5 9 1819 1864 1909 i 9 54 2000, 2o45i 2090 2i35 2181 2226 45 960 982271 23i6 2362 2407 2452 j 2497 2543 2588 2633 2678 45 961 2723 2769 2814 285 9 2904 2949 2994 3o4o 3o85 3i3o 45 962 3i75 3220 3265 33io 3356, 34oi 3446 3491 3536 358i 45 963 3626 36 7 i 3716 3762 3807! 3852 38 97 3942 4392 3987 4o32 45 | 964 4077 4122 4167 4212 4257 43o2 4347 4437 4482 45 ! 965 4527 4572 4617 4662 4707 | 4752 4797 4842 4887 4g32 45 966 4977 5o22 5067 5i 12 5lD7 5202 5247 5292 533 7 5382 45 967 968 5426 5471 55i6 556i 56o6 565 1 5696 5741 5 7 86 583o 45 5875 5920 5965 6010 6o55 ( 6100 6144 6189 6234 6279 45 969 6324 6369 64i 3 6458 65o3 6548 65 9 3 6637 6682 6727 45 970 986772 6817 6861 6906 6951 6996 7040 7o85 7i3o 7175 45 97' 7219 7264 7 3o 9 7353 7 3 9 8 7443 7488 7532 7577 7622 4^ 973 7666 8ii3 8i5 7 77 56 8202 7800 8247 7845 8291 7890 8336 it 8425 8024 8068 8470 85i4 43 45 974 855o 9005 8604 8648 8693 9094 91 38 9539 g583 8 7 3 7 8782 8826 8871 8916 8960 45 973 9049 9i83 9227 9272 93i6 9361 94o5 45 976 945o 9494 9628 9672 9717 9761 9806 985o 44 977 978 9895 9939 9983 ••28 ••72 »ii7 •161 •206 •25o •294 44 990339 o3b3 0428 0472 o5i6 o56i o6o5 o65o 0694 0738 44 979 0783 0827 0871 0916 1359 0960, 1004 1049 1093 1 137 1182 44 980 991226 1270 i3i5 Uo3 1448 1492 1 536 i58o l6 2 5 44 981 1669 1713 1758 1802 1846, 1890 1935 1979 2023 2067 44 982 2 1 1 1 2i56 2200 2244 2288J 2333 2377 2421 2465 25og 44 9 83 . 2554 2598 2642 2686 2 7 3o 2774 2819 2863 2907 3348 2g5i 44 984 2995 3o3g 3o83 3127 3172' 3216 3260 33o4 33 9 2 44 9 85 3436 3480 3524 3568 36i3 3657 3701 3 7 45 3789 3833 44 986 38 7 7 3g2i 3965 4009 4o53i 4097 4493 4537 4Ui 4i85 4229 42 7 3 44 987 43i7 436 1 44o5 4449 458 1 4625 4669 47 1 3 5io8i 5i52 44 988 4757 4801 4845 4889 4933, 4977 5021 5o65 44 989 5196 5240 5284 5328 53 7 2 5416 5460 55o4 5547I 5591 5986) 6o3o 44 990 9956351 5679 5723 5767 58n 5854 58 9 8 5 9 42 638o 44 991 6074 6117 6161 62o5 6249 6293 6337 6424! 6468 44 992 65i2 6555 6599 6643 6687 6 7 3 1 6774 6818 6862 6906 44 993 6949 1 6993 7037 7386 743oj 7474 7080 7124 7168 7212 7255 7209 7736 7343 44 994 7 5i 7 756i 76o5 7648 8o85 7692 8129 7779 8216 44 99 5 7823 7867! 7910 8259 83o3; 8347 86 9 5 8 7 3 9 ! 8782 9i3i 9174 9218 79D4 83 9 o 7998] 8041 8434 8477 8172 44 996 852i 8564 8608 8652 44 997 8826 8869 89 1 3 8 9 56 93o5; 9348 9392 9000 9043 9087 44 998 9261 9435 9479 9522 99 1 3 99 5 7 44 999 N. 9565 9609 9652 9696 9739^ 9783| 9826 9870 43 1 2 3 4 L 5 _ L 6 i ?._. 8 9 ! D. £ TABLE OF LOGARITHMIC SINES AND TANGENTS FOR EVERY DEGREE AND MINUTE OF THE UUADKANT. Kemark. The minutes in the left-hand column of 3ach page, increasing downwards, belong to the de- grees at the top ; and those increasing upwards, in the right-hand column, belong to the degrees below 18 (0 DEGREES.) A TABLE OF LOGARITHMIC Sine 0- 000000 D. Cosine \ D. Tang. D. Cotang. 10-000000 0' 000000 Infinite. 60 I 6-463726 5017.17 000000 -00 6-463796 5017 H i3 • 53627 i 5 9 2 764756 2934- 85 000000 1 "00 764756 2934 83 235244 58 3 940847 2082- 3i 000000 1 '00 940847 2082 3i o59i53 57 4 7.065786 i6i5- J 7 OOOOOOI '00 7.065786 i6i5 17 12-934214 56 5 162696 i3io. 68 OOOOOO "OO 162696 i3io 1 1 1 5 11 83.7304 758122 55 6 241877 11 15- 75 9.999999 -01 241878 54 I 308824 966. 53 999999 .01 3o8825 852 691 175 633 1 83 53 3668i6 852- 54 999999 -01 366817 54 52 9 417968 762- 63 999999 -oi 41797° 762 63 582o3o 5i 10 463725 689. 88 999998 •01 463727 689 88 536273 12-494880 427091 5o 1 1 7-5o5ii8 629- 81 9.999998 •01 7«5o5i2o 629 81 % 12 542906 579. 36 999997 •OI 542909 5 J? 33 i3 577668 536- 4i 999997 •01 577672 536 42 422328 47 14 609853 499- 38 999996 •01 609857 499 s 390143 46 i5 63 9 8i6 467- 438- U 999996 •01 639820 467 438 36oi8c 45 16 667845 81 999995 999995 •01 667849 82 332 1 5i | 44 J2 694173 41 3- 72 •01 694179 4i3 73 3o582i 43 718997 391 • 33 999994 •01 719004 391 36 280997! 42 257516' 41 19 742477 3 7 i- 27 999993 •01 742484 3 7 i 28 20 764754 353 i5 999993 ■01 764761 35i 36 235239 40 21 7-785943 806146 336- 72 9-999992 •01 7-780951 8061 55 336 73 12- 214049! 39 22 321- t 999991 •01 321 76 193845 38 23 82545i 3o8 999990 •01 825460 3o8 06 174540 37 24 843934 295 47 999989 999988 •02 843944 2q5 49 i56o56 36 25 861662 283 88 •02 861674 263 90 138326 35 26 878695 2 7 3 H 999988 •02 878708 2 7 3 .8 1 21292 34 27 895085 263 23 999987 •02 895099 263 2D 1 0490 1 33 28 910879 926119 253 99 999986 •02 91 0894 254 01 089106 32 29 245 3S 999985 •02 926134 245 40 073866 3i 3o 940842 23 7 33 999983 •02 940858 23 7 35 059142 3o 3i 7.955082 229 80 9-9999 82 , '° 2 7 -955 100 229 81 1 2 • 044900 2 32 968870 222 73 999981 •02 968889 222 75 o3 1 1 1 1 33 982233 216 08 999980 •02 982253 216 10 017747 27 34 995198 8-007787 209 81 999979 •02 995219 209 83 004781 26 35 2o3 90 999977 •02 8-007809 020045 203 92 11-992191 979965 25 36 020021 198 3i 999976 •02 198 33 24 37 031919 043 5o 1 io3 02 999975 •02 031945 193 188 o5 9 68o55 23 38 188 01 999973 •02 043527 o3 956473 22 39 054781 i83 25 999972 •02 054809 1 83 27 945191 21 40 . 065776 178 72 999971 •02 o658o6 178 74 934194 20 41 8-076500 174 41 9-999969 •02 8-07653i '74 44 11-923469 9i3oo3 :? 42 086965 170 3 1 999968 •02 086997 170 34 43 097 1 83 166 39 999966 •02 097217 166 42 902783 n 44 107167 162 65 999964 •03 107202 162 68 883o37 16 45 1 16926 1 5 9 08 999963 •03 1 16963 159 i55 10 i5 46 1 2647 1 1 55 66 999961 •o3 i265io 68 873490 14 47 i358io l52 38 999959 -o3 i3585i l52 41 864149 i3 48 144953 149 24 9 999 58 -o3 144996 i53 9 5 2 149 27 855oo4 12 49 153907 146 22 999956 • o3 146 •27 846048 11 5o 162681 143 33 999954 • o3 162727 143 36 837273 10 5i 8-171280 140 54 9-999952 • o3 8-171328 140 57 11-828672 I 52 179713 1 3 7 86 9999 5 ° • o3 179763 i88o36 137.90 135-32 820237 53 187985 i35 29 999948 • o3 811964 7 54 196102 1 132 80 999946 • o3 1 961 56 132-84 8o3844 6 55 204070 i3o 41 999944 -o3 204126 i3o«44 795874 788047 5 56 2ii8 9 5 1 128 10 999942 -o4 21 1953 128-14 4 11 219581 1 125 •3 7 999940 -04 219641 125-90 780359 772800 3 227134 123 •72 999938, >o4 227195 123.76 2 5 9 234557 j 121 ■64 999936 -o4 234621 121-68 765379 r 60 24i855 1,9 •63 999934! -o4 1 241921 i Cotang. 119-67 758079 Cosine 1 *>• Sine |89° D. Tang. M. i? rXES AND TANGENTS (1 DEGREE.) 1 o Sine D. , Cosine | D. Tung. D. Cotang. | 8 "24i855 119-63 9. 999934 1 .04 8-241921 119 67 11-758079 60 i 249033 117-68 999932 j • 04 249102 H7 72 7 50898 I 5g 743835 58 a 256094 u5-8o 999929; • 04 256i65 113 84 3 263042 113-98 9999 2 7| • 04 263 1 1 5 H4 02 736885 37 4 269881 112-21 999923) •04 269936 112 25 730044 56 5 276614 110-30 999922I • 04 276691 I 10 54 723309; 55 6 283243 io8-83 999920 • 04 283323 10b 87 716677 54 2 289773 107-21 999918J • 04 289856 IO7 26 710144I 53 296207 io5-65 999913; •04 296292 io5 70 703708 52 9 302546 io4-i3 99991 3 .04 3o2634 104 18 6 9 7366| 5i to 308794 102-66 999910; • 04 3o8884 102 70 691 1 16 30 is 8- 3 14904 101 -22 9-999907; • 04 8-3i5o46 101 26 II-684954I 4Q 13 321027 99-82 999903; .04 321 122 99 87 678878 48 IJ 327016 98-47 999902; • 04 3271 14 98 5i 672886 47 14 332924 97-14 999899! •o5 333o25 97 19 666 97 5l 46 i5 33o>753 9 5 -86 999897 | •o5 338856 9 5 90 661144I 45 16 344504 94-60 999894! • o5 344610 94 63 6553 9 o 44 \l 35oi8i 93-38 999891! •03 350289 9 3 43 6497 1 1 43 355783 92-19 91 -o3 999888 •o5 355893 36i43o 92 24 644 1 o5 42 x 9 36i3i5 999885 •o5 1 08 638370 41 20 366777 89-90 88-80 999882 •o5 3668 9 5 £ 633io5 40 21 8-372171 9.999879 • o5 8-372292 11-627708 3o 622378 38 22 377499 87-72 9998761 •03 377622 37 77 23 382762 86-67 999873 •o5 382889 86 72 617m 37 24 387962 85-64 999870 • o5 388o 9 2 393234 85 70 611908 36 25 393101 84-64 999867; •o5 84 70 606766 35 26 398179 83-66 999864 • o5 3 9 83i5 83 7i 6oi685 34 2 7 4o3 1 99 82-71 999861 •03 4o3338 82 76 596662 33 28 408161 8o-86 999858 "•o5 4o83o4 81 82 591696 32 2q 4i3o68 999854 • o5 4i32i3 80 9i 586787 3i 3o 417919 79.96 999851 .06 418068 80 02 58i 9 32 3o 3i 8-422717 79-09 78. 23 9.999848 .06 8-422869 427618 $ 14 n .577131 3 32 427462 999844 .06 3o 572382 33 432i56 77.40 76.57 999841 •06 4323i5 77 45 567685 27 34 4368oo 999838 • 06 436962 76 63 563c38 26 35 44i394 75-77 999834 .06 44i36o 75 83 558440 25 36 445941 74-99 99983 1 -06 446110 75 o5 553890 549387 24 37 45o44o 74-22 999827 -06 45o6i3 74 28 23 38 454893 73-46 999823 • 06 455070 73 52 544930 22 3 9 45o3oi 463665 72.73 999820 ■ 06 45q48i 72 79 540319 21 4o 72-00 999816, .06 463849 8-468172 72 06 536i5i 20 4i 8-467985 71-29 9.999812 .06 7i 35 Il-53i828 19 42 472263 70-60 999809 .06 472454 70 66 527546 18 43 476498 69-91 999803 .06 476693 69 98 5233o7| 17 44 480693 69-24 999801 .06 480892 69 6$ 3i 519108 16 45 484848 68-5 9 999797 •07 485o5o 65 5i495o i5 46 488963 67.94 67.31 999793 •07 489170 49325o 68 01 5io83o 14 47 493040 999790 999786 .07 67 38 506750 i3 48 497078 66-69 66-08 .07 497293 66 76 502707 12 49 5oio8o 999782 •07 501298 66 i5 498702 11 5o 5o5o45 65-48 999778; •07 5o5267 65 55 494733 10 5i 8-508974 til 2867 64-89 9.999774 •07 8 -509200 64 96 11-490800 i 52 64-3i 999769 •07 513098 64 39 486902 53 516726 52o55i 63- 7 5 999765 •07 516961 63 82 483o39 7 54 63-i 9 999761 •07 520790 524586 63 26 479210 475414 6 55 524343 62-64 999757 •07 62 72 5 56 528102 62-11 999753 j .07 528349 62 18 47i65i 4 n 53i828 6i-58 999748; •07 532o8o 61 65 467920 3 535523 61-06 999744; •07 535779 61 i3 464221 2 5 9 539186 60 -55 9997- 40 ! •07 53 9 447 543o84 60 62 ;6o553 ; 6o 542819 60-04 999735 1 •07 60 12 456916! Cosino D. Sine | 38° Cotang. D. " Tang~l 20 (2 DEGREES.) A TABLE OF LOGARITHMIC M. , Sine D. Cosine D. Tang. 1 D. Cotang. 11 -456916 60 8-542819 60 -04 9-999735 .07 8-543o84l 60 • 12 i 546422 5 9 - 55 999731 ■07 546691 5 9 62 453309 u 2 549995 553539 5 9 06 999726 -07 500268 u 14 449732 3 58 58 999722J -08 5538i7 66 446i83 57 4 557054 58 11 999717' -08 55 7 336 58 l 9 44 2 664 : 56 5 56o54o 57 65 999713 -08 560828 5 7 73 439172 55 6 563999 ^ l 9 999708' -08 564291 57 27 430709! 54 432273; 53 I 56743i 56 74 9991041 *o8 999699 -08 567727 56 82 570836 56 3o 571137 56 38 428863| 52 9 574214 55 87 999694 -08 999689 J -08 574520 55 £ 425480] 5 1 10 577066 55 44 577877 8-58i2o8 55 422I23i 5o II 8.580892 55 02 9-9996801 -08 55 IC 11-418792 % 12 584193 54 60 999680! -08 5845 1 4 54 68 4i5486 i3 587469 54 *9 999675! -08 587795 54 27 412205 % 14 590721 53 79 999670' -08 59 1 00 1 53 87 408949 i5 5;3 9 48 53 3 9 999665J .08 094283 53 il 405717 402 5o8 45 16 597102 53 00 999660! -08 597492 53 44 \l 6oo33a 52 61 999655J -o8 6oo6tj 52 70 399323 43 6o348 Sine [8T° Cotai tg. D. Tang. M. SINES AND TANGENTfc ^3 degrees/ 21 nsr Sine I D. Cosine D. T;mg. D. Cotang. j 8-718800 4o-o6 9-999404 -ii 8-719396 40-17 1 1 • 280604 60 I 721204 3 9 84 999398 • 11 721806 39-95 278194 5o 2 7 5 79 6: 58 2 723595 3 9 62 9 99 3oi • 11 724204 3g -74 3 720972 3 9 41 999384 • 1 1 726588 39-52 273412 57 4 728337 39 10 999378 • ii 728959 39 -3o 271041I 56 5 730688 38 98 999371 • ii 73i3i 7 39-09 268683 55 6 733027 38 77 999364 •12 733663 38- 89 38-68 266337, 54 7 735354 38 57 999357 •12 735996 264004 53 8 737667 38 36 99935o •12 7383i7 38-48 26l683 52 9 739969 38 16 999343 999336 •12 740626 38-27 259374 5i 10 742239 37 96 •12 742922 38-o 7 257078 5o ii 8-744536 3? 76 9-999329 •12 8-745207 3 7 -8 7 11-254793 4 2 12 746802 37 56 999322 •12 747479 3 7 -68 252521 48 i3 749055 37 37 9993 1 5 •12 749740 37-49 25o26o 47 14 751297 37 n 999308 •12 751989 37-29 24801 1 46 i5 753528 36 98 999301 -12 754227 3-7-10 ^45773 45 16 755747 36 79 999294 •12 736403 36-92 243547 44 17 75 79 n5 36 61 999286 •12 758668 36-73 24i332 43 18 760101 36 42 999279 • 12 760872 36-55 239128 42 19 762337 36 24 999272 •12 763o65 36-36 236q35 4i 20 7645 11 36 06 999260 • 12 765246 36-i8 234754 4o 21 8- 7 e66 7 5 35 88 9-999257 •12 8-767417 36-oo n-232583 3 9 22 768828 35 70 999200 •13 769378 35-83 23o422 38 23 770970 35 53 999 2 42 •13 771727 35-65 228273 37 24 773ioi 35 35 999235 •13 773866 35-48 226i34 36 25 773223 35 18 999227 -13 775 99 5 35-3i 224oo5 35 26 777333 35 01 999220 •i3 77S114 780222 35-14 221886 34 27 779434 34 84 999212 • i3 34-97 34- 80 219778 33 28 781524 34 .67 999205 • i3 782320 217680 32 29 7836o5 34 5i 999197 •i3 784408 34-64 215592 3i 3o 7 856 7 5 34 3i 999189 • i3 786486 34-47 2i35i4 3o 3i 3- -787736 34 18 9-999181 • i3 8- 7 88554 34-3i 1 1 «2i 1446 29 32 789787 791828 34 02 999174 •i3 790613 34-i5 20938T 28 33 33 86 999 1 66 -i3 792662 33-99 33-83 207338 27 34 79385 9 33 70 999158 • i3 794701 205299 26 35 7 9 588i 33 54 999 1 5o • i3 796731 33-68 203269J 25 36 797894 •33 39 999142 • i3 798752 800763 33-52 201248 24 ll 799*97 33 23 999i34 • i3 33-3 7 199237 23 801892 33 08 999126 -13 802765 33-22 197235 22 39 8o38 7 6 32 93 Q99 1 '8 • i3 8o4758 33-07 195242 21 4o 8oo852 32 78 9991 10 • i3 806742 32-92 193208 20 4i 8-807819 32 63 9-999102 • i3 8-808717 32-78 11 • 191 283 J 19 189317 18 42 809777 32 49 999094 •14 8io683 32-62 43 811726 32 34 999086 •14 81 2641 32-48 187359 17 44 8 1 366 7 32 10 999077 • 14 8i458g 32-33 i854n! 16 45 815099 32 o5 999069 •14 816529 32-o5 1 8347 1 | i5 46 817522 3i 91 999061 •14 8 1 846 1 181 539j 14 3 8i 9 436 3i 77 999053 •14 820384 3i- 9 i 179616 i3 821343 3i 63 999044 •14 822298 3i -77 177702 1 12 49 823240 3i 4q 999030 •14 824205 3i-63 175795; 11 173897 10 5o 825i3o 3i 35 999027 •14 826103 3i-5o 5i 8-827011 3i 22 9-999019 •14 8-827992 3i -36 11 -1720081 170126! 8 52 828884 3i 08 999010 •14 829874 3i-23 53 830749 3o f, 999002 •14 83i-48 3i-io 168252 7 54 832607 3o 998993 •14 8336 1 3 3o-o6 166387 6 55 834456 3o 69 998984 •14 835471 3o-S3 164529 5 56 836297 838i3o 3o 56 998976 998967 •14 837321 30-70 162679! 4 u 3o 43 •i5 83 9 i63 3o-57 160837 3 83 99 56 3o 3o 998908 • i5 840908 3o-45 l5q002| 2 59 841774 3o 17 998950 -i5 842825 3o-32 1 57 1 75J I 6o 843585 3o 00 99 8 94i Sine • i5 844644 80-19 i55356| Cosino _D 86° QolMlg. _ Tang. 1 M. _, 2 (i , 3EGREE6 .) A TABLE OF LOGARITHM) M. Sine | 8-843585 D. i 3o-o5 Cosine 1 9-998941! D. Tang. 3o- 19 Co tang. 1 •i5i 8-844644 ii-i55356 60 i 845387 29-92 998932! • i5 846455 30-07 153545, 5o 1 51740! 58 2 847183 29-80 99 8 923, -i5 848260 29-95 3 848971 20-67 998914: •i5 85oo57 29-82 1 4994 3 l 5 7 4 85oi5i 29-55 998905 998896 •i5 85i846 29.70 1 481 54 56 5 85a 525 29-43 •i5 853628 29-58 146372 55 6 834291 29-31 998887 .i5| 8554o3 29.46 1 443971 54 7 856049 29-19 998878 •i5 837171 29-35 142819 s 53 141008, 52 8 867801 29-07 28-96 998869 -i5 858 9 32 29-23 9 85 9 546 99S860 -i5 860686 29-11 139314' 5i 10 86i283 28-84 998801 •i5 862433 29 -cc 1 37567 : 5o 1 1 8-863oi4 2 8- 7 3 9-998841 • i5 8-864n3 28-88 11 135827! 49 1 34094 1 48 12 864738 28-61 998832 •15, 865906 28.77 i3 866455 28 -5o 998823 .16 86 7 632 28-66 132368 47 M 868 1 65 28-39 28-28 998813 •16 869351 28.54 130649 46 i5 869868 998804 •16 871064 28-43 128936 45 16 871665 28-17 998795 ■16! 872770 28-32 127230 44 »2 873255 28-06 998785 .16 874469 28-21 I2553i 43 18 8 7 4 9 38 27-95 998776 ■16 1 876162 28-11 123838 42 '9 876610 27-86 998766 •16 877849 28-00 I22l5l 41 2t> 878285 2 7 - 7 3 998757 •16 879529 27-89 I 2047 1 40 21 8-879949 27-63 9-99 8 747 .16 8-881202 27.79 27.68 II I 18798 3 9 22 881607 27-52 998738 •i6i 882869 117131 38 23 883258 27-42 998728 ■ i6j 88453o 27.58 116470 37 24 884903 886542 27 -3i 998718 •i6| 886i85 27-47 ii38i5 36 25 27-21 998708 •16 88 7 833 27.37 1 12167 35 26 888174 2711 998699 •16' 889476 27-27 no524 34 ^ 889801 27-00 998689 •16 891112 27.17 108888 33 891421 26-00 26-80 998679 •i6| 892742 27-07 107268 32 29 8 9 3o35 998669 •17, 8 9 4366 26-97 io5634 3i 3o 894643 26-70 998659 -17 8 9 5 9 84 26-87 104016 3o 3i 8-896246 26-60 9-998649 •17; 8-897696 26-77 11 -102404 29 32 807842 26-5i 998639 •171 899203 26-67 100797 28 33 899432 26-41 998629 •17 900803 26-58 099197 27 34 901017 26-3i 9986 1 9 - 1 -7 1 902398 •17J 903987 •17 905370 26-48 097602 26 35 902696 26-22 998609 26-38 096013 25 36 904169 26-12 998399 26-29 094430 24 37 905736 26 -o3 998589 998078 •17I 907147 26-20 092853 23 38 907297 25-o3 .17 908719 26-10 091281 22 3 9 908853 25-84 99 8568 •17 910285 26-01 089715 21 4o 910404 25-75 998558 •17 911846 25 -92 088 1 54 20 4i 8-911949 913488 25-66 9-998548 - 1 -7 j 8-913401 25-83 1 1 -086599 \l 42 25-56 998537 -171 914951 25-74 o85o4o o835o5 43 916022 25-47 998527 •17 916495 25-65 17 44 9i655o 25-38 99 85i6 •18! 918034 25-56 081966 16 45 918073 25-29 998506 •18 9 i 9 568 25-47 25-38 080432 i5 46 919391 25-20 998405 •18 921096 078904 14 47 921 io3 25-12 998485 •18 922619 25-3o 077381 i3 48 922610 25-o3 998474 •18 924136 25-21 075864 12 49 924112 24-04 998464 •18 926649 25-12 074351 11 5o 920609 24-86 998453 •18 927156 25-o3 072844 10 5i 8-927100 24-77 9-998442 •18' 8-928658 24-95 24-86 n -071342 I 52 9 2858 7 93oo63 24-69 998431 •18 93oi55 069845 o68353 53 24-60 998421 -i8| 9 3i647 24-78 I 54 93 1 544 24-52 998410 .181 9 33i34 24-70 066866 55 933oi5 24-43 998399 998388 •18 934616 24-61 065384 5 56 934481 24-35 .18! 936093 24-53 06390-7 4 ii 935942 24-27 998377 .18! 9 3 7 565 24-45 o62435 3 937398 9 3885o 24-19 998366 •18 939032 24-37 060068 2 59 24-u 998355 •181 940494 •18I 941932 85°, Cotansr. 24 -3o 069306 i 1 1 o58o48' 6o 940296 24 -o3 D. 998344 24-21 D. _.. ■ Cosine Sine L T an £- : M. SINES AiND TANGENTS. (5 DEGREES.) 23 u~ Sine D. Cosine D. | Tang. D. Cotang. j 8 940296 941738 24 o3 9-998344 •19! 8-941952 24-21 11 '038048' 60 | I 23 % 998333 l 9\ 943404 24 i3 056596 3 2 Q43i74 23 998322 19 944852 24 o5 o55i48 3 944606 23 79 998311 19. 946295 23 97 ©537o5 57 4 946034 23 7* 998300 l 9 947734 23 82 o52266 56 5 947456 23 63 998289 19 949168 23 ©5o832 55 6 948874 23 55 998277 '9 950597 23 74 049403 54 7 95o2«7 23 48 998266 19 952021 23 66 047979 046559 53 8 95169^ 23 40 998255 19 953441 23 60 52 9 953ioo 23 32 99 82 43| 19 954856 23 5i 045 1 44 5i 10 9 54499 23 25 998232! 19 936267 23 44 043733 5o ii 8 .955894 957284 23 '7 9.998220 '9 8-957674 23 37 11-042326 % 12 23 10 998209 IQ 939075 23 29 040925 039327 i3 958670 23 02 998 1 971 998186 19 960473 23 23 47 14 960052 22 9^ 19 961S66 23 14 o38i34 46 id 961429 22 88 998174' 19 963255 23 07 036745 45 16 962801 22 80 99 8i63 19 964639 23 00 o3536i 44 17 96417c 22 73 998101 ' '9 966019 22 9 3 033981 43 18 965534 22 66 998139 20 9 6 7 3 9 4 968766 22 86 o326o6 42 19 966893 22 5 9 998128 20 22 79 o3i234 41 20 968249 22 52 9981 16 20 97oi33 22 7i 029867 il-0285o4 40 2l 8-969600 22 44 9-998104, 20 8 -.071406 22 65 18 22 970947 22 38 $XI 20 972835 22 i 1 027145 23 972289 973628 22 3i 20 974209 22 5i 025791 37 24 22 24 99806S; 20 97556o 22 44 024440 36 25 974962 22 '7 998056 20 976906 22 37 023094 35 26 976293 22 10 998044! 20 978248 22 3o 021762 34 2 7 977619 22 o3 998032 20 979586 22 23 020414 33 28 978941 21 97 998020 20 980921 22 17 019079 32 o9 980259 q8i5 7 3 21 90 998008 20 982251 22 10 017749 3i 3o 21 83 997996, 9. 997985 j 20 983577 22 04 016423 3o 3i 8-982883 21 77 20 8-984899 21 97 1 1 -013101 3 3 2 984189 21 70 997972, 20 986217 21 91 013783 33 985491 21 63 9979 5 9I 20 987532 21 84 OI2468 2- 34 986789 21 57 997947 20 988842 21 78 i i i 58 26 3d 9 88oS3 21 5o 997935 21 990149 21 7i ooo85i 008549 25 36 989374 21 44 997922 21 99i45i 21 65 24 ll 990660 21 38 997910 21 992750 21 58 007250 23 38 991943 21 3i 997897 21 994045 21 52 oo5955 22 3 9 993222 21 25 997885, 21 993337 21 46 oo4663 2! 4o 994497 21 19 997872 21 996624 21 40 003376 20 4i 8-995768 21 •12 9-997860 21 8-997908 21 34 11 -002092 \l 42 997036 21 .06 997847: 21 999188 21 27 000812 43 998299 21 00 997835 21 9-000465 21 21 20-999535 17 44 999500 20 % 997822 997809, 21 001738 21 i5 998262 16 45 9-000816 20 21 003007 21 09 996993 i5 46 002069 20 •82 997797! 21 004272 21 o3 995728 14 47 oo33i8 20 76 997784' 21 oo5534 20 97 994466 i3 48 oo4563 20 "O 997771 ! 21 006792 20 9i 993208 12 49 oo58o5 20 6^ 997758; 21 008047 20 85 991953 1 1 5o 007044 20 -58 99774^: 21 009298 20 80 990702 10 5i 9-008278 20 52 9-997732 1 21 9-010546 20 74 10-989454 I 5s 0095 1 20 •46 9977J9 21 01 1790 20 68 988210 53 010737 20 •40 997706 21 oi3o3i 20 62 986969 7 54 011962 20 •34 997693, 22 014268 20 56 985732 6 55 01 3 1 82 20 •29 997680! 22 oi55o2 20 5i 984498 5 56 014400 20 •23 997667 | 22 016732 20 45 983268 i 57 oi56i3 20 •n 997634; 22 017959 20 40 982041 58 016824 20 • 12 997641 < 22 oi 9 i83 20 33 980817 2 ? 9 oi8o3i 20 -06 997628 2 2 020403 20 28 979597 1 6c 019235 20 •OO 997614; 22 021620 20 23 978380 Cosine D. Sine [84°i Cotang. I). Tacsr. MiT 24 (G DEGREES.) A TABLE OF LOGARITHMIC M. Sine J). Cosine D. Tang. I). Cotang. 9-019235 20-00 9-997°i4 • 22 9-021620 20-23 10-978380 ~6(T i 020435 19 9 5 997601 • 22 0*2834 20 •n 977166 & 3 02I632 r 9 89 997588 •22 024044 20 1 1 975g56 3 022825 l 9 84 997574 • 22 02525l 20 06 974740 973545 57 4 024016 *9 78 997561 • 22 . 026455 20 00 56 5 025203 19 73 997547 • 22 02i655 >9 95 972345 55 6 026386 l 9 67 997534 •23 028852 '9 90 971 148 14 7 027567 J 9 62 997520 -23 o3oo46 •9 85 969954' 53 8 028744 »9 5 7 9975o7 •23 o3i237 19 79 968763 j 52 9 029918 '9 5i 997493 997480 -23 o32425 >9 74 967575 Si 10 031089 19 47 -23 033609 '9 69 966391 5o ii 9>o32257 >9 41 9-997466 •23 9-034791 ] 9 64 10-965209 40 12 o3342i «9 36 9974^2 -23 033969 *9 58 96403 1 48 i3 o34582 !9 3o 997439 -23 037144 '9 53 962856 47 14 035741 J 9 25 997425 -23 o383i6 '9 48 96 i 684 46 i5 036896 >9 20 99741 1 -23 o3 9 485 '9 43 9605 1 5 45 16 o38o48 *9 i5 997397 •23 o4o65 1 '9 38 9 r )n349 958187 44 17 039197 '9 10 997383 •23 041813 '9 33 43 iS 040342 ;g o5 997369 •23 042973 19 28 937027 42 •9 041485 99 99735:) •23 044 1 3o '9 23 935870 41 20 042625 18 9 4 997341 •23 045284 •9 18 954716 40 21 9-043762 18 9-997327 •24 9 • 046434 '9 i3 10-953566: 39 22 044895 18 84 9973 13 • 24 047582 •9 08 9 524i8; 38 23 046026 18 79 697299 •24 048727 •9 o3 951273 37 24 047 ' 54 .8 73 997283 •24 049869 o5ioo8 18 98 95oi3i 36 25 048279 18 70 997271 •24 18 $ 948992; 35 9 47856| 34 26 049400 18 65 997257 •24 o52i44 18 3 050D19 18 60 997242 •24 053277 18 84 946723, 33 o5i63d 18 55 997228 •24 054407 18 79 945593; 32 29 052749 18 5o 997214 •24 o55535 18 74 9444651 3 1 3o o53859 18 45 997199 •24 o56659 18 70 94334i! 3o 3i 9-004966 18 4i 9-997183 •24 9.057781 18 65 10-942219 29 32 056071 18 36 997170 •24 058900 18 n 941100J 28 33 007172 18 3i 997 1 56 •24 060016 18 939984I 27 938870J 26 34 058271 18 27 997141 •24 061 i3o 18 5i 35 059367 18 22 997127 •24 062240 18 46 937760I 25 36 060460 18 17 997112 •24 o63348 18 42 936652 j ?4 37 o6i55i 18 i3 997098! •24 o64453 18 37 935547 23 38 062639 18 08 997083 ; •25 o65556 18 33 934444 22 3 9 063724 18 04 997068 : •25 o66655 18 28 933345 21 4o 064806 17 99 997053! •25 067752 18 24 932248 20 4i 9-065385 17 o4 9-997039! •25 9-068846 18 «9 io-93ii54 19 42 06696; 17 00 997024 •25 069938 18 i5 930062 18 43 o68o36 1/ S6 (-,.,7009 •25 071027 18 10 928973 927887 17 44 069 1 07 •7 81 ';o6994: •25 0721 1 3 18 06 16 45 070176 17 77 9o6«)79 •25 073197 18 02 926803 i5 46 071242 »7 72 996964, •25 074278 '7 97 925722 14 47 072306 '7 68 996049, •25 o 7 5356 17 o3 924644 1 3 48 073366 17 63 996934 •25 076432 17 89 923568 12 P 074424 17 5 9 996919 •25 077505 •7 84 922495 11 5o 075480 '7 53 996904! 9.096889! •25 078576 17 80 921424 10 5i 9-076533 •7 5o •25 9-079644 !7 76 10-920356 * 52 077583 078631 [T 46 096874 ' •25 080710 n 72 919290 53 H 42 996838! •23 081773 H 3 918227 I 54 079676 17 38 996843' •25 oS2833 H 9.1 7 1 67 55 080719 '7 33 996828 •25 083891 H 59 916109 5 56 081739 17 20 23 9968 1 2 .26 084947 17 55 9i5o53 4 57 082797 o83832 17 996797 .26 086000 •7 5i 914000 3 58 17 21 996782 26 087050 17 47 9i2g5o 2 5 9 084864 H 17 996766, • 26 088098 H 43 911902' 1 6o 085894 H .3 996751 j '26 089144 17 38 9io856| Cosine D . Sine 8 3° Cotang. j D. Tang. J M. BINES AND TANGENTS. (7 DEUREES.j 25 M. Sine D. Cosine D. Tang. 1 I>- Cotang. | 9-080894 i 7 -i3 9-996751 •26 1 9-089144 17-38 10-9108561 60 i 086922 17-09 996735 • 26 090187 17-34 90081 3! 5q 2 087947 17-04 996720 .26 091228 i7-3o 908772 58 3 088970 17-00 996704 -26 092266 17-27 907734 57 4 089990 16-96 996688 •26 093302 17-22 906698 5o 5 091008 i6-q2 9Q6673 •26 ! 094336 17-19 905664' 55 6 092024 16-88 996607 •26 095367 17-13 904633! 34 3 093037 16-84 99664 1 •26 096395 17-11 90360OI 53 094047 16-80 996625 -26 097422 17.07 902578J 52 9 095o56 16-76 996610 •26 098446 17-03 901 554! 5i !0 096062 16-73 996094 •26 099468 16-99 i6- 9 5 ooo532 5o io-8go5i3 49 898496 48 !1 9*097065 16-68 9 -996578 •27 9-100487 12 098066 16. 65 996562 •27 ioi5o4 16-91 13 099065 16-61 996546 •27 102519 16-87 897481, 47 M 100062 16.57 996030 •27 io3532 16-84 896468 46 i5 ioio56 16.53 9965 1 4 •27 104542 16-80 895458, 45 16 102048 16.49 996498 •27 io555o 16-76 894450 44 >7 io3o37 16-45 996482 •27 io6556 16-72 893444 43 18 104025 ».6-4i 996465 •27 107559 16-69 892441 42 •9 io5oio 16-38 996449 •27 io856o i6-65 891440 4i 20 io5qq2 i6-34 996433 •27 109559 1661 890441 10-889444 40 21 9- 106973 i6-3o 9-996417 •27 9- 1 ioo56 i6-58 39 22 107951 16-27 996400 •27 1 1 1 55 1 16.54 888449 38 23 10S927 16-23 996384 •27 U2543 i6- 5o 887457 37 24 109901 16-19 99 6368 •27 n3533 16-46 886467 36 25 1 10873 16-16 99635i •27 U452I i6-43 835479 884493 35 26 1 1 1842 16- 12 996335 •27 1 i55o7 16-39 34 27 1 1 2809 16-08 99 63 1 8 •27 1 1 649 1 16. 36 8835o 9 33 28 1 13774 i6-o5 090 3o2 • 28 1 17472 16-32 882528 32 29 .i4 7 3 7 16-01 996285 -28 11S452 16-29 16-25 881548 3i 3o 1 1 56 9 8 I5-Q7 996269 .28 1 19429 88o5 7 i 3o 3i 9- n 6656 15.94 9-996252 .28 9-120404 16-22 10-879596 3 32 iit6i3 15.90 996235 •23 121377I l6-l8 878623 33 118067 i5-87 996219 .28 122343 i6-i5 8 77 652 8 7 6683 27 34 119519 1 5 • 83 996202 •28 123317. 16- 1 1 26 35 1 20469 i5-8o 99&i85 -23 124284 16-07 875716 25 36 121417 l5- 7 6 996168 • 28 125249: 16-04 874751 24 ll 122362 1 5 - 73 996151 .28 126211! 16-01 873780 872828 23 i233o6 1 5-69 996134 .28 127172 15-97 22 39 124248 i5-66 9961 17 .28 i28i3o 1 5 -94 871870 21 4o 125187 i5-62 9961 00 '.28 129087; 15.91 10-87 870913 20 4i 9-126125 i5- 39 9-996083 • 29 9-i3oo4ii 10-869959 19 42 127060 1 5 - 56 996066 •29 i3o994| 15-84 869006 18 43 127993 l5- 32 996049 •29 i3i944| i328g3j i5-8i 868o56 17 44 128925 I5-4Q 996032 •29 •29 15.77 867107 16 45 I2 9 §54 15-45 996015 i3383 9 | 15-74 866161 :5 46 1.^0781 i5-42 993908 • 29 i34784| 1 5 • 7 1 8652i6 U 47 131706 i5-3 9 995980 ■29 135726 l5.6 7 864274 i3 43 i3263o 15-35 995963 • 29 1 3666 7 15-64 863333 12 49 i3355i i5-32 995946 .29 i376o5i i5-6i 862395 861458 11 5o 134470 15.29 995928 • 29 138542 15-58 10 5i 9-135387 i363o3 l5-20 9-995911 9 9 58 9 4 .29 9-139476 i555 io-86o524 I 52 15-22 .29 1 40409 | i5-5i 809591 53 i3t2i6 i5- 19 995876 •29) i4i34o' i5-48 85866o 7 54 i38i28 i5-i6 995859 •29! 142269 1 i5-45 857731 6 55 139037 I5-I2 995841 •29! 143196! i5-42 8568o4 5 56 139944 13-09 995823 .29 i44i2i| 1 5 -39 805879 4 u Uo85o i5-o6 995806 •29 i45o44 i5-35 854956 3 141 754 i5-o3 9 9 5 7 88 • 29 145966 146385, i5-32 854034 2 59 142655 i5-oo 995771 •20 10-29 853u5j 1 oo 143555 14-96 995753 ' .29 147803 l5-20 852197; Cosine D. Sine !82°| Cotung. 1 p: 1 _Tang. _ JL 20 (8 DEGKEUS.) A TABLE OF LOGARITHMIC Sine o- 143555 D. Cosine D. | Tang. D. Cctang. 14-96 9-995753 •3o o- 147803 i5-26 I0'852io7 85i28a 00 i 144453 14 9 3 995735 -3o 148718 i5 23 1 2 145349 14 90 995717 •3oj 1 4963 2 i5 20 85o368 3 146243 14 87 995699 995681 ■ 3o i5o544 i5 17 849456 n 4 I47I36 14 84 •3o i5i454 i5 14 848546 5 148026 14 81 995664 • 3o 1 52363 i5 1 1 847637 55 6 148915 14 78 995646 ■3o 153269 i5 08 846 7 3i 54 J 149802 14 75 995628 -3c 1 54 1 74 i5 o5 845826 53 1 30686 14 72 99 56 10 •3o i55o77 i5 02 844923 5j 9 i5i569 14 69 995591 -3o 155978 14 99 844022 5i IO i5245i 14 66 99.5573 • 3o 1 568 77 14 96 843i23 5o ii j i5333o 14 63 9-995555 995537 • 3o o- l5 777 3 14 9 3 10-842225 48 12 1 54208 14 60 -3o 1 5867 1 14 90 841329 i3 i55o83 14 5 7 993319 • 3o 1 59565 14 87 840435 47 U 155957 14 54 Q955oi .3i| 160457 • 4 84 83 9 543 46 i5 1 5683o 14 5i 993482 • 3i i6i347 14 81 838653 45 16 157700 1 5856o 14 48 995464 • 3i 162236 14 79 837764 44 \l 14 45 993446 • 3i' i63i23 14 76 836877 43 i5g43D 14 42 995427 •3i ! 164008 14 73 835992 42 '9 i6o3oi 14 3 9 995409 -3i 164892 14 70 83 5 1 08 41 20 161164 14 36 993390 -3i i65 77 4 14 67 834226 40 21 9- 162025 14 33 9-993372 • 3i 9- 166634 • 4 64 10-833346 U 2 2 162885 14 3o 995355 ■ 3i; 167532 14 61 832468 -.3 163743 14 27 995334 •3i 168409 U 58 83 1 5 9 i ll 24 164600 14 24 9953i6 • 3i 169284 14 55 830716 2 5 165454 14 22 995297 • 3i| 170157 14 53 829843 828971 35 26 1 663o 7 14 l 9 993278 • 3i| 17 1029 14 30 34 2 I 167 1 59 14 16 995260 •3i| 171899 14 47 828101 33 28 168008 14 i3 995241 .32! 172767 1 4 4i 827233 32 ^9 168856 14 10 993222 •32 i 7 3634 14 42 826366 3i | 3o 169702 14 07 993203 -32 174499 14 3 9 8255oi 3o 1 3i 9-170547 14 o5 9-995184 •32 ! 9. 175362 14 36 10-824638 29 823776 28 ' 32 I 7 i38 9 14 02 995i65 -.32 176224 14 33 I 33 172230 1 3 99 995146 .32] 177084 14 3i 822916! 27 34 173070 i3 96 993127 •32| 117942 14 2S 822o58l 26 1 35 173908 i3 94 995108 -32 '78799 179653 14 25 821201 25 36 174744 i3 91 995089 •32 14 23 820345 24 37 i 7 55 7 8 i3 88 993070 -32 180308 14 20 819492 818640 23 38 17641 1 i3 86 995o5i -32 i8i36o 14 17 22 3 9 177242 i3 83 0g5o32 -32 18221 i 14 i5 817789 21 40 178072 i3 80 9930i3| -32 183039 14 12 816941 20 4i 9-178900 i3 77 9- 99 499 3 •32 9. 183907 14 09 10-81609.3 ;g 42 179726 i8o55i 13 74 99-4974 -32 184752 14 07 8i5248 43 i3 72 994955 .32 185597 186439 14 04 8i44o3 17 44 181374 i3 69 994935 • 3a! 14 02 8i356i l6 45 [82196 i3 66 9949 1 6 •33; 187280 i3 99 812720! 13 46 i83oi6 i3 64 99489 6 •33 188120 i3 96 811880 14 47 183834 i3 61 994877 .33; i88 9 58 i3 93 811042! i3 48 18465 1 i3 5 9 994857 • 33 189794 i3 s 810206J 12 49 185466 i3 56 994838 • 33 190629 1 3 809.371 11 5o 18628c 1 3 • 53 994818 •31 191462 i3 86 8o8538j io 5i 9- 187092 i3 5i 9.994798 • 33 9* 192294 i3 84 10-^077061 9 806876 1 8 5a 187903 i3 •48 994779 994739 • 33 193124 1 3 81 53 188712 i3 46 • 33 193953 i3 79 806047! 7 54 189519 I9c325 .3 • 43 994789 • 33 194780 i3 76 8o522o! 6 55 i3 41 994H9 • 33 195606 •3 74 804394 ' 5 56 191130 i3 ■38 994700 • 33 196430 i3 7i 803570, 4 n 1919.33 i3 • 36 994680 • 33 197253 i3 69 802747J 3 192731 i3 • 33 994660 .33; 198074 i3 66 801926 2 5 9 ■ 393534 1 3 •3o 994640 • 33! 198894 i3 54 801 106 I 6o ic.4332 ii 58 994620 • 33i i997i3 i3-bi 800287 O 1 Cosine E » Sine 1 Bl°l C< ;tang. D. Tang/ M SINES AND TANGENTS. (9 DEGREES.) 27 M, Sine D. Cosine 1 D. Tang. D. Ootang. g (94332 i3- 28 9-994620 33 9-i997i3 i3 61 •0-800287 60 l 1961 ag i3- 26 994600 33 200529 i3 ^ 7994- 1 5o 2 193920 i3 23 99458o, 33 201340 i3 56 7 9 8655 58 3 196719 i3- 21 994360 34 202159 i3 ¥ 797841 57 4 197011 i3- 18 994040 34 202971 i3 52 797c 20 06. 796*18, 00 5 198302 i3 '6 994oi9 ! 34 203782 i3 49 6 1 9909 1 1 j i3 99 i499 34 204592 i3 47 795408 54 7 199879 i3 11 9944 79 34 2o54oo i3 45 794600 53 8 2oc666 i3 08 994439 994438 34 206207 i3 42 793793 52 9 *>oi45i i3 06 34 207013 i3 40 792987 31 10 202234 i3 04 994418 34 207817 9.208619 i3 38 792183 5o ii 5 203017 i3 01 9.994397 34 i3 35 io-79i38i % 12 203797 12 99 994377 34 209420 i3 33 790580 i3 204077 12 96 994357 34 2IO?20 i3 3i 789^80 788982 47 14 2o5354 12 94 994336 34 2iioi8 i3 28 46 i5 2o6i3i 12 87 9943 1 6 34 2 : 1 8 1 5 i3 26 788i85 45 16 206906 207679 12 12 994295 994274 994254 34 35 212611 2i34o5 i3 i3 24 21 787389 786593 44 43 208432 12 85 35 214198 i3 19 785802 42 >9 209222 12 82 994233 35 214989 i3 17 785011 41 20 209992 12 80 994212 35 215780 i3 i5 784220 40 21 9-210760 12 78 9-994191 35 9-2i6568 i3 12 10-783432 IS 22 2Il526 12 7 5 9941 7 1 35 217356 218142 i3 10 782644 23 212291 2i3o55 12 73 9941 5o 35 i3 cS 7 8i858 37 24 12 71 994120 35 218926 i3 c5 781074 36 25 2i38i8 12 68 994108 35 219710 i3 c3 780290 35 26 214579 215338 12 66 994087 35 2204Q2 i3 CI 779308 778728 34 2 7 12 64 994066 35 221272 12 09 33 28 216097 12 61 994045 35 222o52 12 ?7 777948! 32 I** 216854 12 5 9 994024 35 222830 12 ?4 777170 3x 3o 217609 9-2i8363 12 57 994003 35 2236o6 12 ?2 776394 3o 3i 12 55 9-993981 35 9.224382 12 90 10-775618 3 32 219116 12 53 993960 35 225i56 12 88 774844 33 219868 12 5o 993939 993918 35 225929 12 86 774071 27 34 220618 12 48 35 226700 12 84 7733oo 26 35 221367 12 46 9 9 38 9 6 36 227471 12 81 772529 25 36 222II5 12 44 993875 36 228239 1279 771761 24 ll 222861 12 42 993854 36 229007 12-77 770993 23 2236o6 12 3 9 9 9 3832 36 229773 23oo3g 11-76 770227 22 3q 224349 12 37 99381 1 36 il- 7 3 760461 768698 10-767935 21 4o 225092 12 35 993789 36 23l302 12-71 20 4i 9-225833 12 33 9-993768 3o 9-232o65 12-69 \l 42 226573 12 3i 993746 36 232826 12.67 767174 43 22731 1 12 28 993723 36 233586 ii-65 766414 17 44 228048 12 26 993703 36 234345 12-62 765655 l6 45 228784 12 24 993681 36 235io3 12-60 764897 i5 46 229018 12 22 993660 36 235859 .2-58 764141 i4 % 2302D2 12 20 9 9 3638 3o 2366i4 2-56 763386 i3 230984 12 18 993616 36 237368 2-54 762632 12 49 23l 7 l4 12 16 993594 37 238120 12-52 761880 11 5o 232444 12 14 993572 37 2388 7 2 i2-5o 761128 10 5i 9-233172 12 12 9-993500 3^ 9-239622 12-48 10-760378 I 52 233899 12 09 993528 37 240371 12-46 759629 51 234620 12 07 9935o6 37 241 118 12-44 758882 I 54 235349 12 o5 993484 37 24i865 12-42 758i35 55 236073 12 o3 903462 37 242610 12-40 757390 5 56 236795 12 01 993440 37 243354 12-38 756646 4 u 237015 ii 99 993418 37 244097 12-36 755903 3 238235 11 97 993396 37 244839 12-34 755i6i 2 J9 238953 " 9 5 993374 3^ 245579 12-32 754421 1 6o 239670 11 93 99335 1 •3 7 246319 12-30 75368i Ocftine _P Sine 80° Cotang. r >. _ Tang" M7j 28 (10 DEGREES.) A 1ABLE OF LOGARITHMIC 60 M. Sine D Cosine D. Tang. D. Cotang. o 9-239670 11 9 3 9»99335i I 7 9- 240319 12 -3o io-75368i i 240386 1 1 % 993329 •3-1 247057 12-28 752943 5a 752206! 58 2 241101 11 993307 •37 247794 12-26 3 241814 n 87 993285 i 7 24853o 12-24 75i47o 57 4 242526 11 85 993262 I 7 249264 12-22 7507.36 56 5 243237 11 83 993240 :ll 24999 s 25o73o 12-20 75ooo2 55 6 243947 1 1 81 993217 993195 I2-I8 740270 748539 54 I 244656 11 79 • 38 25i46i 12-17 53 245363 1 1 ]l 993172 -38 262191 I2-l5 747809 5a 9 246069 1 1 246775 11 993i49 • 38 252920 1213 747080 5i 10 73 99 3i2 7 • 38 233648 12-11 746352 5o II 0-247478 11 248181 11 7i 9-993104 • 38 9-254374 12-09 10-745626; 49 744900 48 12 69 993081 • 38 255ioo 12-07 12 o5 i3 248883 1 1 ti 993059 •38 255824 744176 % 14 249583 1 1 993o36 • 38 256547 12-03 743453 ID 250282 11 63 9930 1 3 • 38 257269 I2«0I 742731 45 16 250980 11 61 992990 • 38 257990 258710 12-00 742010 44 17 251677 ! 11 it 992967 • 38 II.98 741290 43 18 252373 11 992044 • 38 259429 II.96 740571 42 l 9 253067 11 56 992Q2I • 38 260146 11 -94 73 9 854 41 20 253761 n 54 992898 • 38 26o863 11.92 739137 10-738422 40 21 9-254453 11 52 9-992875 • 38 9-261578 \\r 9 ll 22 255i44 11 5o 992852 • 38 262292 737708 23 255834 11 48 992829 I 9 263oo5 U.87 736995 736283 37 24 256523 11 46 992806 19 263717 u-85 36 25 25721 1 11 44 992783 • 3 9 264428 n-83 735572 35 26 257898 11 42 992759 I 9 2 65 1 38 n-8i 734862 34 2 258583 11 4i 992736 • 3 9 265847 :::?? 734i53 33 239268 n 3 9 992713 i 9 266555 733445 32 29 259951 11 37 992690 •3 9 267261 n.76 732739 732o33 3i 3o 26o633 1 1 35 992666 •3 9 267967 u-74 3o 3i 9-26i3i4 11 33 0-99^043 •3 9 9-268671 11.72 io-73i32o 730623 3 32 261994 11 3i 992619 I 9 269375 11.70 33 262673 11 3o 992596 •? 9 270077 11.69 729923 27 34 26335i n 28 992572 •3 9 270779 :::!? 729221 728521 26 35 264027 11 26 992549 992523 .39 271479 272178 25 36 264703 11 24 I 9 11.64 727822 24 37 265377 11 22 992501 •3 9 272876 11-62 727124 23 38 266o5 1 11 20 992478 .40 273573 1 1. 60 726427 22 3 9 266723 11 «9 992454 .40 274269 u-58 725731 21 40 267395 11 [I 992430 .40 274964 I!-55 725o36 20 41 9- 268o65 11 9-992406 •40 9-275658 10-724342 \l 42 268734 11 i3 992382 •40 276351 u-53 723649 43 269402 1 1 11 992359 .40 277043 11. 5i 722957 17 4i 270069 1 1 10 992333 •40 277734 n-5o 722266 16 45 270733 11 08 99231 1 •40 278424 11.48 721576 i5 46 271400 11 06 992287 •40 2791 i3 w.% 720887 14 % 272064 11 o5 992263 •40 279801 280488 720199 i3 272726 11 o3 992239 .40 1 1-43 71961 2 718826 12 49 2/3388 11 01 992214 •40 281174 n-41 11 5o 274049 10 5 992190 .40 28i858 11-40 718142 iO 5i 9 274708 10 9-992166 •40 9-282542 11. 38 10 717458 I 52 275367 10 96 992142 •40 283225 n-36 716775 53 276024 10 94 992117 •41 283907 •)84588 n-35 716093 7 54 276681 10 92 992093 •41 u-33 7i54i3j 55 277337 10 % 992069 •41 285268 ii-3i 714732 5 56 277991 10 278644 10 992044 ■41 285947 286624 n-3o 7 1 4o53 4 57 87 992020 •41 11.28 713376 3 58 279297 10 86 991996 •41 287301 11.26 712690 2 5 9 279948 10 280599 10 84 991971 •41 287977 11-25 71202J i 60 82 991947 •41 288652 11-23 711348 Cosine D Sine 790 Cotang. D. ~Tang/ M. SIKE8 AND TANQEN T TS. (11 DEGREES.) 29 ^7 Sine i I). Cosine I). Tang. i D. Cot'ang. 1 o 9 280D99 10-82 9-9Qigi7 •41 9-288652 ! 11-23 .0-711348 60 i 281248 10 Si 991922 •41 289326 , 11-22 710674: 5q 7 1 0001 58 2 281897 10 79 99189-7 •4i i 289999 1 I -20 3 282044 10 77 991873 •4» 1 290671 11-18 709329 57 7o8658, 56 4 283190 10 76 991848 •41 , 291342 n 17 5 283836 10 74 991823 •4i i 292013 11. i5 707987I 55 707318 54 6 284480 10 72 991799 •4i 292682 11-14 I 285124 10 7i 991774 •42 293350 11-12 7o665o 53 280766 10 69 991749 •42 294017 11 -ii 705983 5a 9 286408 10 67 991724 •42 294684 11-09 7o53i6 5i 10 287048 10 66 991699 •42 295349 9-296013 11-07 70465i 5o n 5 287687 10 64 9-991674 •42 n-o6 10-703987 % i? 268326 10 63 991649 •42 296677 1 1 -o4 7o3323 i3 288964 10 61 991624 •42 297339 n-o3 702661 47 U 289600 10 u 991599 •42 ' 298001 II -01 701999 7oi338 46 i5 290236 10 991574 •42 29S662 11-00 45 16 290870 10 56 99 '549 •42 299322 10.98 700678 44 3 29004 10 54 99i524 •42 299980 10-96 700020 43 292137 10 53 991498 991473 •42 3oo638 10-95 699362 42 '9 292768 10 5i •42 301295 10-93 698705 41 20 293399 10 5o 99U48 •42 301961 10-92 698049 4c 21 9-294029 294658 10 41 9-991422 •42 9-302607 10-90 10-697393 3q 22 10 46 991397 •42 3o326i 10-89 696739 38 23 295286 10 45 991372 •43 3o3oi4 3o4567 10-87 696086 37 24 295913 10 43 991346 •43 io«86 695433 36 25 296539 10 42 99i32i •43 3o52i8 10-84 694782 35 26 297164 10 40 991295 •43 3o586g io-83 6941 3 1 34 27 297788 10 39 991270 •43 3o65i9 io-8i 693481 33 28 298412 10 37 991244 •43 307168 10-80 692832 32 2Q 299034 10 36 991218 •43 3o 7 8i5 10.78 692185 3i 3o 299655 10 34 991 193 •43 3o8463 10-77 691537 3c 3i 9-300276 10 32 9-991167 •43 9-309109 10-75 10-690891 ll 32 300895 10 3i 99"4i •43 309754 10-74 690246 33 3oi5i4 10 29 991 1 1 5 •43 310398 10-73 689602 27 34 302132 10 28 99 1 090 •43 3no42 10-71 688g58 26 35 302748 10 26 991064 • 43 3n685 10-70 6883 1 5 25 36 3o3364 10 25 99io38 • 43 3i2327 10.68 687673 24 ll 3o3o79 3o4393 10 23 991012 • 43 312967 10-67 68 7 o33 23 10 22 990986 •43 3i36o8 10-65 6863 9 2 22 39 3o5207 10 20 990960 •43 3i4247 10-64 685753 21 40 3o58i9 10 l 9 990934 •44 3 1 4885 10-62 6S5 1 1 5 20 4i 9-3o643o 10 5 7 9-990908 •44 9-3i5523 io-6i 10-684477 19 42 307041 10 16 990882 •44 3 1 6 1 59 1 • 60 683841 18 43 3o765o 10 14 99oS55 •44 316793 10-58 683205 H 44 308209 10 i3 990829 •44 3 17430 io-57 682570 16 45 308867 10 1 1 99o8o3 •44 318064 io-55 681936 68i3o3 i5 46 309474 10 10 990777 •44 318697 io-54 14 s 3 1 0080 10 08 990750 •44 319329 io-53 680671 i3 3 io685 10 °7 990724 •44 3 1 996 1 io-5i 680039 679408 678778 12 $ 3:1289 10 o5 990697 •44 320592 io- 5o 11 5c 3 1 1 8 9 3 10 04 990671 •44 321222 10-48 10 5i c-3i2 495 10 o3 9-990644 •44 9-32i85i 10-47 10-678149 9 52 3 1 3097 313698 10 01 990618 •44 322479 io-45 677521 6 53 10 00 990591 •44 323io6 io-44 676894I 7 54 314297 9 98 990065 •44 323733 10-43 676267 6 55 314897 9 97 99o538 •44 324358 10-41 675642 5 50 3 1 5495 9 06 99o5u .45 3249S3 io-4& 6 7 5oi 7 4 1 816092 3i668 9 9 94 990485 • 45 325607 10-39 6 7 43 9 3 3 9 9 3 990458 .45 32623i io-37 673769 2 59 317284 9 9' 990431 •45 326853 io-36 671147 672526 1 60 3i 7 8 79 9 90 990404 •45 327475 CojEansr. io-35 Cosine D Sine 7 8° ' 1). Tang. IT" 30 (1* I DEGREES.) A TABLE OF LOGARITHMIC it o Sine D. | Cosine | IX | Tang. D. | Cotang 60 9-317879 9.90 9-99040*' -45' 9-32747^ io-35 10-672526 i 3i8473 9-88 990378 -45, 328095 10-33 671905 U 2 319066 9-87 99o35i -4f i 328 7 i5 10-32 671285 3 3i 9 658 1 9-86 99o324j -4' > 32 9 334 io-3o 670666 n 4 320249 9.84 990207) -4f ! 329953 10-20 10-28 670047 5 320840 9-83 990270 -451 33o57o 669430 6688 1 3 55 6 32i43o 9-82 990243 -4i 33 1 1 87 10.56 54 I 322019 9-80 99021' .45 1 33i8o3 10-25 668197 667382 53 322607 9-79 990188 •45 3324i8 10-24 52 9 323194 1 J23780 9-77 9901 61 •45 J 333o33 10-23 666967 666354 5i IC 9-76 990134 •45 333646 10-21 5o ii 9-324366 9-75 9.990107 ■46 ! 9-334259 10-20 10-665741 a i-i 324950 325534 9-73 990079 .46 334871 io- 19 665 1 29 6645 1 8 1 3 9.72 990052 •46 335482 I0-I7 47 14 326117 9.70 990025 989997 .46 336093 io- 16 663907 46 i5 326700 9-69 •46 J36702 .o-i5 663298 662689 45 16 327281 9.68 989970 •46 33 7 3u io- 13 44 :i 327862 9-66 989942 •46 337919 10-12 662081 43 328442 9-65 989915 •46 338327 10- II 661473 42 '9 32902 j 9.64 989887 •46 339i33 10- 10 660867 41 20 329399 9-62 989860 •46 339739 10-08 660261 40 ?i 9-330176 9-61 9-989832 •46 9- 34o344 10-07 10-659656 & :2 33ot53 9-60 989804 •46 340948 34io52 io- 06 659052 23 33i32 9 33i9o3 9-58 989777 -46 10-04 658448 37 24 9-5 7 989749 •47 342i55 10 -o3 65 7 845 36 25 332478 9-56 989721 •47 342757 10-02 657243 35 26 333o5i 9-54 989693 •47 343358 10-00 656642 34 27 333624 9-53 989665 •47 343g58 9'99 656o42 33 28 334195 9-52 989637 •47 344558 9.98 655442 32 n 9 334-766 9-5o 989609 •47 345i57 9'97 654843 3 1 3o 33533- 9-4Q 9-48 9 8 9 582 •47 345755 9.96 654245 3o 3i 9-335906 9-989553 •47 9-346353 9-94 10-653647 3 32 336475 9.46 989525 •47 346949 347043 9-93 653o5i 33 337043 9-45 989497 •47 9-92 652455 27 34 337610 9.44 989469 •47 34*141 9-91 65i85 9 26 35 338176 9-43 989441 •47 348735 ?:g 65 1 265 25 36 33S742 9-4i 989413 •47 349329 650671 24 37 3393o6 9.40 989384 •47 349922 9.87 650078 23 38 339871 9-39 989356 •47 35o5i4 9-86 649486 22 39 340434 9-37 989328 •47 35i 106 9-85 648894 21 4o 340996 9-36 989300 •47 351697 9-352287 9-83 6483o3 20 4i 9-341008 9-35 089271 •47 9.82 10-647713 \l 42 342119 9-34 ' 989243 •47 352876 9-81 647124 43 342679 9-32 989214 •47 353465 9-80 646535 1 44 343239 9-3i 989186 •47 354053 9-79 645947 64536o r6 45 343797 9-3o 9 8 9 i5 7 989128 •47 354640 9-77 9.76 i5 46 344355 9-29 •48 355227 644773 14 S 344912 9.27 989100 •48. 3558i3 9-75 644187 i3 345469 9-26 989071 -48 3563o8 356982 357566 9-74 643602 12 49 346024 9-25 989042 •48 9 . 7 3 643ei8 :i 5o 346579 9-24 989014 •48 9-71 6424341 10 5i 9 347134 9-22 9-988985 •48 9-358149 9-70 io-64i85i, 641269! 8 52 347687 9-21 988956 •48 358 7 3i 9-69 I 9.68 53 348240 9- 20 988927 •48 3593i3 640687 7 640107 6 54 348792 9.19 988898 -.48 359893 9.67 55 J 349343 9.17 988869 1 .48 360474I 9.66 639526 5 56 | 349893 9-16 988840' .48 36io53: 9-65 638947 1 4 57 35o443 9- 15 98881 1 .49 36i632 9-63 638368 3 58 ! 350992 9-14 988782! .49 362210 9-62 637790] 2 5o 35i 540 9-i3 988753 .49 1 362787 9-61 6372i3| 1 6o 352088 9. 11 988724 -49 1 363364: 9.60 Th j 636636 1 Cosine | D. Sine 17° Cotang^ 1 Tang. 1, M. j SINES AND TANGENTS. (13 DEGREES. ) 3 M. Sine __^_ Cosine D. Tang. D. Cotang. 9-352088 9. 11 9-988724 "^49 9-363364 9-60 io-b36636 "60" i 352635 9 10 988695 .49 363940 3645 1 5 9 u 636o6o % 2 353i8i 9 3 988666 •49 9 635485 3 353726 9 9 88636 .49 365ooo 9 57 634910 57 4 354271 9 07 988607 .49 365664 9 55 634336 56 5 3548i5 9 o5 988578 •49 366237 9 54 633763 55 6 355358 9 04 9 88548 988519 •49 3668 10 9 53 633190 54 I 355901 9 o3 •49 367382 9 52 632618 53 356443 9 02 988489 •49 367953 9 5i 632047 52 9 356o84 357624 I 01 988460 •49 368D24 9 5o 631476 5i IO 99 988430 •49 369094 9 4 2 48 630906 5o ii 9- 358o64 8 98 9-988401 •49 9-369663 n io-63o337 % 12 3586o3 8 97 988371 •49 370232 9 40 629768 i3 359141 8 96 988342 •49 370799 9 45 629201 628633 47 14 359678 8 9 5 9 883 1 2 •5o 371367 9 44 46 15 36o2i5 8 93 988282 00 3 7 i 9 33 9 43 628067 45 16 360702 8 92 988252 .50 372499 9 42 627501 44 \l 361287 8 9i 988223 .50 373o64 9 41 626936 626371 43 361822 8 1 988193 .50 373629 9 40 42 *9 362356 8 988163 -50 374193 9 u 625807 41 20 362889 8 9 88i33 •5o 3747^6 9 625244 40 21 9-363422 8 87 9 • 988 1 o3 •50 9-375319 9 11 10-624681 M 22 363954 8 85 988073 •5o 37588i 9 6241 19 623558 23 • 364485 8 84 988043 •5o 376442 9 34 37 24 365oi 6 8 83 9 88oi3 •5o 377003 9 33 622997 36 25 365546 8 82 987983 •5o 377563 9 32 622437 35 26 366075 8 81 987953 -50 378122 9 3i 621S78 34 27 3666o4 8 80 987922 987892 •5o 378681 9 3o 62 1319 33 28 367 1 3 1 8 79 -50 379239 9 8 620761 32 I 9 367659 8 77 987862 -50 379797 9 620203 3i 3o 368 1 85 8 76 987832 •51 38o354 9 27 6 1 9646 3o 3i 9-368711 8 75 9-987801 •51 9-380910 9 26 10-619090 6i8534 \l 32 369236 8 74 987771 -51 38i466 9 25 33 369761 8 73 987740 •51 382020 9 24 6179S0 27 34 370285 8 72 987710 •51 382575 9 23 617425 26 35 370808 8 71 987679 •51 383 1 29 9 22 616871 25 36 37i33o 8 70 987649 •5i 383682 9 21 6i63i8 24 U 37i852 8 69 987618 •5i 384234 9 20 615766 23 372373 8 67 987588 •5i 384786 9 IO 6i52i4 22 39 372894 8 66 987557 •5i 385337 9 10 6i4663 21 4o 373414 8 65 987526 •5i 385888 9 17 614112 20 4i 9-3 7 3 9 33 8 64 9.987496 •5i 9-386438 9 i5 io-6i3562 \l 42 374452 8 63 987465 •5i 386987 9 14 6i3oi3 43 374970 8 62 987434 •5i 387336 9 i3 612464 •7 44 375487 8 61 987403 •52 388084 9 12 611916 16 45 376003 8 60 987372 •52 38863i 9 1 1 6n36 9 i5 46 376019 8 5? 987341 -52 389118 9 10 610822 14 47 377035 8 987310 •52 389724 9 a 610276 i3 48 377549 8 57 987279 •52 390270 9 609730 12 i 9 378063 8 56 987248 •52 3 9 o8i 5 9 07 609185 608640 1 1 5o 378577 8 54 987217 •52 391360 9 06 IC 5i 9-379089 8 53 9-987,86 •52 9-391903 9 o5 10-608097 \ 52 379601 38ou3 8 52 987155 •52 392447 9 04 607553 53 8 5i 987124 -52 392989 9 o3 60701 1 7 54 380624 8 5o 987092 •52 3 9 353i 9 02 606469 6 55 38u34 8 % 987061 •52 394073 9 01 605927 5 56 38i643 8 9S7030 .52 394614 00 6o5386 4 £ 382.52 8 47 98699S •52 3931 54 8 9Q 604846 3 382661 8 46 986967 •52 393694 8 98 6o43 06 2 59 383 1 68 8 45 986936 •52 396233 8 97 603767 1 6o 383675 8 44 986904 Sine •52 76^ 396771 Cotang. 8 96 603209 " Cosine D. 1). Taiifi M 25* 32 (14 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine | D. | Tang. D. Cotang. 60 9-3836 7 5 8-44 9-986904 -52 9 -39677 1 8-96 IO-6o322C. i 384182 S-43 986873. -53 397301; 8.96 602691 602164 IS a 384687 8-42 986841 -53 397846 8-95 3 385i 9 2 8-41 986809 1 -53 986778! -53 3 9 8383 8-94 601617 57 4 3856 9 7 8-4o 398910 399433 8- 9 3 601081 56 5 386201 8-3 9 986746 -53 8-92 600545 55 6 386704 8-38 9867 1 4 1 ■■& 399990 8-91 600010 54 I 387207 S-3 7 986683 -53, 400024 8-90 8-89 8-88 590476 53 387709 8-36 9 8665 1 -53 {oio58 598942 52 rj 3882 10 8-35 986619 •53 401591 598409 1 5i 10 3887 1 1 8-34 986587 •53 402124 8-87 597876 5o II 9389211 8-33 1 9-986555 •53 9-402656 8-86 10-597344 % 12 38971 1 8-3s 9 86523 -53 403187 403718 8-85 5 9 68 1 3 i3 390210 8-3i 986491' '53 8-84 596282 47 14 390708 8-3o 986439' -53 404249 8-83 595751 46 i5 391206 8-28 986427J -53 404778 8-82 595222 45 16 391703 8.27 986395. -53 4o53o8 8-81 594692 44 «7 392199 8-26 986363 '54 4o5836 8-8o 594164 43 1 8 392695 8- 2 5 98633 1 1 -54 4o6364 8-70 8-78 593636 42 19 393191 8-24 986299 -54 406892 D93 1 08 41 20 393685 8-23 986266 -54 407419 9-407945 8-77 8.76 592581 40 21 9-394I79 8-22 9-986234 '54 io-592o55 3 9 22 394673 8-21 986202 -54 408471 8. 7 5 591529 38 23 395166 8-20 986169 -54 408997 409021 8.74 59 1 oo3 37 24 3 9 5658 8-19 8.18 986137 -54 8-74 590479 589955 36 25 39600 986 1 041 -54 4ioo45 8- 7 3 35 26 396641 8.17 986072 -54 410569 8-72 58 9 43 1 34 a 397132 5"7 986039 •34 41 1092 8,7, 588 9 o8 33 39-7621 8.16 986007 •54 4n6i5 8-70 588385 32 29 3981 1 1 8-i5 985974 •54 412137 8-69 8-68 58 7 863 3i 3o 398600 8.14 985942 •54 412658 58 7 342 3o 3i 9-399088 8-i3 9-985909 985876 • 55 9-413179 8-67 io-58682i 3 32 399575 8-12 • 55 413699 8-66 5863oi 33 400062 8- n 985843 • 55 414219 414738 8-65 585 7 8i 27 34 4oo54q 8-io 98581 1 • 55 8-64 585262 26 35 4oio35 8-09 8-o8 9 85 77 8 • 55 413257 8-64 584743 25 36 401 520 985745 • 55 415775 8-63 584225 24 ll 4o2oo5 8-07 8-o6 985712 • 55 416293 8-62 583707 23 38 402489 9 856 79 -55 416810 8-6i 583190 22 3 9 402972 8-o5 9 85646l -55 417326 8-6o 582674 21 4o 4o3455 8.04 9856i3; «55 417842 8-5 9 8 -,58 582i58 20 4i 9 403938 8-o3 9- 9 8558o, -55 9 -4i 8358 io-58i642 \l 42 404420 8-02 985547 -55 4i88 7 3 8-57 581127 43 404901 4o5382 8-oi 9855i4| '55 419387 8-56 58o6i 3 17 44 8-oo 980480! -55 419901 8-55 580099 57 9 585 16 45 4o5862 7-99 7.98 985447 -55 42041 5 8-55 i5 46 4o634i 985414 * 36 420927 8-54 579073 14 % 406820 7-97 9 8538o -56 421440 8-53 578560 i3 407299 7-96 985347 -56 421952 8-52 578048 12 49 407777 7- 9 5 9 853i4! -56 422463 8-5i 5 77 537 II 5o 408254 7-94 985280 -56 422974 8-5o 577026 10 .)i 9 408731 7-94 9-985247! -56 9-423484 8-40 8-48 10-576516 8 5a 409207 7.93 9 852i3 -56 428093 576007 53 409682 7-92 985 1 8o: -56 4245o3 8-48 575497 574989 "» 54 410107 7-91 985146 -56 423011 8-47 6 55 4io632 7-90 9 85n3 -56 425519 8-46 574481 5 56 411106 7-88 980079! -56 426027 8-45 573973 4 11 41 1579 985043; -56 426534 8-44 573466 3 4I2052| 7.87 985ou : -56 ! 427041 8-43 5 7 2 9 5o 572453 2 5 9 4i2524 ! 7-86 984978J ' 56 i 427347 8-43 1 60 41 2996 j 7-85 984944 56 428052 8.42 571948; Cosine ' D 1 Sine 75°| Cotang. D. Tancr. 1 M.J m: SINES AND TANGENTS. (15 DEGREES.) S3 9 10 II 12 i3 14 i5 16 «7 18 19 20 21 22 23 24 23 26 11 II 3i 32 33 34 35 36 u 39 40 41 42 43 44 45 8 & 5i 52 53 54 55 56 Suio 9-412996 413467 4i3938 414408 414878 4i5347 4i58i5 416283 416751 417217 417684 ;-4i8i5o 4i86i5 419079 419544 420007 420470 42oo33 421J95 421857 4223i8 9-422778 423238 42^697 424106 424613 425073 42553o 425987 426443 426899 9-42733- 427809 428263 428717 429170 429623 430075 43o527 430978 431429 9-431879 432329 432778 433226 433675 434122 434569 435oi6 435462 435qo8 5*436353 436798 437242 437686 438129 438572 439014 439456 439897 44o338 Cosine _IX_ 7-85 7-84 7-83 7-83 I Cosine 1 D. 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7' 7' 7' 7- 7- 7-65 7-64 7-63 7-62 7.61 7-60 7-60 7-5 9 58 57 56 55 54 53 52 52 5i DO 7-49 7-49 7-48 7-47 7-36 7-36 7-35 7-34 9-984944; 984910' 984876! 984842 ! 984808! 984774 984740 084706 ! 984672! 984637I 984603 9845351 984500] 984466 984432 984397 984363 984328 984294 984209 j- 984224 984190 984155 984120 984085 984050 98400 983981 983946 98391 1 (.983875 983840 9 838o5 983770 983730 980700 98366. 983629 983594 9835o8 •983523 983487 983452 983416 98338i 983345 983309 983273 983238 983202 •983166 983 1 3o 983094 983oo8 983022 982986 982950 982914 982878 982842 D. I bine 1 V 'V •37 '57 V 57 •57 •57 •57 •57 :U -58 •58 •58 •58 •58 •58 •58 ■53 •58 •58 •58 -58 •58 •58 •58 • 58 I 9 •o 9 .59 • 3 9, • 5 9 • 5 9 •5 9 ■59 '?' .59 '?9. 09, .60 6o| 60 60 60 60 60 6o| 60 60 60 60 6o, 74° Tang. I^TTT "^- .- i-428o52 42855 7 429062 429066 430070 430073 431075 43 1 577 432079 43i58o 433o8o •43358o 434080 434579 435078 435576 436073 436570 437067 43i563 438o59 438554 439048 439043 44oo36 440529 441022 44i5i4 442006 442497 442988 443479 443968 444458 444947 445435 445923 44641 1 446898 447384 447870 9-448356 448841 449326 449810 400294 400777 45i26o 45i743 452225 452706 9-453187 453668 404148 454628 455io7 455586 456o64 456542 457019 457496 Cotang. D. 8-42 8.41 8.40 8-3 9 8-38 8-38 8-3 7 8-36 8-35 8-34 8-33 8-32 8-32 8-3i 8-3o 8-29 8-28 8-28 8-27 8-26 8-25 8-24 8-23 8-23 8-22 8-21 8-20 8-19 8-i 9 8.18 8.17 8-i6 8-16 8-i5 8-14 8-i3 8-12 8-12 8-n 8-io 8-09 8-09 8-o8 8-07 8-o6 8-06 8-o5 8-04 8-o3 8-02 8-02 8-oi 8-oo 7-99 7-99 7.98 7-97 7-96 , 7.96 1 7-95 j 7-94_ D. Cotang 10-571948 57U43 570938 57043 569930 569427 568 9 25 568423 567921 567420 566920 o- 566420 565920 565421 564922 564424 563927 56343o 562933 562437 561941 io-56i446 560952 560457 559964 559471 558978 558486 557994 5570o3 057012 10-556521 556o32 555542 555o53 554565 554077 553589 553 1 02 5526i6 552i3o io-55i644 55 1 1 59 550674 550190 549706 540223 548740 548207 547775 54729 10 -5468 1 3 546332 545852 540372 544892 544414 543936 543458 542981 542604 1 T»h&_J!L 60 u n 55 54 53 52 5i 5o 8 47 46 45 44 43 42 41 40 12 37 36 35 34 33 32 3i 3o 3 27 26 25 24 23 22 21 20 13 n 16 i5 14 i-3 12 11 10 34 (16 DEGREES.) A TABLE OF LOGARITHMIC M. Sine 1). Cosine J). Tang. D. Cotang. "6o~ 9-44o338 7-34 9-982842 .60 9-457496 7-94 io-54?5o4 i 440778 7 ■ 33 982805 •6o 457073 7-93 542027 it 2 441218 7 32 982769 •61 45844Q 458926 7.93 54i55i 3 44i658 7 3i 982733 •61 7-92 541075 57 4 442096 442535 7 3i 982696 •61 459400 7-91 540600 56 5 7 3o 982660 ■61 459875 7-9° 540125 55 6 442973 7 29 982624 •61 460349 7-88 539651! 54 I 4434IO 7 28 982587 •61 460823 539177 53 538703! 5a 443847 7 27 982551 •61 461297 9 444284 7 27 982514 •61 461770 7.88 53823o 5i 10 444720 7 26 982477 •61 462242 7-87 537758, 5o n 9-445i55 7 25 0-982441 •61 9-462714 7.86 :a 537286 49 12 445590 7 24 982404 •61 463 1 86 7-85 5368i4 48 i3 446025 7 23 982367 •61 463658 7-85 536342 47 14 446459 446893 7 23 982331 •61 464129 7-84 5358 7 i 46 i5 7 22 982294 982267 •61 464599 7-83 5354oi 45 16 447326 7 21 ■6l 465069 7-83 53493i 44 17 447759 20 982220 •62 46553o 7.82 53446i 43 18 448191 7 20 982183 -62 466008 7.81 533992 533524 42 19 448623 7 IO 982146 •62 466476 7.80 41 20 449054 7 l8 982109 •62 466945 7.80 533o55 40 21 9-449485 7 17 9-982072 •62 9-4674i3 ?3 io-53258 7 li 22 4499i5 7 16 982035 •62 467880 468347 532120 23 45o345 7 16 981998 •62 7-78 53 1 653 li 24 450775 7 i5 981961 -62 468814 I'll 53n86 . 25 45 1 204 7 14 981924 981886 •62 469280 7-76 530720 35 26 45i632 7 1 3 •62 469746 7-75 53o254 34 \l 452o6o 7 i3 981849 •62 4702 1 1 7 . 7 5 529789 33 452488 7 12 981812 •62 470676 7-74 529324 32 \9 45291 5 7 11 981774 •62 471141 7-73 5 2 885 g 3i 3o 453342 7 10 981737 •62 471605 7- 7 3 5283 9 5 3o 3i 9-453768 7 10 9-981699 •63 9-472068 7.72 10-5279.32 2 2 32 454194 7 3 981662 •63 472532 7.71 527468 28 33 454619 7 981625 •63 472995 473457 7.71 527005 27 34 455o44 7 07 981587 •63 7.70 526543 26 35 455469 7 07 981549 •63 473919 47438i 7 .6 9 526081 25 36 4558 9 3 7 06 981512 •63 ?:S 5256iq 24 ll 4563 1 6 7 o5 981474 •63 474842 525i58 23 456739 7 04 981436 •63 4753o3 7 .6 7 524697 524237 22 3 9 457162 7 04 981399 •63 475763 7 -b 7 21 40 457584 7 o3 9 8i36i •63 476223 7.66 523777 20 41 9- 458oo6 7 02 9-98i323 •63 9-476683 7-65 10.523317 •9 42 458427 7 01 981285 • 63 477 l 42 7-65 522858 18 43 458848 7 01 981247 •63 477601 7-64 522399 '7 44 459268 7 00 98,1 209 •63 478059 7 .63 52 1 94 1 16 45 459688 6 $ 981171 • 63 478517 7-63 52U83 i5 46 460108 6 98u33 •64 478975 7.62 521025 14 47 460527 6 98 981095 981007 • 64 479432 7.61' 520568 i3 48 460946 46 1 364 6 97 .64 479889 48o345 7.61 520III 12 49 6 96 981019 • 64 7.60 519655 1 1 5o 461782 6 95 980981 .64 480801 7 .5 9 519199 10-518743 10 5i 9-462199 6 9 5 9-980942 • 64 9-481257 ?:S t 52 462616 6 94 980904 980866 • 64 481712 518288 53 463 o3 2 6 9 3 • 64 482167 7-57 5 i 7 833 1 54 463448 6 93 980827 .64 482621 m 517379 516925 6 55 463864 6 92 980789 .64 483075 5 56 464279 6 9 1 980750 .64 483529 7.55 5 1 647 1 4 n 464694 6 90 980712 -64 483982 7.55 5i6oi8 3 465 1 08 6 90 980673 • 64 484435 7-54 5 1 5565 2 59 465522 6 & 9 8o635 -64 484887 7-53 5i5n3 1 60 465935 6 980596 • 64 48533 9 7-53 514661 Cosine P. Sine 73° Cotang. D. Tang. SrNEb AND TANGENTS. (17 DEGREES. ) 35 XL Sino D. Cosine D. | Tan£. D. Cotang. _ zn 9-465q35 6-88 9.980596 98o558 • 64 9-485339 7-55 lo-5i466i 5o i i 466348 6-88 .64 485791 7-52 514209 5i3 7 58 u 2 466761 6-87 980519 65 486242 7 .5i 3 467173 6-86 980480 • 65 486693 7 .5i 5i33o7 57 4 467585 6-85 980442 •65 487143 7«5o 512857 56 5 467996 6-85 980403 • 65 487593 7-49 512407 55 6 468407 6-84 980364 •65 488043 7-4Q 7-48 5i 1957 54 7 468817 6-83 98o325 • 65 488492 5n5o8 53 8 469227 6-83 980286 •65 488941 7-47 5uo59 52 9 469637 6-82 980247 •65 489390 7-47 5io6io 5i 10 470046 6-8i 980208 • 65 489838 7.46 5ioi62 5o ii 9'470455 6-8o 9-980169 • 65 9-490286 7.46 10-509714 % 12 470863 6-8o 980130 • 65 490733 7.45 509267 5o882o 5o8373 i3 471271 471679 6-79 6-78 980091 980002 •65 • 65 49 1 1 80 491627 7-44 7-44 % i5 472086 6-78 980012 • 65 492073 7-43 507927 45 16 472492 6-77 979973 .65 492519 7.43 507481 44 12 472898 6.76 979934 979895 • 66 492965 7-42 5o7o35 43 4733o4 6-76 • 66 493410 7.41 506590 42 19 473710 6-75 979855 •66 493854 7.40 5o6i46 4i ?.o 474i 1 5 6-74 979816 • 66 494299 9-494743 7.40 5o57oi 40 21 9'4745iq 474923 6-74 9-979776 .66 7.40 io-5o52D7 39 22 6- 7 3 979737 • 66 495i86 ]'M 5o48i4 38 23 475327 6-72 979697 • 66 49563o 504370 37 24 473730 6-72 979658 • 66 496073 7-37 503927 36 20 476i33 6.71 979618 • 66 49651 5 7-37 5o3485 35 26 476536 6-70 979579 • 66 496957 7-36 5o3o43 34 27 476938 477340 6-69 979539 • 66 497399 7-36 5o26oi 33 28 6-69 6-68 979499 979409 • 66 497841 7-35 5o2i5o 501718 32 ?9 477741 .66 498282 7-34 3i 3o 478142 6-67 979420 • 66 498722 7-34 501278 3o 3i 9-478542 6-67 9-979380 • 66 9-499 163 7-33 io-5oo837 3 32 478942 6-66 979340 • 66 499603 7-33 5oo3o7 33 479342 6-65 979300 .67 5ooo42 7-32 499968 27 34 479741 6-65 979260 .67 5oo48i 7-3i 499319 26 35 480140 6-64 979220 .67 5oog2o 7-3i 499080 498641 25 36 480539 6-63 979180 - 67 1 5oi359 7-3o 24 3 7 480937 48i334 6-63 979 '4o .67, 5oi 79 7 •67 502235 7-3o 498203 23 38 6-62 979100 7-29 497765 22 3 9 481731 6-6i 979059 •67J 502672 7.28 497328 21 4o 482128 6-6i 979010 •671 5o3i09 7.28 496891 20 41 9-482525 6-6o 9-978979 •671 9 5o3546 7-27 10-496404 IO 42 482921 6-5 9 978939 •67 503982 7-27 496018 10 43 4833 1 6 6-5 9 6-58 978898 .67 5o44i8 7-26 495582 17 44 483712 978838 • 67 5o4854 7. 2 5 495146 16 45 484107 6-57 978817 .67 5o5289 7-2D 4947 1 1 i5 46 4845oi 6-57 978777 .67 5o5724 7-24 494276 14 47 484895 485289 6-56 978736 •67 5o6i59 7-24 493841 i3 48 6-55 978696 97 8655 • 68 506393 7-23 493407 12 t 9 485682 6-55 • 68 507027 7-22 492973 1 1 5o 5i 486075 6-54 978615 • 68 507460 7-92 492340 10 9-486467 6-53 9-978574 -68 ! 9.507893 -68 5o8326 7-21 ir 492107 I 52 486860 6-53 978533 7-21 491674 53 48 7 25i 6-5a 9784Q3 •68, 508759 7-20 491 241 7 54 487643 6-5i 978452 •68 509191 7- 19 490809 6 55 488o34 6-5i 97841 1 •68 509622 ?:a 490378 5 56 488424 6-5o 978370 •68 5ioo54 489946 489515 4 ^ 488814 6-5o 978329 •68 5io485 7.18 3 58 489204 6-4Q 978288 •68 5io 9 i6 7.16 489c 84 2 ^ 9 489J93 480982 6- 48 978247 •68 5n346 488654 1 60 6-48 978206 •68 72° 511776 7.16 488224 Conine D. Sine Cotang. _D. _Tan£ L _ .il a 36 (18 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine | 1). Tang. D. Cotang. 60 9-489982 490371 6-48 9-978206' »68 9-5i 1776 7.16 10 -488222! i 6-48 97 8i65! -68 512206 7.16 487794 58 2 490759 6-47 978124! -68 5i2635 7 -i5 487365 3 491 147 6-46 978083, -69 5 13064 7-14 486o36 486O07 57 4 49i535 6-46 978042! -69 5 1 3493 7'i4 56 5 491922 49230S 6-45 978001 1 -69 D13921 5i4349 7-13 486079 55 6 6-44 977959 -69 977018, -69 7 -i3 48565i 54 7 492695 493081 6-44 5i4777 7-12 485223 53 8 6-43 977»77 -69 J15204 7-12 484796 52 9 493466 6-42 977835 .69 5i563i 7-u 484369 483 9 43 5i IC 49385 1 6-42 977794 9. 977752 •69 5 1 6057 7-10 5d ii 9-494236 6.41 -69 9-516484 7-10 io-4835i6 8 12 494621 6-41 9777 1 1 .69 516910 5i 7 335 7-09 483090 i3 495oo5 6-40 977660 977628 -69 7-09 7-08 482665 47 14 495388 6-3 9 •69 517761 5i8i85 482239 48181O 46 i5 495772 6-3 9 6-38 977086 •69 7-08 45 16 496154 977544 •70 5i86i 7-07 481390 44 '7 496537 6-37 977503 •70 519034 7-06 480966 43 18 496919 497001 6-3 7 977461 •70 519458 7-06 480042 42 19 6-36 977419 •70 5i 9 88 2 7-o5 480118 4i 20 497682 6-36 977377 9-977335 •70 52o3o5 7-o5 479695 40 21 9-498064 6-35 •70 9-520728 7-04 10-479272 3 9 22 498444 6-34 977293 •70 52ii5t 7-o3 470849 38 23 498825 6-34 977201 •70 52i573 7-o3 478427 37 14 499204 6-33 977209 •70 521995 7-o3 478005 36 »5 499084 6-32 977167 •70 522417 7-02 477583 35 (6 499963 6-32 977125 •70 522838 7-02 477162 34 n 5oo342 6-3i 977083 •70 52325q 7.01 476741 33 18 500721 6-3i 977041 •7° 52368o, 7-01 476320 32 >9 501099 6-3o 976999 •70 524100 7-00 475900 3i lo 501476 6-29 976907 •70 524020 6-99 475480 3o ti 9-5oi854 6-29 6-28 9-976914 976872 97683o •70 9-524939 6.90 6.98 10-475061 3 \2 D0223l •71 52535 474641 53 502607 6-28 •71 520778 6-98 474222 27 U 502984 6-27 976787 *7' 52619-7; 6.97 4738o3 26 )5 5o336o 6-26 976745 •71 5266i5 6-97 473385 25 16 5o3735 6-26 976702 •71 527033 6.96 472967 472049 24 *7 5o4i 10 6-25 976660 'V 52745i! 6-96 23 58 5o4485 6-25 976617 •7' 5278681 6- 9 5 472132 22 *9 504860 6-24 976074 •71 528285 6- 9 5 47i7i5 21 4o 5o5234 6-23 976532 •71 528702 6-94 471298 20 4i 9-5o56o8 6-23 9-976489 •7' 9-529119' 6- 9 3 10-470881 '2 i2 5o5 9 8i 5o6354 6-22 976446 •7' 529035 6- 9 3 470465 18 43 6-22 976404 •7 l 029950 6- 9 3 470050 '7 44 506727 6-21 976361 •7i 53o366 6-92 469634 16 45 507099 6-20 6 7 63i8 -71 530781 6-91 469219 i5 46 507471 6-20 g-]62-j5 j -71 531196 6.91 468804 14 47 5o 7 843 6.19 9762321 -72 53i6n 6-90 46838o 467970 i3 48 5o82i4 6-19 976189; -72 532025 6.90 12 49 5o8585 6-i8 976146- -72 532439' 532853 6- £9 467561 11 5o 5o8o56 9-509326 6.18 ! 976io3 i -72 6-89 467147 10 5i 6-i 7 : 9- 976060' -72 9 533266 6-88 10-466734 I 5s 509696 6-i6 j 976017! -72 533679 6-88 466321 53 5ioo65 6-i6 I 975974 .72 534092 6-87 465908 7 54 5io434 6-i5 975930; .72 97 588 7 .72 5345o4 6-87 4654o6 6 55 5io8o3 6-i5 5349161 535328 6-86 465o84 5 56 5i 1 172 6-14 975844J -72 6-86 464672 4 2 5u54o 6-i3 975800! -72 535 7 3 9 ! 6-85 464261 3 511907 6-i3 975757 1 -72 536i5o< 6-85 46385o 2 5 9 612275 6-12 975714 .72 53656i i 6-84 46343a 463028 1 6o 512642 6-12 975670 -72 536972! 6-84 Cosino _©.__ Sine J71^°_ Co tang. 1 D. _l Tang.__ mT BINES AND TANGENTS. (19 DEGREES. ) 3 1 M. Sine D. Cosine | D. Tang. 1 D - Cotang. o 9; 5 1 2642 6-12 9-975670 .73 9-536972 6-84 io-463o28 60 i 5i3ooq 6- 11 975627 .73 53 7 38 2 6-83 462618 fg 2 5i337$ 6 11 975583 .73 537792 538202 6-83 462208 3 5i374i 6-io 975539 •73 6-82 461798 57 4 514107 6-09 975490 •73 5386i 1 6-82 461389 56 5 5i4472 6.09 6-oB 975452 .73 539020 6-8i 460980 55 6 5i4837 975408! .73 97 5365 .73 539429 6- 81 460371 54 I 5l5202 6-o8 53 9 837 6-8o 460163 53 5i5566 6-07 97 532i .73 9752771 -73 540245 6-8o 459755 52 9 5 1 5930 6-07 6-o6 54o653 6-79 45o347 438939 5i 10 516294 975233 .73 541061 6- 7 Q 6-78 5o ii 9.516637 6-o5 9-975189 .73 975145, -73 9.541468 10-458532 % 12 517020 6-o5 541875 6-78 458i25 i3 517382 5-o4 975101 .73 542281 6-77 457719 47 14 517745 6-o4 9 7 5o57 .73 542688 6-77 4573i2 46 i5 518107 6-o3 975oi3 •73 543094 6.76 456906 45 16 518468 6-o3 974969 974925 •74 543499 6-76 4563oi 44 n 518829 6-02 •74 543900 5443 1 6-75 436095 43 18 519190 6-oi 974880 •74 6- 7 5 455690 455285 42 '9 5i955i 6-oi 974836 •74 544715 6-74 4i 20 51991 1 6-oo 974792 •74 545119 6-74 45488i 40 21 9.520271 6-oo 9-974748 •74 9-545524 6.73 10-454476 ll 22 52o63i 5-99 974703 •74 545928 6-73 454072 23 520990 5.99 974659 •74 54633i 6.72 453669 453263 37 24 52i349 5- 9 8 974614 •74 546735 6-72 36 25 521707 5- 9 8 974570 •74 547 1 38 6-71 452862 35 26 522066 5-97 97452D •74 547540 6-71 45246o 34 27 522424 5.96 974481 •74 547943 548345 6-70 452057 33 28 522781 5- 9 6 974436 •74 6.70 45 1 655 32 ?9 5 2 3i38 5- 9 5 974391 •74 548747 6-69 45i253 3i 3o 523495 5- 9 5 974347 •75 549149 6-69 6-68 43o85i 3o 3i 9-523852 5-94 9-974302 •75 9- J49550 10-430430 2 32 524208 5- 9 4 974257 •75 549921 6-68 45oo4o 449648 33 524564 5- 9 3 974212 •75 55o352 6-67 27 34 524920 5- 9 3 974167 •75 55o752 6-67 449248 26 35 526275 5-92 974122 •75 55i i52 6-66 448848 25 36 52563o 5-91 974077 • 75 55i552 6-66 448448 24 ll 525o84 526339 52669J 5-91 974o32 .75 55l 9 32 6-65 448048 23 5-90 973987 -75 55235i 6-65 447649 22 39 5-90 5-8 9 973942 973897 • 7 5 552750 6-65 44725o 21 4o 527046 - 7 5 553i49 6-64 44685i 20 4i 9-527400 5-8 9 5-88 9-973832 •75 9-553548 6-64 IO-446452 :? 42 527753 973807 • 7 5 553946 6-63 446o54 43 528io5 5-88 973761 -75 554344 6-63 445656 17 44 528458 5-87 973716 .76 554741 6-62 445259 16 45 528810 5-8 7 973671) .76 555i3 9 6-62 444861 i5 46 529161 5-86 973625 .76 555536 6-6i 444464 14 % 5295i3 5-86 9735So| -76 555933 556J29 6-61 444067 i3 529864 5-85 973535 •76 6-6o 443671 12 i 9 53o2i5 5-85 973489 •76 556723 6 60 443275 1 5o 53o565 5.84 973444 •76 5571 2 1 6-59 442870 . > 10-442483 9 442087 8 441692 j 7 441 298 S 6 f 1 9-53o9i5 5.84 9-973398 .76 9-557517 6 5 9 5a 53i265 5-83 973352 •76 55791I 5583o8' 6-5 9 53 53i6i4 5.82 973307 .76 6-58 54 53 1 963 53 2 3i2 5.82 973261 .76 538702 6-58 55 5.8i 9732i5 •76 559097. 6-57 44oqo3| 5 56 53266i 5.8i 973169 .76 539491 55 9 885: 6-57 440 5oo 4401 1 5 4 tl 533009 5.8o 978124 -76 6-56 3 533357 5.8o 973078 .76 560279! 560673! 6 56 439721 439327 43S934 2 J 9 533704 i$ 973o32| -77 6-55 1 6o 534o52 972986! .77 56 1 066 6-55 Cosine D. _Siiie [ 7

° Cotang. D. Tang. | M. 1 SINES AND TANGENTS. (21 DEUREES.] 39 M. o S3 no D. Cosine D. Tang. D. Cotang. 9-554329 554658 5 48 9-970152 "87 9-384177 6-29 io-4i5823 6c i 5 48 970103 .81 584555 6 3 4i5445 5 9 2 554987 5553 1 5 5 47 970055 .81 584932 585309 6 4i5o68] 58 3 5 47 970006 .81 6 28 414691! 57 4 555643 5 46 969957 .81 585686 6 27 4U3i4 56 5 555971 5 46 969909 .81 586062 6 27 413938! 55 6 556299 5 45 969860 • 8i 586439 6 27 4i356i| 54 7 556626 5 45 96981 1 • 8i 586813 6 26 4i3i85 53 8 556g53 5 44 969762 .81 587190 6 26 412810 52 9 557280 5 44 969714 .81 587066 6 25 412434 5i 10 557606 5 43 969665 .81 587941 6 25 412009 5o n 9-557932 5 43 9*969616 .82 9 -5883 16 6 25 10-411684 49 12 558258 5 43 969067 .82 588691 6 24 4n3o9 48 i3 558583 5 42 969018 .82 589066 6 24 410934 47 U 558909 5 42 969469 .82 589440 6 23 410060 46 i5 55g234 5 4i 969420 .82 58 9 8i4 6 23 410186 45 16 55o558 5 4i 969370 .82 5 9 oi88 6 23 409812 44 )l 55 9 883 5 4o 969321 .82 590062 6 22 409438 43 560207 5 40 969272 • 82 590935 6 22 409065 42 *9 56o53i 5 i 9 969223 .82 59i3o8 6 22 408692 4i 20 56o855 5 ll 969173 • 82 591681 6 21 408319 40 21 9-56ii78 5 9-969124 .82 9-592054 6 21 10-407946 H 22 56i5oi 5 38 969073 .82 592426 6 20 407074 23 561824 5 37 969025 968976 .82 592798 6 20 407202 I 1 24 562146 5 3 7 .82 593170 6 19 406829 36 25 562468 5 36 968926 968877 •83 593542 6 l 9 4o6458 35 26 562790 5 36 •83 593914 6 18 406086 34 2 7 563 1 1 2 5 36 968827 •83 594285 6 18 4057 1 5 33 28 563433 5 35 968777 •83 5g4656 6 18 4o5344 32 2 9 563755 5 35 968728 •83 595027 6 17 404973 3i 3o 564075 5 34 968678 •83 5 9 53 9 8 6 n 404602 3o 3i 9-564396 5 34 9-968628 •83 9-595768 6 n 10-404232 2Q 32 564716 5 33 968578 •83 5 9 6i38 6 16 4o3862 28 33 565o36 5 33 9 685 2 8 •83 5 9 65o8 6 16 403492 27 34 565356 5 32 968479 •83 596878 6 16 4o3i22 26 35 565676 5 32 968429 •83 597247 6 i5 402753 25 36 565995 5 3i 968379 •83 597616 6 i5 402384 24 37 5663U 5 3i 968329 •83 597985 5 9 8354 6 i5 40201 5 23 38 566632 5 3i 968278 •83 6 U 401646 22 3 9 566951 5 3o 968228 .84 598722 6 14 401278 21 4o 567269 5 3o 968178 •84 599091 6 i3 400909 20 4i 9-56 7 58 7 5 29 9-968128 .84 9 -5-99459 6 i3 10 -400041 IO 42 567904 568222 5 2 § 968078 • 84 599827 6 i3 400173 l8 43 5 28 968027 •84 600194 6 12 399806 '7 44 56853g 5 28 967977 •84 6oo562 6 12 399438 16 45 568856 5 28 967927 967876 • 84 600929 6 11 399071 398704 i5 46 569172 5 27 • 84 601296 6 11 14 47 569488 5 ll 967826 • 84 601662 6 11 3 9 8338 i3 48 569804 5 967775 • 84 602029 6 10 397971 12 49 570120 5 26 967725 • 84 602395 6 10 397605 11 5o 570435 5 25 967674 • 84 602761 6 10 397239 10 5i 9-570751 5 25 9-967624 • 84 9-6o3i27 6 09 10-396873 8 %2 571066 5 24 967573 • 84 6o3493 6o3858 6 09 396507 53 57i38o 5 24 967522 .85 6 3 396142 1 54 571695 5 23 967471 .85 604223 6 395777 6 55 572069 5 23 967421 • 85 6o4588 6 08 395412 5 56 5 7 2323 5 23 967370 •85 604953 6o53i7 6 07 395047 4 & 572636 5 22 967319 •85 6 07 3 9 4683 3 5729D0 5 22 967268 •85 6o5682 6 C7 3 9 43i8 2 59 5 7 3263 5 21 967217 •85 606046 6 06 393954 393090 1 6o 573575 5-21 967166 •85 606410 6 06 Cosine D. Sine G80 Cotang. D. ~T^7~ 26 40 (22 DEGREES.) A TABLE OP LOGARITHMIC M. Si no D. Cosine | D. Tan*. D. | Cotang. 9-573575 573888 5-21 9-967166 -85 9 • 6064 1 6- 06 10-393590 60 i 5-20 9671 1 5 .85 60677c 6-o6 393227 5? 2 574200 5-20 967064 .85 607137 6-o5 3 9 2863 3 574512 5-19 9670 I C • 85 607500 6-o5 392500 57 4 574824 5-19 966961 • 85 607863 6-04 39213" 56 5 570136 5-19 5-i8 966910 9 6685 g 966808 • 85 608225 6-04 391775 55 6 575447 • 85 6o8588 6-04 391412 54 7 575708 5.i8 • 85 608950 609312 6-o3 391050 53 8 576069 5- 17 966756 • 86 6-o3 390688 52 9 576379 576689 5.17 966705 • 86 609674 6-o3 390326 389964 5i 10 5-i6 9 66653 • 86 6ioo36 6-02 5o ii 9-576999 577309 577618 5-i6 9-966602 .86 9-610397 610739 6-02 10-389603 % 12 5-i6 96655o • 86 6-02 389241 38888o i3 5-i5 966499 • 86 611120 6-oi 47 14 577927 5-i5 966447 • 86 61 1480 6-oi 388520 46 i5 578236 5-14 966395 .86 611841 6-oi 388io 9 45 16 578545 5-i4 966344 .86 612201 6- 00 387799 387439 44 !Z 5 7 8853 5-i3 966292 .86 6i256i 6-00 43 579162 5-i3 966240 .86 612921 6-oo 387079 42 !9 579470 5-i3 966188 .86 6i328i 5-99 386719 41 20 579777 5-12 966136 .86 6i364i 5-99 5.98 38635 9 40 21 •y-58oo85 5-12 9-966085 .87 9-614000 io-386ooo ll 22 580392 5. II 966o33 .87 6i435 9 5- 9 8 385641 23 580699 5. II 960981 .87 614718 5- 9 8 385282 37 24 58 1 oo3 5.11 965928 960876 •87 616077 5-97 384Q23 384565 36 25 58i3i2 5- 10 •87 6i5435 5-97 35 26 58i6i8 5-io 965824 .87 6 i 5793 5-97 384207 34 27 581924 5.09 965772 .87 6161O1 5.96 383849 33 28 582229 5.09 965720 •87 616509 5-96 383491 32 o 9 582535 5-09 5-o8 9 65668 .87 616867 5.96 383i33 3i 3o 582840 9 656i5 .87 617224 5- 9 5 3S2776 3o 3i 9-583145 5.o8 9 -965563 .87 9-617582 5. 9 5 10-382418 3 32 583449 5.07 9655i 1 .87 617939 618295 6i8652 5. 9 5 382061 33 583754 5.07 965458 .87 5.94 38i 7 o5 2 34 584058 5.o6 965406 :S 5.94 38i348 35 58436i 5.o6 9 65353 619008 5.94 380992 38o636 25 36 584665 5.o6 9653oi .88 619364 5. 9 3 24 u 584968 5-o5 965248 .88 619721 5. 9 3 380279 23 585272 5-o5 965195 .88 620076 5. 9 3 379924 37 9 568 22 3 9 585574 5-o4 965i43 .88 620432 5-92 21 4o ■ 5858 77 5-04 965090 9-965007 .88 620787 5-92 379213 20 4i 9-086179 5-o3 .88 9-621142 5-92 10-378858 \l 42 586482 5-o3 964984 .88 621497 621852 5.91 3785o3 43 586783 5-o3 96493 1 964879 .88 5.91 378148 17 44 587085 5-02 .88 622207 5.90 377793 16 45 58 7 386 5-02 964826 .88 62256i 5-90 377439 i5 46 687688 5oi 964773 .88 622915 5.90 377085 14 % 587989 5-oi 964719 .88 623269 623623 i'b 376731 i3 588289 5-oi 964666 .89 5.89 376377 12 P 5885 9 o 5-oo 964613 .89 623076 5.89 376024 11 5o 588890 5«oo 964560 .89 62433o 5.88 375670 10 5i 9*589190 589489 4-99 9-964507 .89 9-624683 5.88 10-375317 8 52 4-99 964454 .89 625o36 .89' 625388 5.88 374964 53 58 97 8o 5 9 oo88 1 4- 90 4-98 964400 5.87 374612 1 54 964347 .89 625 7 4i 5.87 374259 6 55 590387 4-98 964294 .89 626093 5.87 373907 373555 5 56 5 9 o686 4-97 964240 .89! 626445 5.86 4 u 590984 4-97 964187 .89 1 626797 5.86 3732o3 3 591 2 82 4-97 964i33 •8 9 i 627149 5-86 372851 2 & 591580 4-96 964080 .89 1 627501 5-85 372499 1 . 372148! 6o 59187b' 4-96 964026 •89 627802 5-85 1 Cosine D. i Sine G7°, Coking, i D. Tang. J M^ SINES AND TANGENT8. (23 DEGREES.) 41 M. Sine 1 D ' Cosine D. | Tung. 1 D. | Cotanj;. 1 o 9.591878 . 4-96 9-964026 •89 9-627852 5-85 10-37214^ 60 i 592176 ! 4-9 5 963972 .89! 628203 5-85 371797 5 9 58 2 592473 4-93 963919 963865 •89! 62855/j 5-85 371446 3 592770 4-9 5 •90J 628903 5-84 371095 57 4 593067 5 9 3363 4.94 963811 | -90, 629255 5-84 370745 56 5 4-94 963757 1 -go! 629606 5-83 370394 55 6 593659 4- 9 3 963704 •90! 629956 5-83 370044 54 I 593955 4- 9 3 96363o •9c 63o3o6 5-83 36969/ 53 594251 4-93 963596 .90 63o656 5-83 36 9 344 52 9 594547 594842 4-92 963542 • 90 63ioo5 5-82 368995 5i 10 4.92 963488 ■9c 63i355 5-82 368645 5o ii 9.590137 4-9" 9-963434 ■ 90 9-631704 5-82 10-368296 49 12 590432 4-91 963379 .90 632053 5-8i 367947 48 i3 595727 4-91 963323 ■90 6324oi 5- 81 367399 47 14 396021 4.90 963271 •90J 63275o 5-8i 367230 46 i5 5963i5 4.90 4.89 963217 .90 633o 9 8 5-8o 366902 45 16 596609 596903 963 1 63 .90 633447 5-8o 366353 44 \l 4.89 963 l 08 •91 633795 5-8o 3662o5 43 597196 4.89 963o54 •91 634U3 5-79 365857 42 *9 597490 4-88 962999 •91 634490 5.79 3655 10 41 20 597783 4-88 962943 •91 634838 5.79 5.78 365i62 40 21 9.598075 4-8 7 9-962890 9 62836 • 9 f 9-635i85 io-3648i5 3 9 22 598368 4-8 7 • 9 ! 635532 5.78 364468 38 23 5 9 866o 4-8 7 962781 .91 635879 5- 7 8 364121 ll 24 598932 4-86 962727 •91 636226 5.77 363774 25 599244 4-86 962672 •91 636572 5.77 363428 35 26 599336 4-85 962617 •91 636919 5.77 363o8i 34 27 599827 4-85 962562 •91 637265 5.77 362735 33 28 600118 4-85 962508 •91 63 7 6n 5.76 36238 9 32 29 600409 4.84 962453 •91 637956 638302 5-76 362044 3i 3o 600700 4-84 962398 •92 5.76 36i6 9 8 3o 3i 9 . 600990 4.84 9-962343 .92 9-638647 5.75 io-36i353 3 32 601280 4-83 962288 •92 638992 6393J7 5.75 36 1 008 33 601370 4-83 962233 •92 5.73 36o663 27 34 601860 4-82 962178 •92 639682 5.74 36o3i8 26 35 602 1 30 4-82 962123 •92 640027 5-74 359973 25 36 602439 4-82 962067 •92 640371 5.74 359629 24 ll 602728 4.81 962012 •92 640716 5.73 359284 23 6o3oi7 4.81 901957 •92' 641060 5- 7 3 358940 22 3 9 6o33o5 4-8i 961902 •92J 641404 5.73 358596 21 40 603394 4-8o 961846 •92 641747 5-72 358253 20 41 9 .6o3882 4-8o 9-961791 •92, 9-642091 5-72 10-357909 18 4T 604170 4-79 961735 •92 642434 5-72 357566 43 604457 4-79 961680 •9 2 , 642777 5-72 357223 »7 44 6o4745 4-79 961624 •g3 643120 5.71 35688o 16 45 6o5o32 4-78 961569 •93 643463 5.71 356537 i5 46 6o53i9 4-78 9 6i5i3 • 9 3 ; 643806 5-71 356194 14 s 6o56o6 4-78 961458 •93 644148 5-70 355852 i3 600892 4-77 961402 •931 644490 5-70 3555io 12 49 606179 4-77 961346 •93 644832 5-70 355i68 11 5o 606465 4-76 961290 •93 645174 5-69 354826 10 5i 9-606751 4-76 9-961235 •93 9-6455i6 5-69 io.354484 Q 52 607036 4-76 961179 961123 • 9 3; 645857 5-69 354U3 8 53 607322 4-75 •931 646199 5-69 5-68 3538oi I 54 607607 4-75 961067 •q3' 646340 35346o 55 607892 4-74 96101 1 •93' 646881 5-68 353no 352778 5 56 608177 4-74 960955 •93 647222 5-68 i U 608461 4-74 960899 960843 •93 647562 5-67 352438 i 608745 4- 7 3 •94' 647903 5-67 352097 2 5 9 60Q029 609313 4- 7 3 960786 •94! 648243 5-6 7 351737 1 60 4-73 960730 •94 648583 5-66 35i4i7 1 Cosine I). Sine GG C I Cotang. 1 D. ""rltn"^" 42 (24 DEQREB8.) A rABLK OF LOOARITHMTO M. o Sino D. Cosine D. Tang. D. Cotang. 60 9»6o93i3 4-73 9-960730 •94 9-648583 5-66 io-35i4i7 2 609597 609880 4 4 72 72 960674 960618 •94 •94 648923 649263 5 5 66 66 351077 350737 is 3 610164 4 72 960561 •94 649602 5 66 350398 57 4 610447 4 7* 96o5o5 .94 649942 5 65 35oo58 56 5 610729 4 7i 960448 .94 65o28i 5 65 3497>9 55 6 611012 4 70 960392 •94 65o62o 5 65 34938o 54 I 61 1294 4 70 96o335 .94 650959 5 64 349041 348703 53 61 1576 4 70 960279 .94 651297 65i636 5 64 52 9 6u858 4 69 960222 .94 5 64 348364 5i IC 612140 4 69 960165 ■94 651974 5 63 348026 5o ii 9.612421 4 69 9-960109 - 9 5 9*6523i2 5 63 10-347688 % is 612702 4 68 960052 .95 65265o 5 63 34735o i3 612983 4 68 939995 .95 652988 5 63 347012 47 U 6i3264 4 67 959938 .95 653326 5 62 346674 46 i'i 6i3545 4 67 959882 .95 653663 5 62 346337 45 ri 6i3825 4 67 959825 .95 654000 5 62 346000 44 »! 6i4io5 4 66 959768 .95 654337 5 6i 345663 43 iB 6i4385 4 66 9597 1 1 .95 654674 5 61 345326 42 »9 614665 4 66 959654 .95 655ou 5 61 344989 4i 20 614944 4 65 959596 9-959539 .95 655348 5 61 344652 4o 21 9-6i5223 4 65 .95 9-655684 5 60 io-3443i6 is 2J 6i?5o2 4 65 959482 .95 656o2o 5 60 343980 23 6i5 7 8i 4 64 959425 .95 656356 5 60 343644 37 24 616060 4 64 9D9368 .95 656692 5 5 9 3433o8 36 23 6i6338 4 64 95q3io .96 657028 5 5 9 342972 35 26 616616 4 63 959253 .96 657364 5 5 9 342636 34 27 616894 4 63 959195 .96 657699 658o34 5 5 9 3423oi 33 28 617172 617450 4 62 9D9138 .96 5 -58 341966 3"2 29 4 62 959081 .96 658369 5 58 34i63i 3i 3o 617727 4 62 939023 .96 658704 5 58 341296 3o 3i 9-618004 4 61 9-958965 .96 9-659o3o 65937J 5 58 10-340961 it 32 618281 4 61 958908 .96 5 5 7 340627 33 6i8558 4 61 95885o .96 659708 5 57 340292 33 9 958 27 34 6i8834 4 60 938792 .96 660042 5 5 7 26 35 6iqiio 4 60 938734 .96 660376 5 5 7 339624 25 36 619386 4 60 938677 .96 6607 1 5 56 339290 338967 24 3i 619662 4 5 9 958619 .96 661043 5 56 23 38 619938 4 J 9 938361 .96I 661377 5 56 338623 22 3g 6202l3 4 & 9385o3 .97 661710 5 55 338290 21 . 4o 620488 4 938443 •07! 662043 7 5 55 337967 20 4i 9-620763 4 58 9-958387 •97 9-662376 5 55 10-337624 » 42 62io38 4 5 7 95832 9 •97 662709 5 54 337291 336958 43 62i3i3 4 57 958271 •97 663o42 5 54 1*7 44 621587 4 5 7 9582i3 •97 6633 7 5 5 54 336625 l6 45 621861 4 56 958 1 54 •97 663707 5 54 3362 9 3 i5 46 622135 4 56 958096 •97 664039 5 53 335 9 6i 14 47 622409 4 56 958o38 •97 664371 5 53 335629 i3 48 622682 4 55 957979 •97 664703 5 53 335297 12 49 622956 4 55 957921 •97 665o35 5 53 334965 1 1 5o 623229 4 55 95 7 863 •97 665366 5 52 334634 10 5i 9-6235o2 4 54 9-957804 •97 9 -6656 9 7 5 52 io-3343o3 i 52 623774 4 54 957746 .98 .98 666029 5 52 333971 53 * 624047 4 54 957687 66636o 5 5i 333640 7 54 624319 4 53 957628 .98 666691 5 5i 333309 6 55 624591 4 53 957570 .98 667021 5 5i 332979 332648 5 56 624863 4 53 93751 1 • 9 8, 667352 5 5i 4 U 625i35 4 52 957452 • 9 8 ; 667682 5 5o 3323i8 3 625406 4 52 957393 .98 668oi3 5 5o 331987 2 5 9 625677 4 52 957335 - 9 8 ! 668343 5 5o 33i65t 1 6o L 625948 Cosine 4 5i 937276 Sine . 9 3 ; 668672 5 5o 33 1 328 ] [>. «35=> Cotang. ~] 0. Taru£.__ SINES AND TANGENTS. (25 DEGREES.) 43 M. Sino 1). Cosine | D. Tang. L>. Cotimg'. io-33i327 33009S 9-6a5g48 4-5i 9.957276 .98 9.668673 5-5o 60 i 626219 4 5i 957217; .98 669002 5 49 U 2 626/490 4 5i 9 5 7 i58 . 9 8 66 9 332 5 49 33o668 3 626760 4 5o 957099 • 9 8 669661 5 % 33d3.?9 5 7 4 627030 4 5o 95 7 040 ; .98 669991 5 330009 56 5 627300 4 5o 956981 .98 670320 5 48 329680 55 6 627570 4 49 966921 9 56862 •99 670649 5 48 329351 54 I 627840 628109 4 49 •99 67097] 671306 5 48 329023 53 4 49 9368o3 •99 5 47 328694! 52 9 628378 4 48 956744 •99 67 1 634 5 47 328366 5i IO 628647 4 48 9 56684 •99 671963 5 47 328037 5o ii 9-628916 4 47 g. 936625 •99 9-672291 5 47 ,0.327709 % 12 629185 4 47 936566 •99 672619 5 46 32738i i3 629453 4 47 9565o6 •99 672947 5 46 32 7 o53 326726 47 14 629721 4 46 936447 •99 673274 S 46 46 i5 629989 4 46 9D6387 •99 673602 5 46 3263 9 8 45 16 630257 4 46 956327 •99 673929 5 45 326071 44 H 63o524 4 46 9 56268| •99 674257 5 45 325743 43 18 630792 4 45 9362081 •00 6 7 4584 5 45 3254i6 42 x 9 63 1 039 4 45 936148 i •00 674910 5 44 325090 41 20 ■ 63i326 4 45 936089 1 •00 673237 5 44 324763 40 21 9-63i5g3 4 44 9-936029 1 • 00 0.673504 5 44 I0 . 324436 3 9 22 63i85o 632123 4 44 955969 i ■00 675890 5 44 324110 38 23 4 44 955909 1 •00 676216 5 43 323784 37 24 632392 4 43 955849 i •00 676543 5 43 323457 36 25 632638 4 43 933789! i •00 676869 5 43 323i3i 35 26 632923 4 43 933729'] •00 677194 5 43 322806 34 27 633 1 89 4 42 955669'! •00 677320 5 42 322480 33 28 633434 4 42 955609 1 ■00 677846 5 42 322i54 32 29 633719 4 42 95554811 •00 678171 5 42 321829 3i 3o 633984 4 4i 955488! 1 •00 678496 5 42 32i5o4 3o 3i 9-634249 4 4i 9.955428 1 •01 9.678821 5 4i 10.321179 29 32 6343i4 4 40 955368! 1 •01 679146 5 4i 320854 28 33 634778 4 4o 955307 1 •01 679471 5 4i 320529 27 34 635o42 4 40 955247 1 •01 679795 5 4i 320205 26 35 6353o6 4 3 9 955 186! 1 •0, 680120 5 40 319880 25 36 63)370 4 3 9 9DD 1 26i J •0. 68o444 5 40 319556 24 3 2 635834 4 3 2 955o65 1 •01 680768 5 40 319232 23 38 636097 4 38 955oo5 1 •01 681092 5 40 318908 3 1 8584 22 3 9 636360 4 38 954944 1 954883 1 •01 681416 5 39 21 4o 636623 4 38 •01 681740 5 39 318260 20 4i 9-636886 4 37 9-954823 1 •01 9.682063 5 39 io.3i79 3 7 IQ 42 637148 4 37 954762 1 •01 68 2 387 5 3 9 317613 l8 43 63741 1 4 37 954701 1 •0. 6S2710 5 38 317290 17 44 637673 4 37 954640 i •01 683o33 5 38 316967 16 45 637 9 35 4 36 954579 1 9 545i8 1 •c: 683356 5 38 316644 i5 46 638i 9 7 638438 4 36 •02 683679 5 38 3i632i 14 47 4 36 954457 1 • 02 6S400 1 5 37 315999 i3 48 638720 4 35 954396 1 •02 6H4324 5 37 315676 12 49 638981 4 35 954335 1 •02 684646 5- 37 3 1 5354 11 5o 639242 4 35 954274 ' •02 684968 5- 37 3i5o32 10 5i 9-6395o3 4 34 9-934213 1 ■02 9-685290 5 36 jr>.3l47IO 9 52 639764 4 34 954i52 1 ■02 6856 1 2 5 36 3U388 8 53 640024 4 34 954090 1 •Oi 685 9 34 5 36 i 1 4066 7 54 640284 4 33 954029 1 •02 686255 5- 36 3 1 3745 6 55 640544 4 33 9 53 9 68 1 •02 6865 77 5 35 3i3423 5 56 640804 4 33 953906 1 •02 686898 5- 35 3i3io2 4 57 641064 4 32 953845 1 •02 687219 5- 35 312781 3 58 641324 4 32 953783 1 •02 687540 5- 35 3 1 2460 2 5 9 641584 4 32 953722 1 •o3 68-1861 5- 34 3i 2 139' 1 3u8i8| 60 641842 4-3i 95366o 1 •o3 688182 5- 3/ Cosine J =^_. Sine [C M°, J 5 \ Tang. 1 M. 14 (26 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine i~o3 Tang. D. Cotnng. 9-641842 4-3i 9-953660 9.688182 5.34 io-3ii8i8 60 i 642101 4-3i 933599 953537 i-o3 688502 5 •34 3 1 1498 % 2 64236o 4-3i i-o3 688823 5 •34 311177 3 642618 4-3o 933473 i-o3 689143 5 •33 310837 57 4 642877 4-3o 9534i3 i-o3 689463 5 • 33 3io537 56 5 643 1 35 4-3-0 953352 i-o3 689783 5 -33 310217 55 6 643393 4-3o 953290 q53228 i-o3 690103 5 •33 3098971 54 I 64365o 4-29 i-o3 690423 5 ■ 33 309377, 53 3oo258' 52 3o8 9 38: 5i 643908 4-29 q33i66 i-o3 690742 5 •32 9 644i65 4-29 q53io4 i-o3 691062 5 •32 m 644423 4-28 933042 1 -o3 691381 5 -32 308619! 5o 1 1 9 • 644680 4 28 9-952980 1.04 9-691700 5 • 3i io-3o83oo % 12 644936 4 28 932918 [•04 692019 5 ■ 3i 307981 .3 640193 4 27 932855 1-04 6 9 2338 5 •3i 307662 47 H 645400 4 27 932793 [•04 692656 5 3i 307344 46 i5 643706 4 27 952731 [■04 692975 5 3i 307025 45 16 643962 4 26 932669 1-04 693293 5 3o 306707 3o6388 44 \l 646218 4 26 9526061 [•04 693612 5 3o 43 646474 4-26 952544 1.04 693930 5 3o 306070 42 »9 646729 4-25 932481! [•04 694248 5 3o 3o5752 41 20 646984 4-25 932419! [•04 694566 5 29 3o543 i 40 21 9-647240 4-25 9-932356 [•04 9-694883 5 29 io-3o5iJ7 3 9 22 647494 4-24 952294 [•04 695201 5 29 3o4799 38 23 647749 4-24 95223i • 04 6 9 55i8 5 29 304482 37 24 648004 4-24 952168 -o5 6 9 5836 5 2 304164 36 25 648238 4-24 932106' • o5 696153 5 3o3847 35 26 648312 4-23 952043 •o5 696470 5 28 3o353o 34 27 648766 4-23 9 5i 9 . Sine ^62° Cotang. D Tang. 1 M. 46 (28 DEGREES.) A TABLE OP LOGARITHMIC M. Sino D. Cosmo | D. Tang. 1 D. Cotang. ._- o 9-671609 3.96 9-945o35 1. 13 9.725674 5-o8 10-274326 60" i 671847 3. 9 5 945868 1. 1 2 725979 726284 I 5- 08 274021 5 5 9 672084 3. 9 5 945800 i- 1 2 ' 5-07 273716 58 3 672321 3. 9 5 945733 i- 1 2 72658S | 5#0 7 273412 5 7 4 672558 3. 9 5 945666 1 • 1 2 726892 5-07 273108 56 5 672795 673032 3-94 945598 I -12 727197 5 -07 272803 55 6 3.94 94553i 1. 1 2 727501 ! 5.07 272499 54 I 673268 3.94 945464 I-K 727805 5.06 272195 53 67350D 3.94 945396 I • K 728109 1 5. 06 271891 271588 52 9 6 7 3 7 4i 3. 9 3 945328 i - 13 728412 : 5-o6 5i 10 673977 3. 9 3 94526l I-K 728716 5.o6 271284 5o ii 9-674213 3- 9 3 9-945I93 I-K 9.729020 5- 06 10-270980 49 12 674448 3-92 945i25 i-k 729323 ! 5-o5 270677 48 i3 674684 3-92 945o58 no 729626 ' 5-o5 270374 47 14 674919 3-92 944990 1. 1 3 729929 73o233 1 5-o5 270071 46 i5 6 7 5i55 3-92 944922 1 - 13 944854 1- 1 3 5-o5 269767 45 16 675390 3-91 73o535 5-o5 269465 44 \l 675624 3-91 944786 i- 1 3 73o838 5-o4 269162 43 675859 3gi 944718 1 -i3 731141 5-04 26885 9 41 »9 676094 3-91 94465o 1 . i3 73i444 i 5-04 268556 4i 20 676328 3-90 944582 1 • 14 73i746 5-04 268254 40 21 9-676562 3 90 9-9445i4 1 -14 9-732048 5 /°i 10-267952 3 9 22 676796 6770.50 3.90 944446 1-14 73235i 5-o3 267649 38 23 3.90 944377 1 -14 732653 5-o3 267347 37 24 677264 3.^9 944309 1. 14 732955 5-o3 267045 36 25 677498 3-89 944241 1 1 -14 733257 5-o3 266743 35 26 677731 3-89 3.88 944H 2 1 • 14 733558 5-o3 266442 34 *7 677964 944104 1 -14 73386o 5-02 266140 33 jS 678197 3-88 944o3 6 1 -14 734162 5-02 265838 32 29 6?843o 3-88 943967 1.14 734463 5-02 26553 7 3i 3o 678663 3-88 943899 1 .14 9-94383o;i -14 734764 5-02 265236 3o 3i 9-678895 3-8 7 9- 735o66 5-02 10-264934 29 32 679128 3-8 7 943761 ji. 14 735367 5-02 264633 28 33 679360 3-8 7 943693 1 - 1 5 735668 5-oi 264332 27 34 679092 3-8 7 943624 1 - 15 735969 5-oi 264031 26 35 679824 3-86 943555 1 - 15 736269 5-oi 263731 25 36 68oo56 3-86 943486 1. 1 5 736570 5-oi 26343o 24 37 680288 3-86 943417 1 • 1 5 736871 5-oj 263129 23 38 680D19 3-85 943348 1 - 15 737171 5-oo 262829 22 . 3 9 680750 3-85 943279 1 -i5 737471 5-oo 262529 21 4o 680982 3-85 9432io 1 - 1 5 737771 5-oo 262229 20 4i 9-68i2i3 3-85 9-943i4i 1 • i5 9-738071 5-oo 10-261929 :? 42 68i443 3.84 943072 1 • 1 5 738371 5- 00 261629 43 681674 3-84 943oo3 1 ■ i5 738671 4-99 261329 17 44 68i 9 o5 3-84 942934 1 • 1 5 738971 4.99 261029 16 45 682135 3 -84 942864 1 - 15 739271 4-99 260729 1 5 46 682365 3-83 942795 i- 16 739570 4.99 26o43o 14 4 I 6825 9 5 3-83 942726 i- 16 739870 4.99 26oi3o i3 48 682825 3-83 942656 1 - 16 740169 740468 4.99 25983ij 12 49 683o55 3-83 942587|i-i6 4.98 259532 11 5o 683284 3-82 942517 1 • 16 740767 4-98 1 » s5o233 10 10-258934 9 258635, 8 5i 9-6835i4 3-82 9-942448 i- 16 9.741066 4-98 ; 52 683743 3.82 942378.1-16 74i365 4- 9 8 53 683972 3-82 9423o8!i-i6 741664 4-98 258336' 7 54 684201 3-8i 942239'i-i6 741962 4-97 258o38 6 55 68$43o 3.8i 942169 i- 16 742261 4-97 257739! 5 56 6846581 3-8i 942099 1 - 16 742559 742858 4-97 257441' 4 n 684887 | 3-8o 942029 1 -16 4-97 257142: 3 685n5j 3.8o 941959 1 - 16 743 1 56 4-97 256844' 2 5q 685343! 3-8o 941889 1-17 743454 4-97 4-96 D. 256546! 1 6o 6855 7 1 | 3-8o 941819 1-17 743752 256248! 1 Cosine 1 D. Sino Gl° Cctang. ^an^T KL SINES AND TANGENTS. (29 DEGREES. ) 47 M. Sine D. Cosine | D. Tang. 1 D- Cotang. | 9-685571 3-8o 9-941819 1-17 9-743752 4 96 10-256248 60 i 685799 a- 79 941749 117 744o5o 4 96 255 9 5o 5o 255652 58 2 686027 3-79 941679 1-17 744348 4 96 3 686254 3-79 941609 117 744645 4 96 255355 5 7 4 686482 £:? 94i539 1-1-7 744943 4 96 255o57| 56 5 686709 941469 1-17 745240 4 96 2547601 55 6 686 9 36 3- 7 8 94 1 398 1-17 745538 4 9 5 2 04462 1 54 I 687163 3- 7 8 94i328 117 745835 4 95 254i65 53 687389 3.78 941258 1-17 746i32 4 95 253868 52 9 687616 3-77 94i 187 1-17 746429 4 95 253571 5i IO 687843 3-77 94i 1 17 I- 17 746726 4 95 253274 5o ii 9-688069 3-77 9 • 94 1 046 1 - 18 9-747023 4 94 10-252977 A l 12 68829D 3.77 940Q75 1. 18 7473i9 4 94 25268i 48 i3 6&8021 3-76 940005 1-18 747616 4 94 252384 47 i4 688747 3.76 940834 i- 18 747913 4 94 202087 46 i5 688972 3.76 040763 1- 18 748209 4 94 201791 45 16. 689198 3.76 940693 1 - 18 7485o5 4 93 25i495 44 17 689423 3.75 940622 1. 18 748801 4 93 25i 199 43 1 8 689648 3- 7 5 94o55i 1-18 749097 4 93 25o9o3 42 19 689873 3- 7 5 940480 1 • 18 7493o3 4 9 3 250607 4i 20 690098 3- 7 5 940409 1- 18 749689 4 93 25o3u 40 21 9 690323 3-74 9-94o338 1. 18 9-7499 8 5 4 93 iO'25ooi5 u 22 690548 3.74 940267 1 -18 750281 4 92 249719 23 690772 3-74 940 1 96 1. 18 750576 4 92 249424 37 24 690996 3 74 940120 1. 19 750872 4 92 249128 248833 36 25 691220 3.73 940054 1-19 751167 4 92 35 26 691444 3- 7 3 939982 1-19 701462 4 92 248538 34 27 691668 3- 7 3 93991 1 i- 19 751757 4 92 248243 33 28 691892 3.73 Q 3 9 84o 1-19 752o52 4 9' 247948 32 29 6921 1 5 3.72 939768 1-19 752347 4 9i 247653 3i 3o 692339 3. 7 2 939697 1-19 752642 4 9' 247358 3o 3i 9-692562 3.72 9-939625 1-19 9-752937 4 9i 10-247063 2 32 692780 3.71 939554 I- 19 75323i 4 9i 246769 33 693008 3.71 939482 1-19 753526 4 9' 246474 27 34 693231 3.71 939410 1-19 753820 4 9° 246180 20 35 693453 3.71 939339 I 19 7541 i5 4 90 245885 25 36 693676 3-70 939267 I- 20 754409 4 90 245591 24 37 693898 3-70 Q39195 1 -20 754703 4 90 245297 23 38 694120 3-70 939123 I- 20 754997 4 90 245oo3 22 39 694342 3-70 93oo52 1-20 755291 4 5 244709 2444i5 21 40 694564 3.69 938980 1-20 755585 4 20 41 9-694786 3-6g 9.938908 938836 1-20 9 -755S 7 8 4 *9 10-244122 18 42 695007 3.69 1-20 756172 4 89 243828 43 695229 3-6 9 3-68 938763 I- 20 756465 4 8 9 243535 17 44 695450 938691 I '20 756759 4 89 243241 16 45 695671 3-68 938619 1-20 757052 4 a 242948 i5 46 695892 3-68 9 38547 1-20 757345 4 242655 14 41 696 1 1 3 3-68 938475 1-20 757638 4 88 242362 i3 48 696334 3-6 7 938402 I -21 757931 4 88 242069 12 49 696554 3.67 93833o I -21 758224 4 88 241776 11 5o 696775 3.67 938258 I - 21 7585i7 4 88. 24U83 10 5i 9-696995 3.67 9-938i85 I -21 9-758Sio 4- 88 IO-24HQO 52 697215 3-66 9381 i3 I -21 759102 4- 87 240898 8 53 697435 3-66 938040 I • 21 759395 4- 87 24o6o5 7 54 697654 3-66 937967 I -21 759687 4- 87 24o3i3 6 55 697874 3-66 937895 I-2I 709979 4- 87 240021 5 56 698094 3-65 937822 I-2I 760272 4- 87 239728 4 57 6 9 83 1 3 3-65 937749 I -21 760564 4- 87 239436 3 58 698532 3-65 937676 I-2I 76o856 4- 86 239144 2 5 9 698751 3-65 937604 I-2I 761 148 4- 86 238852 1 60 698970 3-64 937531 I-2I 761439 4- 86 23856i Tang. Cosine D. Sine GOO Cotang. _J X 18 (30 DEGREES.) A TABLE OP LOQAKITHMIC M. Sine IX Cosine | D. Tang. D. Cotang. 60 9-698970 3 64 9*93753i'i >2i 9-761439 4-86 io-23856i i 699189 3 • 64 937458 1 •22 761731 4-86 238269 n 2 699407 3 -64 9 37385 1 •22 762023 4-86 237977 3 699626 3 64 937312^ •22 762314 4-86 237685 57 4 699844 3 67 937238 1 •22 762606 4-85 237394 56 5 700062 3 oJ 937165 1 •22 762807 4-85 237103 55 6 700280 3 63 937092 1 •22 763i88 4-85 236812 54 I 700498 3 63 937019 1 •22 7634-9 4-85 236521 53 700716 3 63 936946 1 •22 763770 4-85 2362 3o 52 9 700933 3 62 936872 1 •22 764061 4-85 235939 5i IO 701 :5i 3 62 936799 1 9-936725 1 •22 764352 4-84 235648 5o ii 9-70i368 3 62 •22 9-764643 4.84 i3-23535 7 % 12 7oi585 3 62 936652 1 •23 764933 4-84 235067 i3 701802 3 6i 936578 1 •23 765224 4-84 234776 47 14 702019 3 61 9365o5 1 •23 7655i4 4-84 234486 46 15 702236 3 61 93643i 1 •23 7658o5 4-84 234195 45 16 702432 3 61 936357 1 •23 766095 4-84 233905 2336 1 5 44 \l 702669 702885 3 60 936284 1 •23 766385 4-83 43 3 60 936210 1 •23 766675 4-83 233325J 42 *9 7o3ioi 3 60 936 1 36 1 •23 766965 4-83 233o35| 41 20 703317 3 60 936062 1 •23 767255 4-83 232745' 40 21 9 -7o3533 3 5 9 9 -935988 1 •23 9-767545 4-83 10-232455 39 232i66 : 38 22 703749 3 5 9 935qi4!i 935840 1 •23 767834 4-83 23 703964 3 5 9 •23 768124 4-82 231876; 37 24 704179 704395 3 5 9 935766 1 ■24 768413 4-82 23i587i 36 25 3 tl 035692(1 ■24 768703 4-82 2312971 35 20 704610 3 9356i8!i • 24 768992 4-82 23 1 008: 34 3 704825 3 58 935543[i •24 769281 4-82 230719 1 33 7o5o4o 3 58 935469^1 •24 769570 4-82 23o43o 32 20 703204 3 58 935390 1 • 24 769860 4-8i 23oi4o! 3i 3o 705469 3 57 935320 1 • 24 770148 4-8i 22Q852J 3o 3i 9«7o5683 3 57 9-935246 1 •24 9.770437 4-81 io. 2 2 9 563l 29 32 705898 3 57 935171 1 •24 770726 4-8i 229274 28 33 7061 12 3 57 935097 1 •24 771015 4-81 228985 27 34 706326 3 56 935022 1 •24 77i3o3 4-8. 228697 26 35 706539 3 56 934948 1 •24 771592 4-8i 228408 25 36 706753 3 56 934873 1 •24 771880 4-8o 228l 20' 24 s 706967 3 56 934798; 1 •25 772168 4-8o 227832 23 707180 3 55 934723|i •25 772457 4-8o 227543 22 3 9 707393 3 55 934649 1 •25 772745 4-8o 227255 21 4o 707606 3 55 934574' 1 •25 773o33 4-8o 226967 20 41 9.707819 3 55 9-934499; ] •25 9-773321 4-8o 'O.226679 \l 42 708032 3 54 934424(1 •25 773608 4-79 226392 43 708245 3 54 934349;' •25 773896 4-79 226104 '7 44 708458 3 54 934274,1 •25 774184 4-79 2258i6 16 45 708670 3 54 934!99 !l •25 774471 4-79 225529 i5 46 708882 3 53 9341 23 ( I •25 774759 4-79 225241 14 47 709094 3 53 934048 1 •25 775046 4-79 224954 i3 48 709306 3 53 933973 1 •25 775333 4-79 4-78 224667 12 49 709518 3 53 933M(i • 26 775621 224379 11 5o 709730 3 53 933822 1 ! • 26 775908 4-78 224092 10 5i 9-709941 3 52 9-933747 1 • 26 9.776195 4-78 «o 2238o5 % 52 7ioi53 3 52 933671 1 • 26 776482 4-78 2235i8 53 7io364 3 52 933596' 1 ■ 26 776769 777055 4-78 22323l 7 54 710075 3 52 93352oi 1 • 26 4-78 222945 6 55 710786 3 5i 933445 1 • 26 777342 4 78 222658 5 56 710997 3 5i 933369 1 • 26 777628 4-77 222372! 4 u 711208 3 5i 933293 1 • 26 777915 4-77 222085j 3 7*1419 3 5i 933217 1 • 26 778201 4-77 221799; 2 59 711629 3 5o 933i4i 1 • 26 778487 4-77 2 2 I 5 1 2 i I 6o 71 1839 3 5o 933o66 1 • 26 778774 4-77 221226, Cosine D. Sine £ 9° Cotaiig. D. Taiig._ M.J SINES AND TANGENTS. (31 DEGREES. ) 4 Sine D. | Cosine | D. | Tang. D. Cotang. 9-711839 3-5o 1 9-933o66|i-26| 9-778774 4-77 10-221226 ~6o~ 2 7i2o5o 712260 3-5o 3-5o 932990 1-27 9 32oi4ii-27 9 32838{i-27 j 779060 779346 4-77 4-76 220940 220654 5c 58 3 712469 3-49 779632 4-76 220368 57 4 712679 3-49 9327621 -27 779918 4-76 220082 56 5 712889 713098 3-49 q32685| 1 - 27 780203 4-76 219797 55 6 3-49 932609 1 -27 780489 4-76 2 1901 1 , 54 I 7i33o8 3-4C 3-48 9325331-27 1 780775 4-76 2IQ225! 53 7i35 1 7 9324571-27 , 781060 4-76 218940; 52 9 713726 3-48 93238o 1 -27 781346 4-75 2i8654| 5i IO 713935 3-48 9323o4 1 -27 78i63i ' 4-75 2183691 5o ii 9-714144 3-48 9 932228 1-27 9-781916 ; 4-73 10 218084 49 12 714352 3-47 9321 5i 1 -27 782201 4-7 = 2H799 48 i3 7i456i 3-47 932075 1 -28 782486 4-75 217514 47 14 714769 714978 3-47 931998,1 -28 782771 4-75 217229 46 i5 3-47 931921 ji -28 783o56 4-75 216944 45 16 7 1 5 i 86 3-47 93i845 1-28 783341 4-75 216659 44 17 715394 3-46 931768 1-28 783626 4-74 216374 43 18 715602 3-46 93 1691 1-28 783910 4-74 216090 42 19 7 1 5809 3-46 931614 1-28 784195 4-74 2i58o5 4i 20 716017 3-46 93 1 537 1 -28 784479 4-74 2I552I 40 21 9-716224 3-45 9-931460 1 -28 9-7B4764 4-74 io-2i5236 ll 22 716432 3-45 931383^-28 785048 4-74 214952 23 716639 3-45 93i3o6 1 -28 7 85332 4-73 214668 u 24 716846 3-45 93i 229J1 • 29 7856i6 4-73 214384 25 717053 3-45 93ii52 1 -29 785900 4-73 214100 35 26 717259 3-44 931075,1 -29 786184 4-73 2i38i6 34 a 7H466 3-44 930998 1-29 786468 4-73 2i3532 33 717673 3-44 93092 j 1 -29 93o843 1-29 786752 4- 7 3 213248 32 29 717379 3-44 787036 4- 7 3 2 1 2964 3i 3o. 7i8o85 3-43 930766 1 -29 787319 4-72 212681 3o 3i 9-718291 3-43 9.9306881-29 9-787603 4-72 10-212397 3 32 718497 3-43 930611 1 -29 7878S6 4-72 2121 14 33 718703 3-43 9 3o533 1.29 788170 4-72 2ii83o 27 34 718909 3.43 93o456 1 -29 788453 4-72 2 I I D47 26 35 7i9"4 3-42 93o378 1-29 7 88 7 36 4-72 21 1264 25 36 719320 3-42 93o3oo 1 -3o 789019 4-72 2 1 098 1 24 37 719525 3-42 93o223 1 -3o 789302 4-71 2 1 0698 23 38 719730 3-42 93oi45 i-3o 7S 9 585 4-71 2 1 04 1 5 22 3 9 719935 3-41 930067 1 -3o 789868 4-71 210132 21 4o 720140 3-4i 929980 1 -3o 7901 5 1 4-71 209849 20 4i 9-72o345 3-4i 9-92991 1 ■ 1 -3o 929833 1 -3o 9-790433 4-71 IO-209567 \i 42 720549 3-4i 790716 4-71 209284 43 720754 3-40 929755^ -3o 790999 79 1 28 1 4-7i 209OOI H 44 720958 3-4c 929677,1.30 4-71 208719 16 45 721162 3-40 929D99 i-3o 79 1 563 4-70 208437 i5 46 72i366 3-4o 929D21 1 -3o 791846 4-7o 208 1 54 14 47 72157c. 3.40 929442 1 -3o 792128 4-70 207872 i3 48 721774 3-3 9 929364 1 -3i 792410 4-70 207 D90 12 i 9 721978 3.39 929286 1 - 3 1 792692 4-70 2 ">73o8 11 5o 722181 3-39 929207 1 -3i 792974 4-70 207026 10 5i 9-722385 3-3 9 9-9291291 -3i 9-793256 4-70 10-206744 i 52 722588 3.39 3-38 92905c 1 -3 1 928972 i-3i 79 3538 4-6 9 206462 53 722791 793819 4-6 9 206l8l I 54 722994 3-38 928893 1 -3 1 794ioi 4-6 9 205899 55 723197 3-38 928815, i-3i 794383 4-6 9 205617 5 56 723400 3-38 928736' 1-3 I! 794664 4-6 9 205336 4 Si 7236o3 3.37 928637 1. 3 1 794945 4-6 9 2o5o55 3 7238o5| 3-3 7 928378,1 -3i ! 795227 4-69 4-68 204773 2 5o | 724007 3.37 928499 i-3i 7955o8 204492 1 6o 1 724210 3.37 ! 928420J1 -3i Sine |58° 795789 4-68 20421 1 Cosine D. Cotang. D. | Tang. M. 50 (32 DEGREES.) A TABLE OF LOGARITHMIC M. o Sine D. Cosine | T>. Tang. D. Cotang. 9-724210 3.37 9-928420 1-32 9 -795789 4-68 10-204211 60 i 724412 3-37 928342 1 ■32 796070 4-68 203930 u 2 724614 3-36 928263 1 •32 796351 4-68 203649 3 724816 3-36 9 28i83ji •32 796632 4-68 203368 u 4 725017 3-36 928104 1 •32 796913 4-68 203087 5 72D219 3-36 928o25'i •32 797I94 4-68 202806 55 6 723420 3-35 927946 1 •32 797475 4-68 202525 54 2 723622 3-35 927867 1 •32 797755 4-68 202245 53 725823 3-35 92778-7 1 •32 798036 4-67 201964 52 9 726024 3-35 927708 1 ■32 7 9 83 1 6 4-67 201-684 5i 10 726225 3-35 927629 1 •32 798596 4-67 201404 5o ii 9-726426 3-34 0-927549 1 •32 9.798877 4-67 IO-20I 123 % 12 726626 3-34 927470 1 •33 799157 4-67 200843 i3 726827 3-34 927390 1 • 33 799437 4-67 2oo563 47 14 727027 3-34 927310 1 ■33 7997H 4-67 200283 46 i5 727228 3-34 927231 1 •33 799997 4-66 200003 45 16 727428 3-33 927 1 5i ; 1 •33 800277 4-66 199723 44 \l 727628 3-33 927071 ji •33 8oo557 4-66 199443 43 727828 3-33 926991 1 •33 8oo836 4-66 199164 42 19 728027 3-33 9269I ! I •33 801 1 16 4-66 I9S884 4i 20 728227 3-33 92683i Ii •33 801396 4-66 198604! 40 21 9-728427 3-32 9-926701 j 1 •33 9-801675 4-66 io- 198325 39 198045 38 22 7 2 86 26 3-32 o2667i'i •33 8 [ 9 55 4-66 23 728825 3-32 926591 j 1 •33 802234 4-65 197766! 37 24 729024 3-32 9265 1 11 •34 8o25i3 4-65 197487J 36 25 729223 3-3i 92643i|i ■34 802792 4-65 197208 35 26 729422 3-3i 92635i 1 ■34 803072 4-65 196928, 34 196649 33 2 I ■72Q621 3-3i 926270! 1 •34 8 335i 4-65 28 729820 3-3i 926190 1 •34 8o363o 4-65 196370 32 29 73ooi8 3-3o 9261 ioj 1 •34 803908 4-65 196092 3i 3o 730216 3-3o 926029:1 •34 804187 4-65 i 9 58i3 . 3o 3i 9-73o4i5 3-3o 9-925949 1 925868 1 ■34 9 • 804466 4-64 io- 195534 3 32 73o6i3 3-3o •34 8o4745 4-64 195255 33 73o8i 1 3-3o 925788 1 •34 8o5o23 4-64 194977 194698 27 34 731009 3-29 925707)1 •34 8o53o2 4-64 26 35 73 i 206 3-29 920626' 1 •34 8o558o 4-64 194420 25 36 731404 3-29 925545 1 •35 8o585 9 4-64 194141 24 12 731602 3-29 925465 1 • 35 8o6i3 7 4-64 i 9 3863 23 731799 3-29 3-28 92538411 ■ 35 8064 1 5 4-63 193585 22 3 9 731996 92o3o3 1 •35 806693 4-63 193307 21 4o 732193 3-28 925222 1 • 35 806971 4-63 193029 20 4i 9-732390 73258 7 3-28 9-925141 1 •35 9-807249 4-63 10-192751 a 42 3-28 92^060 1 • 35 807527 4-63 • 192473 43 732784 3-28 924979 ' • 35 807805 4-63 192195 12 44 732980 3-27 924897 1 •35 8o8o83 4-63 191917 45 733177 3-27 924816 1 -35 8o836i 4-63 191639 :5 46 733373 3.27 924735 1 ■36 8o8638 4-62 191362 14 47 733569 3. 2? 924654 1 •36 808916 4-62 191084 i3 48 7 33 7 65 3-27 924572 1 •36 809193 4-62 190807 12 49 733961 3-26 924491 ! 1 •36 809471 4-62 190529 11 5o 734157 3 • 26 924409 1 ■36 809748 4-62 190252 10 5i 9-734353 3-26 9-924328:1 -36 9-8ioo25 4-62 10-189975 8 52 734549 3-26 924246 1 •36 8io3o2 4-62 189698 53 734744 3-25 924164 1 1 •36 8io58o 4-62 189420 1 54 734939 3-25 924083 1 -36! 8io85 7 4-62 189143 188866 6 55 735 1 35 3-25 924001 1 •36 8iii34 4-6i 5 56 73533o 3-25 923919 1 • 36i 81 1410 4-61 i885 9 o 4 12 735525 3-25 923837 1 • 36! 81 1687 4-6i i883i3 3 735719 3-24 923755 j • 3 7| •37, 81 1964 4-6i i88o36 2 59 735914 3-24 923673 1 812241 4-61 187739 1 6o 736109 3-24 923591 j 1 37; 812517 4-6i 1 874831 Cosine D. Sine |5T°i Cotang. D. Tang. 1 M. 6INEE AND TANGENTS. (33 DEGREES. ) 51 [■* bine D. Cosine D. Tang. D. Cotang. 9-736109 7363o3 3-24 9.923591 t -3 7 9-8i25i7 4-6i 10-187482 60 I 3 24 923509 i-3 7 812794 4-6i 187206 3 2 736498 3 24 923427! t -3 7 813070 4-6i 1 86930 3 736692 3 23 923345! [.3 7 8i3347 4-6o 186653 57 4 736886 3 23 923263 1-37 8i3623 4-6o 186377 5b 5 737080 3 23 923181 ; •37 8 1 38 99 4-6o 1 86101 55 6 737274 3 23 923098 -3 7 814175 4-6o 185825 54 5 737467 3 23 923oi6j -3 7 8i4452 4-6o 185548 53 737661 3 22 922933I -3 7 814728 4-6o l852 7 2 52 9 737855 3 22 922851' •3 7 810004 4-6o 184996 5i IC 738048 3 22 922768. -38 815279 9 .8i5555 4-6o 184721 5o II 9*738241 3 22 9-9226861 -38 4-5g 10-184445' 49 n 738434 3 22 922603, -38 8i583i 4-5 9 1 841 69 48 :3 738627 8 21 922D20 1 -38 816 107 4-5 9 i838 9 3 47 14 738820 3 21 922438 -38 8i6382 4-5 9 i836i8 46 i5 739013 3 21 922355 -38 8i6658 4-5 9 183342 45 16 739206 3 21 922272 -38 8i6 9 33 4-5 9 183067 44 17 73 9 3 9 8 3 21 922189 -38 817209 4-5 9 182791 43 18 739590 3 20 922106 -38 817484 4-5 9 i825i6 42 »9 739783 3 20 922023 -38 817759 8i8o35 4-5o 182241 4i 20 739070 3 20 921940 9-921857 -38 4-58 181965 40 21 9-74oi67 3 20 -3 9 9-8i83io 4-58 10-181690 ll 22 74o359 3 20 921774 .39 8i8585 4-58 i8i4i5 23 74o55o 3 •9 921691 • 39 818860 4-58 181140 37 24 740742 3 J 9 921607 • 39 8i 9 i35 4-58 i8o865 36 2D 740934 3 19 921524 • 39 819410 819684 '4-58 180590 35 26 741 125 3 19 921441 • 39 4-58 i8o3i6 34 27 74i3i6 3 921357 1 .39 819959 4-58 1 8004 1 33 28 741 5o8 3 l8 921274 1 • 39 820234 4-58 179766 32 29 741699 3 l8 921190 1 • 39 82o5o8 4-57 179492 3i 3o 741889 3 l8 92 1 1 07 1 • 3o 820783 4-57 179217 10-178943 3o 3i 9-742080 3 18 9-921023 1 •3 9 9-821057 4-57 a 32 742271 3 18 920939 1 920856 1 • 40 82i332 4-57 178668 33 742462 3 17 • 40 821606 4.57 178394 3 34 742652 3 17 920772 I • 40 821880 4-57 178120 35 742842 3 17 920688I1 .40 822154 4-5 7 177846 25 36 743o33 3 •7 920604 1 • 40 822429 82270J 4-57 177571 24 37 743223 3 17 920520J1 .40 4-57 177297 23 38 7434 1 3 3 16 92o436|i • 40 822077 4-56 177023 22 39 7436o2 3 16 920352 1 ! .40 82325o 4-56 176750 21 I 4o 743702 3 16 920268I1 • 40 823524 4-56 176476 20 4i 9-7^982 3 16 9-920184 I .40 9-823798 4-56 io- 176202 » 42 744i 7 1 3 16 920090.1 • 40 824072 4-56 175928 43 74436i 3 i5 920015 I .40 824345 4-56 175655 17 44 74455o 3 i5 9 1 9Q3 1 J 1 919846! 1 •41 824619 4-56 I7538i 16 45 74473o 744928 3 i5 •41 824893 4-56 175107 i5 46 3- i5 919762 1 .41 825i66 4-56 H4834 14 a 743117 3- i5 91967711 •41 825439 825 7 i3 4-55 1 7456i i3 7453o6 3- 14 919593 1 •41 4-55 174287 12 49 7454o4 3- 14 919508! 1 •4i 820986 4-55 1 740 1 4 11 5o 745683 3- 14 9194241 •41 826259 4-55 173741 10 5i 9-745871 3- 14 9-919339 1 • 41 9-826532 4-55 10-173468 I 52 746o5o 746248 3- 14 919254 1 •41 8268o5 4-55 173195 51 3- i3 919163 1 •41 827078 4-55 172922 7 54 746436 4- i3 919080 1 •41 827351 4-55 172649 6 55 746624 3- i3 919000:1 91891 5) 1 9i883oji •4i 827624 827897 4-55 172376 5 56 746812 3- i3 •42I 4-54 172103 4 ^ 746999 3- i3 •42 828170 4-54 171830 3 58 747187 3- 12 9i«745!i •42 828442 4-54 I7i558 2 59 747374 3- 12 918659,1 •42| 828715 4-54 171285 I Co 747562 3-12 9185741 •421 828987 Cotang. i 4-54 171013 . Cosine D. Sine |5G°! D Tang. M. 62 (34 ■ DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. 1 Tang. D. Cotang. j 9-747562 3-12 9-918574 1-42 918489 1-42 9-828987 4-54 10-171013 60 i 747749 3-12 829260 4-54 170740 & 2 7479 3 ^ 3-12 918404 1*42 829532 4-54 170468 3 748123 3. II 9i83i8 1-42 829805 4-54 170195 57 4 7483io 3. II 918233 1-42 830077 4-54 169923 56 5 748497 3. II 918147 1-42 83o34g 4-53 1 6965 1 55 6 748683 3. II 918062 1-42 83o62i 4-53 169379 54 I 748870 3. II 917976 1-43 83o8 9 3 4-53 169107 53 749056 3-io 917891 1-43 83 1 1 65 4-53 168835 5a 9 749243 3-io 917805 1-43 83i437 4-53 168563 5s 10 749429 9-749615 3-io 917719 1-43 83i7og 4-53 1 68291 5o ii 3-io 9-917634 1-43 9-831981 4-53 io- 168019 3 12 749801 3-io 917548 1-43 832253 4-53 167747 i3 749987 3-09 917462 1-43 83 2 5 2 5 4-53 167475 47 M 750172 3-09 9 i 7 3 7 6 1-43 832796 4-53 167204 46 i5 75o358 3-09 917290 1-43 833o68 4-52 166932 45 16 75o543 3-09 917204 i-43 83333 9 4-52 1 6666 1 44 \l 750729 3-09 3.08 917118 1.44 8336n 4-52 i6638o 43 7D0914 917032 1-44 833882 4-52 166118 42 *9 751099 3-o8 916946 916859 1-44 834 1 54 4-52 165846 41 20 751284 3-o8 1-44 834425 4-52 165575 40 21 9-751469 3-o8 9-916773 1-44 9-834696 4-52 io- i653o4 $ 22 75i654 3-o8 916687 1-44 834967 835238 4-52 i65o33 23 7 5i83 9 752023 3-o8 916600 1.44 4-52 164762 37 24 3-07 9i65i4 1-44 835509 4-52 1 6449 1 36 25 752208 3-07 916427 1-44 835 7 8o 4-5i 164220 35 26 752392 3-07 9i634i 1-44 836o5i 4-5i 163949 34 11 752576 3-07 916254 1-44 836322 4-5i 163678 33 732760 3-07 916167 1-45 8365 9 3 4-5i 163407 32 ?9 752944 3-o6 916081 1-45 836864 4-5i !63i36 3i 3o 753i28 3-o6 915994 1-45 837134 4-5i 162866 3o 3i 9-7533i2 3-o6 9-915907 915820 1-45 9-8374o5 4-5i 10-162595 3 32 753495 3-o6 1-45 837675 4-5i 162325 33 753619 3-o6 915733 1-45 83 79 46 4-5i 162054 27 34 753862 3-o5 9 1 5646 1-45 8382i6 4-5i 161784 26 35 754046 3-o5 915559 1-45 83848 7 4-5o i6i5i3 25 36 754229 3-o5 915472 1 -45 838 7 5 7 4-5o 161243 24 37 754412 3-o5 9 1 5385 1-45 839027 4-5o 160973 23 38 754595 3-o5 910297 1-45 839297 83 9 568 4-5o 160703 22 3 9 754778 3-04 91D210 i-45 4-5o i6o432 21 4o 754960 3-o4 9i5i23 1-46 83 9 838 4-5o 160162 20 41 9-755i43 3-04 9-9i5o35 1-46 9-840108 4-5o io- 159892 \l 42 755326 3-o4 9M948 1-46 840378 4-5o 159622 43 7555o8 3-04 914860 1-46 840647 4-5o 159353 17 44 755690 3-o4 914773 1 -46 840917 4-49 1 59083 i588i3 16 45 755872 3-o3 914685 1 -46 841 187 4-49 i5 46 756o54 3-o3 914598 1-46 841457 4.49 158543 14 % 7 56236 3-o3 914510 1-46 841726 4-49 158274 i3 756418 3-o3 914422 1-46 841996 4-49 1 58oo4 12 i 9 7566oo 3-o3 9U334 1-46 842266 4-49 157734 11 5o 756782 3-02 914246 1-47 842535 4-49 157465 10 5i 9-756963 3-02 9-914158 1-47 9-842805 4-49 10-157195 * 52 757144 3-02 914070 1-47 843074 4-49 156926 53 757326 3-02 913982 1-47 843343 4-49 i5665 7 156388 7 54 757507 757688 3-02 913894 1.47 843612 4-49 4-48 6 55 3-oi 913806 1-47 843882 i56n8 5 1 56 757869 3-oi 913718 1-47 8441 5 1 4-48 155849 4 1 B 758o5o 3-oi 9i363o!i-47 844420 4-48 i5558o 3 75823o 3-oi 913541 1 1 -47 844689 844958 4-48 i553ii 2 5 9 7584i 1 3-oi 9i3453 1 -47 4-48 i55o42 1 1 6o 7585 9 i 3-oi 9i3365|i-47 845227 4-48 i54773 1 Oosino D. Sine 1 55° Cotang. D. J?™&^. mJ BINES AND TANGENTb. (35 DEGREES. ) 53 M. Sine D. Cosine D. Tang. D. Cotang. f 9-758591 3-oi 9-9i3365 i-47 9-845227 4 48 io- : 54773 60 i 7 58 77 2 3 OO 913276 47 845496 4 48 i545o4 5I 2 758 9 52 3 00 913187 48 845764 4 48 154236 3 759132 3 00 913099 48 846o33 4 48 153967 57 4 759312 3 00 9i3oio 48 846302 4 48 i536 9 8 56 5 759492 3 00 912922 48 846570 4 47 1 5343o 55 6 759672 2 99 912833 43 846839 4 47 i53i6i 54 I 759852 2 99 912744 48 847107 4 47 152893 53 76003 r 2 99 912655 48 847376 4 47 152624 52 9 76021 1 2 99 912566 48 847644 4 47 i52356 5i 10 760390 2 9 912477 48 847913 4 47 152087 5o II 9-760569 2 9-912388 48 9-848181 4 47 io-i5i8i9 % 12 760748 2 98 912299 49 848449 4 47 i5i55i 1 3 760927 2 98 912210 49 848717 848986 4 47 i5i283 47 14 761106 2 98 912121 49 4 47 i5ioi4 46 i5 761285 2 98 9i2o3i 49 849254 4 47 1 50746 45 16 761464 2 98 91 1942 49 849522 4 47 150478 44 «7 761642 2 97 9ii853 49 849790 85ood8 4 46 l5o2IO 43 18 761 82 1 2 97 911763 49 4 46 14994a 42 '9 761999 2 97 91 1674 49 85o325 4 46 149675 4i 20 762177 2 97 911584 49 85o5 9 3 4 46 149407 40 21 9-762306 2 97 9-911495 1 49 9-85o86i 4 46 10-149139 14887 1 3 9 22 762534 2 96 911405 1 49 85i 129 85i3 9 6 4 46 38 23 7627:2 2 96 91 i3i5 1 DO 4 46 143604 37 24 762889 2 96 911226 DO 85 1 664 4 46 148336 36 25 763067 2 96 9in36 5o 85i 9 3i 4 46 148069 35 26 763245 2 96 91 1046 5o 852199 852466 4 46 147801 34 27 763422 2 96 910956 910866 DO 4 46 147534 33 28 763600 2 9 5 5o 852 7 33 4 45 U7267 32 29 763777 2 9 5 910776 DO 853ooi 4 45 146099 3i 3o 763954 2 9 5 910686 1 DO 853268 4 45 146732 3o 3i 9-764131 2 9 5 9-91059611 DO 9-853535 4 45 10-146465 2 5 32 764308 2 9 5 9io5o6 DO 853802 4 45 146198 28 33 764485 2 94 910415 5o 854069 4 45 14593 1 27 34 764662 2 94 9io325 5i 854336 4 45 145664 26 35 764838 2 94 910235 5i 8546o3 4 45 145397 25 36 765oi5 2 94 910144 5i 854870 4 45 i45i3o 24 3- 765191 2 94 91005411 5i 855i37 4 45 144863 23 38 760367 2 94 909963 1 5i 855404 4 45 144596 22 39 765544 2 9 3 909873 Dl 855671 4 44 144329 21 4o 765720 2 9 3 909782 5i 855 9 38 4 44 J 44062 20 4i 9.765896 2 9 3 9-909691 DI 9-856204 4 44 10-143796 19 42 766072 2 9 3 909601 DI 856471 4 44 U352o 18 43 766247 2 9 3 909510 1 DI 856737 4 44 143263 H 44 766423 2 9 3 909419 1 DI 857004 4 44 142996 16 45 766598 2 92 90932811 52 857270 4 44 142730 i5 46 766774 2 92 909237 1 D2 857537 4 44 142463 14 a 766949 2 92 909146,1 D2 85 7 8o3 4 44 142197 i3 767124 2 92 9ooo55 1 52 858069 4 44 141931 12 49 767300 2 92 908964 1 52 858336 4 44 141664 11 5o 767475 2 9i 908873 1 D2 858602 4 43 141398I 10 io-i4n32 9 5i q- 767649 2 9 1 9-908781 1 52 9-858868 4 43 5> 767824 2 91 908690 1 D2 859134 4 43 140866 8 53 767999 768173 2 91 908399' 1 D2 859400 4 43 140600 I 54 2 91 9085071 1 52 859666 8D9932 4 43 Uo334 55 768348 2 90 908416! 1 53 4 43 140068 5 56 768522 2 90 908324 1 53 860198 4 43 139802 4 n 768697 2 9 c 9oS233;i 53 860464 4 43 139536 3 768871 2 90 908 1 41 ' 1 53 860730 4 43 139270! 2 59 769045 2 90 908049 1 53 86099$ 4 43 1 39005 1 138739 Co 76921? Cosine 2 90 00795811 53 86i2Si Cotaog 4-43 ~ D. Sine - 154° T X Tan*. ,2«kj 54 (30 DEGREES.) A TABLE OF LOGARITHMIC "MT Sine D. Cosine | D. Tang. D. Cotaag 60 o 9-769219 2-90 9-9079581-53 9-861261 4-43 10-138739 138473 i 7 6 9 3 9 3 2 ^ 9 907866, 1 -53! 86i52 7 4 43 3 2 769366 2 89 907774:1 907682' 1 53 1 861792 53 j 862058 4 42 138208 3 769740 2 89 4 42 137942 n 4 769913 2 89 907590-1 53 862323 4 42 137677 5 770087 2 & 907498 1 53 86258 9 4 42 137411 55 6 770260 2 90740611 53 862854 4 42 137146 54 1 7 ■770433 2 88 907314 Ii 54 863iiq 863385 4 42 i3688i 53 6 770606 2 88 907222 1 54 4 42 i366i5 52 9 770779 2 88 907129 1 54 86365o 4 42 i3635o 5i 10 770952 2 88 907037 54 8639 1 5 4 42 i36o85 5o ii 9-771125 2 88 9-906945 54 9-864180 4 42 io-i3582o % F2 771298 2 87 906852 54 S64445 4 42 135555 13 771470 2 87 906760 54 864710 4 42 135290 % U 771643 2 87 906667 54 864975 4 41 i35o25 i5 771815 2 87 906575 54 865240 4 41 134760 45 16 771987 2 87 906482 54 8655o5 4 41 134495 i3423o 44 \l 772109 2 S 906389 55 865770 4 4i 43 77233i 2 86 906296 55 866o35 4 41 i33g65 42 19 7725o3 2 86 906204 55 8663oo 4 4i 133700 41 20 772675 2 86 9061 1 1 55 866564 4 4i 133436 40 21 9-772847 773018 2 86 9-906018 55 9-866829 4 4i 10-133171 ll 22 2 86 9o5o25 55 867094 4 4i 132906 23 773190 2 86 9o5832 55 86 7 358 4 4i 132642 37 24 77336i 2 85 905739 905645 55 867623 4 4i 1 323 77 36 25 773533 2 85 55 867887 4 4i i3?n3 35 26 773704 2 85 9o5552 55 8681 52 4 4o 1 3x848 34 27 7738 7 5 2 85 905459 55 868416 4 4o i3i584 33 28 774046 2 85 9o5366 56 868680 4 4o i3i32o 32 I 9 774217 2 85 905272 56 868945 4 4o i3io55 3i 3o 774388 2 84 9o5i7Q 9-9o5o85 56 869 2 00 4 4o 1 30794 3o 3i 9-774558 2 84 56 9-869473 4 4o io-i3o527- 3' 32 774729 2 84 904992 56 869737 4 40 i3o263 33 774899 2 84 904898 56 870001 4 4o 1 29999 129735 27 34 775070 2 84 904804 56 870265 4 40 26 35 775240 2 84 9047 1 1 904617 56 870529 4 40 129471 25 36 77 5 4io 2 83 56 870793 871057 4 40 129207 24 ll 77558o 2 83 904523 56 4 40 128943 23 775750 2 83 904429 904335 57 8 7 i32i 4 40 128679 128415 22 09 775920 2 83 57 8 7 i585 4 4o 21 4o 7760QO 2 83 904241 57 871849 4 3 9 I28i5i 20 4' 9-776259 2 83 9-904147 57 9-872112 4 3 9 10-127888 \l 42 776420 776598 2 82 904053 5 7 872376 4 3 9 127624 43 2 82 903959 57 872640 4 3 9 1 27360 17 44 776768 2 82 903864 57 872903 4 3 9 127097 16 45 776937 777106 2 82 903770 57 873167 4 3 9 126833 i5 46 2 82 903676 57 873430 4 3 9 126570 14 % 777275 2 81 9o358i 57 873694 873967 4 39 i263o6 i3 777444 2 81 903487 57 4 3 9 1 26043 12 49 777613 2 81 903392 58 874220 4 39 125780 n 5o 777781 2 81 903298 58 874484 4 39 I255i6 10 5i 9 777950 2 81 9«9o32o3 58 9-874747 4 39 IO-I25253 i 52 778119 2 81 903 1 08 58 875010 4 3 1 24990 53 778287 2 80 9o3oi4 58 875273 4 124727 I 54 778455 2 80 902919 902824 58 8 7 5536 4 38 1 24464 55 778624 2 80 58 875800 4 38 124200 5 56 778792 2 80 902729 58 876063 4 38 123937 4 S 778960 2 80 902634 58 876326 4 38 123674 3 779128 2 80 902539 5 9 876589 4 38 1 234i 1 2 59 779295 2 79 902444 5 9 8 7 685 1 4 38 123149 1 60 779463 2.79 902349 5 9 877114 4-38 122886 Cosine D. Sine 53°l Cotang. D. Tang. blNKS AND TANGENTS. (37 DEGREES.) 55 6 Sine 1 D ' Cosine i D. Tang. i D - Cotang. 60 9-779463 a •79 9-902349' i-5 9 9-877114 4-38 10-122886 77 9 63i 2 •79 902253 i-5 9 877377 4-38 122623 tl 2 779798 2 "9 902158 1 i-5 9 877640 4-38 122360 3 779966 2 79 902063 i-5 9 877903 4-38 122097 5 7 4 78oi33 2 •79 901967 i-5 9 8 7 8i65 4-38 I2i835 56 5 78o3oo 2 78 901872! i-5 9 878428 4-38 121672 55 6 780467 2 78 901776, i.5 9 878691 4-38 121309 54 7 780634 2 78 901681J i-5 9 878933 4-3 7 121047 53 8 780801 2 78 90 1 585; i.5 9 879216 4-37 120784 52 9 780968 2 78 901490 i-5 9 879478 4-3 7 120522 5i 10 781 i34 2 •78 901394 i-6o 879741 4-37 I 202D9 DC ii 9«78i3oi 2 •77 9-901298 i-6o 9-88ooo3 4-37 IO-II9997 3 12 781468 2 •77 901202; [•60 880265 4-3 7 1 19735 i3 i8/634 2 77 90II06| i- 60 880528 4-37 I I9472 % 14 781800 2 77 9OIOIO: i- 60 880790 4-3 7 IIG2I0 i5 781966 2 77 900914' 90081 8 | i-6o 881002 4-3 7 1 1 8948 45 16 782132 2 77 i-6o 88i3i4 *-3 7 1 1 8686 44 \l 782298 2 76 9OO722! [•60 881576 4-3 7 118424 43 782464 2 76 900626 i-6o 881839 4-3 7 118161 42 19 782630 2 76 900529 900430 [•60 882101 4.37 1 17899 1 17637 4i 20 782796 2 76 [•61 882363 4-36 4o 21 9-782961 2 76 9-900337 [-61 9-882625 4-36 10-117375 ii 22 783i2 7 2 76 900240 •61 882887 883i48 4-36 1 171 i3 23 783292 783458 2 75 900144 [•61 4-36 1 1 6852 37 24 2 75 900047 •61 8834io 4-36 1 16590 36 25 783623 2 75 899951 899854 •61 883672 4-36 1 16328 35 26 7 83 7 88 2 75 -6i 883 9 34 4-36 1 16066 34 11 783 9 53 2 75 899757 •61 884196 884457 4-36 n58o4 33 7841 18 2 75 899660 •61 4-36 1 1 5543 32 29 784282 2 74 899564 •61 884719 4-36 u528i 3i 3o 784447 2 74 899467 -62 884980 4-36 Il5o20 3o 3i 9-784612 2 74 9-899370 -62 9-885242 4-36 10-114758 It 32 784776 2 74 899273 •62 8855o3 4-36 1 14497 33 784941 2 74 899176! •62 885 7 65 4-36 1 14235 27 34 7°5 1 o5 2 74 899078 898981 •62 886026 4-36 1 13974 26 35 785269 2 73 •62 886288 4-36 113712 25 36 785433 2 73 898884 -62 886549 4-35 ii345i 24 u 785597 2 73 898787 •62 886810 4-35 113190 23 785761 2 73 898689 -62 887072 4-35 1 1 2928 22 39 785925 2 73 898592 •62 887333 4-35 1 1 2667 21 40 786089 9-786252 2 73 898494 •63 887594 4-35 1 1 2406 20 41 2 72 9-898397 •63 9.887855 8881 16 4-35 10-112145 % 42 786416 2 72 898299 •63 4-35 1 1 1884 43 786579 2 72 898202 •63 888377 4.35 111623 \l 44 786742 2 72 898104 •63, 88863 9 4-35 1 1 i36i 45 786906 2 72 898006 •63' 888900 4-35 IIIIOO i5 46 787069 2 72 897908 -63 889160 4-35 1 10840 14 47 787232 2 7i 897810 •63 889421 4-35 1 1 0579 i3 48 787395 7 8 7 55 7 2 7' 897712 •63 1 889682 4-35 iio3iS 12 49 2 7i 897614 •63i 889943 4-35 1 1 0057 11 5o 787720 2 7i 897516 •63! 890204 4-34 109796 10 5i 9.787883 2 7i 9-897418! •64! 9-890465 4-34 10-109535 9 52 788045 2 7* 897320 •64 890725 4-34 109275 8 53 788208 2 7i 897222 •64 890986 4.34 1090141 7 54 788370 2 70 897123 •64! 891247 4-34 108753 55 7 88532 2 70 897025 •64; 891507 4-34 io84o3 1082J2 5 56 788694 788856 2 70 896926 •64 891768 4-34 4 57 2 70 896828 ] •64! 892028 4-34 10-1972 3 58 789018 2 70 896729 •64 892289 4-34 107711 2 5 9 789 1 80 2 70 896631!] •64 892549 4-34 1 0745 1 1 60 789342 2-69 896532 1 •64! 892810 4-34 107190 D. Siue N Cotang. i D. Tiiiiar. M. 56 (38 DEGREES.) A TABLE OF LOGARITHMIC to. Sine D. Cosine D. _Tang. D. Cctang. * 9-789342 2-69 9-896532 1-64 9-892810 4-34 10-107190 1 06930 60 I 789504 2-69 896433 1-65 893070 4-34 u 2 789665 2-69 896335 1-65 8 9 333 1 4-34 1 06669 3 789827 2-69 896286 1-65 893591 8 9 385i 4-34 106409J 57 4 7S9988 2-69 896137 1-65 4-34 1 06 1 49 56 5 79° '49 2-69 2-68 8 9 6o38 1-65 8941 1 1 4-34 io588g 55 6 7903 10 8 9 5o3 9 895840 1-65 894371 4-34 io562o io5368 54 I 790471 2-68 1-65 894632 4-33 53 790632 2-68 895741 1-65 894892 895152 4-33 io5io8 52 9 790793 790954 2-68 895641 1-65 4-33 104848 5i 10 2-68 895542 1-65 895412 4-33 104588 5o ii 9-79iii5 2-68 9-895443 1-66 9-895672 4-33 10-104328 % 12 791275 2-67 8o5343 1-66 895932 4-33 104068 i3 79U36 2-67 895244 i-66 896192 896402 4-33 io38o8 47 U 791396 791757 2-67 895145 1-66 4-33 io3548 46 i5 2-67 895045 1-66 8967 1 2 4-33 io3288 45 16 791917 2-67 894045 1-66 80697 1 4-33 103029 44 3 792077 2-67 894846 1-66 897231 4-33 102769 43 792237 2-66 894746 1-66 897491 897751 4-33 102509 42 19 792397 2-66 894646 1-66 4-33 102249 4i 20 792537 2-66 894546 1-66 898010 4-33 101990 10-101730 40 21 9.792716 2-66 9-894446 1-67 9-898270 4-33 ll 22 792876 2-66 894346 1-67 8 9 853o 4-33 101470 23 793o35 2-66 894246 1-67 898789 4-33 101211 ll 24 7 9 3 1 o5 793354 2-65 894146 1-67 899049 899308 4-32 1 0095 1 36 25 2-65 894046 1.67 4-32 100692 35 26 79^5i4 2-65 893946 1-67 899568 4-32 1 0043 2 34 3 793673 2-65 893846 1-67 899827 4-32 1 00 1 73 33 793832 2-65 893745 1-67 900086 4-32 099914 32 29 793991 2-65 893645 1.67 900346 4-32 099654 3i 3o 79400 2-64 893544 1-67 900605 4-32 099395 10-099136 098876 3o 3i 9- 7943o8 2-64 9 -8 9 3444 i-68 9.900864 4-32 It 32 794467 2-64 893343 1.68 901124 4-32 33 794626 2-64 893243 1-68 90i383 4-32 098617 ll 34 794784 2-64 893142 i-68 901642 4-32 098358 35 794942 2-64 893041 1-68 901901 4-32 098099 25 36 790101 V64 892940 892839 i-68 902160 4-32 097840 24 37 795259 2-63 1-68 902419 4-32 097581 23 38 790417 2-63 892739 i-68 902679 902938 4-32 097321 22 3 9 79 55 7 5 2-63 892638 i-68 4-32 097062 21 4o 795733 2-63 8 9 2536 i-68 903197 4-3i 096803 20 4 1 9.795891 2-63 9-892435 1.69 9-9o3455 4-3i 10-096545 ■9 42 796049 2-63 892334 1-69 903714 4-3i 096286J 18 43 796206 2-63 892233 1-69 903973 4-3i- 096027 \l 44 796364 2-62 892132 1-69 904232 4-3i 090768 45 796521 2-62 892030 1.69 904491 4-3i 095509 i5 46 796679 2-62 891929 1-69 904700 4-3-1 095250 14 % 7 9 6836 2-62 891827 1.69 905008 4-3i 094992 i3 796993 2-62 891726 1.69 905267 4-3i 094733 12 49 797i5o 2-6l 891624 1.69 9o5526 4-3i 094474 11 5o 797307 2-6l 891523 1-70 905784 4-3i 094216 10 5i 9-797464 2- 6l 9-891421 1.70 9-906043 4-3-1 10-093957 § 52 797621 2-6l 891319 1-70 906302 4-3i 093698 53 797777 2-6l 891217 1.70 9o656o 4-3i 093440 7 54 797934 2-6l 891115 1-70 9068 10 4-3i 093181 6 55 798091 2- 6l 891013 1-70 907077 4-3i 092923 5 56 798247 2-6l 89091 1 | 890809' /•70 907336 4 3i 092664 4 u 798403 2-6o 1-70 907594 907852 4-3i 092406 3 798560 2-60 890707! 1-70 4-3i 092148 2 59 798716 2-60 890605 1.70 9081 11 4-3o 091889 60 798872 2-6o 890503! 1.70 908369 4-3o 091 63 1 Cosine D. Sine |tfl°! Cotang. D. ~Tang7~| m7 SINES AND TANGENTS. (39 DEGREES." ) 5*3 M. o Sine D. Cosine j D. Tang. D. Cotang. ! 9-798872 2-60 9 -89o5o3! 1 -70 9-908360 4-3o 10-091631 60 i 799028 2 60 890400(1 7i 908628 4 3o 091372 5o 091 1 14 58 a 799184 2 60 890298 1 71 908886 4 3o 3 799339 2 5 9 890195 1 7i 909144 4 3o 090856 j 57 4 799495 2 5 9 89009-3 ; 1 7i 909402 4 3o 090598 56 5 799601 2 5 9 809990' I 7i 909660 4 3o 090340 55 6 799806 2 5 9 8898881 7i 909918 4 3o 090082 54 7 799962 2 5 9 889785' I 7i 910177 4 3o 089823 53 8 8001 17 2 5 9 889682 ' I 71 910435 4 3o 089565 52 9 800272 2 58 889579)1 71 910693 910931 4 3o 089307 | 5i 10 800427 2 58 889477,1 7i 4 3o 080049 5o u 9-8oo582 2 58 9-889374 I 9-911209 4 3o 10-088791 o88533 % I 12 800737 2 58 889271 I 72 91 146- 4 3o 1 l3 800892 2 58 88gi68ii 72 911724 4 3o 088276 47 U 801047 2 58 889064' 1 72 911982 4 3o 088018 46 i5 801201 2 58 88S96 1 ; I "2 912240 4 3o 087760 45 16 80 1 356 2 5 7 888858 i ^2 912498 4 3o 087502 44 \l 8oi5ii 2 5 7 8887551 1 72 912756 4 3o 087244 43 80 1 665 2 57 88865i 1 1 72 9i3oi4 4 29 086086 42 »9 801819 801973 2 57 888548|i 72 913271 4 29 086729 41 20 2 57 888444 ' 1 "3 913529 4 29 086471 4o 21 9-802128 2 57 9-88834i| 1 73 9-9*3787 4 29 10-086213 ll 22 802282 2 56 888237 1 73 914044 4 29 085956 23 802436 2 56 888i34 1 -3 9U3o2 4 29 085698 37 24 802589 802743 2 56 888o3o|i 73 9U56o 4 29 085440 36 25 2 56 887926! 1 73 9U817 4 29 o85i83 35 26 802897 2 56 8878221 -3 915075 4 29 084925 34 27 8o3o5o 2 56 88771811 -3 915332 4 29 084668 33 28 8o32o4 2 56 887614 1 73 915590 4 29 084410 32 & 8o3357 2 55 887510 1 73 9i5847 4 29 084 1 53 3i 3o 8o35u 2 55 887406 1 74 916104 4 29 083896 io-o83638 3o 3i 9-8o3664 2 55 9-887302 1 "4 9-916362 4 29 ll 32 8o38i 7 2 55 887198!] 74 916619 4 29 o8338 1 33 803970 2 55 887093 1 1 8869^9 1 886885! 1 74 916877 4 29 o83i23 27 34 804123 2 55 74 9Hi34 4 29 082866 26 35 804276 2 34 74 917391 4 29 082609 25 36 804428 2 54 886780:1 -4 917648 4 29 o82352 24 h 8o458i 2 54 886676 1 -4 917905 4 3 082095 23 38 804734 2 54 8865 7 i|i 74 9i8i63 4 081837 22 3 9 804886 2 54 886466 1 U 918420 4 28 o8i58o 21 4o 8o5o39 2 54 886362 1 7 5 018677 4 28 o8i323 20 4! 9- 800191 2 54 9.886257 1 V 9-918934 4 28 1 • 08 1 066 19 42 8o5343 2 53 886i5 2 !i 73 919191 4 28 080809 18 43 8o5495 2 53 886047! 1 75 919448 4 28 o8o552 17 44 8o5647 2 53 885942 ji 88583 7 1 75 919705 4 28 080295 o8oo38 16 45 800799 8059D1 2 53 75 919962 4 28 i5 46 2 53 885 7 32 75 920219 4 28 079781 14 47 806 1 o3 2 53 885627 75 920476 4 28 079524 i3 48 806254 2 53 885522 75 920733 4 28 079267 12 49 806406 2 52 885416 1 13 920990 4 28 0190101 11 5o 806557 2 52 8853iiji 76 921247 4 28 078753 10 5i 9-806709 2 52 9-8852o5'i 76 9-92i5o3 4 28 10-078497 I >2 806860 2 52 885iooi -6 921760 4 28 078240 53 80701 1 2 52 884Q94 1 -6 Q220T7 4 28 077983 7 54 807163 2 52 8848891 -6 922274 4 28 077726 6 55 807314 2 52 8847S3 1 76 Q22530 4 28 077470 5 56 807465 2 5i 884677 1 -6 922187 4 28 077213 4 a 807615 2 5i 884572 1 "6 923044 4 28 076956 3 807766 2 5i 884466 1 76 923300 4 28 076700 2 5, 807917 2 5i 88436o| 1 76 923557 4 27 076443 1 * 808067 2-5l 884204 1 77 9 238i3 4 27 076187 Tang. _M. Coeine D. Sine !oO c Cotang. D. 5b (4( J DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. | Tang. D. Cotang, o 9 • 808067 808218 2-5l 9-884254 1-77J 9-923813 4-27 10-07618-] 60 i 2-5l 884148 1 -77 92407c 4-27 07593c sii 2 8o8368 2-5l 884042 x -77 92432- 4-27 075673 3 8o85i 9 2-5o 883 9 36 1.77 924583 4-27 075417 5 7 56 4 8o366 9 2-5o 883829 1.77 92484c 4-27 07516c 5 808819 2-5o 883723 1.77 925096 4-27 074904 55 6 808969 2-5o 883617 *-77 9253D2 1 4-27 074648 54 7 809119 2-5o 8835io 1 -77 925600 925865 4-2 7 074301 53 8 809269 2-5o 883404 1.77 4-2 7 074135 52 9 809419 2-49 883297 1.78 926122 4-2 7 073878 5i IC 809669 9-809718 2-49 883191 1.78 926378 4-27 073622 5o i j 2-49 9-883o84 1-78, 9-926634 4-2 7 I C -073366 49 12 809868 2-49 882077 882871 1 -78: 926890 1 4-2 7 07311c 48 i3 810017 2-49 1-78 927147 ; 4-2 7 072853 47 14 81 01 67 2-49 2.48 882764 1.78 927403 4-2 7 072597 46 i5 8io3t6 882657 1.78 927609 927910 4-27 072341 45 16 8io465 2.48 88255o i. 7 » 4-27 072085 44 17 810614 2-48 882443 1.78 928171 4-27 071829 43 18 810763 2-48 882336 1-79 928427 4-27 071573 42 •9 8 1 09 1 2 2-48 882229 1.79 928683 4-2 7 071317 41 20 8 r 1 06 1 2-48 882121 1.79 928940 4-2 7 071060 40 - 21 9-811210 2-48 9-882014 I.79 9-929196 929452 4-2 7 10 070804 u 22 8u358 2-47 881907 1.79 4-2 7 070548 23 81 1D07 2-47 88 1 799 1.79 929708 4-2 7 070292 37 24 8n655 2-47 881692 1.79 929964 4-26 070036 36 25 81 1804 2-47 88 1 584 1.79 930220 4-26 069780 35 26 81 1952 2-47 881477 1.79 93o475 4-26 069525 34 27 81 2 1 00 2-47 88i36 9 1.79 930731 4-26 069269 069013 33 28 812248 2-47 2-46 881261 i-8o 930987 4-26 32 2 9 8i23g6 881 1 53 i-8o 93i243 4-26 068757 3i 3o 812044 2-46 881046 i-8o 931499 4-26 o685oi 3o 3i 9-812602 2-46 9-88o 9 38 1. 80 9-931755 4-26 10-068245 29 32 812840 2-46 88o83o 1. 80 932010 4-26 067990 28 33 812988 2.46 880722 i-8o 932266 4-26 067734 27 34 8i3i35 2-46 88o6i3 1. 80 932522 4-26 067478 26 35 8i3283 2-46 88o5o5 1.80 932778 4-26 067222 25 36 8i343o 2-45 8S0397 1. 80 933o33 4-26 066967 24 3 7 8i35 7 8 2-45 880289 1. 81 933289 4-26 0667 ' ' 23 38 8i3725 2-45 880180 1. 81 933545 4-26 o66455 22 3 9 8i38 7 2 2-45 880072 1. 81 9338oo 4-26 066200 21 4o 814019 2-45 879963 1. 81 934o56 4-26 065944 20 4i 9-814166 2-45 9-879855 1. 81 9-9343ii 4-26 1 • 065689 I? 42 8i43i3 2-45 £79746 1.81 934567 4-26 065433 43 814460 2-44 879637 1. 81 934823 4-26 065177 17 44 814607 2-44 879529 1. 81 935078 4-26 064922 16 45 8i4753 2-44 879420 i-8i 935333 4-26 064667 i5 46 814900 2-44 8793 n i-8i 935589 4-26 06441 1 14 47 81 5o46 2-44 879202 1-82 935844 4-26 064 1 56 i3 48 SiSigS 2-44 8790931 1-82 936 1. .10 4-26 063900 12 49 8i5339 2-44 878984I 1-82 936355 4-26 063645 11 5o 8 1 5485 2-43 878875 9-878766 1.82 936610 4-26 063390 io-o63i34 10 5i 9-8i563i 2-43 1-82 9-936866 4-25 3 52 815778 2-43 8 7 8656 1.82 937121 4-25 062879 53 815924 2-43 8785471 1-82 937376 4-25 062624 7 54 816069 2-43 878438 1.82 937632 4-25 062368, ( 55 8i62i5 2-43 878328 [•82 937887 4-25 0621 i3 5 56 8i636i 2-43 878219 [-83 938142 4-25 06 1 858 4 ^7 816007 2-42 878109 [-83 938398 9-38653 4-25 061602 3 58 816602 2-42 877999 -83 4-25 061347 2 5 9 8167981 2-42 877800 [-83 9 38 9 o8 4-25 061092 060837 Tnv.g. 1 6o 816943 Cosine | 2-42 8777S0 [-83 939163 4-25 I). Sine 49° Cotansr. ~~i)7~ BiJNES AND TANGENTO. (41 DEGREES. ) 59 p£" Sine D. Cosine | D. Tang. D. Cotang. c g. 816943 2-42 9.877780 i-83 8776701-83 9939163 4-25 i 0-060837 60 i 817088 2 •42 939418 4-25 o6o582 U 2 817233 2 •42 877560! i-83 939673 4-25 060327 3 817379 2 .42 877450' 1 -83 939928 4-25 060072 57 4 817524 2 •41 877340 i-83 94oi83 4-25 059817 55 5 817668 2 •41 877230 1.84 940438 4-25 359562 55 6 817813 2 •41 877120 1-84 940694 4-25 059306 54 I 817958 2 •41 877010 1-84 940949 4-25 03905 1 058796 058542 53 8i8io3 2 •41 876890I1.84 876789I1.84 941204 4-25 52 9 818247 2 •41 941458 4-25 5i 10 8i83o2 o-8i8536 2 •41 87667s 1 1- 84 9417U 4-25 058286 5o ii 2 .40 g. 876568 I.84 9-941968 4-25 io-o58o32 % 13 818681 2 .40 876457i I- 84 942223 4-25 057777 i3 818825 2 .40 876347 j I- 84 942478 4-25 057522 47 14 818969 8191 i3 2 .40 8 7 6236| 1. 85 942733 4-25 057267 46 ; 5 2 40 876125,1-85 942988 4-25 057012 45 16 819257 2 40 876014 i-85 943243 4-25 056757 44 17 819401 2 .40 875904 i i-85 943498 4-25 o565o2 43 18 819545 2 39 875793!l85 875682! 1. 85 943752 4-25 056248 42 19 819689 2 39 944007 4-25 055993 4i 20 819832 2 39 875571(1-85 944262 4-25 055738 4o 21 9.819976 2 39 9-87545911.85 87534811.85 9.944517 4-25 10-055483 3 9 22 820120 2 39 944771 4-24 055229 38 23 820263 2 3 9 875237 i-85 945026 4-24 054974 37 24 820406 2 a 875126 i-86 945281 4-24 0547 1 9 36 25 82o55o 2 875oi4|i-86 945535 4-24 054465 35 26 820693 82o836 2 38 874903 i-86 945790 4-24 054210 34 3 2 38 874791 1 1 -86 946045 4-24 053955 33 820979 2 38 874680! i-86 946299 946554 4-24 053701 32 29 821122 2 38 8745681 1- 86 4-24 053446 3i 3o 82126D 2 38 874456 i-86 946808 4-24 o53ig2 3o 3i 9-821407 2 38 9.87/34411-86 9-947063 4-24 10.052937 2 32 82i55o 2 38 874232 ji -87 9473i8 4-24 052682 33 821693 2 37 874i2iji-87 947572 4-24 052428 27 34 82i835 2 37 874009 1-87 947826 4-24 032174 26 35 821977 2 ? 7 87389611.87 948081 4-24 051919 25 36 822120 2 37 873784! 1-87 948336 4-24 031664 24 u 822262 2 37 873672J1.87 948590 4-24 o5i4io 23 822404 2 37 873560 1-87 948844 4-24 o5u56 22 39 822546 2 37 873448 1 1. 87 949099 4-24 030901 21 4o 822688 2 36 873335 1.87 949353 4-24 050647 20 4i 9-822830 2 36 9-87322311 .87 9 949607 4-24 io-o5o393 \l 42 822972 2 36 8731 10 i-88 949862 4-24 o5oi38 43 823i 14 2 36 872998 872885 1.88 950116 4-24 049884 17 44 823255 2 36 i-88 950370 4-24 049630 16 45 823397 2 36 872772 i-88 9D0625 4-24 049375 i5 46 823539 2 36 87265911.88 950879 4-24 049121 048867 14 8 82368o 2 35 872547 1-88 95u33 4-24 i3 823821 2 35 872434 1-88 9 5i388 4-24 048612 12 4 9 823963 2 35 872321 1-88 951642 4-24 048358 11 5o 824104 2 35 872208 1.88 951896 4-24 048104 10 5i 9-824245 3 35 9-872095 1-89 8719811-89 87 1 8681 1. 89 9-952i5o 4-24 10-047850 I 52 824386 2- 35 9524o5 4-24 o475 9 5 53 824527 2- 35 952659 95291J 4-24 047341 I 54 824668 2- 34 871755 1. 80 4-24 047087 046833 55 824808 2- 34 871641 1-89 953167 4-23 5 56 824949 2- 34 871528 1-89 953421 4-23 046579 046325 4 u 825090 82523o 2- 34 871414 1-89 953675 4-23 3 2- 34 871301 1.89 953929 954i83 4-23 04607 1 2 5 9 825371 2- 34 871187 i-8o 4-23 045817 1 6o 8255n 2-34 871073:1.96 954437 4-23 045563 Cosine D. Sine |48q Cotang. J ' D. ! Tang. &L 60 (42 DEGREES.) A TABLE OF LOGARITHMIC ~M.~ Sine D. Cosine | D. Tang. D. Cotang. 9-8255u 2-34 9-871073,1 -90 9-954437 4-23 io-o45563 60 i 82565i 2 33 870960 1 •90 954691 4-23 045309 n 2 82^791 2 33 870846 1 .90 954945 4-23 o45o55 3 825g3i 2 33 870732 1 •90 955200 4-23 044800 57 4 80607 1 2 33 870618,1 • 9 c 955454 4-23 044546 56 5 82621 1 2 33 870504 1 ■90 . 9 55 707 4-23 044293 55 6 82635i 2 33 870390 1 •90 955961 4-23 044039 54 I 826491 2 33 8702761 •90 9562i5 4-23 043785 53 826631 2 33 870161 ; 1 •90 956469 4-23 o43 53 1 52 9 826770 2 32 870047J1 •91 956723 4-23 043277 5i 10 826910 2 32 869933 1 •91 956977 4-23 o43o23 5o ii 9-827049 2 32 9.869818; 1 •91 9-957231 4-23 10.042769 % 12 827189 827328 2 32 869704!! .91 957485 4-23 o425iS i3 2 32 86 9 589 1 .91 9 5 77 39 957993 4-23 042261 47 14 827467 2 32 869474I1 .91 4-23 042007 46 i5 827606 2 32 869360' 1 •9' 958246 4-23 o4h54 45 16 827745 2 32 86 9 245ii .91 9585oo 4-23 041 5oo 44 n 827884 2 3i 86gi3o!i .91 9 58 7 54 4-23 041246 43 18 828023 2 3i 86901 5 1 .92 959008 4-23 040992 040738 42 19 828 162 2 3i 86890011 •92 959262 4-23 4i 20 8283oi 2 3i 868785] 1 •92 959516 4-23 040484 4o 21 9.828439 2 3i 9-86867011 •92 9 -959769 4-23 10- 04023 I ll 22 828578 2 3i 8685551 1 •92 960023 4-23 039977 23 828716 2 3i 868440 1 •92 960277 4-23 o39723' 37 24 828855 2 3o 868324 1 •92 96o53 i 4-23 039469 36 2D 828993 8291 3i 2 3o 868209 1 86809J 1 •92 060784 4-23 039216 35 26 2 3o •92 961038 4-23 o38 9 62 34 3 829269 2 3o 867978 1 867862 1 •93 961 291 4-23 038709 33 829407 2 3o . 9 3 961545 4-23 o38455 32 29 829545 2 3o 867747 1 •93 961799 962052 4-23 o382oi 3i 3o 829683 2 3o 86 7 63 1 1 .93 4-23 037948 3o 3i 9-829821 2 29 9- 867*51 5 1 ■ 9 3 9-962306 4-23 10-037694 3 32 829959 2 29 867399 1 867283 1 - 9 3 962560 4-23 037440 33 830097 2 29 •93 962813 4-23 037187 27 34 830234 2 29 867167 1 • 9 3 963067 4-23 036933 26 35 83o372 2 29 867051 1 ■93 963320 4-23 o3668o 25 36 83o5o9 2 2 9 866935 1 •94 963574 4-23 o36426 24 37 83o646 2 29 86681911 866703 1 ■94 963827 4-23 o36i73 23 38 830784 2 8 ■94 964081 4-23 035919 o35665 22 39 830921 2 866586 1 .94 964335 4-23 21 40 83io58 2 28 866470 1 •94 964588 4-22 o354i2 20 41 9 -83 1 195 83i33 3 2 28 9-866353 1 •94 9 • 964842 4-22 io-o35i58 \l 42 2 28 866237 1 •94 965095 4-22 o349o5 43 83i46g 2 28 866120 1 •94 965349 4-22 o3465i 17 44 83 1 606 2 28 866004 1 • 9 5 965602 4-22 034398 16 45 831742 2 28 865887 1 •95 965855 4-22 o34U5 i5 46 831879 2 28 865770 1 • 9 5 966105 4-22 033891 14 47 832oi5 2 27 865653! 1 ■ 9 5 966362 4-22 033638 i3 48 832i52 2 27 865536ji 865419 1 ■ 9 5 966616 4-22 033384 12 49 832288 2 27 • 9 5 966869 4-22 o33i3i 11 5o 832425 2 27 8653o?ji • 9 5 967123 4-22 032877 10 5i 9-83256i 2 2 7 9-865i85 1 • 9 5 9-967376 4-22 10 032624 52 8326 97 2 27 865o68 1 • 9 5 967629 4-22 032371 8 53 832833 2 11 864950 1 864833 1 •9D 967883 968136 4-22 032117 7 54 832969 2 .96 4-22 o3i864 6 55 833io5 2 26 8647 1 6 1 1 .96 968389 4-22 o3i6u 5 56 833241 2 26 8645981 .96 968643 4-22 o3i357 4 n 8333 77 2 26 864481:1 .96 968896 4-22 o3no4 3 8335i2 2 26 864363 j 1 •96! 969149 4-22 o3o85i 2 59 833648 2- 26 864245 1 •96! 969403 4-22 030597 1 60 833783 2- 26 864127 1 .96 969656' 4-22 o3o344 (losme P. Sine 14?°' Cotang. 1 D. Tarur. mT 5INE8 AND TANGENTS. (43 DEGREES. ) 61 M. Sine K Cosine D. Tang. D. Cotanor. 9-833783 2-26 9-864127 1-96 9-969656 4-22 io-o3o344 ~6o~ i 833qi9 2 •25 864010 1 .96 969909 4 •22 030091J 5o 02 9 838| 58 2 834054 2 •25 8638 9 2 1 •97 970162 4 22 3 834i8 9 834325 2 •25 863774 1 •97 970416 4 22 029584 57 4 2 2C 863656 1 •97 970669 4 22 029331 56 5 83446o 2 25 863538|i •97 970922 4 22 029078 55 6 834595 834730 2 25 863419 1 •97 971175 4 • 22 028825 54 7 2 25 8633oi 1 •97 97U29 4 22 028571 53 8 834865 2 25 863 1 83 1 •97 971682 4 22 0283i8 52 9 834999 2 24 863o64 1 ■97 971935 4 22 028065 5i 10 835i34 2 24 862946 1 .98 972188 4 22 027812 5o ii 9.835269 8354o3 2 24 9-862827 1 .98 .98 9-972441 4 22 10-027559 49 12 2 24 862709 1 972694 4 22 027306 48 i3 835538 2 24 8623901 .98 972948 4 22 027052 47 U 835672 2 24 862471 1 .98 973201 4 22 026799 46 i5 835807 2 24 862353 1 .98 973454 4 22 026546 45 16 835g4i 2 24 862234' 1 .98 973707 4 22 026293 44 n 836o 7 5 2 23 862II3I .98 973960 4 22 026040 43 18 836209 836343 2 23 861996 I .98 9742i3 4 22 025787 42 19 2 23 861877:1 .98 974466 4 22 025534 41 20 836477 2 23 86i 7 58 1 •99 974719 4 22 025281 40 21 9. 8366 n 2 23 9-86i638 1 •99 9-974973 4 22 10.025027 3 9 22 836 7 45 2 23 86i5i 9 1 ■99 975226 4 22 024774; 38 23 806878 2 23 861400 1 •99 975479 4 22 024521 37 24 837012 2 22 861280 1 •99 975732 4 22 0242681 36 25 837146 2 22 861161:1 •99 975985 4 22 024015 35 26 837279 3 22 861041 1 •99 976238 4 22 023762 34 27 837412 2 22 860922 ' 1 860802 1 ■99 976491 4 22 023509 33 28 83 7 546 2 22 •99 976744 4 22 023256 32 29 837679 2 22 860682 2 •00 976997 4 22 o23oo3 3i 3o 837812 2 22 86o562 2 •00 97725o 4 22 022750 3o 3i 9-837945 2 22 9-860442 2 •00 9-9775o3 4 22 10-022497 29 022244 28 32 838o 7 8 2 21 86o322 2 •00 977756 4 22 33 8382 1 1 2 21 860202 2 •00 978009 4 22 021991 27 34 838344 2 21 860082*2 •00 978262 4 22 021738 26 35 - 838477 2 21 859962 2 ■00 9785i5 4 22 021485 25 36 8386io 2 21 859842^2 •00 978768 4 22 021232 24 37 838742 2 21 859721 2 •0, 979021 4 22 020979] 23 38 838875 2 21 859601 2 •01 979274 4 22 020726! 22 3 9 839007 2 21 859480 2 •01 979527 4 22 020473; 21 40 839140 2 20 859360 2 ■01 979780 4 22 020220 20 4i 9-839272 2 20 9-859239^ •01 9-93oo33 4 22 10-019967; 19 OI97U l8 4? 83 9 4o4 2 20 85gi 192 •01 980286 4 22 43 83 9 536 2 20 858998 -o4 985343 4-21 014657 3 842i63 ' 2-17 856568 >-o4 985596 4-21 014404 u 4 842294 : 2-17 856446 >-o4 983848 4-21 014152 5 842424 ; 2-17 856323 >-o4 986101 4-21 013899 55 6 842555 ; 2-17 856201 • > • 04 986354 4-21 013646 54 2 842685 I 2-17 856078 >-o4 986607 4-21 013393 53 8428i5 i 2.17 855o56 ' 855833 >-o4 986860 4-21 oi3i4o 52 9 842946 2.17 ^•04 987 1 1 2 4-21 012888 5i 10 843076 ! 2-17 855 7 u >-o5 98 7 365 4-21 012635 5o u 9-843206 2-l6 9-855588 >-o5 9-987618 4-21 10-012382 8 12 843336 2-l6 855465 ' >-o5 987871 4-21 012129 i3 843466 2- l6 855?/ t2 ' >-o5 988123 4-21 01 1877 47 U 8435g5 2-l6 855219 ' >-o5 988376 4-21 01 1624 46 i5 843725 2-l6 855096 : >-o5 988629 4-21 011371 45 16 843855 2-l6 854973 : 85485o ' j.o5 988882 4-21 011118 44 13 843984 2-l6 •o5 989134 4-21 010866 43 844H4 2-l5 854727 : • 06 989387 4-21 oio6i3 42 «9 844243 2-l5 8546o3 5 • 06 989640 4-21 oio36o 4i 20 844372 2-l5 854480 5 • 06 989893 4-21 010107 4o 21 9-844502 2-l5 9-854356: • 06 9-990145 4-21 10-009855 ll 22 84463i 2-l5 854233 : • 06 990398 990631 4-21 009602 23 844760 2-l5 854109 : • 06 4-21 009349 ll 24 844889 845oi8 2-l5 853 9 86 2 853862 : • 06 990903 4-21 009097 25 2-l5 ■ 06 991 1 56 4-21 008844 35 26 845 1 47 2-l5 853 7 38 : • 06 991409 4-21 008591 34 27 845276 2-14 8536i4 2 •07 991662 4-21 oo8338 33 28 8454o5 2-14 853490 : ■07 991914 4-21 008086 32 ?° 845533 2-14 853366 : •07 992167 4-21 007833 3i 3o 845662 2-14 853242 2 • 07 992420 4-21 007580 3o 3i 9-845790 2-14 9-853n8 2 ■ 07 9-992672 4-21 10-007328 ll 32 845919 2-14 852994 2 85286o 2 •07 992925 4-21 007075 33 846047 2-14 • 07 993178 4-21 006822 27 34 846175 2-14 852745 : •07 99343o 4-21 006570 26 35 8463o4 2-14 852620 2 • 07 993683 4-21 006317 25 36 846432 2-l3 852496 2 • 08 993936 4-21 006064 24 ll 84656o 2-l3 852371 2 .08 994189 4-21 oo58n 23 846688 2-l3 852247 2 • 08 994441 4-21 oo5559 22 3 9 846816 2-l3 852122 2 .08 994694 4-21 oo53oo 21 4o 846944 2-l3 851997 2 9-851872 2 • 08 994947 4-21 oo5o53 20 4i 9-847071 2-l3 • 08 9-995199 995452 4-21 10-004801 a 42 847199 2-l3 85i 7 47 2 • 08 4-21 004548 43 847327 2-l3 85i622 2 • 08 995705 4-21 004295 3 44 847454 2-12 85 1 497 2 • 09 99 5 9 57 4-21 004043 45 847582 2- 12 85i372 2 • 09 996210 4-21 003790 oo3537 oo3285 i5 46 8477 9 847836 212 85 1 246 2 • 09 996463 4-21 14 47 212 85i 1 21 j2 •09 9967 1 5 4-21 i3 43 8479°4 2-12 85099612 • 09 996968 4-21 oo3o32 12 49 848091 2-12 85o87o ! 2 • 09 997221 4-21 002779 11 5o 848218 2-12 85o745,2 •09 997473 4-21 002527 10 5i 9-848345 2-12 9-85o6io 2 • 09 9.997726 4-21 10-002274 % 52 848472 2 • 1 1 85o49j,2 •10 997979 4-21 002021 53 8485 99 2-II 850368,2 •10 998231 4-21 001769 ooi5io I 54 848726 2-II 85o242j2 • 10 998484 4-21 55 8488D2 2-II 85ou6 2 •10 998737 4-21 001263 5 56 848979 2- 1 1 84999012 849864 2 • 10 998989 4-21 OOIOII 4 n 849106 2 • 1 1 • 10 999242 4-21 000758 3 849232 2-II 849738 2 • 10 999495 4-21 ooo5o5 2 A 84935o 849485 211 2-II 84961 I [2 849485 2 • 10 • 10 999748 10-000000 4-21 4-21 D. ooo253 10- 000000 1 w Cosine D. Sine [450 Cotang. Tang.