GIFT OF Dr. Horace Ivie Digitized by the internet Arcliive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofgeometOOdavirich ELEMENTS OF GEOMETRY TRIGONOMETRY, APPLICATIONS IN M^^^lJRAl'f^^^^ BY CHARLES DAVIES. LL. D. AOmOU OF FIRST LESSONS IN ARITHMETIC, ELEMENTARY ALGEBRA, FttACTICAL MATHEMATICS FOR PRACTICAL MEN, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOMETRY, SHADES, SHADOWS, AND PERSPECTIVE, ANA- LYTICAL GEOMETRY, DIFFERENTIAL AND INTEGRAL CALCULUa A. S. BARNES & COMPANY, NEW YORK AND CHICAGO. GIFTOF _ , QAh-Z9 DAVIES' MTHEMATICS. rl!^ And Only Thorough and Complete Mathematical Series. ii-^l BOOK IV. Measurement of Areas and Proportions of Figures, - • 82 — 108 PrDblcms relating to the Fourth Book, . . - . 109—113 Appendix — Regular Polygons, - - - . 113 — US BOOK V. Of Planea and their Angles, - • - 110^120 BOOK VI O! Solids, - 126—162 Appendix, - IG3— 164 CONTENTS. TRTGONOMETRT. Or LoGARiroMa, Of Scales, Definitions, and Exjlanation of TablevB, - llieorcms, Examples, Application to Heights and Distances, - APPLICATIONS OF GEOMETRY. MeNSUBATION of SlTRFACEB, General Principles, Contents of Figures, Mensuration of Solids, General Principles, Solidities of Figures, Mensuration of the Round Bodiis, To find the Surface of a Cylinder, To find the Solidity of a Cylinder, To find the Suiface of a Cone, To find the Solidity of a Cone, To find the Surface of the Frustum of 'a Cone, To find the Solidity of the Frustum of a Cono, To find the Surface of a Sphere, - To find the Surface of a Spherical Zone, To find the Solidity of a Sphere, - To find the Solidity of a Spherical Segment, To find the Solidity of a Spheroid, T:: find the Surface of a Cylindrical Ring, To find the Solidity of a Cylindrical Ring, Pace. 166—17(1 176—181 181—189 189—193 193—201 202— 21 n 211 211-213 218—239 289 289—240 240-241 248 248—249 249—260 250—261 251--262 253 264 266 266—266 266—267 268 269—260 260-261 261—208 ELEMENTARY BOOK I. DEFINITIONS AND REMARKS. 1. Extension has three dimensions, length, breadth, anJ thickness. Geometry is the science which has for its object: Ist. The measurement of extension ; and 2dly, To discover, by means of such measurement, the properties and relationa 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. 6. A Curve Line is one which changes is direction at every point. The word lirie 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 nny direction, will exactly coincide in its whole lensfth. 8. A Curved Surface has length and breadth without thick- Dees, and like a curve line is constantly changing its direction 9. A Solid or Body is that- which has length, breadth, and ihickness. Length, breadth, and thickness are called dimcn- 10 GEOMETRY. IDefinitior s. eiaas.; II'«rtc|e^. ^" feC'lid; has three dimensions, a surface two and a line one. A poiiiv' lias. Ho dimensions, but position only 10. Geometry treats of lines, surfaces, and solids. 11. A Demonstration is a course of reasoning which estalv hshcs a truth. 12. 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- sitions. 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 two straight lines are called the sides of the angle, and the common point of intersection, the vertex. Thus, the part of the plane included P between AB and AC is called an angle : ^^ AB and A C are its sides^ and A its vertex. J^ fn An angle is generally read, by placing the letter at the vci tox 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 andther when it BOOK I II Definitions B C J) B C 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 DEC. 21. When two lines are perpendicular to each other, the angles which they form are called right angles. Thus, DBA and DBC RTG 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 circle is a curve line all the points of which arc equally distant from a certain point within called the centre. Thus, if all the points of the curve A£B are equally distant from the centre C, this curve will be the circumference of a circle. 25. Any portion of the circumference, OS AED, 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 circum^'erence, as ACB is called a semicircumference ; and one quarter of the circumference, as -4C IS called a quadrant 12 GEOMETRY. Definitions. 28. The circumference of a circle is used for the measuro« 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 CjE, then, the angle E CB will be equal to 40 degrees. If CF were drawn through the point of division marked 80, the angle BCF would be e(|ual 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 level, are called hor'zontal lines. 34. Lines which are perpendicular to the horizon, or to the wafer level are cl^Ued vertical lines B O O K 1 . 13 Defin it ions, or PLANE FIGURES. 35. A Plane Figure is a portion of a plane terminated on a]1 9ide8 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 perinuter of the polygon. Hence, the perimeter of a polygon is the sum of all its sides. 37. A polygon of three sides is called a triangle. 38. A polygon of four sides is called a quadrilateral. 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. 3 14 GEOMETRY De f init ions. 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 dodocagon. 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, which has its three sides all unequal. Fourth. A right angled triangle, which has one right angle. In the right angled triangle ABCy the side A C, 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 baae and per-^ pendicular to the base. Thus, CD is the altitude of the tri angle ACB BOOK I u Definitions. 48. There are three kinds of quadrilaterftla. 1. The trapezium^ which has none of )tB sides parallel. 2. The trapezoid^ which has only two of its sides parallel. mx 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 is an equilateral rhomboid. 8. The rectangle^ which is an equian- galar parallelogram. 4. The square^ which is both equilat- eral and equiangular. 16 GEOMETRY. Of Axioms. 60. A Diagonal of a figure is a line which joins the vertices of two angles not adjacent. 61. The base of a figure is the side on which it is supposed U) stand ; and the altitude is a line drawn from the opposite side or angle, perpendicular to the base, AXIOMS. 1. Things which 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, ooin- ddo 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 tvxy equal parts. Let ADBE be the circumference of a circle, and A CB a diameter : then will the part^Di? 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 ' oincide with the curve ADB, or else there would be some point in the curve AEB or ^ Z)5, unequally distant from the centre C, wliich is contrary to the definition ot a circumference (Def. 21). Hence, the two curves will bo equal (Ax. 13). Corollary 1. If two lilies, 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 semicircumfer* encc, or 130 degrees ; and four right angles by the whole cLr« cumfercnce, or 360 degrees 18 U E M E T R V . Of Angles. D THEOREM II. If one straight line meet another straight linc^ the sum of thA two adjacent angles will he equal to two right angles. [iCt the straight line CD meet the straight line AB^ at the point C; then will the angle DCB\>\\xs the angle DC A be equal to two right angles- A C B About the centre C, with any radius as CB, suppose & 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 seraicir- cumference • hence, the sum of the two angles is equr.l to two right angles (Th. i, Cor. 2). Co7. 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 (7, 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. li DC meets two lines CB, CA, making DCB plus DCA equal to two right angles, ACB will form ohp straight line. Cor. 4. Hence, also, all the anglea which can be formed round any point, as (7, are equal to four right angles. For, the Bum of all the arcs which measure them, is equal to the entire circumference, w-hich is the measure of four right angles (Th. i. Cor. 2). B O K 1 . 19 Of Triangles THEOREM III. Ij ivDO Straight lines intersect each other^ the opposite or ocr- tical angles which they form ^ are equal. Let ihc two straight lines AB and CD intersect each other at the point E : then will the opposite angle A EC be equal to Z)£B, and AED=zCEB. For, since the line AE meets the "c line CD, the angle AEC-{-AED:= two right angles. But 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 equal to the angle CEB. THEOREM IV. Jf two triangles hctve 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 AC 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 ^4 C, of the -^ ^ ^ ^ triangle ACB, to be placed on DF, so that the extremity C Bhall fall on the extremity F: then, since the sides are equaj A will fall on D. But since the angle C is equal to the angle F, the line CB 20 GEOMETRY. Of TriangIo8 will fall on FE ; and since CB is equal to FE, the extremity^ 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.). ^ ^ F> Scholium. Two triangles are said to be equal, when being applied the one to the other they exactly coincide (Ax. 12). Ilence, 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 tnt one, equal to two angles and the included side of the other, each to rach, 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 bo equal to the triangle DEF. For, let the side AB be placed on the side DE, the cxtrem ity A on the extremity D ; and since the sides are equal, the point B will fall on the point E. Then since l E T R Y . Of Parallel Lines. cannoi be equal to B, nor less than B, it follows iliat it musi bo greater TIIEOREJl XII. If a straight line intersect two parallel lints, the alternate angi:^ will be equal. I f two parallel straight lines, A B CD, are intersected by a third line GN, the p/^ angles AEF and EFD are called alternate ^ yC-"'" ^ angles. It is required to prove that these C '/' f D IT 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 ^^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^^FsolongasFii meets EB. But since we have supposed EFD to be greatei than AEF, it folloAVS tliat 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 I nee, the alternate angles AEF and EFD are equal. Cor. U a line be perpendicular to one of two parallel lines, it will also be pcr- pendiculAT to the other B O K 1. 27 Of Parallel Lino THEOREM XIII. Conversely, — If a line intersect two straight lines, making thi alternate angles equal, those straight lines mil be parallel. fict tlic line LF meet the lines AB, CD. making the angle AEF equal to the fto.^lo EFD : then will the lines AB and CD be j)arallcl. 7/:^\~~7) For, if they arc not parallel, suppose G through the point F the line FO to be drawn parallel to AB. Then, because of the parallels AB, FGy the alternate angles, AEF and EFG will be equal (Th. xii). Rut, by nypoihesis, the angle AEF is equal to EFD: hence, the angle EFD id 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 tc 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 tlie line EF cut the two parallels A B CD . tlien will the outward angle y EGB be cquai to the inward opposite an- A ^/^^ ^ glo EHD ; and the two inward angles, C ~/^^^ 7 nCH and GllD, will be equal to two ^^ Tight angles. 28 GEOMETRY Of Parallel Lines First. Since the lines AB, CD, are parallel, the anglo AGH IS equal to the alternate angle GHD E (Til. xii) ; but the angle AGH is equal ^ y^' to the opposite angle EGB : hence, the C /H D angle EGB is equal to the angle EHD y (Ax. 1). Secondly. Since the two adjacent angles EGB and BGli are equal to two right angles (Th. ii) ; and since tlie 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, makmg the sum (jf the two inward angles less than two right angles, those hnes will not be parallel, but will meet if sufficiently produced. THEOREM .\v. All straight lines which are parallel to the same line, are parallel to each other. Let the lines AB and CD be each par- allel to EF: then will they be parallel to each other. For. let the line GI be drawn perpen- dicular to EF : then will it also be per- pendicular to the parallels AB CD (Th. fii Cor.V < 7 A Ii C D E I f' B O K c . 29 Of Trianglca. Then, since the lines AB and CD are porpiiiJiciilar to the line Gly ihcy will be parallel to each other (Th. xiii. Cor). THEOREM XVI. If one side of a tnangle be produced, the outward angle will be C(]ual to the sum of the inward opposite angles. In the triangle yl 5 C, let the side AB be produced to D : then will the outward /^v ^ angle CBD be equal to the sum of the in- y^ \ ward opposite angles A and C. ^ ^ jy For, conceive the line BE to be drawn parallel to the side AC. Then, since BC meets the two j)a- rallels A C, 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 tnangle the sum of the three anglts is equal to two righ angles. Let ABC be any triangle: then will t;ic sum of the three angles C ^ 4- J5-hC= two right angles. ^/ \ For, let the side AB be produced \o D A j] Then, the outward angle CBD Z.A+C {Th. 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, 's equal to two right angles -^ (Th.ii): hence A \-B-^C=U\'o 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 XVIII. 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 from the same point, thost which arc nearest the perpendicular are less than those which are more remote. Let ^ be a given point, and BE a straight line. Suppose AB to be drawn perpendiculai to DE, and suppose the -fjlique lines AC and AD also to be D O O K I 3i Of Triangles drawn: Then, AD will be shorter than ciihcr of the oblique lines, and AC will be less than AD First. Since the angle Z?, in the triangle .4 CB, is a righ angle, the angle C will be acute (Th. xvii. Cor. 3) : and since llie greater side of every triangle is opposite the greater angle (Th. xi), the side AC will be greater than AB. Stcondbj. Since the angle ACB is acute, the adjacent angle ACD will be obtuse (Th. ii) : consequently, the angle 2) is acute (Th. xvii. Cor. 3), and therefore less than the an^lo ACD. 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. // two right angled triangles have the hypothenuse and a sid^ of the one equal to the hypnlhemtse and a side of the other, the reirMtning parts will also he rr/ual, each to each. I>ct the two rijzlit auiilcd trianoles or? '^ ABC and DEI'\ have the hypothe- nuso AC equal to DF, and the side A B equal to DE : then will the re- maining parts be equal, each to each. ^ For, if the sido BC is equal to EF, the correspond! rig an- gles of the two triangles will be equal (Th. viii). If the sides Ere unequal, suppose BC to be the greater, and take a part, EG equal to EF, and draw AG. Then, in the two triangles ABG and DEF the angle B ig equal to the angle E, the side AB io the side DE, and the side f^G to the sido EF: hence, the two triangles are equal in al! respects (Th. iv) and consequently, the side ^G is ei -h 5 +C-fZ)+£z= four right angles. For, each interior angle, plus its exte- rior angle, as A-{-a, is equal to two right ingles (Th. ii). But there are as many exterior as interioi 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) : thcit 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 paraRclogram are equals 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 tw^o equal triangles. For, since the figure is a parallelogram, the sides AB^ DC are oarallel. as also the sides AD, BC. Nonv, since thf R n o K 1 . 3o Of Parallelograms. parallels arc cut by the diagonal DBy the alternate angles will be equal (Th. xii) : that is the angle ADB-DBC and BDC-ABD. Hence the two triangles ADB i5Z)C, having two angles in the one equal to two angles in the other, will have their third angles ccjual (Th. xvii. Cor. 1), viz. the angle A equal to the angle C, and these are two of the opposite angles of the parallelogram. Also, if to the equal angles ADB, DBC, we add the equals BDC, ABD, the sums will be equal (Ax. 2) : viz. the whole 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 eijual 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 anglo, 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 bo equal to two right angles. THEOREM XXIV. If the opposite sides of a quadrilateral, are equal, each to each^ the €fjual sides wUl^e parallel, and the figure will be a |» rallelogram. m GEOMETRY Ol Parallelograms Lei 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 81) 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, ia 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 ^ *he remaining sides will also he equal and parallel, and the figure will be a parallelogram. Let ABCD be a quadrilateral, having the sides AB, CD, equal and parallel: then will tlie figure be a parallelogram. For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, GJnce ^S 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 ang.t of the one, equal to two sides and the included angle of the Other: 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 io BC (Th xiii Tlierei'bre, the figure ABCD is a parallelogram. B O K 1 37 Of Parallelograms THEOREM XXVI. TJu two diagonals of a parallelogram divide each other into tqua. parts, or mutually bisect each otlter. Lc* A BCD be a parallelogram, and ACf Bl) Its two diagonals intersecting at E. Then will AE = EC and BE=ED. A Comparing the two triangles AED and BEC, we find the side AD— EC (Th. xxiii), the angle ADE = EBC and EAD=ECB : hence, the two triangles are equal (Th. v) : therefore, AE^ the side opposite ADE, is 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. ^ Whence it follows that the angles AEB and BEC are equal. Therefore, the diagonals of a rhombu biBOct each other at right angles. G E O M E T K y . BOOK II, OF THE CIRCLE DEFINITIONS. 1. The ciicumferencc of a circle is a curve line, all the points of whicli are equally distant from a certain point within called the centre. 2. The circle is the space bounded by this curve line. 3. Everystraightline,CA,CZ>,C£, drawn ^ K^ from the centre to the circumference, is called a radius or semidia7neter. Every / line which, like AB^ passes through the centre and terminates in the circumfe- rence, is called a dia?neter. 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. 39 D e fi n i t i on s 7. A sector is liie part of the circle in- cliidod between an arc and the two radii drawn through its extremities. Thus, CA 5 is a sector 8. A straight line is said to be in- m scribed in a circle, A\iien 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. 10. An inscribed triangle is one which has its three angular points in the circmiiference. Thus, ABC is an inscribed triangle. U 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 iho polygon. 40 GEOMETRY Definitions 12 A secant is a line which meets the circumference in two points, and lies ^nartly within and partly without the circle. Thus ^ S is a secant. iM 13. A tangent is a line which has but one point in conunon 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- lences have but one point in common BOOK II 41 Of the Circle. THEOREM I. A diameter is greater than any other chord. Let AD l>c any chord. Draw the radii CA, CD to its extremities. We shall then have A G-\- CD greater than AD (Book I. Th. X*). But AC-\-CD is equal to the diameter AB : hence, tlie diameter AB is greater than AD, THEOREM II. If from tkc centre (f a circle a line be drawn to the middle oj a chord, I . It mil be perpendicular 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 ACD^ DCB, have the three side s of the one equal to the tliree sides of the *NoU. NMien reference if made fronr. one theorem to another, in the same Book, the number of ihe thoorem refertftd to is aloue c^iven • but when the theorem rcferrol to is found ir a preceding Book, the number of the Book is also w'ven. 42 G E O 31 E T R y . Of th le. other, each to each: viz. AC equal to CB, being radii, AD equal to DB, by hypothesis, and CD common: hence, tlie corresponding angles are equal (I Jock I. Th, viii) : that is, the angle CDA equal to CDB, and the angle ACD equal to the angle DCB. But, since the angle CDA is equal to the angle CDB, the radius CE is perpendicular to tbe chord AB (Bk. 1. DeL 20). Secondly. Since the angle ACE is equal to 5 Ci?, 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 o( fhe circle. Also, a line drawn through the centre of the cir- cle and perpendicular to the chord, bisects it. tiieore:\i III. If more than tiro equal lines can he drawn from any point witfan a circle to the circiunfejence^ 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 11 43 Of the Circle Then, since llic two triangles DAE and DEB have the side AE equal to EB, AD equal to DB, and DE coi^mon, ihey wiU be equal in all respects ; and consequently, the angle DEA is equal to the angle DEB (Bk. 1. Th. viii) ; and therefore, DE is perpendicular to AB (Bk. I. Dei 20) But if DE bisects AB at right angles, it wih 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 EDy DF, it will be found at tlieir common inter- section D. THEOREM IV. Ant/ chords which are equally distant from ike centre of a arcltt, are equal. i^ei AB and ED be two chords equally distant from the centre C: then will the two chords AB, ED he 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, anH the side CF equal to CG, will have the third side BF equal tc EG (Bk. I Th. xix) But, BF is the half of BA and EG the half cf DE (Th. ii. Cor); hence BA is equal to DE (Ax 6). 44 GKORIET RY. Of the Circl THEOREM V. A line which is perpendicular to a radius at its extremity^ is tangent to the circle. Let 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 greatei 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 between the centies of two circles is equal to the sum of their radii, the two circles will touch each other externally. BOOK 1 r 45 Of the Circle ^ Let C and D be the two centres, and suppose the distance between them to be equal ♦.o the sum of tlie radii, that is, to CA-\ AD The circumferences of the circles will ey idently have the point A common, and they will have n«j oilier. Because, if they had two points common, that, is if ihey 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. // the distance between the centres of two circles is equal to the difference of their radii, the two circles will touch each otk'^ interna II I/. Let C and D be ilie centres of two circles at a distance from each other equal to AD- A C= CD. Fr Now, it is evident, as in the last theo- 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, the distance between their centres : therefore, they cannot have two poir.ts in common when the difference of their radii id equal to the distance between their centres : hence, they are tangent to each other. Sch If two circles touch each other, either externally oi internally, their centres and the point of contact will be in thf same straight line 46 G E O I\I £ T R y Of the Circle THEOREM VIII A n angle at the circumference of a circle is measured by half i it arc that subtends it Let BAD be an inscribed angle : llicn will it be mccisiircd by half the arc BED, which subtends it. For, through the centre C draw the diameter ACE, and draw the radii BCj CD. Then, in the triangle ABC, the exte- rior angle BCE is equal to tlie sum of the interior angles B and A (Bk. 1. 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 BAC. But, the angle BCE is measured by the arc BE, which euhtends it ; and consequenily, 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 Ly half the arc BED, which subtends it. Cor. 1. Hence, if an angle at the centre^ and an angle nt the circumference, both stand on the same arc, the angle at the ccntie will be double the ansle at the circumference. Cor. 2. If two angles at the circumference stand on equaj arcH thev will he equal to each other. BOOK II 47 Of the Circle THEOREM n. A U angles at the arcumfcrcncc^ which stand upon *.hf same art are equal to each other. F.et the angles BA C,BDC, BFC, liavo their vertices in ihe circumference, and 3tand on the same arc BEC : then will they be equal to each other. For, each angle is measured by lialf the arc BEC (Th. viii) ; hence, the an- gles are all equal. 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 yiZ)C, that is, by a quadiant, which measures a right angle CBk 1. Th. i. Cor. 2). THEOREM XI. If a quadrilateral he inscribed in a circle, the sum of either two 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. A \J For, the angle A is measured by half ^^- — ' ^^ the arc DCB, wliich subtends it (Th. viii) ; »'.^ Z?\ 4S a E M E T R y . Of the Circle. and the angle C is measured by half the arc DAB, which subtends it. Hence, iho sum of the two angles, A and C. is meaiiured by half the entire circumference. Bat half the entire circumference is the measure of two right angles ; therefore, tho sum of the opposite angles A and C is equal to two rigbl angles. In like manner, it may be shown, that the sum of the wo angles B and D is equal to two right angles THEOREM XII If the side of a quadrilateral, inscribed in a circle, be pro- duced out, the exterior angle will be equal to the inward opposiit angle Let the side BA, of the quadrilateral aBCD be produced to E, then will the outward angle DAE be equal to the in- ward opposite angle C. E For, the angle DAB plus the angle C, is equal to two right angles (Th. xi). But DA B plus DAE is also equal to two right angles (Bk. 1. Th. ii). Taking from each the common angle DAB, and we shall have tlie angle DAE equal to the interior opposite angle C. THEOREM XIII. Two parallel chords intercept equal arcs. BOOK I ! 49 Uf the Circle Let the chords A B and CD be parallel : then will the arcs AC and BD be equal For^ draw the line A D. Then, because ihc hnes AB and CD are parallel, the ailcinate angles ADC and DAB will be equal (Bk. I. Th. xii). IJut the angle ADC is measured by half the arc AC, and the angle DAB by half the arc BD (Th. viii) : hence the two arcs A C and BD are themselves equal. THEOREM XIV. The angle formed by a tangent and a chords 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 At 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 iT (Bk. I, Th. i. Cor. 2). But the angle DAC being at the circumference, is measure 1 by half the arc DC: hence, by the addhion of equals, the two angles BAD and DAC, or the entire angle BAC w'lU be moas- lucd by half the arc AMDC. It may be shown, by taking the difference between the tAvo angles DAE and DAC, that the angle CAE is measured by lialf the arc AC included between its sides. 5 60 G E O iM E T il Y . Of the Circle. THEOREM XV. If a tangent and a chord are parallel to each other, they will intercept equal arcs. Let the tangent ABC he parallel to the chord DF: then will the intercepted arcs I)B, 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 beinor an ande at the circumference will be measured by half the arc BF (Th. viii). But since the angles are 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 AFC, 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 f.ngle DEB will be equal to the angle FAB (Bk 1. Th. xiv), and the arc FD to the arc AC. But the ang^le FA B is meas- ured by half the arc FDB, that is, by half the sum of lh») arcs FD, DB. Now, since FD is equal to ^C,it follows tliat the angle DEB, ox its equal AEC, will be measured by lialf tlu' Slim of the arns DB and A G BOOK II. Of the Circle. THEOREM XVU. The angle formed without a circle by the intersection cj two secants is measured by half the difference of the intcrcifted arcs. Let the two secants DE and EB inter- sect eaeh 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 arc 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 difTerence of the intercepted arcs DFB and CA THEOREM XVIir. An angle formed by two tangejits is measured by half tliC 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 CDA be measured by half tlic difference of the intercepted arcs CEA and CFA. For, draw the chord AF parallel to the tangent CD. Then, because the lines CD -iud AF arc parallel, the angle BAF will bo equal to the angle BDC (Bk. I. Th. xiv). But the inerle BAF, formed by a tanfjent and a chord, is measured by 52 G E O iU E T R Y . Of the Circle half the arc AF, that is, by half the difference of CFA and CF. But since the tangent DC and the chord AF 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 chord 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 63 Of the Circle. follows tliat the sum of the two angles A and B will be equil to one hundred and twenty degrees. But the triangle CA B is isosceles: hence, the angles at the base are equal (Bk. I. I'h. vi) : hence, each angle is equal to sixty degrees, and consequently, the side A Bis equal io AC or CB (Bk. I. Th vi). PROBLEIMS RELATING TO THE FIRST AND SECOND BOOKS. The Problems of Geometry explain the methods of con structmg 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 5a, Ac, which tum eftcily about a common joint at b. The legs of the dividers •'!)4 G E O 31 E T R Y Problems are extended or brought together by placing the forefinger on the joint at 6, and pressing the thumb and fingers a^ainf^t the logs PROBLEM 1. On any line, as CD, to lay off a distance equal to A B, Take up the dividers with the fhumb and second finger, and place the forefinger on the joint at 6. 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 tliis distance wiU evi- dently be equal to AB. TROBLEM 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 leg until it reaches to B. Tlien, turn the di- viders around the leg at C, and the other leg will descrilv? the required circumference BOOK II 55 P robl cms. 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 straight and smooth. PROBLEM III. 7 draw a straight line throvgh two given potrJs 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. Jy' PROBLEM IV. To bisect a given line: that is, to divide it uito two equal parts, l,Gi AB be the given line to be di\ided. With ^ as a centre, and radius greater than half of AB, describe an arc IFE. Then, with 2i as a centre, and an equal radius BI describe the arc IHE. Join the points / and E by the line IE. the point Z), where it intersects AB, will be the middle point of the Une AB. 56 G E O M E T fl Y . Problems. For, draw the radii AI, AE BI, and BE. Then, since these radii are equal, the triangles AIE • nd BJE have all the sides of the one equal to the corresponding sides of the other ; hence, their corres ponding angles are equal (Bk I. Th. viii) ; that is, the angle A IE is equal to ihe angle BfE Therefore, the two triangles AID and BID, have the sidi A I— IB, tlie angle AID = BID, and ID common: honcp thcv are equal (Bk. I. Th. iv), and AD is equal to DB. PROULExM V. To bisect a given angle or a given ate. f^et 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 ACE into two equal parts, and also bisect the arc AEB at E. 'For, draw the radii AD and BD. Then, in the two triangles ACD, CBD,Me have AC=CB, AD = BD and CD common : hence, the two triangles have their corrrg. ponding angles equal (Bk I. Th. viii), and consequently, A CD is equal to BCD. But since ACD 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 i)-. BOOK II. 67 Problems. PROBLEM VI. At a given vomt in a straight line tc erect a perpendicular to tht line. Let A be the given point, and BC the given line. From A lay off any two distances, AB and AC, equal to each other Then, from the points B and C, as centres, with a radius greater than AB, describe two arcs intersecting each other at D ; draw 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 5 C. (Bk. IDef. 21). SECOND METHOD. When 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 ^, 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 to BA, since CAD is an angle in a dcmicirclo (Bk. 11. Th. x). 68 GEOMETRY Problems PROBLEM VII. Frrni a given point without a straight line to let fall a perpen dicular on the line. Let A be the given point, and BD the given line From the point ^ 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,vvith 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 DAF 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. I. Th. iv) : hence, ^i^jG will be perpendiculai lo BD. SECOND METHOD Wlien the given point A is nearly opposite the extremity of the line. Draw A C, to any point C of the line BD. Bisect ^C at P. Then, with P as a centre and PC as a ra- dius, describe the semicircle CD A ; draw ^D, and it will be perpendicular to CD since CD A is an angle in a semicircle (Bk. II. '1 h. x). B O O K I I . 59 Problems. PROBLEM VI II. At a given point in a given line, to make an angle eq^ial to i given angle Lot A bo ilio given point, AE Uio given line, and IKL the given angle. From the vertex /{", aa a ccnlre, -^ ^ with any radius, describe llic arc /L, terminating in the two sides of the angle : and draw the chord IL. From the point ^4, as a centre, with a distance AE, equal to KI, describe the arc DE : then with £, 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. 1. 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 /^ g RC the given line. With A as a centre, and any ra- dius greater than the shortest dis- " lance 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 AF and draw AD, and it will b^ the required parallel. 60 GEOMETRY. B- Problem s. For, since the arcs AF and ED are equal, the angles EAD and AEFy wliich they measure, are equal : hence, the line AD is parallel to BC (Bk 1. Th xiii). PROBLEM X. Two angles of a triangle being given or known ^ to find the. thir^i Draw the indefinite line DEF. At any point, as E, make the angle DEC equal to one E of the given angles, and then CEH 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 i^ 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 Aj B, and C, be the given fiides. Draw DEj and make it equal lo the side A. From the point D, as a centre, with a radius equal ro the ^*" B*»cond side B, describe ar. ajc o B O O K 1 1 . 6] Problems. from £ as a centre, with the third side C, describe another arc intersecting the former in F: draw DF and FE: then will OFF bo the required triangle. For, the three sides are respectively equal to the three lines A. B and C. PROBLEM XII. TTie adjacent sides of a parallelogram, nnth Che angle which rh^iy contain, being given, to descrthe the narallelo^gram Let ^ and ^ be the given sides ,. and C the given angle. / Draw the line DFJ and make it i/. /£ equal to A. At the point B make ^' ' /^ the angle FJBF equa\ to the angle C. Make the side DF equal to B. Then describe two arcs, one from i^ as a centre, with a radius FG equal to DF^ tht other from F^ as a centre, with a radius FO equal to DF. Through the point G, the point of intersection, draw the lines EG and FG, and DEGF will be the required parallelogram. For, in the quadrilateral DFGE, tlie opposite sides DE and FG are each equal to A : the opposite sides DF and EG arc each equal to B, and the angle EDF is equa. to C. But, since the opposite sides are equal, they are also parallel (Bk. 1 Th. xxiv), and therefore the figure is a arallclogram PROBLEM XIII. To describe a square on a given line. G r>2 G E I\I E T R Y . Problems. Let AB he the given line. At the point B draw 5 C perpendicu- lar to AB, by Problem VI, and then make it equal to AB. Tlien, wiili yl as a centre, and ra- dius equal to AB, describe an arc ; and with C as a centre, and the same radius ^iB, describe another arc; and through D, their point of intersection, draw AD and CD : then will ABCD be the required square. For, smce 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. 1. 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 f^iven an^le. At B lay off an angle, ABC, equal to E, by Prob. VIIL and make BC equal to AB. Then, with A and C tis centres, and a radius equal io 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 par-'dlol iBk. 1. Th. xxiv). But they are each equal to AB, and the BOOK II «3 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. 11. Th. ii. Cor.). Then bisect DE at C, and C will be the centre of the circle. PROBLEM XVI. Tn describe the circumference oj a circle through three given points not in the same straight line. Let A, 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 jjass through the pohits A, B, and C. For, the two right angled triangles OA P and OBP have the side AP equal to the side BP. OP common, and the included 64 GEOMETRY. Pro bl ems. 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 0(7 is equal to OB. Hence, a circumference described with the radius OA, will pass through the points B and C. 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. Through a given point in the circumference of a circle, to drau a tangent line to the circle. Let A be the given point j^ Through A, draw the radius ^C 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. Th. ^) PROBLEM XVIII. Thro*jgh a given point mthout the circumference, to draw a tangent hne to the circle. BOOK 11. 65 Prob 1 ems Lcl C be ihe ceuirc of the circle, and A ilie given point williout the circle. Join A and the centre C, and on A C as a il.ameter, describe a circumference. Through the points B and D where / the two circuijiferences intersect each / oilier, draw the lines AB and AD: \ these lines will be tangent to the circle »\'hose 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 lines AB, AD, are each perpendicidar to a radius at its extremity, they will be tanc^enl to the circle (Bk. II. Th. v). PnOBLEM XIX To inscribe a circle in a given triano^lc. Let ABC be the given tri- angle. Bisect the angles A and B by the lines AG and BO, meet- ing at the point O. From O, let fall the perpendiculars O/), 0£, 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/« 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 1) But the two triangle-s havt, a common side AO, hence, they are eqnnl (Bk. I. Th v), and consequently, OD is equal to OF 6* 66 GEOIMETRV. P r o I) 1 e In a similar manner, it may be proved that OE and OD are equal . hence, ilie three per- pendiculars, OD, OF, and OE, are all eq\ial. • Now, if with 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 (Rk. II. Th. v). Hence, the circle will be inscribed in the triangle. PROBLEM XX. To inscribe an equilateral triangle m a circle. Through the centre C draw any diam- eter, as ACB. From i? as a 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 (sach 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 tire 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 consc* quently, the third angle ADE, is equal to sixty degrees BOOK e*? Problems, Tlierefore, tlie triangle ADE is equilateral (Bk. I. Th. vl Cor. 2). PROBLEM XXI. To inscribe a regular hexagon in a circle. Draw any radius, as -AC. Then ap- ply the radius A C around the circum- ference, and it will give the chords AD, DE, Eh\ FG, GH, and HA, which m ill be the sides of the regular hexagon. For, ^^ _^n A l) the side of a hexagon is e(|iiai to ihe radius (iik. II. Th. xix). PROBLEM XXII. To inscribe a square tn 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, bein|j 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, DAB, is a right angle, being an angle in a semicircle (Bk. II. Th x) : hence, the figi'TP ABCD is a square (Bk. I. Dcf 48) 68 G E I\l E T R Y Problems. Sch. If we bisect the arcs AB^ BCf CD, DA, and join the points, we shall liave a reorular octagon in- scribed in the circle. If we again bisect the arcs, and join the points of bisection, we shall have a regulai polygon of sixteen sides. PROBLEM XXIII. To describe a square about a given circle. E H ; D I Jl r>raw 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 FGIHhe the required square. For, since ACDG is a parallelogram, the opposite sides arc 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 that 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. Hence the figure GIHF is a square (Bk I. Def. 48). GEOMETRY. BOOK III. OF RjrriOB AND PROPORTIONS. DSP1N1TI0N8. 1. Ratio is the quotient arising from dividing one quantity by another quamity of the same kind. Thus, if the numbers 3 and 6 have the same unit, the ratio of 3 to G will be. expressed by And in general, if A and B represent quantities of the same kind, the ratio of yl to 5 will be expressed by B A 2. If there be four numbers, 2, 4, 0, IG, lijiviiig such values that the second divided by the first is equal to the fourth di- vided by the third, the numbers are said to be in proportion. A.nd in general, if there be four quantities -4, B, C, and D hftvijig such values that B D A=C' then, A is said to have the same ratio to By that C has to D, or. the ratio of A to Z? is equal to the ratio of C to D When 70 GEOMETRY Of Uatios and ProportionB. four quantities have this relation to each other, they are said to bo 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 ^ to 5 is equal to the ratio of C to D, we write the quantities thus : A '. B :: G : 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, of inversely, when the consequents are made the antecedents and the antecedents the consequents. Thus, if we have the proportion 3 : 6 :: 8 : 16. thfi inverse proportion would be 6 : 3 :: 16 : 8. BOOK III Of Ratios and Proportions. 6. Quantities are said to be in proportion by allernattcfi, oi altcrnateli/, when antecedent is compared with antecedent and consequent with consequent Thus, if we have the proportion 3 : 6 : : 8 : IG, tlic ahernate proportion would be 3 : 8 : : G : 10. 7. Quantities are said to be in proportion by CLfnposition, when the sum of tlie antecedent and conse(iucnl is compared cither with antecedent or consequent. Thus, if we have the proportion 2 : 4 : : 8 : IG, the proportion by composition would be 2-f4 ; 4 ;: 8-}-lG : IG; that is, 6 : 4 : : 24 : IG. 8. Quantities are said to be in proportion by division^ when the difference of the antecedent and consequent is compared either wi^tli llie antecedent or consequent. Thus, if we have the proportion 3 : 9 : : 12 : 36, tlie proportion by division will be 9 — 3 : 9 :: 3G — 12 : 36; Ihal is, G : 9 : : 24 : 36. 9. Equimultiples of two or more quantities are the products? which arise from nudliplying the quantities by the same number. Thus, if we have any two numbers, as 6 and 5 and multiply 72 GEOMETRY Of Ratios and Proportions them both by any number, as 9, the equimultiples will be 54 find 45 ; for 6x9 = 54 and 5x9 = 45. .Alio, 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- ciprocalbj 2)'>'oportional^ or inversely proportional, when one increases in the same ratio as tlie 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 diminish, and their product would not be changed. THEOREM I. // four quantities are in proportion, the product of the two ea trcmcs will be equal to the product of the two rrteana If we have the proportion A : B '.'. C ', D wo have, by Def. 2, Il_D A~ C Uid 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 : 00 which gives 2>60=:10x\2 = 120 B O O K I I 1 . 78 Of Ratioa and Proportions. THEOREM II. If four quantities are so related to each other ^ that the product of ttro of them is equal to the product of the other two ; thm rxto of them may be made the means, and the other txDO the CM.trcines of a proportion. Let -4, i?, C, and D, have such vahics that BxC=AxD Divido both sides of the equation by A and we have A Then divide both sides of the last equation by C, and we have B_D 7v~c hence, by Dcf. 2, we have A : B :i C '. D. Sch. The general truth may be verified by the numbers 2x18 = 9x4 which give 2 : 4 : : 9 18 THEOREM III. y three quantities are in proportion, the product of the two extremes will he 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 an«l by clearing the equation of its fractions, vrt bave '7 74 GEO M E T R V . Of Ratios and Prop or t ione. Sch. The proposition may be verified by the numbers 3 : 6 : : 6 : 12 which give 3x12 = 6x6=36 THEOREM IV. Ij four quantities are in proportion, they will he in proportion when taken alternately. Let A : B '. : C '. D Then, by Def. 2, we have B_B A~C Multiplying both members of this equation by — , we have B C_D A~B and consequently, A I 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. [f thcie be two sets of proportions, having an antecedent and a wnsequent in the one, equal to an antecedent and a consequent in the other; then, the remaining terms will he proportional. If we have ^ : ^ : . C . A and A : B : E : F . then we shall have B O O K I I I 76 O f Ratios and Propor iona. B D B F A=C ^"^ A = E Hence, by Ax. 1, we have D_F C~E and consequcnily, C : D :: E '. F Sch. The proposition may bo verified by the following proportions, 2 : 6 : : 8 : 24 and 2 : 6 : : 10 : 30 which give 8 ' 24 : . 10 : 30. THEOREM VI. J/ four quantities are in proportion, tlie.y will be in proportion when taken inversely. If we have the proportion A : B :. C '. D we have, by Th. I, AxD=zBxC, or BxCz=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. sffouT t^antities are in proportion^ they wUl be in proportion oy composition. 76 GEOMETRY Of Ratios and Propoi ion a. Let US suppose that we have A . B : : C : D we shaL then have AxD = BxC. To each of these equals, add 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 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 YIII. If four quantities are m proportion^ they will he 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)xDr=z{C-D)xB; and by separating the factors by Th. II, we have, A-B : B '. : C-D : D. Sck The proposition may be verified by the proportion, 24 • 8 : : 48 : 16 { BOO K I I I. 77 Of Ratios and Proportions We have, by division, 24-8 : 8 : : 48-16 : 16; that i8, 16 : 8 : : 32 : 10; in which the ratio is one-half. THEOREM IX. Eqiial multiples of two quantities have the same ratio aa Uu. quantities themselves. It wo have the proportion A : B ". C I D we shall ha\e B_D a" C Now, let M be any number, and by it multiply the iiu« merator and denominator of llie first member of the equation which will not change its value : we shall then ha\e M^B D UxA" C and her.ce we have ' My.A '. 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 multiplj-ing the first antecedent and consequent by any number, as 6, we have 30 : 60 : : 12 : 24, V ^vhich the ratio is still 2. 78 GEOMETRY. Of Ratios and Proportions. THEOREM X. Ij four quantities are proportional^ and one antecedent and iLs wnsequnht be augmented by quantities which have the same ratio as the antecedent and consequent, the four quantities will stUl he in proportion liCt us take the proportions A X B X : C I D, 2iii^ A I B I : E \ F, which give AxD-BxC and AxF=zBaE; adding these equals we have Ax{D-\-F) = Bx{C-{-E); and by Th. II, we have A : B '.: C^E : D+F in which the antecedent C and its consequent D, are augmeni- 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 15 and 30, which have the same ratio, we have 9 : 18 : : 20+15 : 404 30 that is, 9 : 18 : : 35 • 70, in ^^hjch 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 unll still bg in pvportion B O K I I I . 79 Of Ratios and Proportiona. Let us lake the proportions A : B : : C : D, Bind A : B : : E : F. which give AxD=BxC and AxF=BxE. By subtracting these equalities, we have Ax{D-F) = Bx(C-E)i and by Th. II, wc obtain A : B : : C-E : D-F, in which the antecedent and consequent, C and Z>, are dimin- ished by E and P, which have the same ratio Sch. The proposition may be verified by tlie proportion, 9 : 18 : : 20 : 40, for, by diminisliing 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. Jf wc have several sets of proportions^ having the same ratio,, any antecedent unll he to its consequent, as the sum of the onto cedents to the sum of the consequents. If we have the several proportions, D which gives AxD=BxC F which gives A x F=Bx E H which gives A x H=Bx O We shall then have, by addition, Ax[D-]-F+H)=:Bx(C-rE+G)', and consequently, by Th II. A : B ;: C-\-E-\-G : D-^F-^Il. A B : , C A ■ B : : E A B ' : G 80 GEOMETRY. Of Ratios and Proportions Sch. The proposition may be verified by the following proportions : viz. 2 • 4 : : G ; 12 and 1 ; 2 : : 3 : 6 Then, 2:4:: 6-f-3 : 12-f6; 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 : D, it gives n_D A~C Then, if we square both members, we have and if we cube both members, we have B^ D^ and then, changing these equalities into a proportion, we have for the first, A^ : B' :: C^ : 23^ and foi the second A^ B^ : • C^ : D Soh. We may verify the proposition by the proportion, 2 : 4 : : 6 : 12, and by squaring each term we have, 4 : IG : . 36 • 144 BO O K 1 I I . 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' : 4' :• 6= • 12^ tiiat is, 8 : 64 : ' 2.6 • 1721 in wliich the ratio is 8. THEOREM XIV. If we have two sets of proportional quantities, the products oj the corresponding terms will be proportional. Lot U8 take the proportions, A ' B : : C : D which gives B_D A~C F_H E~G E : F : : G : H which gives Multiplying the equalities together, we have BxF DxH AxE^CxG md this by Th. II, gives AxE : BxF :: CxG : DxH. Sch. The proposition may be verified by the followmg proportions : '8 ; 12 : : 10 : 15, and 3 : 4 : : 6 : 8 ; we shftll then have 24 : 48 : : 60 : 120 whi'jh are proportional, the ratio being tK GEOMET R Y. BOOK IV 01^ THE MEASUREMENT OF AREAS, AND THB PROPORTIONS OF FIGURES. DEFINITIONS. 1 Similar figures, are those which have the angles of tlie 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 oi angles. 3. A polygon which has all its angles equal, each to each, and all its sides equg,l, 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. Il 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. 83 Of Parallelograms, lyd.-3 1t«t. 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. 7. The area of a figure is the measure of its surface. The unit of tlie number which expresses the area, is a square, the side of wliich is the unit of length. 8. Figures have equal areas, when they contain the same measuring unit an equal number of times. 9. Figures 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. Parallelograms which have equal bases and equal altitudes, arc equivalent. Place the base of one parallel- ogram on that of the other, so that AB shall be the tommon base of the two parallelograms ABCD and ABEF. Now, since the par- allelograms have tlic 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. jaiii), AD is equal to BC. Also, DC and FE are each equal to AB : and consequently, they are equal to each ;^4 G E O .'\l E T R Y . Of Triangles and Parallelogramj. olhei (Ax. 1 ). To each, add ED : p ^ p ^. then will CE bo equal to DF. \~ ^^7" V But since the line FC cuts the \ /\ / tw'O parallels CB and DA, the \Z___a/ 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 | 7 j 7 rectangle, having equal bases and / equal ahitudes, are equivalent V. THEOREM II, Triangles which have equal bases and ".qual altitudes, an equivalent. Place the base of one triangle F D,_E C on that of the other, so that ABC \ \\/ ^^ and ABB shall be iv?o trian- \ / /0\ / jjrles, having a common base AB, y^^— -\\ ° ° . A B 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 SD, formiii^ the two parallelograms BE and BF Then BOOK IV 85 (Jf Triangles and Parallelograms. since these parallelograms have a common base and equal altitudes, they will be equivalent (Th. i). But the triangle ABC is lialf the parallelogram BE (Bk. 1 Th. xxiii); and >1j5D is half the equal parallelogram BF. hence, the triangle ABC Is equivalent to the triangle ABD, THEOREM 111. If a triangle and a parallelogram have equal bases and equal altitudes^ the triangle xcill be half the parallelogram. Place the base of the triangle on the base of the parallelogram, so tliat 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 tha^ of the parallelogram, th« vertex will be found some where in CD^ or in CI) produced. Now the two triangles ABC and ABEj having the same base AB, 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. i\. Cor. Hence, if a trijlngle and a rect- angle have equal bases and equal alli- uides, 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 the rectnnsile 8G GEOMETRY Of Rectansloa. D C . "—^ /' / / / / ' / / \-4 ^-i THEOREM IV. Rcctar,gles which are described on equal lines are equivalent liCt BD and FHhe two rectangles, having the sides AB, BC, equal to the two sides £F, 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 tqual lines are equal. For a square is but a rectangle having its sides equal. THEOREM V. l^wc rectangles having equal altitudes are tc each other as theif 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. D F V 1 \ 1 1 i 1 A 1 5 '5 For, suppose the base ^^ to be to the base EB,rs any two numbers, say tl e numbers 4 and 3 T>et AE be then divided B O K 1 V . 87 Of Rectangles. into four equal parts, and EB into three equal parts, and tlirougli the points of division draw parallels to AD We Bl'.all 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 will 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 whoso bases are whole numbers : hence, AEFD : EBCF : i AE i EB. THEOREM VI. Any two rectangles are to each other as the products of thtit bases and altitudes. l.et ABCD and AEGF be H D two rectangles : then will ABCD : AEGF • : ABxAD : AFxAE For, having placed the two rectangles so that BAE and G DA F 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, arc to each other w their bases AD and AF. Thus, we have the proportions ABCD : AEHD : : AB : AE, AEHD : AEGF : ; AD : AF, 6S> GEOMETKY. Of Bectangles 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 H D G ABCD AEGF ABXAD : AExAF, 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^ and tht several parts of another line any way divided^ is equivalent to tJu 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 FII, parallel to the line AD : then will the rectangle AO D G H C 1 i ; i ' b B K I V . 89 Of Areas of Parallelograms, be equal to the rectangle of ADxAE ; EH will be equal to EGxEF,oTtoADxEF; and FC willhe equal io FHxFB or to AD X FB. But the rectangle ^iCis equal to the sum of tbe partinJ rectangles : hence, ADxAD=0=ADxAE-{-ADxEF+AD>.Fn. THEOREM Mil. The area of any parallelogram is equal to the product of its base by its altitude. Let ABCD be any parallelogram, and BE its altitude : then will its area be f^ T~7' equal io ABx BE. For, draw AF perpendicular to the base AB, and produce CD to F. Tlien, the parallelogram BD and the rectangle BF^ having the samr base and altitJide are equivalent (Th. i. Cor.). But the arei of the rectangle BF is equal to the product of its base AB h^ the altitude AF (Th. vi. Sch.): hence, the area of the paral lelogram is equal to ABx BE. Cor. Parallelograms of equal bases are to each other as then altitudes ; and if their altitudes are equal, they arc to each other as their bases. For, let B be the common base, and C and D the altitudes of two parallelograms. 'I'hen, by the theorem, theii areas arc to each other, as B\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 wM be to each other as AxC : BxC: that is, as A : F 90 G E O 1\I E T R r . Areas cf Triangles and Traperoids. THEORExM IX The area of a triangle is equal to half ii^ product of its base hi. its altitude. Let ABC ha any triangle and CD its altitude : then will its area be equal to half the product of AB x CD. For, through B draw BE parallel to A C, and through C draw C£ parallel to ^Z? ; 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 ABhy 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 o( ABxCD Cor. Two triangles of the same altitude are to each othei 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. 7 he 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) ROOK IV. 91 Of Rectangles. For, produce AB iiniil BE is ecjutil to DC, and complete Jlio rectangle AF ; also, draw BII perpendicular to AD. Then, the rectangle AC will be equivalent to BF, since they have equal bases and equal altitudes (Th iv). The diagonal BC viill divide the rectangle Gil into two equal triangles; and hence, the trapezoid A BCD will be equivalent to the trapezoid BEFC ; and consequent ly, the rectangle AF, is double the trapezoid ABCD. But the rectangle AF \s equivalent to the product of ADxAE; that is, to CGx{AB-{-DC)\ and consequently the trapezoid ABCD is equal to half that product. THEOREM XI. If a line he divided into two parts, the square described on the whole line is equivalent to the sum of the squares described on the two parts, together with twice the rectangle cnntaijicd by the parts Let the line AB he divi'led into iv/o parts at the point E: then wnl the scjuarc described on ABhQ equivalent to the two squares described on AE and EB, to- gether with twice the rectangle contained by AE and EB : that is JjL ^^ E R AEi' ■AE^-\-EB'^^2AExEB. Foi let AC be a square on A B, and AF a square on AE and produce the sides EF and GF to // and /. Then since EH is equal to AD, being the opposite side o( a rectangle, it is also equal to AB ; and GI is likewise equai u^ AB If therefore, from these equals we take av^ ay EF and 92 GEOMETRY Of Rectangles GF, there will remain FH equal to FI, and each will be equal to HC or IC ; and since the angle at i^ 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, D H B 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. 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 of algebra, thus, THEOREM XII. 7 Vie square acscrihcd on the hypothenuse of a right angleJ triangle, is equivalent to the sum of the squares described on tht other two sides. BOOK 17. 93 Of Right Angled Triangles. Let BA C be a right an- gled triangle, right angled at A: tlicn will the square dc- Bcribcd on the hypothenuse BCj be equivalent to the two squares described on I^A and AC. Having described the squares £Gy JJL, and AIj let fall from A, on the hy- pothenuse, the perpendicular AD, and produce it to JS; then draw the diagonals AF, CIT. Now, the angle ABF is made up of the right angle FBC and the angle CBA ; and the angle CBII is made up of the right angle ^i?//and the same angle CBA: hence, the angle ABF is equal to CBII. But FB is equal to BC, being sides 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 the included angle of the other, each to each, are equal (Bk. I. Th. iv). Since the angles BAC and BAL are right angles, as Mso the angle ABII, it follows that CAL is a straight line parallel to BH. (Bk. T. Th. ii. Cor. 3). Hence, the square HA and the triangle HBC stand on the same base and be- tween the same parallels; therefore the triangle is half the tquare (Th. iii. Cor.). For a like reason, the triangle ABF is half the rectangle BE. But it has already been proved that the triangle ABF ia equal to the triatigle CBH : hence, the rectangle BE, which is double the former, is equivalent to the square BL, which ifl double the latter (Ax. 6). 94 G E i\I E T R Y Of Right Angled Tr' angles. In the same manner it may be proved, that the rect- angle DG is equivalent to the square CK But the two rectangles DEy 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 I wo sides. Cor. Hence, the square of either side of a right angled triangle is equivalent to the square of the hypothenuse diminijshed by the square of the other side. That is, in the light angled triangle ABC ab'^oIc^-bc' or BC^OAC^-AB^ Sck. The last theorem may be illustrated by de- scribing a square on the hy- pothenuse BC, equal to 5, also on the sides BA, AC, 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 C B O K I V . 95 Of Triangle Sides cut Proportional ly. THEOREM XIII. I) a line be drawn parallel to the base of a tnangle, it will divide the other two sides proportionally. Let ABC be any triangle, and DE a firaight line drawn parallel to the base BC: then will AD : DB :- AE : EC. For, draw BE and DC. Then, ilie two triangles BDE and DCE have the same base DE, and the same altitude, B since tlieir vertices B and C, lie in the lini BC parallel to DE : hence, they are equivalent (Th. ii). Again, the triangles ADU 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 h»ve 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. III. Th v) we have AD : BD : : AE : EC. Cor. The sides AB, AC, are also proportional to ihe part* AD, AE, or to BD, CE. For, by composition (Bk. III. Th. vii), we have AD-\-BD : BD :: AE+EC : EC. TTien, by alternation (Bk. 111. Th. iv). AB : AC : : BD: EC, hence, also, AB : AC : : AD : AE 96 GEOMETRY. Proportions of Triancrles, THEOREM XIV. A line which bisects the vertical angle of a triangle divider the base into two segments which are proportional to the adjacent nde. Let AC£ be a triangle, hav- ing the angle C bisected by the Hue CD: thet will AD : r>0 :: AG : CB. For, draw £E parallel to CD and produce AC to E. Then, since CB cuts the two parallels CD, EB, the alternate angles BCD and CBE are equal (Bk. I. Th. xii) : hence, CBE is equal to angle ACD. Buf, since AE cuts the two parallels CD, BE, the angle ACD is equal to CEB (Bk. I. Th. xiv) : consequently, the angle CBE is equal to the angle CEB (Ax. I) : hence, the bide 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 : 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 DEFhe 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 : DF B C E BOOK IV 97 Proportions of Triangles. For, on the sides of the larger triangle DEF, make Dl equal to -4C and DG equal to AB, and join IG. 'I'hen tho two triangles ABC and DIG^ having two sides and the in* oluded angle of tlie one equal to two sides and the included angle of the other, each to each, will be equal (Hk. I Th. iv) Hence, the angles / and G are equal to C and />, and conso .]uently, to the angles F and E: therefore, /G is parallel to £F(Bk. I. Th. xiv, Cor. 1). Now, in the triangle DEF, since IG is parallel to the base, we have (TIi. xiii). DG : DI : : DE : DF, iliai i.s, AB ' AC :: DE : DF. THEOREM XVI. Two triangles which have their sides proportional are equiarir- gular and similar. Let BAC and EDF be two triangles l.a\ in^- BC . EF : : AB : ED, and BC : EF : : AC : DF; then will they have the cor res- ^ ponding angles eq'i:il, \\7... the angle Bz^E, A = D and C=F. For. at the point E make FEG equal to tho angle B, find at F make the angle E FG equal to the an^le C . Tlien will the angle at G be equal to A, and the two triangles BAC and EGF will be equiangukr (Bk. I Tli xvii. Cor 1). Tlicrcfore, by the last theorem, we shall have BC : EF :: AB : EG ; 9 98 (5 E O M E T R \' Proportions of Triangles, but by hypothesis, BC : EF :: AB : DE: hence, EG is equal to ED. By the last theorem we also liave BC : EF :: and by hypothesis, BC : EF : : AC AC DF, hence, FG is equal to DF. Therefore, ilie triangles DEF and EGF, liaving 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. Sck. By Theorem XV, it appears that if the corresponding angles of two triangles are equal, each to each, the correspond- mg 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 c(]ual, but the sides are not proportional ; and the angles of a parallelogram or quadrilateral, may be varied at pleasure, witlioui altering the lengths of the sides. THEOREM XVII. ij two triangles have an angle in the one equal to an angle in the other, and the sides containing these angles propo?'tional,thf two triangles unll be eq'iiangttlar and similar. BOOK IV. 99 Proportion a of Ti i angles Let ABC and DEF be two tri- angles having the angle A equal to (ha angle D, and Ali DE : : AC , DF; ihon will ilic two triangles be similar. For, lay olT^G equal to DE, and through G draw GfipaT' allel to BC. Then the angle AG I will be equal to the angle ABC (Bk. I. Th. xiv) ; and the triangles AGI smdABC will be equiangular. Hence, we shall have AB : AG :: AC : AL Out, by hypothesis, we have AB '. DE '.'. AC \ DF, and by construction, ^G is equal to DE; therefore, Al u equal to DF, and consequently, the two triangles AGI and DEF are equal in all their parts (Bk. 1. Th. iv). But the tri- angle ABC is similar to AGI, consequently it is similar to DEF TIinOREM -Win. ij from the rigid angle of a right angled triangle, a perpfK- diculiir be let fall on the hi/polhcnuse, then I. The two partial triangles thus formed will be similar ta fticA other and to the whole triangle. Ix. Either side including tJie right angle will be a mean pro- pt>rtiorMl between the hypothcnuse and the adjacent segment. III. The perpendicular will be a mean proportional hetuven tkt segments of the hypothmuse 100 v the circumference. O O K IV. 109 Pr olems. PROBLEMS RELATING TO THE FOURTH BOOK. PROBLEM I. To divide a lijie into any proposed nujnbcr 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 ADj 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 AC (Th. xiii) : that is, into equal parts. PROBLEM II. To find a third proportional to two given tines. Let A and B be the given lines. Make AB equal to A^ and draw ACt making an angle with it. On AC lay off ^C equal to 5, and join BC . then lay off AD, also equal to B and through D draw DE parallel to BC : then will AE be the third proportional sought. For, since DE is parallel to BC, we have (Th. xiii) AB '. AC :: AD or AC : AE- ♦hcrcfore, AE is tlie third proportional sought JO IK) GEOMETRY Problems PROBLEM III. To find a fourth proportional to the lines A, B, and C. Place two of the lines forming an angle with eacli other at A ; that is, make AB equal to A, and AC equal B ; also, lay off AD equal to C. Then join BC, aiul through D draw DE parallel to BC, and AE will be the fourth proportional sought. For. since DE is parallel to BC, we have AB : AC :: 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 4 C, and it will be the mean proportional sought (Th. xviii. Cor). PROBLEM V. To make a square which sJiall be equivalent to the mm of two 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 AC and the square described on A C will be equivalent to the squares on A and B (Th. xii). BOOK IV. Ill P r ob 1 e ms. E a PROBLEM VI. 7V make a square which shall be equivalent to the differcncJB be tween two given squares. l-ct A and B be the sides of will be parallel to each other B D 124 GEOMETRY Of Planes TIIEOREiM XI. If two angles^ not situated in the same plane^ have their stdes paraUel and lying in the same direction, the angles will li equal. Lot tlie angles ACE and BDF have tlin sides A C 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 AB, CD, and EF ; also, draw AE, BF. Now since A C 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 A B 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. I. Th. xxv), and conse- quently, AE is equal to BF. Hence, the two triangles ACE and BDF have the three sides of the one equal to the throe sides of the other, each to each, and therefore the angle ACJl is equal to the angle BDF (Bk. I. Th. viii). THEOREM XII. If two planes are parallel, a straight line which is perpendicular to the one will also be perpendicular to the other. BOOK V I2d Of Planes. { A'. \ 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 tlie plane PQ^ and through the lines AB, BC, suppose the plane ABC to be drawn, intersecting ^ the plane MN in the line AD : then, the intersection AD will be parallel to BC (Th. ix). But since AB is perpendicular to the plane iO/", it will be perpendicular to the straight hne 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 ; hence, it is perpendicular to the plane (Def. 1). Cor. It from any point as H^ any oblique lines, as IIEF, HDC, be drawn, the parallel planes will cut these lines proportionally. For, draw HAB perpendicular to the plane MN : then, by the theorem, it will also be perpendi- cular to P^. Then draw AD, AE, BC, BF. Now, since AE, BF, \re tlie intersections of the plane FliB, with the two parallel planes MN, PQ, they are parai- k'l (Th ix.) ; and so also are AD, BC. Then, HA : HB : HE : HF, and HA \ HB '. HD : EC, hence, HE : HF . : HD : HC GEOMETRY. BOOK VI. OF SOLIDS. DEFIXITIOXS 1. Evcrj^ solid bounded by planes is called a. polycdron. 2. The planes which bound a polyedron are called faces. 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 w^ertices 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 ilie pentagon ABODE, terminates in an equal and parallel pentagon FGHIK, which is called the vpper base. The side faces of the prism are the parallelograms DH, DK, EF, AG, and BH. These are called the convex, or /a^era/ surface of th(x ori 3m BOOK V 127 Of the Prism 5. The aliitiidc of a prism is the distance between its upper and lower bases : tliat is, it is a line drawn from a point of iho upper base, perpendicular, to the lower base 6, A right prism is one in which the edges AF, BG, EK, HC, and D/, 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. F \ : i i y] 1 A G \ 7 D 1 i C 7. A prism whose base is a triangle, is called a triangular prism ; if the base is a quadrangle, ii is called a quadrangidai prism ; if a pentagon, a pcntaj][onal prism ; if a hexagon a hexagonal jirism ; (Vc. 8 A prism whose base is a paralielo- graut, ami all of whose faces are also parallelojrrams, is called a parallelopipe- don. If all the faces are rectangles, it is called a rectangular parallelopipedon. 4 ::^ 9, If the faces ol the rectangidar par- allelopipedon are squares, the solid is called a cube: hence, the cube is a prism bounded by bIx equal squares 128 G E O IVI E T R Y Of the Pyramid 10. A pyramid is a solid, formed by several triangles united at the same point S, and terminating in the difTcr- ent sides of a polygon ABODE. The polygon ABODE ^ is called the base of the pyramid ; the point *S, is called the vertex, and the triangles ASB, BSO, OSD, DSE, and ESA, form its lateral, or convex surface. 1 1 . A pyramid Avhose base is a triangle, is caOed a tnun^ gular pyramid; if the base is a quadrangle, it is called a quadrangular p}Tamid ; if a pentagon, it is called a petagonaj pjramid; if the base is a hexagon, it is called a hexagonal l>>Tamid; &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 ^-^ABODE. 13. When the base of a pjTamid is a regular polygon, a;ul 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. ]29 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 heigiit of the P}Tamid S'-ABCDE. 15. If from the pyramid S—ABCDE the pyramid S — abcde be cut off by a plane parallel to the base, the remain- mg 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 sold, described by the revolution of a rectangle, AEFD, about a fixed side, EF. As the rectangle AEFD, turns around ihe 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 A E, is called the lowei base of the cylinder, and the circle described by Z)F, is called vhc upper base. vso GEOMETRY Of the Cylinder The immovable line EF is called the axis 3f 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, FG, which is double the revolving rectangle DF!. 18. If 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 Bide FC, of the rectangle MC, describes the lower base, the equal side MPj 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. 131 Of the Cone 20. A cone is a solid, described by the revolution of a right angled triangle, ABC, about one of its sides, CD The circle described by the revolving side, AB^ is called the base of tlie 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, C5, which remains fixed, is called the axisy 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,ihe 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 pjTamid S — ABCD is inscribed in the cono 132 G E O ]\1 E T Jl Y Of the Sphere, 23. Two cylinders are similar, when the iiameters of theii ba^cs are proportional to their altitudes. 'M. Two cones are also similar, when the diameters of tlieii bases are proportional to their altitudes. 25. A sphere is a solid terminated by a curved surface, all tlie points of which are equally distant from a certain poml 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 lino passing through the centre, end terminated by the circumfer ence. Thus. AD is a diameter BOOK VI 133 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 ro- rolves ; and the points at which the axis meets the surface, are called the poles. 31. A plane is tangent to a sphere when it has but one point in com- ^^ mon with it. Thus, ^i3 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 G E M E T li Y Of the P I ; s in . 34. A spherical segment is a portion of the solid sphere in- chided between two parallel planes. These parallel planes arc 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 !i\-een the parallel planes which form its bases THEOREM I. Tlic convex surface of a right prism is equal to (he perimeter of %ts base ?nultiplicd Inj its altitude. Let ABODE— K be a right 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 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 surftice of the prism, is equal to {AB-\-BC-\-CD-hDE-\-EA)xAF; that is, equal to the perimeter of the base of the prism nmhi plied by its altitude. THEOREM II. T%€ convex surface of a cylinder is equal to the circumference oj its base multiplied by its altit'ude BOOK VI. 135 Of the Prism Let DB be a cylinder, and AB the diameter of its base : the convex sur- face will then be equal to the altitude AD mullij lied by the circumference of the base. For, suppose a regular prism to be insc/ibed 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 cyhnder. But its convex surfjice is still equal to the perimeter of its base multiphed by its altitude : hence, the convex surface of a cylin- der is equal to the circumference of its base nmltiplied by its al- titude. THEOREM III. In every prism the sections formed by planes parallel to the bast are equal polygons. Let AG he any prism, and IL a sec- lion made by a plane parallel to the base AC: then will the polygon IL be equal to A C, For, the two planes AC, 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- 136 GEOMETRY Of the Pyramid. M K 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. I. 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 A C. Sch. It was shown in Definition 18, that the section of a cylinder, by a plane parallel to the base, is a circle equal to the base. THEOREM IV. If a pyramid be cut by a plane parallel to the base, I. The edges and altitude vnll be divided proportionally. II. The section will be a polygon similar to the base. Let the pyramid S—ABCDE, of which SO is the altitude, be cut by the plane abcde parallel to the base : then will, Sa : SA : : Sb : SB, and the same for the other edges ; and the polygon abcde will be s jnilar to the base ABCDE. First. Since the planes ABC and abr B O O R V 1 . 137 Of the Pyramid we parallel, their intersections, AB, ab, by the plane SAB^ will also be parallel (Bk. V. Th. ix) ; hence, the trVangloa SABj sab, are similar, and we have SJ Sa : : SB : Sb ; for a similar reason, mc 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, :3=27; and so on: hence, if the sides of a i^crios of cubes are to each other as the numbers 1, 2, 3. Sic, the Qubes themselves, or their solidities, will l>e as the num- bers 1, 8, 27, Slc. 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 equivaUiii 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. '^ £ \ IT ^ B C r For, since their bases are equivalent they will contain the same number of units of surface (Bk. IV. Def. 9). Now for e.ich unit of height there will be one tier of equal cubei in each solid, and since the altitudes are equal, the number o tiers ill each solid will be equal : hence, the solidities will bt equal, and therefore the solids will be equivalent. Cot Hence, we conclude, that the solidity of a prism < cylinder is equal t > the area of its base mtdtiplied by i\ ; altitude. us GEOMETRY. ()f Triangular I'yramids. THEOREM XV. Two triangular pyramids, hiving equivalent bases and equal altitudes, are equivalent, or equal in solidity. Let their equivaleul bases, ABC, abc, be situated in the same plane, and let AT be their common altitude. If they arc not equivalent, let S — abc he 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, cry, 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 wilj be respectively equivalent, namely DEF to def, GHI to gki, &c. (Th. v. Cor.) BOOK VI. 149 Of T r i angular Pyramida. This being granted, upon the triangles ABCj DEF, GUI, (fee, taken as bases, construct extejrior prisms hanng ioj edges the parts AD, DG, GK, &c., of the edge SA ; in like manner, on bases dcf, 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 5 — ABC will be greater than the pyramid; while the sum of the interior prisms of the pyramid tlian any point of the other. Cor. 4. The centre of a small circle, and that of the sphere, are in the same straight line, perpendicular to the plane of the small circle Cor 5. Small circles are the less the fartlior they lie from 56 G E O M E T R y Of the Sphere the centre of the sphere ; for the greater CO is, the less is the chord AB, the diameter of the small circle AMB. THEOREM XXI. Everp plane perpendicular to a radius rt its extremity is law gent to the sphere. Let FAG be a plane perpen- dicular to the radius OA^ at its extremity A. Any point M, in this plane, being assumed, and OM, AM, being drawn, the an- gle 0AM will be a right angle, and hence, the distance OM will be greater than OA. Hence, the point M lies without the sphere ; and as the same can be showTi for every other point of the plane FA G, this plane can have no point but A common to it and the surface of the sphere ; hence it is a tangent plane (Def. 31). Sch. In the same way it may be shown, that two spheres have but one point in common, and therefore touch each other, when the distance between their centres is equal to the sum, or the difference of their radii ; in either case, the ontrcs and the point of contact lie in the same straight line. THEOREM XXII. If a regular semi-polygon he revolved about a line passing through the centre and the vertices of two opposite angles^ the surface described by its perimeter will be equal to the axis multi vlied by the circumference of the inscribed circle. BOOK VI 157 Of t h e Sphere Suppose the regular semi-polygon ABODE to be revolved about the line AF as an axis: then will the surface described by its perimeter be equal to AF multiplied by the circumference of the inscribed circle. From E and Z), the extremities of one of the equal sides, let fall the per- pendiculars EH, DI, on the axis AFy and from the centre O, draw ON per- pendicular to the side DE: ON will then be the radius of the inscribed circle (Bk. IV. Prob. x). Ijct us first find the measure of the surface described by one of the equal sides, as DE. From N, the middle point of DE, draw NM perpendicular to the axis AF, and tlirough E, draw EK, parallel to it, meet- ing MN in S. Then, since EN is half of ED, NS will be half of DK (Bk. IV. Th. xiii) : and hence, NM is equal to half the sum oi EH+DL But, since the circumferences of circles are to each other as their diameters (Bk. IV. Th. xxiv), or as their radii, the halves of the diameters, we shall have the circumference do- scribed by the point N, equal to half the sum of the circum- ferences described by the points D and E. But in the revolution of the polygon the line ED describes the siurface of the frustum of a cone, the measure of which is equal to DE multiplied into half the sum of the circumfe- rences of the two bases (Th. ix) ; that is, equal to DE into the circumference described by the point N 14 158 GEOMETRY Of the Sphere But, the triangle ENS is similar to SNT (Bk. IV. Th. xviii), and also to EDKy and since TNS is similar to ONM, it follows that EDK and ONM aro similar ; hence, ED : EK or /// : : ON : NM, or ED : HI : : circumference ON : circumference MN. consequently, E D X circumference MN= HI X circumference ON, that is, ED multiplied into the circumference of the circle de- scribed with the radius NM, is equal to HI into the circum* fcrence of the circle described with the radius ON^. But the former is equal to the surface described by the line ED in the revolution of the polygon about the axis AF; hence, the latter is equal to the same area ; and since the same may be shown for each of the other sides, it is plain that the surface des- cribed by the entire perimeter is equal to (FH-\-HI+IP-\-PQ+QA)xcirf. 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, multiplisd by the circumference of the inscribed circle. For, the sur- face described by DE is equal to ///x circumference ON. and the surface described by DC is equal to /Pxcircumfe- BOOK VI . 159 Of the Sphere. ronce ON: hence, the surface described by JE^D+DC, is equal (o (///H-/P)x circumference ON, ot equal to HP xciicum' ference ON. THEOREM XXIII. The surface of a sphere is equal to the product of its diameter hy the circumference of a great circle. JiCt ABODE be a semicircle. In- jcribe in it any regular semi-polygon, and from the centre 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 -A^x cir- cumference OF (Th. xxii) ; and this will be true whatever bo the number of sides of the polygon. But if the number of sides of the polygon be indefinitely increased, its perimeter will coincide ^v^th the circumference ABODE, the perpen- dicular OF will become equal to OE, and the surface de- scribed 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 ^J^^xcircum ference OE. Cor Since the area of a great circle is equal to the product of 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 circl'DS. 160 GEOMETRY. Of the Zone THEOREM XXIV. The surface of a zone is equal to its altitude multiplied by the circumference of a great circle. For, tho surface described by any |X)rtion of the perimeter of the in- scribed polygon, as BC+CD is equal to £//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 £i/xcircmnference OA. Sch. 1. When the zone has but one base, as the zone do scribed by the arc ABCD, its surface will still be equal to the altitude AE multiplied by the circumference of a great 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 tho diameter of the si here. THEOREM XXV. The solidity of a sphere is equal to one third of the product yf the surface multiplied hy the radius. For, conceive a polyedron to bo inscribed in the ?phore. BOOK V i . 16] 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 tliird of the product of its base by its altitude (Th. xvii). But if we suppose the faces of the polyedron to be conlinu- ally diminished, and consequently, the number of the pyra- mids to be constantly increased, the polyedron will fmaily become the 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 th3 product of the bases by the common altitude, which y, ill then be equal to the radius of 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 Beniicirclc will describe the sphere, vhilc the half square will describe the cylinder circumscribed about that sphere. The altitude AD, of the cylinder, is equal to the diameter 14* 162 G K ]\I E T R Y Of the Sphere. PQ; llic base of the cylinder is equal to llie great circle, since its diameter ^i3 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 ilie 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 ('J1i. 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, oi by two-thirds of the diameter , hence, the sphere is two-thirds of the circimiscribirg cyhndcr. BOOK V 163 Appendix. APPENDIX OP THE FIVE REGULAR POLYEDRONS- A regular polyedron, is one wliose faces are all equal poly- gons, and whose polycdral angles are equal. There are fi^e Ruch solids. 1. The Tetraedron, or equilateral p}Tamid, is a solid bounded t>y four equal triangles. 2. The kexacdron or cube, is a solid, boiir dnd by six equal squaros. r 3 The oclaedron, is a solid, bounded by eight equal og.. ther, and their sum is the logarithm of the product of the numbers (Art. 3). The term sum is to be understood in its a'gebraic seniw; 172 TRIGONOMETRY. Of Logarithms. therefore, if any of the logarithms have negative characteristicaaj the diflference between their sum and that of the positive characteristics, is to be taken ; the sign of the remainder u that of the greater sum. EXAMPLES. 1. Multiply 23.14 by 5.062. log 23.14 = 1.364363 log 5.062 = 0.704322 Product 117.1347 .... 2.068685 2. Multiply 3.902, 597.16 and 0.0314728 together. log 3.902 = 0.591287 log 597.16 = 2.776091 log 0.0314728 = 2.497936 Product 73.3354 .... 1.865314 Here the 2 cancels the + 2, and the 1 carried from the deci- mal part is set down. 8. Multiply 3.586, 2.1046, 0.8372, and 0.0294, together, log 3.586 = 0.554610 log 2.1046 = 0.323170 log 0.8372 = 1.922829 log 0.0294 = 2.468347 Product 0.1857615 . . r.268956 In this example the 2, carried from the decimal part^ cancols 2, and there remains 1 to be set down. DIVISION OF NUMBERS BY LOGARITHMS. 1 1 . When it is required to divide numbers by means of tbeif lof^rithms, we have only to recollect, that the subtraction of INTRODUCTION. 173 Of Logarithms. logarithms corresponds to the division of their numbers (Art. 3). Uence, if we find the logarithm of the dividend, and from it oub- tract the logarithm of the divisor, the remainder will be the loga- rithm of the quotient. This additional caution may be added. The diflference of tho logarithms, as here used, means the algebraic difference; so that, if the logarithm of the divisor have a negative characteristic its sign must be changed to positive, after diminishing it by the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of tlie dividend is negative, it must be treated as a negative number. EXAMPLES. 1. To divide 24163 by 4567. log 24163 = 4.383151 W 4567 = 3.659631 Quotient 5.29078 .... 0.723520 2. To divide 0.06314 by .007241 log 0.0G314 = 2.800305 log 0.007241 = 3.859799 Quotient . . 8.7198 . . . . 0.940506 Here, 1 carried from the decimal part to the 3 changes it to 2, which being taken from 2, leaves for the characteristic 3 To divide 37.149 by 523.76 log 37.149 = 1.569947 log 523.76 = 2.719133 Quotient . . 0.0709274 . 2.850814 15* 174 TRIGONOMETRY. Of Logarithm B. 4. To divide 0.7438 by 12.9476 log 0.7438 = 1.871456 log 12.9476 = 1.112189 Quotient . . 0.057447 . . 2".7592C7 Here, the 1 taken from 1, gives 2 for a result, as set down. ARITHMETICAL COMPLEMENT. 12. The Arithmetical complement of a logarithm is the num- ber which remains after subtracting the logarithm from 10. Thus, . . 1^—9.274687 = 0.725313 Hence, 0.725313 is the arithmetical complemem Df 9.274687. 13. We will now show that, the difference betiveen two loga- rithms is truly found^ by adding to the first logarithm the arithmetical comjdement of the logarithm to be subtracted^ and then diminishing the sum by 10. Let a = the first logarithm b = the logarithm to be subtracted and c = 10 — 6 = the arithmetical complement of 6. Now the difference between the two logarithms will be eX' pressed by a — b. But, from the equation c = 10 — 6, we have c-10 = —b hence, if wo place for— 6 its value, we shall have a—b = a-\-c—\0 which agrees with the enunciation. When we wish the arithmetical complement of a logarithm, we may write it directly from the table, by subtracting the le/\ INTRODUCTION. 176 Of Logarithms. hand figure from 9, then proceedinrj to the riyht^ subtract each Jifjure from 9 till we reach the last siynifcant fiijure, which must be taken from. 10: this will be the same as taking th( logarithm from 10. EXAMPLES. 1. From 3.274107 take 2.104729. By common method. By arith. comp. 3.274107 3.274107 2.104729 its ar. comp. 7.895271 Diff. 1.1G9378 Sum 1.1 G 93 7 8 after sub- tracting 10. Ilenr^, to perform division by means of the aritlimetical com* plenviiit we have the following RULE. To the logarithm of the dividend add the arithmetical cofu- plemenf of the logarithm of the divisor : the sum after subtracts inq in, icill be the logarithm of the quotient. EXAMPLES. 1. Divide 32 7.5 by 22.07. log 327.5 . . . 2.515211 log 22.07 ar. comp. 8.65G198 Quotient . . 14.839 .... 1.171409 2. Divide 0.7438 by 12.9476. log 0.7438 T.871456 leg 12.9476 ar. comp. 8.887811 Quotient . . 0.057447 . . . 2.759267 170 TRIGONOMETRY. Description of Instruments. In this example, the sum of the characteristics is 8, from which, taking 10, the remainder is 2. 3. Divide 3Y.149 by 523.76. log 37.149 1.589947 log 523.76 ar. coup. 7.280867 Quotient . . 0.0709273 . . . 2.850814 SECTION II. OF SCALES. SCALE OF EaUAL PARTS. t , I .1 a ..T.4.5 -g .7 .8 .97P J= I I I [ 1 t I I I 1 —1— J. 2, J a \ if 14. A scale of equal parts is formed by dividing a lino ol a giver length into equal portions. Ii, for example, the line ab of a given length, say one inch, be divided into any number of equal parts, as 10, the scale thus formed, is called a scale of ten parts to the inch. The line a6, which is divided, is called the unit of the scale. This unit ib laid off several times on the left of the divided line, and its extremities marked, 1, 2, 3, , and then divide it into 90 equal parts, each part is called a degree. Through C, and each point of division, let a chord be drawn, and let the lengths of these chords be accurately laid otF on a Bcale : such a scale is called a scale of chords. In the figure, the chords are drawn for every ten degrees. The scale of chords being once constructed, the radius of the circle from which the chords were obtained, is known ; for, the chord marked 60 is always equal to the radius of the circle. A scale of chords is generally laid down on the scales which belong to cases of mathematical instruments, and is marked cho. To lay offj at a given point of a line, with the scale nf chords^ an angle equal to a given angle. Let AB be the line, and A the given point. Take from the scale the chord of 60 de- grees, and with this radius, and the point A S A as ti centre, describe the arc BC. Then take from the acale 180 TllIGONOMETRY DeBcription of Instruments. the chord of the given angle, say 30 degrees, and with this Hne as a radius, and ^ as a centre, describe an arc cutting BC m C Through A and C draw the line AC^ and BAC will be the re- quired angle. SEMICIRCULAR PROTRACTOR. C 17. This instrument is used to lay down, or protract angles* It may also be used to measure angles included between lines already drawn upon paper. It consists of a brass semicircle ABC divided to half degrees The degrees are numbered from to 180, both ways; that is, from ^ to ^ and from B to A. The divisions, in the figure, are only made to degrees. There is a small notch at the mid« die of the diameter AB^ which indicates the centre of tie pro- tractor. GUNTERS' SCALE. 18. This is a scale of two feet m length, on the faces of which a variety of scales is marked. The face on which the TRIGONOMETRY- 181 Definitions. divisions of inches are made, contains, however, all the scales necessary for laying down lines and angles. These are, the scale of equal parts, the diagonal scale of equal parts, and the EcaJe of chords, all of which have been described. PLANE TRIGONOMETRY. DEFINITIONS AND EXPLANATION OF TABLES. 19. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remain- ing parts can be obtained, either by geometrical construction or by trigonometrical computation. 20. Plane Trigonometry explains the methods of computing the unknown parts of a plane triangle, when a sufficient num- ber of the six parts is given. 21. For the purpose of trigonometrical calculation, the cir- cumference of the circle is supposed to be divided into 360 ftqual parts, called degrees ; each degree is supposed to be di- fided into 60 equal parts, called minutes ; and each minute into 00 equal parts, called seconds. Degrees, minutes, and seconds, are designated respectively IB IK2 TRIGONOMETRY Definitions. by the characters ° ' ". For example, ten degrees^ eighteen minutes^ and fourteen seconds^ would be written 10° 18' 14" If two lines be drawn through the centre of the circle, al right angles to each other, they will divide the circumference intc four equal parts, of 90° each. Every right angle then, as EOA, is measured by an arc of 90° ; every acute angle, as BOA, by an arc less than 90° ; and every obtuse angle, as FOA, by an arc greater than 90°. 22. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc UB is the complement of AB. The sum of an arc and its complement is equal to 90°. 23. The suirplement of an arc is what remains after subtracting the arc from 180°. Thus, 6^i^ is the sup- plement of the arc AEF. The sum of an arc and its sup- plement is equal to 180°. 24. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB. 25. The cosine of an arc is the part of the diameter inter- cepted between the foot of the sine and centre. Thus, OD is the cosine of the arc AB. 26. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AQ \% the Ungent of the arc AB. TRIGONOMETRY 183 Definitions. 27. The secant of an arc is the hne drawn from the ceDti^ of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, 0(' -s the secant of the arc AB, 28. The four Hnes, BD, OD, AG, OC, depend for tlicir Nalacs on the arc AB and the radius OA; they are thus designated : sin AB for BJ) cos AB for OD tan AB for AC sec AB for OC n V B tA \ T^X, G r \ 1 \ f X 29. If ABE be equal to a quad- rant, or 90°, then EB will be the complement of AB. Let the lines ET and IB be drawn perpendicular to OE. Then, ET, the tangent of EB, is called the cotangent of AB : IB, the sine o^ EB, is equal to the cosine of AB ; OT, the secant of EB, is called the cosecant of AB. In general, if A is any arc or angle, we have, cos A — sin (90° — ^) cot A = tan (90° — ^) cosec A = sec (90^ — ^) 30. If we take an arc ABEF, greater than 90°, its sme will be FH ; 0^ will be its cosine; ^^ its tangent, and OQ its secant. But FH is the sine of the arc GF, which is the supplement of AF, and OE is its cosine : hence, the sine of 184 TRIGONOMETRY. Definitions. an arc is equal to the sine of its supplement; and the cosiru of an arc is equal to the cosine of its supplement* Furthermore, ^ ^ is the tangent of the arc AF, and OQ it its secant: GL is the tangent, and OL the secant of the sup- plemental arc GF. But since ^^ is equal to GL, and OQ to OL, it follows that, the tangent of an arc is equal to the tannent of its supplement; and the secant of an arc is equal to the secant of its supplement.* Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have been calculated for evefy minute or second of the quadrant, and arranged in a table ; such a table is called a table of sines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the values of the sines, cosines, tangents and cotangents of all the arcs of a quadrant, divided to minutes or seconds. If the sines, cosines, tangents, and secants are known for area less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20°, we also know the sine of its supplement 160^ ; for the two are equal to each other. TABLE OF LOGARITHMIC SINES. 31. In this table are arranged the logarithms of the nume- rical values of the sines, cosines, tangents, and cotangents of all * These relations are between the numerical values of the trigonometrica] lines; the algebraic signs, which thoy have in the diifcrent quadrants, art? Dot considered. TRIGONOMETRY 185 Uses of tho ToblcB. the arcs of a quadrant, calculated to a radius of 10,000,000,000. The logarithm of this radius is 10. Iii the first and last hori- zontal lines of each page, are written the degrees whose sines, cosines, , after the column cosine, and of the column i>, between tho tangents and cotangents. The column D between the columns tangents and cotangents, answers to both of thcst columns. TRIGONOMETRY. 197 Ubob of the Tables. Now, if it were required to find the logarithmic sine of an arc expressed in degrees, minutes, and seconds, we have only to find the degrees and minutes as before; then, multiply tJie corresponding tabular difference by the seconds, and add the pro duct to the number first found, for the sine of the given arc Thus, if we wish the sine of 40° 26' 28". The sine 40° 26' 9.811952 Tabular difft-rence 2.47 . Number of seconds 28 . Product . . 69.16 to be added 69.10 Gives for tlie sine of 40° 26' 28" 9.812021. The decimal figures at the right are generally omitted in the final result ; but when they exceed five-tenths, the figure on tlie left of the decimal point is increased by 1 ; this gives the nearest approximate result. The tangent of an arc, in which there are seconds, is found in a manner entirely similar. In regard to the cosine and co- tangent, it must be remembered, that they increase while the arcs decrease, and decrease as the area are increased ; conse- quently, the proportional numbers found for the seconds, musl be subtracted, not added. EXAMPLES. 1. To find the cosine of 3° 40' 40" The cosine of 3° 40' ... 9.999110 Tabular difference .13 . Number of seconds 40 . Product 6.20 to be subtracted 6^ Gives for the cosine of 3° 40' 40" . 9.999105 188 TRIGONOMETRY. T) BCB of the TabloB. 2. Find the tangent of 37° 28' 31" Ans. 9.884692. 8. Find the cotangent of 87° 57' 59" Ans. 8.550356. CASE II. To find the degrees, minutes and seconds^ answering to any given logarithmic sine, cosine, tangent or cotangent. 35. Search in the table, and in the proper column, and if the number be found, the deo^rees will be shown either at the top or bottom o^" the page, and the minutes in the side columns, either at the left or right. But, if the number cannot be found in the table, take from the table the degrees and minutes answering to the near- est less logarithm, the logarithm itself, and also the corres- ponding tabular difference. Subtract the logarithm taken from the table from the given logarithm, annex two ciphers to the remainder, and then divide the remainder by the tabular dif- ference : the quotient will be seconds, and is to be connected with the degrees and minutes before found ; to be added for che sine and tangent, and subtracted for the cosine and co- tangent. EXAMPLES, 1. Find the arc answering to the sine 9.880054 Sine 49° 20', next less in the table 9.879963 Tabular difference . . . 1.81)91.00(50" Hence, the arc 49° 20' 50" corresponds to the given sid*: 9 880054. 2. Find the arc whose cotangent is . 10.008688 cot 44° 26', next less in the table . 10.008591 Tabular difference . . . 4.21)97.00(28" TRIGONOMETRY. 189 Theoroms. Hence, 44° 26'-23" = 44° 25' 37'' is the arc answering to the given cotangent 10.008688. 8. Find the arc answering to tangent 9.979110, Ans. 43° 37' 21". 4 Find the arc answering to cosine 9.944599. Ans. 28° 19' 46". 86. We shall now demonstrate the principal theorems of Plane Trigonometry. THEOREM I. The sides of a plane triangle are proportional to the sines of their opposite angles. Let ABC be a triangle; then will CB : CA :: sin A : sin B. For, with ^^ as a centre, and AD equal to the less side -5(7, as a radius, describe the arc DI: and with B as a centre and the equal radius BC^ ^ Kl L describe the arc CL'. now BE is the sine of the angle A^ and CF is the sine of B^ to the same radius AD ov BC. But by similar triangles, AD \ DE \\ AC \ CF. But AD being equal to BCj we have BC : sin ^ : : ^C : sin B, or BC : AC : : sm A '. sin B. By comparing the sides AB ^C, in a similar manner, we should find, AB -. AC \ '. ^m C : sin B. 190 TRIGONOMETRY. Theorems. THEOREM II. In any triangle, the sum of the two sides containing eithei angle, is to their diference, as the tangent of half the sum of the two other angles, to the tangent of half their differerxc. IjQt AC B be a triangle: then will AB-\-AC:AB--AC:'.i2in\{C-{-B) : tan i(C-5). With ^ as a centre, and a radius AC the less of the two given sides, let the semicircle IFCE be de- scribed, meeting AB in /, and BA produced, in E. Then, BE will be the sum of the sides, and BI Hieir difference. Draw CI ^w — Ti^. But since the diflference of the squares of two lines is equivalent to the rectangle contained by their sum and diflference (Davies* Legendre, Bk. IV, Prop, x,) we havo, AC" — A^ = {AC -\- AB) . {AC—AB) ind 'C^ — 'dF= {CD +DB).{CD — DB) therefore, (CD + DB) . (CD — DB) = (AC-]'AB).(AC-A/i] hence, CD -^DB : AC + AB : :AC—AB:CD — DB. 192 TRIGONOMETRY Theorems. THEOREM IV. In any right-angled plane triangle, radius is to the tan^ gent of either of the acute angles, as the side adjacent to thi ktde opposite. Let CAB be the proposed triangle, j^ and denote the radius by B : then will Ji : tan C::AC : AB. For, with any radius as CD describe ^^ the arc DH, and draw the tangent DG. From the similar triangles CDG and CAB we have CD :DG :: CA: AB', hence, E: tan C :: CA: AB. By describing an arc with i? as a centre, we could show in the same manner that, B ' tan B ::AB : AC. THEOREM V. In every right-angled plane triangle, radius is to the cosine of either of the acute angles, as the hypothenuse to iJie side adjacent. Let ABC be a triangle, right-angled at B then will R : cos A'.'.AC : AB. Ur For, from the point yl as a centre, with y. nny radius as AD, describe the arc DF, which will measure the angle A, and draw DE perpendlculai to AB : then will AE be the cosine of A. The triangles ADE and ACB, being similar, we have AD '.AE '.'.AC '.AB: that is, E : oos A :: AC : AB. TRIGONOMETRY. 193 Application B. Remark. The relations between the sides and angles of plane triangles, demonstrated in these five theorems, are suf- ficient to solve all the cases of Plane Trigonometry. Of the i\\ parts which make up a plane triangle, three must be given, and at least one of these a side, before the others can be de- termined. If the three angles are given, it is plain, that an indefi- nite number of similar triangles may be constructed, the angles of which shall be respectively equal to the angles that are given, and therefore, the sides could not be de- termined. Assuming, with this restriction, any three parts of a trian* gle as given, one of the four following cases will always be pre sonled. I. When two angles and a side are given. II. When two sides and an opposite angle are given, III. When two sides and the included angle are given, IV. When the three sides are given. CASE I. When two angles and a side are given. Add the given angles together and subtract their sum from ISO degrees. The remaining parts of the triangle can then be found by Theorem I. KXAMPLES. 1. In a plane triangle ABCj there are given the ang'e A = 58° 07', the angle B ^ 22° 37 , and the side AB = A i08 J ards. Required the other parts. 17 194 TRIGONOMETRY. Application B. GEOMETRICALLY. Draw an indefinite straight line AB^ and from the d'^ale ct ecjual parts lay off AB equal to 408. Then at A lay off aE angle equal to 58° 07', and at B an angle equal to ?2° 37', and draw the lines AC and BC : then will ABC be the tri- angle required. The angle C may be measured either with the protractor or the scale of chords (Arts. 16 and 17), and will be found equal to 99° 16'. The sides AC and BC may be measured by re- ferring them to the scale of equal parts (Art. 2). W« *hall 6nd AC = 158.9 and BC = 351. yards. TRIuOyOMfiTKICALLT BY LOGARITHMS. To the angle . . ^ = 58° 07' Add the angle . B = 22° 37' Their sum = 80° 44' taken from . . 180° 00' leaves C . . 99° 16' which. exceeding 00" we use its supplement 80° 44'. To find the side BC. As sin C 99° 16' ar. comp. . 0.005705 : sm A 58° 07' ; 9.928972 : : AB 408 .... 2.610660 ; BC 361.024 (after rejecting 10) 2.545337 Remark. The logarithm of the fourth term of a propoition IS obtained by adding the logarithm of the second term to th.il of the third, and subtracting from their sum the logarithm of the first term. But to subtract the first term is the same ae TRIOONOMETRT 195 Application B. to .'idd its arithmetical complemen* and reject IC from the sum (Art. 13) : hence, the arithmetical complement of the first term added to the logarithms of the second and third terms, micus ten, will give the logarithm of the fourth term. To find side AC, As sin C 99° 16' : sin B 22° 37' : : AB 408 : AC 158.976 ar. comp. 0.005705 9.584968 2.610660 2.201333 2. In A triangle ABC^ there are given A = 38° 25', B = 67° 42', and AB = 400 : required the remaining parts. Ans. C = 83° 53', BC = 249.974, AC = 34C.04 CASE II. When two sides and an opposite angle are given. In a plane triangle ABC^ there are given AC = 216, C^ = 117, the angle ^ = 22^* 37', to find the other paita. GEOMETRIC ALLY. Draw an mdefinite right line ABB' : from any point as Aj draw AC making BAC = 22° 37', and make AC = 216. With as a centre, and a radius equal to 117, the other given side, describe the arc B'B; draw B' C and BC: then will cither of the triangles ABC ot AB Cy answer all the condi- tiona of the question. 196 TRIGONOMETRY. Applicati ons. TRIGONOMETRIC ALLY. To find the angle B, As 5(7 117 . ar. comp. . . 7.931814 ', AC 2\Q 2.334454 ; : sin ^ 22° 37' 9.584968 : 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'Cj 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 hne ABB', on the same side of the point A, in but one point, and then there will be only one triangle an- Bv/ering the conditions. If the side CB is equal to the perpendicular Cd, the arc 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 m 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. Ans. If the angle opposite the side 50 is acute, it is equal U) 41° 28' 59" ; the third angle is then equal to 106° 31' 01", and the third side to 72.368. If the angle opposite the side TRIGONOMETRY. 19T Applications. 50 is obtuse, it is equal to 138° 31' 01", the third angle to 9** 28' 69", and the remaining side to 12.436. CASE III. When the two sides and their included angle are given. Let ABC he a triangle; AB, BC, the given sides, and B tbe given angle. Since B is known, we can find the sura of the two other angles : for A-\- C = 180° — B and \{A + C) = J(180° - B) We next find half the diflfereuce of the angles A and C by Theorem ii., viz. BC + BA'.BC-BA\'. tan \{A + C) : tan i(A - C): in which we consider BO 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 0, by adding it to the half sum, ^(A -f C), we obtain tlie greater angle, and by subtracting it from half the sum, we obtain the less. That ia 1{A + C) H- l(A -C) = A, and i{A-{-C)-i(A-.C)= C. Having found the angles A and (7, the third side AO may be found by the proportion. sin A : sin B : : BC : AC. EXAMPLES. 1. In the triangle ABC, let BC = 540, AB = 450, and the included angle B = 80° : required the remaining parts. 17* 198 TRIGONOMETRY Applications. GEOMETRICALLY. Draw an indefinite right line BC and from any point, as /?, lay oflf a distance BC = 540. At B make the angle CBA — 80° : draw BA and make the distance B A -- 450 1 draw AC\ then will ABC be the required triangle. TRIGONOMETRICALLY. BC + BA z= 540 + 450 = 990 ; and BG — BA = 640 — 450 = 90. ui + e = ISO*' — ^ = 180° — 80° = 100^ and therefore, i(^4- (7)=i(100°) =60* \0 To find \[A—C). As BC -\- BA 990 . ar. comp. . 7.004366 BC—BA 90 . . . . 1.954243 tan ^{A + (7) 60° . . . . 10.076187 ^ tan ^(A—C) 6° 11' . . . 9.034795 Ifence, 50° + 6° 11' = 56° 11' = A- and 60° — 6° 11' = 43° 49' = C. To find the third side AC, As sin C 43° 49' . ar. comp. . . 0.159672 : sin B 80° ... . . 9.993361 :: AB 450 .... . 2.653213 AC 640.C82 .... . 2.80623C 2. Given two sides of a plane triangle, 1686 and 960, and iheir included angle 128° 04' : required the other parts. Ans. Angles, 33° 34' 39"; 18° 21' 21"; side 2400. TRiGONOMEl RY. 199 Applications. CASE IV. Having given the three sides of a plane triangle, to find die angles. Let tali a perpendicular from the angle opposite the greater tide, dividing the given triangle into two right-angled triangles : then find the diflerence of the segments of the bfise by Theo- rem iii. Half this diflerence 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,-4(7 = 34 and AB = 26 : required the angles. B GEOMETRICALLY, With the three given lines as sides construct a triangle as in Bk. 11. 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 Tliat is, 40 : 59 : : 9 : ^^ ^ ^ = 13.27c 40 40 + 13.275 Then, And = 26.6375 40 CD 40 — 13.275 13.3625 = BD. 200 TRIGONOMETRY. ApplicationB. In the triangle J) AC, to find the angle DAC, Ab AC 34 . . ar. comp. , 8.468521 DC 26.6375 .... 1.42549S sin i> 90° 10.00000 sin DAC 51'> 34' 40" . . . 9.8940 N In the triangle BAD, to find the angle BAD. As AB 25 ar. comp. . 8.602060 BD 13.3625 . . . 1.125887 sin D 90° ... . 10.000000 sin BAD 32° 18' 35" . . . 9.727947 Hence 00° — i>^(7 = 90° — 51° 34' 40" = S8° 25' 20" = C and 90° — BAD = 90° — 32° 18' 35" = 57° 41' 25" = B and BAD + DAC= 51° 34' 40" + 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 OF 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 sum of the squares of the two othei 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. Afl BC 250 . ar comp. 7.602060 : AC 240 ... 2.380211 : : sin ^ 90° ... 10.000000 sin B 73° 44' 23" . 9.982271 Bttt C = 90° — i? = 90° — 73° 44' 23" = 16° 15' 37" : Or C may be found from the proportion. Afl CB 250 ar. comp. . 7.602060 AC 240 ... . 2.380211 B 10.000000 008 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 ... . 2.380211 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 ; BC = 479.979 ; C = 36° 62'. 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 sid*'. and the other oblique, but lying directly above or below the first 202 TRIGONOMETRY Applications. APFLICATION TO HEIGHTS AND DISTANCKS. PROBLEM I. To determine the horizontal distance to a point which is inar. cessible by reason of an intervening river. Let ^ be the point. Measure along the bank of the river ^ hori- zontal base line AB^ ana select the fttations A and -5, in such a manner that each can be seen from the other, and the point C from both of them. ^^S Then measure the horizontal angles a CAB and CBA^ with an instrument adapted to that purpose. Let us suppose that we have found AB — 600 yarda, CAB = 51° 35' and CBA = 64° 51'. The angle C = 180° — (A + B) = 57° 34\ To find the distance BC, As sin C 51° 34' ar. comp. 0.073649 : sin A 57° 35' ... . 9.926431 • • AB 600 .... 2.778151 • BC 600.11 yards. To find the distance AC, 2.778231 Ab sin 57° 34' ar. comp. 0.073649 sin B 64° 51' ... 9.956744 : : AB 600 . ; . 2.778151 AG 643.94 yards. . 2.808544 TBJGONOMETRY. 203 ApplioatiouB. PROBLEM II. To determine the altitude of an inaccessible object above c given horizontal plane. FIRST METHOD Suppose 2) to be the inaccessible j) object, and BC the horizontal plane "'''"'' Js\ from which the altitude is to be r» --'"i^ ^-*^^^^'/^ estimated: then, if we suppose DC V /' / j j; to be a vertical line, it will repre- \ 'vC'-'' sent the required distance. X^''"' Measure any horizontal base line, as BA\ and at the ex- tremities B and A, measure the horizontal angles CBA and CAB. Measure also, the angle of elevation DBC. Then in the triangle CBA there will be known, two angles and the side AB\ the side BC can therefore be determined. Having found BC^ we shall have, in the right-angled triangle DBC^ the base BC and the angle at the base, to find the per- pendicular 7) (7, which measures the altitude of the point D above the horizontal plane BC. Let us suppose that we have found BA — YSO yards, the horizontal angle CBA =41° 24', the horizontal angle CAB = 96° 28', and the angle of eleva- tion J)BC^\0° 43'. In the triangle BCA^ to find the horizontal distance BC, The angle ^C^ = ] 80° — (41° 24' + 96° 28') = 42° 08'= ( As sin C . 42° 08' ar. comp. . 0.173369 : sin ^ . 96° 28' . . . . 9.997228 : : ^j5 . 780 .... 2.892095 : BC . 1165.29 .... 8.062692 ^04 TRIGONOMETRY, Applioations. In the right-aDgled triangle DBO^ to find DC. As R ar. comp. . , 0.000000 tan DBC 10° 43' ... 9.277043 BG 1155.29 . . . 3.062692 DC 218.64 . . . 2.339735 Remark I. It might, at first, appear that the solution which we have given, requires that the points B and A should be in the same horizontal plane ; but it is entirely independent of such a supposition. For, 'the horizontal distance, which is represented by BA^ is the same, whether the station A is on the same level with B^ above it, or below it. The horizontal angles CAB and CBA are also the same, so long as the point C is in the verti- cal line D C. Therefore, if the horizontal line through A should cut the vertical line D C, at any point as E^ above or below C, AB would still be the horizontal distance between B and A, and AE which is equal to AC^ would be the horizontal dis- tance between A and C. If at A, we measure the angle of elevation of tie point D wo shall know in the right-angled triangle DAE, the base AE, and the angle at the base ; from which the perpendicular D^ can be determined. TRIGONOMETRY. 208 Application B. Let us suppose that we bad measured the angle of elevation DAE, and found it equal to 20° 15'. First: In the triangle BAC, to find AC or its equal AE, As sin C 42° 08' ar. comp. . 0.1733G9 : sin B 41° 24' ... 9.820406 :: AB 780 ... 2.892095 AC 768.9 . . . 2.885870 In the right-angled triangle DAE, to find DE. A3 R ar. comp. . . 0.000000 tan A 20° 15' ... 9.5G6932 AE 768.9 . . . 2.88587 DE 283.66 . . . 2 .452802 Now, since DC is less than DE, it follows that the station B is above the station A, That is, DE - DC= 283.66 — 218.64 = 65.02 = EC, which expresses the vertical distance that the station B is above the station A. Remark II. It should be remeqjbered, that the vertical dia tance which is obtained by the calculation, is estimated from a horizontal line passing through the eye at the time of ob- Bcrvation. Hence, the height of the instrument is to be added, in order to obtain the true result. SECOND METHOD. When the nature of the ground will admit of it, measure a base line AB in the direction of the object D. Then mea sure with the instrument the angles of elevation at A and B. Then, smce the outward angle DBC is equal to the sum 18 206 TRIGONOMETRY. Applications. of the angles A and ABB, it follows, that the angle ADB is equal to the diflference of the angles of ele- /^ — ^c vation at A and B» Hence, we can find all the parts of the triangle ADB. Having found DB, and knowing the angle DBC, we can find the altitude DC, This method supposes that the stations A and B are on the same horizontal plane ; and therefore can only be used when the line AB is nearly horizontal. Let us suppose that we have measured the base line, and the two angles of elevation, and !AB = 9T5 yards, A = 15° 36', DBC= 27° 29'; required the altitude DC. First: ADB = DBC - A = 27° 29' - 15° 36' = 11° 58' In the triangle ADB, to find BD. As sin D 11° 53' ar. comp. . 0.686302 sin^ 15° 36' . . . 9.429623 AB 975 ... . 2.989005 DB 1273.3 . . . 3.104930 In the triangle DBC, to find DC. AS B ar. comp. . 0.000000 sin B 27° 29' ... 9.664163 DB 1273.3 . . . 3.104930 DC 587.61 . . . 2.76909a TRIGONOMETRY. 207 Applicationn. PROBLEM III. To determine the perpendicular distance of an object below a given horizontal plane. Suppose C to be directly over , ^^^ tlie given object, and A the point P^^^^ yX \ through which the horizontal plane is supposed to pass. Measure a horizontal base line ^^/^ AB, and at the stations A and B ^^ conceive the two horizontal liues '""^ AC, BC^ to be drawn. The oblique lines from A and B to the object will be the hypothenusca 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 the}' 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 tho angles of depression C'AC, C'BC Let us suppose that we have AB = 672 yards BAC --= Y2° 29' found I ABC = 39° 20' C'AC =2r 49' , C'BC = 19° 10' First: In the triangle ABC, the horizontal angle ACB =a 180« - (AA- B) = 180° - 111° 49' = 68° 11'. 208 TR3GONOMETR7 Applications. Ab sin C 68° 11' . sin A 72° 29' j: ab 672 BO 690.28 To find the horizontal distance AC. As sin 68° 11' ar. comp. , 0.032276 : Bin B 39° 20' ... 9.801973 J : AB 672 ... 2.827369 AO 458.79 . . . 2.661617 To find the horizontal distance BC, ar. comp. , . 0.032275 . 9.979380 . 2.8273C9 . 2.839024 In the triangle CAC, to find C(7', Ab B . ar. comp. . . 0.000000 tan C'AC 27° 49' . . . . 9.722315 AO 458.79 . . ; . 2.661617 CC 242.06 .... 2.383932 In the triangle C7^(7", to find 00" As i? . ar. comp. . . 0.000000 ; tan 0"B0 19° 10' . . . 9.541061 .: BO 690.28 .... 2.839024 : 00" 239.93 . . 2.380083 TRIGONOMETRY 209 Applications. Hence also, CC - CC" = 242.06 - 239.93 = 2.13 yards, which is the height of the station A above station £. 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 26° 30' : required the distance between them. Ans. 173.656 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 were measured, and also the included angle ACBi it was found that CB = 672 yards (7i4 = 688 yards ACB = 55° 40'; required the distance AB. Ans. 592.967 yards. 3. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, 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 tLe height of the towei. Ans. 83.998 feet. 18* 210 TRIGONOMETRY. Applications. 4. Wanting to know the horizon- tal distance between two inaccessi- ^-Q*^ ble objects E and TT, the following caeasnreraents were made, f AB — 636 yards BAW = 40° 16' Mz. ^ WAE = 5V° 40' ABE = 42° 22' EBW= 71° 07' required the distance EW. A71S. 939.634 yards. 5. Wanting to know the liorizontal distance between two inaccessible objects A and B, ' and not finding any station from which both of them could he 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, equal to 200 yards, and from J) a distance DE equal to 200 yards, and the follow- ing anglef taken, {AFC = 83° 00' BBE = 54° 30' ACD = 63° 30' BDC = 156° 25' ACF= 64° 31' BED = 88° 30' Ans. AB = 345.467 yards. APPLICATIONS OF GEOMETRY. MENSURATION OF SURFACES. DEFINITIONS. I 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, (fee, is called the measuring unit ; 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. Sur\'eyors, 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 chaino and the otlier 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 6 square rods ; and therefore, an acre, which is 10 square chains, contains 160 square rods, and a rood contains 40 square rods. Tlie square rods are called perches. 212 APPLICATIONS Mensuration of Surfaces. 5. Land is generally computed in acres, roods, and perchoa which are respectively designated by the letters A, R, P. When the linear dimensions of a survey are chains oi liake the area will be expressed in square chains or square links, and it is necessary to form a rule for reducing this area tc acres, roods, and perches. For this purpose, let }:s form llie following TABLE. 1 square chain =: 1 00 x 1 00 = 1 0000 square links. I acre = 10 square chains = 100000 square links 1 acrer=4 roods=zl60 perches. 1 square mile = 6400 square chains = 640 acres. 6. Now, when the linear dimensions are links, the 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 th*' right hand. If the decimal part be then multiplied by 4, and five places of decimals pointed ofi* 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, (he 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. Wlien both dimensions are in chains, the ])rodnct is re- OF GE O M E TR Y. 213 Mensuration of Surfaces. luccd to acres by dividing by 1 0, or pointing off one dcciraaJ place. From which we conclude : that, I. [f links be mulliplied by links, the product is reduced to ayres by pointing off Jive 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 equal to 1 6.5 X 1 6.5 =272.25 square feet. If the last number be multiplied by 160, we shall have 272.25x160=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 4840 = the number of square yards in an acre PROBLEM I. To find the area of a square, a rectangle, a rhombus, or a paralleh)gram. RULE. Multiply the base by the perpendicular height and the produd 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 214 APPLICATIONS Mensuration of Surfaces. We multiply two sides of the square together, and the product is the area in square feet. Operation. 36x36 = 1290 sq. ft. 2. How many acres, roods, and perches, in a square whose side is 35.25 chains? Ans. 124 A. \ R. \ P 3. Wliat 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. 12 A. R. ^6 P. 5. Whit is the area of a square whose side is 4769 yarde ! 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 ? D A~^ We multiply the base 64, by the perpendicular height 36, and the product is the re- quired area. Operation. 6ix36=2304 sq.ft. 7. What is the area of a parallelogram whose base is If^.gfi 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. b A. \ R OP. 9. What is the area of a parallelogram whose base is 7 ''oot 9 inclies, and altitude 3 feet 6 inches ? Ans. 27 sq.ft. 1' 6' OF GEOMETRY 216 Mensuration of Surfaceo 10. To find the area of a rectangle A BCD, of which the base AB=45 yards, and the altitude AD=:\5 yards. Here we simply multiply the base by the altitude, and the product is the area. B Operation 45xl5rzG75 sq. yds. 11. What is the area of a rectangle whose base is 14 feel 6 inches, and breadth 4 feet 9 inches ? Ans. G8 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. \ 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, wheii the base and altitude are known. RULE. I. Multiply the base by the altitude^ and half the product mil be the area. II. Multiply the base by half the altitude and the product utill he the area (Bk. IV. Th. ix). EXAMPLES. I Required the area of the triangle ABC, whose base yl5 is 10,75 foet, anJ altitude 7,25 feel. 15 216 APPLICATIONS Mensuration of Surfaces. We first multiply the base by the altitude, and then di- ♦ ide the product by 2. Operation. 10,75x7,25 = 77,9375 and 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 aUitude 8.5 chains? Ans. b A. OR. 33 P. 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 ahitude 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 III. To find the area of a triangle when the three sides are known. RULE, }. Add the three sides together and take half their sum, I I . 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 he the required area. OF G E O M E r R Y. 217 Mensuration of Surfacea. EXAMPLES. 1. Find the area of a triangle whose sides are 20, 30, and 10 rods. 20 30 45 45 45 20 30 40 40 2)<)0 45 lialf sum 25 1^^ rem. 15 2d rem. 5 3d rem Then, to obtain the product, we have 45x25x15x5 = 84375; from which we find area= -/84375 =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. 66j. 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 ia 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. A. OR. 38 P. 6. How many square yards in a triangle whose sides are 13, 14, and 15 feet. Ans. 91. 7 What is the area of an equilateral triangle whose side is 25 feet ? Ans. 270.6329 sq. ft. 8. What is the area of a triangle whoso sides are 24, 3G, and 48 yards? Ans 418.282 sq. yds. 218 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. IT. Add the squares together. III. Extract the square root of the sum, which will be the hy- 'pothcnuse of the triangle (Bk. IV. Th. xii). EXAMPLES. 1. In the right angled triangle ABC, we have, ^5 = 30 feet, BC—AQ feet, to find^lC. We first square each side, and then take the sum, of which we extract the square root, which gives A Operation. 30^= 900 40^=: 1600 ^C=-v/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? Ans. 241.86 ft. 3. The side roofs of a house of which the caves 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 ? Ans. 1 7.204 ft. 4. WhU 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 o( a right angled tri- angle are knou n, to find the other side 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 -4 jBC, there are given AC = 50 feel, and ^5 = 40 feet, required the side BC. We first square the hypoth- enuse and the other side, after wliich we take the difierence, and then extract the square root, which gives Z?C=-/900=30 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 pendiciilar 45 yards : what is the base ? Ans. 28 yds. 4 \ ladder 60 feel in len^h, will reach to a windoAv 40 Difl'-^r 900 2'20 A P P L I C A T I O IN S Mensuration of Surfaces feet from the ground on one side of the street, and by tnniiug it over to the other side, it will reach a window 50 feet from the ground : required the breadth of the street. Ans. 77.8875 fi. PROBLEM VI. To find the area of a trapezoid. RULE. Multiply the sum of the parallel sides hy the perpendicular distance between them, and then divide the product by two : the quotient will be the area (Bk. IV. Th. x). EXAMPLES. 1 . Required the area of the trapezoid ABCD, having given AJ5=321.51 feet, DC=2U.24 f^et, and CE=zl7lA6 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. 321.514-214.24=535.75- sum of parallel sides. Then, 535.75x171.16 = 91698.97 and, ?i^?!:?Z = 45849.485 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.l R. 9.956 P. 3 Required the area of a trapezoid whose parallel sides O F G E O M E T R Y . 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 ? Ans. 15 A. R. 33.2 P PROBLEM VII. To find the area of a quadrilateral. RULE. 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 tht area of the quadrilateral. EXAMPLES. 1. Suppose that we have meas- ured the sides and diagonal A C, of the quadrilateral ABCD, and found AT ^5=40.05 chains; CD =29.87 chains, BC =26.27 chains ^D = 37.07 chains, and ^C= 55 chains: required the area of the quadrilateral ^^^ Ans. 10\ A. 1 R 16 P Ii22 APPLICATIONS Mensi. ration of Surfaces. Remark. — Instead of measuring tlie four sides of the quadrilateral, we may let fall the perpendicu- lirs Bbj Dgj on the diagonal AC. The area of the triangles may then be determined by measuring these perpendiculars and diagonal AC. The pendiculars are^Dg — 18.95 chains, and Bb =17.92 chains. 2. Required the area of a quadrilateral whose diagonal is B0.5, and two perpendiculars 24.5, and 30.1 feet. Ans. 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 33i feet ? Ans. 222^2 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. RULE. Multiply half the perimeter of the figure by the perpendicular Let fall from the centre on one of the sides, and the product icill he the area, (Bk. IV. Th. xxvi) OP GEOMETRY 223 Mensuration of Surfaces EXAMPLES. 1. Required the area of the regular pentagon ABiWE, each of whose sides AB, EC, &c., is 25 feet, and the perpendicuhir 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. Operation. ?5^=62.5=half the penny. eter. Then, 62.5x17.2 = 1075 ^^. /^=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. Ans. 688.191 sq. yds. 3. The side of a regular hexagon is 14, and the perpen- licular from the contre on one of the sides 12.1243556: re- ]uired 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 whoae side is 19,38 arid perpendicular 20 feet. Ans. 1 356.6 sq. ft. The following table shows the areas of the ten regular 224 APPLICATIONS 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 sides. Names. Areas. Radius of inscnbcd 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 1 : tabular area : : any side squared : area. Hence, to find the area of a regular polygon, we have the following I Square the side of the polygon. n. Multiply the square so founds by the tahular area set oppo- site the polygc^ 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. OF GEOMETRY 225 Mensuration of Surfaces. 2. What is the area of a pentagon whose side is 25 ? Arts. 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, boiuided 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 tlie intermediate breadths^ and half the svm of the extreme ones III. Multiply this sum by the base line, and divide the produd bif th^ number of equal parts of the base. EXAMPLES. 1 . The breadths of an irregu- lar figure, at five equidistant % places, A, B, C, P, and E, be- ing 8.20 chains. 7.40 chainn. 2'2(i 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 )lK8Q 4)1408.00 8,40 mean of the extremes. 352.00 square chains. 7.40 9.20 10.20 35.20 thd 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 breaoths 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 J : what is the area? Ans. 219.255. PROBLEM X. To find the circumference of a circle when the diameter is known. OFGEOMETRY. 221 Mensuration of Surfaces. RULE Multiply the diameter by 3.1416, and the product will te tht circumference. EXAMPLES. 1. What is the circumference of a circle whoso diamelei is 17? Wc simply muhiply the immbcr 3.1 41 G by the diam- eter anl the product is the circumference Operation. 3.1416x17 = 53.4072, which is the circumference. 2. What is the circumference of a circle whose diameter \& 40 feet? Ans. 125.664/11. 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 knowa RULE. Dimde the circumference by the number 3.1416 and the qxxo^ tient tmll be the diameter. EXAMPLES. 1. The circumference of a circle is 69.1152 yards: whai is the diameter * 228 APPLICATIONS Mensuration of Surfaces. We simply divide the cir- cumference by 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 la 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? ^n;?. 15.915. 5. If the circumference of a circle is 25000.8528, what is tlie diameter ? Ans. 7958. PROBLEM XII. To find the length of a circular arc, when the number ol 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 given decimal by the number of degrees, anl by the radius. Operation. ,01745x30x9 = 4.7115, 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 thera by 60. OF GEOMETRY 229 Mensuration of Surfaces. 2. What is the length of an arc containing 12° IC oi 12^," the diameter of the circle being 20 yards ? Ans. 2.1231 3 What is the length of an arc of 10° 15' or IQio^ in a circle -whose diameter is 68 ? Ans. G.0813. PROBLEM XIII. To find the length of the arc of a circle when the chord and radius are given. RULE. 1. Find the chord of half the arc. n From eight times the chord of half the arCj subtract the chord of the whole arCj and divide the remainder by 3, and the quotient will be the length of the arc^ nearly. EXAMPLES. 1. The chord AB=20 feet, and the radius i4C=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 ^P= 15 feet. Hence, A&-AP=:CP'. that is, 400-225 = 175 and ^/\^E=U:Z2S=CP CD-CP=20-\3.228=6.772=DP. Then Again, hence> Then, 20 AD=V-JF-KRD^=V'225+45.859984 AD 1=1 6.4578= chord of the half arc. 16.4578x8 — 30 ^ ^=33.8874 = arc ADB. 230 A P 1' L 1 C A T I O N S jMeusuration of Surfaces 2. What is the length of an arc the chord of wliicli is 21 feet, and the radius of the circle 20 feet ? Ans. 25.7309 Jt. 3. The cliord of an arc is 16 and the diameter of the ciiclo 20 : what is the length of the arc ? A?is. 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 ? Ans. 55 5. PROBLEM XlV. To find tlie 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 Operation. [6x2i= which is the area. 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 SJ 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 78.')4 OF GEOMETRY. 231 Mensuration o f Snrf ac e s. 6. 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, anJ 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 deamal .7854 EXAMPLES. What is the area of a circle whose diameter is 5 ? We square the diameter, Tvhich gives us 25, and we then multiply this number and the decimal .7854 to- gether. Operation. .7854 5^= 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 iircle whoso diameter is 1| yards ? Ans. 1.069016. 5. What is tlie 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. 23? APPLICATIONS Mensuration of Surfaces. RULE. Multiply the square of the circumference by the decimal .07968, and the product will be the area very nearly EXAMPLES. 1. What is the area of a circle whose circumference ie 3.1416? Operation. We first square the cir- cumference, and then multi- ply by the decimal .07958. 3.1416 =9,86965056 ,07958 area =.7854 + 2. What is the area of a circle whose circumference is 91! Ans. 659.00198. 3. Suppose a wheel turns twice in tracking 16^ 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 P. 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 circmn ference is 7 miles ? Ans. 399300.608. PROBLEM XVII. Having given a circle, to find a square which shall have an equal area. RULE. I. Th6 diameter X .S862= side of an equivalent square II. The circumference x .2821= side of an equivalent square OF GEOMETRY. 233 Mensuration of Surfaceg. EXAMPLES. 1. The diameter of a circle is 100: what is the side of a square of equal area ? Ans. 88. G2. 2. The diameter of a circular fishpond is 20 feet, wlial 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 ? Ans. 56.42. 5. The circumference of a round fishpond is 400 yards : what is the side of a square pond of equal area ? 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 =side of the inscribed square. II. The circumference X .2251 z^side of the inscribed square, 20* 234 APPLICATIONS M ens ti ration of Surfaces. EXAMPLES. 1. Till) diameter AB of a circle is 400 : what is the value of A C, the side of the inscribed square ? ^1 Here, .7071 X 400:^282.8400=^0. 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 oi the inscribed square ? Ans 424.26. 4. The circumference of a circle is 312 feet: what is the side of the inscribed square 1 Ans. 70.2312 ft. 5. The circumference of a circle is 819 yards : what is the side of the inscribed square ? Ans. 184.3569 ijds. 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 the side of an inscribed square ? Ans. 140.6875. TROBLEM XIX To find the area of a circular sector RULE. I . Find the length of the arc by Problem XllX II, Multiply the arc by one half the radius ^ and the product will be the area O F G E O M E T R Y . 235 MensuralioD of Surfaces. EXAMPLES. 1. What is tho area of the circular sector ACB, the arc AB containing 18°, and the radius CA being equal to 3 feet. i First, .01745 X 18 x3 = .94230=lengih AB. Then, .94230 x ll=1.41345=area 2. What is the area of a sector of a circle in which tho 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. ^«^. 2G5.4143. 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 the 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 ts greater than the semicircle, add the two areas together; but if it is less, subtract them, and the result in cithei case, rmll be the area required. 236 APPLICATIONS Mensuration of Surfaces EXAMPLES. 1 . What is the area of the seg- ment ADB, the chord AB =24 feet and CA =20 feet. First, CP=V^-^^ = V'400-144nzl6 Then, PD=CD-CP=20-ie=4. And, ^D=/AP^+PD*=vT444-16 = 12,64911 : 12,64911x8-24 then, arc ADB: Arc ADB=25,7309 half radius = 10 area sector ^Z)5C=257,3090 area C^5=192 :25,7309. ^P=12 CP=16 area C^j5 = 192 65,309= area of segment ADB 2. Find the area of the segment AFB, knowing the following lines, viz: ^5=20.5; PP= 17.17; AF =20; FG=11.5; and C^ = 11.64. A Ar^v FGxS-AF 11.5x8-20 Arc AGF= = =24: and sector ^GF5C=24x 11.64=279.36 : but CP=PP— ^0=17.17-11.64 = 5.53: tI^xGP 20.5x5.53 Then, area7lC5=: 56.6825 O F G E O M E T R Y . 23'? Mensuration of Surfaces. Then, area of sector AFBC=279.3e do. of triangle ABC= 56,6825 gives area of segment AFB=336.0i25 3 What is the area of a segment; the radius of the circl being 10 and the chord of the arc 12 yards ? Arts. 16.324 sq. yds, 4. Required the area of the segment of a circle whose chord is 1 6, and the diameter ol 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 seo-ment 60 : what is the area of the segment ? Ans. 408, nearly. PROBLEM XXI. To find the area of an ellipse. Multiply the two axes trgethcr, 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=ilO feet, and the conjugate axis DE =50 feet. ABxDE=70x50=3500: Then, .7854 x 3500 =2748.9= area. 2. Required the area of an ellipse whose axes are 24 and t \ Ans. 339.2928. 238 APPLICATIONS Mensuration of Surfaces, 3. What is the area of an ellipse whose axes are 80 and 60 ? Ans. 3769.92. 4. What is the area of an ellipse whose axes are 50 and 1? Ans. 1767.15. PROBLEM XXII. To find the 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 thtr d'cimai 7854, and the product will be the area. EXAMPLES. 1. In the concentric circles having the common centre C, we have ^5 = 10 yds., and DE = 6 yards : what is the area of the space in- cluded between them ? 5.4*=10*=100 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 are 20 and 10 f Ans. 235.62. OF GEOMETRY. 239 Mensuration of Solids. 3. If the diameters are 20 and 15, what will be the area in- cluded between the circumferences ? Ans 137.445. 4. If the diameters are IG and 10, what will be the area ia- cluded between the circumferences ? Ans. 122.5224. 5 Two diameters are 21.75 and 9.5 ; required the area o( the circular ring. Ans. 300.6600 6. If the two diameters are 4 and 6, what is ihe area of the rin^? Ans. 15.708 MENSURATION OF SOLIDS. DEFINITIONS. The mensuration of solids is divided into two parts. Ist, The mensuration of the surfaces of solids : and 2d, The mensuration of their solidities. We have already seen that the unit of measure for plane surfaces, is a square whose side is the unit of length (Bk. IV Dcf. 7). 2. A curve line which is expressed by numbers is also re- ferred to an unit of length, and its numerical value is the num- ber of times which the line contains the unit. If then, we suppose the linear unit to be reduced to a Btiftight line, and a square constructed on this line, this square will be the unit of measure for curved surfaces. 3. The unit of solidity is a cubo, whose edge is the unit in which the linear dimensions of the solid are expressed ; and 240 APPLICATIONS Mensuration of Solids. the face of this cube is the superficial unit in which the but- fiace of the solid is estimated (Bk. VI. Th. xiii. Sch). 4 The following is a table of solid measure. 1 cubic foot =1728 cubic inches. 1 cubic yard =27 1 cubic rod =4492^ 1 ale gallon =282 1 wine gallon =231 1 bushel cubic feet, cubic feet, cubic inches, cubic inches. :2150,42 cubic inches. PROBLEM 1. To find the surface of a right prism. RULE. Multiply' the perimeter of the base by the altitude and the pro- duct will be the convex surface : and to this add the area of the bases J when the entire surface is required (Bk. VI. Th. i). EXAMPLES 1. Find the entire surface of the rcgidar prism whose base is the reg- ular polygon ABCDE and altitude AFf when each side of the base is 20 feet and the altitude AF, 50 feet. />~ 1) B C A84-5C-hCD+D£4-E^ = 100; and ^i^=50 : then (AB-\-BC->t- CD-\-DE-^EA) x ^F=convex surface OF GEOMETRY. 24i Mensuration of Solida which becomes, 100x50 = 5000 square feet; which is ihe convex surface. For the area of the end, we have AB X tabular number = area -A5Ci)i?, that is, 2?* X tabular number, or 400 x 1.720477 = 688.1908^ the area ABODE. Then, convex surface = 5000 square feet, lowei base 688.1908 square feet, upper base 688.1908 square feet. Entire surface 6376^816 2. What is the surface of a cube, the length of each sme being 20 feet ? Ans. 2400 sq. ft. 3. Find the entire surface of a triangular prism, whose base is an equilateral triangle, having each of its sides equal to 18 inches, and aliiiude 20 feet. Ans. 91.949 sq. ft. 4. What is the convex su'-face of a regular octagonal prism, ^he side of whose base is 15 and altitude 12 feet? Ans. 1440 sq. ft. 5. What must be paid for lining a rectangular cistern w'th lead at 2d a pound, the thickness of the lead being such as to require lib. for each square foot of surface ; the inner dimen- sions of the cistern being as follows : viz. the length 3 feet 2 inches, the breadth 2 feel 8 inches, and the depth 2 feet 6 inches ? Ans. £2 3s lOf^f. PROBLEM II To find the solidity of a prism. RULE. Multiply the area of the base by the perpendicular height, and the product will he the solidity. 242 APPLICATIONS Mensurat'on of Sol'ds BX AMPLK8. 1. What is the solidity of a reg- ular pentagonal prism vvliose altitude IS 20, and each side of the base 15 feet? To find the area of the base we have by Problem VIII, page 178. .>\ 15^^=225: and 225x1.7204774 = 387.107415=. the area of the base : hence, 387,107415 X20 = 7742.1483 = solidity. 2. What is the solid cimtents of a cube whose side is is-l inches ? Ans. 13824 solid in. 3. How many cubic feet in a block of marble, of which tlie length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches ?. Ans. 2H solid ft. 4. How many gallons of water, ale measure, will a cistern contain whose dimensions are the same as in the last ex- ample ? Ans. 1291^ 5. Required the solidity of a triangular prism ^hose alti- tude is 10 feet, and the three sides of its triangular base 3, 4, and 6 feet. Ans. 60 solid ft. 0. What is the solidity of a square prism whoce heiglit ii> 6{ feet, and each side of the base 1^ fcot? Ans 9| soha ft. OF GEOMETRY 243 Mensuration of Solids. 7. What is the sohdity of a prism whose base is an equi- lateral triangle, each sitle of which is 4 feet, the height of the prism being 10 feet? Ans. G9.282 solid ft. 8. Wliat is the number of cubic or solid feel in a resulai penti.gonal prism of which the altitude is 15 feet and each Rifle of the base 3.75 feet ? Ans. 362.913 PROBLEM III. To find the surface of a regular pyramid. RULE. Multiply the perimeter of the base by half the slant lieight^ and the product will be the convex surface : to this add the area of the base, if the entire surface is required ifik. VI. Th \\\ EXAMPLES. 1. In the regular pentagonal pyramid S—ABCDE, the slant height SF is equal to 45, and each side of the base IS 15 feet : required the convex sur- face, and also the entire surface. 15 X5 = 75 = perimeter of the base, 75x22^ = 1087.5 square feet = area of ctmvex surface. And 15^=225: then 225 x 1.7204774 = 387.1074151^ the area of the base. Ilcnce, convex surface =1687.5 area of the base= 387.107415 Entire surface =2074.607415 square feet 'M4 APPLICA.TIONS Mensuration of Solids 2. What is the convex surface of a regular triangular pyra- mid, the slant height being 20 feet, and each side of the base 3 fed ? Ans. 90 sq. ft 3. What is the entire surface of a regular pyramid whose ftiaiit height is 15 feet, and the base a regular pentagon, ol which each side is 25 feel? Ans. 2012.798 sq. ft PROBLEM IV. To find the convex surface of the frustum of a regidai pjTamid. RULE. Multiply half the sum of the perimeters of the two bases by the slant height of the frustum, and the product will be the con' vex surface (Bk. VI. Th. vii). EXAMPLES. 1. In the frustum of the regular pen- tagonal pyramid each side of the lovi^er base is 30, and each side of the upper base is 20 feet, and the slant height fF is equal to 15 feet. What is the convex surface of the frustum ? Ans. 1875 sq. ft. 2. How many square feet are there in the convex surface of the frustum of a square pyramid, whose slant height is 10 feot, each side of the lower base 3 feet 4 inches, and each evif of the upper base 2 feet 2 'nches ' Ans. 1 10. 3. What is the convex surface oi the frustum of a heptago nai pjTamid whose slant height is 55 feet, each side of the lowoi base 8 feet, and each side of the upper base 4 feet ? Ans. 2310 sq. ft. OF GEOMETRY 245 Mensuration ol Solids. PROBLEM V. To find the solidity of a pjTamid. RULE. Multiply the area of the base hy the altitude and divide the pro- duct hy 3, the quotient will he the solidity (Bk. VI. Tli. xvii). FXAMPLES. 1 What is the solidity of a pyramid Uie area of whose base is 215 square feet and the altitude 50=45 feet? First, 215x45 = 9675: then, 9675 H- 3 = 3225 which is the solidity expressed in solid feet. 2. Required the solidity of a square pyramid, each side ol 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 of which is 14 feet 6 inches, and the three sides of its base 5, 6 and 7 feel? 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* 5240 APPLICATIONS Mensuration of Solids, 6 How many solid feet in a regular hexagonal pyramid whose altitude is 6.4 feet, and each side of the base 6 inches ' Ans. L385G4. 7. How many solid feet are contained in a hexagonal p}T:a- mid the height of which is 45 feet, and each side of the base 10 feet? Ans. 3897.1143. 8. The spire of a church is an octagonal pyramid, each side 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 1 1 inches, and its perpendicular height 41 feet: how many yards of stone does the spire contain' Ans. 32.197353 PROBLEM VI. To hnd the solidity of the frustum of a pyramid. RULE. Add together the areas of the two bases of the frustum and a geometrical mean vroportional 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 feet, the altitude being 1 feet ' O F G E O M E T R Y . 24' Mensuration of Solids First, 16x9=144: then,-v/l'^'* = 12, the mean Then, area of lower base =16 area of upper base = 9 mean of bases = 12 height 7 3) 259 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 inches, 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 3U 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 inchcfl, and each side of the smaller 12 inches ? Ans. 28.5 aibte ft. 7. What is the solidity of the frustum of a regular hexago- tial 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 248 APPLICATIONS Mensuration of Soli do MEASURES OF THE THREE ROUND BODIES. PROBLEM I To find the surface of a cylinder. RULE. 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. iiV EXAMPLES. 1. What is the entire surface of the cyhnder in which AB, the diameter of the base, is 12 feet, and the aUitude 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= 1130.9760=convex surface. Also, 12^=144: and 144 x .7854 = 113.0976 = area base. Then. convex surface =1130.9760 lower base 113.0976 upper base 113.0976 Entire area =1357.1712 of the 2. What is the convex curface of a cylinder, the diameter of whose base is 20, and the altitude 50 feet ? Ans 3141.6 sq.ft. OF GEOMETRY 249 Mensuration of the Round Bodies. 3. Required the entire surface of a cylinder, whose altitude is 20 feet and the diameter of the base 2 feet. Ans. 131.9472 /^ 4. What is the convex surface of a cylinder, the diameter of whose base is 30 inches, and altitude 5 feet? Ans. 5654.88 sq. in. 5. Required the convex surface of a cylinder, whose aJti- tude is 14 feet, and the circumference of the base 8 feet i inches. Ans. 116.6666, 1 3 1 4 5 6 1 7 8 9 /D. 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8621! 8783 ••l5\ 8236' '398 9106 9268 9429 959, 162 269 7752 9914 •559 •720 •881' 1042 1203 i6i 270 43 .364 1525 i685| 1846' 2007 2167 23j8 2488; 2649 2809 161 271 3069 3i3o 32901 345o 36io 48881' 5o48 5207 3770 i^\ S?' i^ 4409 160 :72 4729 5367 6004 09 273 616? 6322 64811 6640' 6798 8067 8226' 8384 6957 7116 72-5; 7433 7592 ;S Vi 7751 7909 8542 8701 8859' 9017 9n5 275 9333 9491 9648 9806 9964 •i22 1224' r3Si, i538 1695 •279 •437, '594 •752 1 58 276 440909 1066 1 852 3009! 2166 2323 1 57 277 2480 2637 2793 2950: 3io6 3263 4357 431 3 4669 4825 34iq! 3576; 3732 3889 'il 278 4045 4ioi 4981 5i37; 5293 7003 1 56 J79_ N. 5604 5760 5915 6071 6226 6382 6537! 6O92 6848 1 55 1^ « 1 _3j _.iJ H n 7_|_8 |_? D. A TADLK OF 1 .OOAKirilMS FROM 1 TO 10,OC.O. 1 n ~^1 3 1 4 5 16 7 7q33 8o38 8242 8 8397 9 8552 "d." 447158 8706 ls6i 7468 7623 777B 1 55 aSi 9015 9J70I 9324i 9478 96331 9787 9941 16^3 1 54 282 450249 04:3 0557 0711, o865| 1018 1172 i326 •479 1 54 283 1786 1 940 2093 2247 2400 .2553 2706' 2839 3012; 3i65 i53 284 33i8 347. 3624 3777, 393c 4082 4235| 4387 4540, 4692 1 53 285 4H45 4997 5no 5302' 5454 56o6 575s' 5910 6062 6214 l52 286 6366 65iS 6670 6S21 6973 7125 7276! 7428 8336 8487' B633 87S9 8940 7579 773. l52 III 7882 80,33 81S4 9091 9242 i5i 9392 9543 9694 9S45 9995^ •146 •296 '447 •597 •748 i5i 389 460898 1048 1198 i348 1499 1649 1799' 1948 2098 224S i5o 190 462398 2548 26971 2847 2997I 3i46 3296 3415 l^i 3744 i5o 2gi 53?3 4042 '& 4340 4490 4639 4788^ 4936 5234 149 29: 5532 5829 5977 6126 6274 6423 6571 6719 'a 293 6868 7016 uti 73121 7460 7608 7756 7904 8790 8938 9085 9233 9380 8o52 8200 294 8347 8495 9960 1438 9527 •998 9675 148 295 9822 •116 •263: •410 •557' •704 •85 1 1 145 147 296 '''.]ll 1 585 1732' 1878 2025 2171 23l8 2464 2610 146 !?J 2903 4362 3o4o 45o8 3io5^ 33411 3487 3633 3779 4653 4799 4944 5090 5235 3925 538i 4071 146 4216 5526 146 299 5671 58i6 5962 6107 6232; 6397 6542 6687 -»55d; 7700I 7844, 79^9' 81 33 8999' 9143, 9287 94311 9575 0433, o582 0725 0869 1012 6832 6976 145 3oo 477»2i 8566 7266 74..' 8278 8422 145 3oi 87.1 8855 97'9 9863 144 302 480007 1443 oi5i 0294 1156 1299 144 3o3 1 586 1729 1872' 2016 2159' 23o2 2445 2588 273. 143 3o4 2874 3oi6 3.59 3302! 3445, 3587 3730 3872 4oi5 4157 143 3o5 43oo 444'2 4585 4727; 4869 5oiil 5t53 5295 5435? 5579' 142 3o6 5721 5863 6oo5, 6147 6289 643o 6572 6714 6855 6997, >42 12 S551 7280 7421 7563 7704 7845; 79S6 8127 8269 8410 141 8692 8833 8974 91 14' 9255 9396 9537 9677 98.8 141 309 9958 ••99 •239 •38o •J2o| •66 r •8oii •g'.i 1081 1222 140 3io 491362 l502 1642 17S2: 1922 2062 2201: 234J 3119' 3319I 3458. 3507 3737 4572; 47" 485o! 49*59 5128 5960; 6099! 6238; 6376' 65 1 5 7344I 7483, 7621 7759' 7897 8724! 8862 8999' 9137! 9275 2481 2621 140 3ii 2760 2900 3o4o| 3876 401 5 139 3l2 4i55 4294 4433; 5267 5406 139 3i3 5544 5683 5822! 6653 6791 \^ 3i4 tt. 7068 7206' 8o35 8173 3i5 8448 8586| 9412 9550 r38 3i6 9687 9824 9962 ••99j ^236 •374; •5ii! ^648 •785 •922 .37 3'7 3id 5oio59 1 196 1 333 1470 1607 1744 1880 2017 2i54 2291 3655 137 2427 2564 2700 2837 2073 4199 4J35 3109I 3246 3382 35i8 1 36 3.9 3791 3927 4o63, 447 1 1 4607 4743 4878 5oi4 1 36 320 5o5iDo 5286 542 1 1 55D71 5693 5828 5964 6099 71B1 73i6 745i 6234 6370 1 36 3ai 65o5 6640 6776, 6911 7046 7586 7721 i35 322 7856 ^11; 8126 8260 8395, 853o! 8664 8799 8934 9068 i35 323 9^03 9471 9606 9740 9874' •••9 ^143 •277 •411 1 34 324 5io545 0679 o3i3i 0947 1081 I2i5] i349 1482 1616 1750 1 34 325 1 883 2017 2l5li 2284' 2418 255i| 2684 281S 2951 3o84 1 33 3:5 3218 335i 3484 36171 3750 3883 4016 4U9 4282 4414 i33 327 328 4548 4681 4813 4946 5079 5211 5344 5476 5609 3741 i33 5874 6006 6.39' 6271 64o3 6535 6668 6800 6932 7064 l32 319 7196 7328 7460, 7V' 7724 8909 9040 7855] 79871 8119 825i 8382 l32 33o 5i85i4 8646 8777 9171 93o3. 9434 9566 9697 i3i 33i «9828 9959 ••90' •221 •353 •48' •6i5| -745 •876 1007 i3i 332 5jii38 ;^;'? 1400 i53o 1661 1792 1022 2d53 2i83 23i4 i3i m 2444 2705; 2835 2966 3096I 32261 3356 3486 36i6 i3o 334 3746 3876 4006 4 1 36 4266 4396 4526! 4656 4785 4qi5, i3o 335 5o45 5i74 5304* 5434 5563 5693 6985 5822: 5951 6081 D-^^io !2g 336 6339 64^9 6=iQs; 6727; 6856 7114 7243 8660 75oi! 129 8788 120 337 fi 7739 7SS3: 8016 8145 8274 8402' 853 1 338 9045 9' 74; 9302 943o 9559 96871 9815 9943 ••72 ia8 i3i)il 128 339 530200 o328 0456: o584 0712I 0840; 0968 1096 1223 8 N. 1 I > 1 3 i 4 i 5 1 6^ 7 9 D. A TABLE OF LOGARITHMS FfloM 1 TO 10,000. K. 1 2 3 4 5 6 7 ! 8 9 ur 340 531479 1607 1734 1862 1990 2117 2245 2372: 25oo 2627 128 341 2754 2882 3009 3i36 3264 3391 35i8 3645 3772 3899 127 342 4026 4i53 4280 4407 4534 4661 4-87 4914 5o4i 5167 127 343 5294 6558 5421 5547 5674 58oo 5927 6o53 6180 63o6 643a 126 344 6685 6811 6937 7063 n 73i5 7441, 7567 & 126 345 7819 7945 807. 8197 8322 8574 8699' 8825 126 346 9076 9202 9327 9432 9578 9703 9829 9934 ••79 i33o •2,14 125 347 540329 0455 o58o 0705 o83o 0955 1080 i2o5: 14^^^ 125 34« 1579 1704 1829 1953 20l8 2203 2327 2452 2576 27.1 125 349 282! 2950 3074 3199 3323 3447 3571 4936; 3820 3944 124 O'JO 544068 4192 43 16 4440 4564 4688 4812 5o6o 5i33 124 35 1 5307 5431 5555 5678 58o2 5925 6049 6172: 6296 6419 124 35; 6543 6666 6789 6913 7o36 7.59 7282 74o5, 8635 m 7652 123 353 7775 7898 8021 8144 8267 8389 85i2 8881 123 354 9003 9126 9249 9371 9494 9616 9739 9861 i 9984 •106 123 355 550228 o35i 0473 0595 0717 0840 0962 1084 1206 1328 122 356 i45o 1572 1694 1816 1938 2060 2l8l 23o3, 2425 2547 122 iil 2668 2790 2911 3o33 3i55 3276 3398 35i9 3640 3762 121 358 3883 4004 4126 4247 4368 4489 4610 47311 4H52 4973 121 359 5094 52i5 5336 5457 5578 5699 5820 5940' 6061 6.82 121 36o 5563o3 6423 6544 6664 6785 6903 7026 8349' 7267 8469 l?J, 120 361 7507 7627 8829 7748 7868 7988 810S 8228 120 36a 8709 8948 9068 9188 93o8 9428 9548, 9667 9787 120 363 9907 ••26 •146 •265 •385 •5o4 •624 •743 •863 •982 119 364 56II01 1221 i34o 1459 1078 1698 18.7 1936 2o55 2174 119 365 2293 2412 253i 2o3o 2769 2887 3oo6 3i25, 3244 3362 119 366 3481 36oo 3718 3837 3933 4074 4192 43ii 4429 4548 ',\t 367 4666 4784 4903 502I 5i39 5257 5J76 5494! 56i2 5730 368 5848 5966 6084 6202 6320 6437 6555 6673: 6791 6909 118 369 7026 7144 7262 7379 7497 7614 7732 7849' 7967 8084 118 370 568202 83i9 8436 8554 8671 8788 8905 9023 9140 9257 •17 371 9374 9491 9608 9725 9842 9959 ••76 •■9^: •309 •426 •17 372 570543 0660 0776 0893 1010 1126 1243 1339' 1476 1% 117 373 1709 2872 1825 1942 2o58 2174 2291 2407 25231 2639 116 374 2988 3io4 3220 3336 3432 3568 3684 1 38oo 3915 116 375 4o3i 4147 4263 4379 4494 565o 4610 4726 4841 1 4957 5072 116 376 5i8S 53o3 5419 5534 5765 588o 5996; 6111 6226 ii5 377 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 ii5 378 U% 7607 7722 7836 7951 8066 8i8i 8295; 8410 8525 1 15 379 8754 8868 8983 9097 9212 9326 9441' 9555 9669 114 3bo 579784 9898 1039 ••12 •126 •241 i355 •469 •583 •697 •81I 114 38i 580925 ii53 1207 i38i 1493 1608 1722 2§58 1 836 1950 114 382 2063 2,77 2291 2404 25i8 26ii 2-45 2972 3o85 114 383 ^3^ 33i2 3426 3539 3652 3765 3879 3992 4io5 4218 ii3 384 4444 4557 4670 4783 4896 5009 5l22 5235 5348 ii3 385 546! 5574 5686 5799 5912 6024 6137 6250 6362 6475 n3 386 6587 6700 6812 6925 7037 7149 7262 7374^ 8496 7486 8608 7599 1 12 387 m'. 7823 7935 8047 8160 8272 8384 8720 1 12 388 8944 90 56 9167 9279 9391 95o3 9615 9726 9838 1 12 389 9950 ••61 •173 •284 •396 •5o7 •619 •730 •8/.2 •953 1 12 390 591065 1176 1287 1399 i5io 1621 1732 1843 ,955 2066 I n 39» 2177 2288 2399 25lO 2621 2732 2843 2954 3o64 3.75 1 1 1 ]g'i 3286 3397 35o8 36i8 3729 3840 3950 4061 4171 41S2 1 11 l9i 4393 45o3 4614 4724 4834 4945 5o55 5i65 5276 538ff no ^1 5496 56o6 3717 5827 5937 6047 6157 6267 6377 6487 no 3c,5 2695 6707 6817 6927 7037 7146 7256 7366, 8462 7476 7586 no 396 78o5 8900 1% 7914 8024 8i34 8243 8353 8572 8681 no 399 9883 600973 9009 •lOI 1191 9119 •210 12Q9 9228 •319 140S nil l5i7 9446 •537 1625 9556 9665 •646 ^755 1734' 1843 '.Hi 1951 109 109 109 N. 2 o_ 1 » 3 4 ^ ^_ 6 7 ! 8_ 9 D. A TiBLE OF LOGARITHMS FROM ] TO 10,000. It N. 400 • "'~' 3 2386 4 i 6 2494 2603 6 -Tr»- 9 |D. 3o36 108 602060 2160 2277 32d3 336i 2711 2819 2928 401 3i44 346g 3577 3686 3-194 3902 4010 4.18 .08 402 4226 4334 4442 455o 4658 ijbb 4874 4982! 5089 5.97 .08 4o3 53o5 54i3 '5521 5628 5136 5844 5931 681 1 6919 7026 6059 6166 7i33: 724. 6274' «o8 404 638 1 6489 6596 6704 7348, .07 84.9! 37 9488, 107 4o5 7455 7562 7669 8633, 8740 8847 7884 }^954 799 1| 8098 82o5j 83.2 406 8526 906 II 9167 9274, 938. 407 ^ 9594 970 1 j 9808 99' 4 ••21 •128, •234 •34 1 1 •447 •554' 107 .617 106 4o3 610660 0767, 0873 1829 1936 0979 1086 1192 1298 i4o5; i5ii 4f>9 1723 2042 2148 2254 236o 2466; 2572 26781 106 410 612784 2.S90 2996 3947 40D3 3l02 3207 33.3 34.9 4473 35251 3630 3736' .06 411 3842 ii^ 4264 4370 458.! 4686 47921 106 412 4897 5oo3, 5 108 5319 5424 5529 5634 5740 5845! io5 4i3 5960 6od5| 6160 6265 6370 6476 658. 6686 s 6895 Jo5 414 415 7000 8048 7io5 7210 8153! 6257 73.5 8362 7420 8466 7525 857. 8676! 8780 7943 io5 8989 io5 416 9093 9198; 9302 9406 95ii 9615 9719 9H24 9928 ••321 104 ^'1 4i8 620106 0240! o344 0448 o552 o656' 0760 0864 0968 .072 104 1 1 76 1280 1 384 1488 1592 .695 »799 .903 2007 2..0 .04 419 2214 23i8 2421 2525 2628 2732 283d 2939 3973 3o42 3.46 .04 420 623249 3353 3456 3559 3663 3766 3S69 4076 4179 io3 421 4282 4385 4488 4591 4695 479« 4901 5oo4 5.07 6.35 52.0 io3 422 53i2 54i5 55i8 5621 5724 5827 5929 6o32 62381 .o3 1 423 6340 6443 6546 6648 6751 6853 6956 & 7.6. 8.85 7263 io3 424 7366 7468 8491 7571 8593 mi ^797 9817 0835 7878 8900 7980 8287 9308 102 425 8389 9002 9.041 9206 102 426 94io 95i2 9613 971 5 9919 ••2. •.23 •224 •326 .02 ^l 630428 o53o o63i 0733 0936 .038 ;;^? 124. .342 102 1444 1 545 1647 1748 1849 .951 2052 2255 2356 10. 429 2457 2559 2660 2761 2862 2963 3o64 3.65 3266 3367 101 43o 633468 457? 5584 3670 3771 4880 3973 4074 4.75 5.82 4276 4376 .00 43 1 5484 4679 i]it 498. 5o8i 5283 5383 100 432 068D 5886 5986 6087 6.87 6287 6388 100 433 6488 6588 6688 6789 6S89 6989 Co 9088 7189 7290 8290 9287 9387 ICO 434 435 1% 9486 l^ a° 7790 8780 9785 & 7990 89H8 8.90 9.88 99 99 436 9586 9686 9885 9984 ••84 •i83 •283 •382 99 43^ 640481 o58i 0680 0779 0879 0978 .970 1077 "77 1276 .375 99 1474 1573 1672 1771 1871 2069 3o58 2168 2267 2366 99 439 2465 2563 2662 2761 2860 2959 3.56 3255 3354 ^ 440 643453 355i 365o 3749 3847 3946 4044 4143 4242 4340 44i 4439 4537 4636 4734 4832 4931 5o29 5.27 5226 5324 98 442 5422 5521 56i9 5717 6698 58j5 59.3 6894 601 1 6.10 6208 63o6 98 443 6404 65o2 6600 6796 6992 7089 8067 9043 ^!?5 7285 98 444 7383 7481 8458 7579 7676 7774 7872 8848 & 8262 98 445 8360 8553 8653 8750 9.40 9237 97 446 9335 9432 9530 9627 9724 982. X2 ••16 •..3 •2.0 97 ni 65o3o8 o4o5 o5o2 0599 0696 0793 0987 1084 1.8. 97 1278 1375 1473 069 1666 .762 18D9 1956 2o53 2.5o 97 449 2246 2343 2440 2536 2633 2730 2826 2923 30.9 3. .6 ^9^ 430 653213 3309 4273 34o5 3502 3598 3695 4658 379. 3888 3984 4080 45 1 mi 4369 4465 4562 4734 485o 4946 5o42 96 452 5235 533 1 5427 5523 56.9 57.5 58io 6?64 6002 96 453 6098 7o56 6194 6290 6386 6482 6577 75i4 6673 6769 7725 6960 96 454 7i52 7247 7343 7438 itr. 7820 K $ 455 8011 8107 8202 8298 925o 8393 8488 8679 8774 456 8965 9060 Qi55 9346 944. 9536 963 1- 9726' 9821 95 n 66??65 ••11 •106 •201 •296 •39. •486 •58. •676 •771 95 0960 io55 n5o 1245 .339 1434 .529' 1623 1718I 95 459 i8i3 1907 2002 2096 2191 2286 238o 2473; 2569' 2663 j 95 N. ' ^.! 5 4 ! 5 1 6 7 i 8 1 9 i r>. A TABLB OF LOGAFITilMS FROM 1 TO 10,000. ri^r 1 I 1 2 j 3 1 4 1 5 1 6 1 7 1 8 I 9 1>. | 460" 662758 7852 ' 2947! 3o4i! 3i35 323o; 3324 3418! 35i2' 3607, 94 | 401 3701 3795 : 38»9' 3783, 4078 4172 4266 436o 4454: 454fe 94 462 4642 4736 1 4830 1 4324' 5oi8 5ii2 1 5862 J 5906; 6o5o 5206 5299 5393, 5487 94 463 5581 5675 : 5769 6143 6237 633i 6424 94 464 65i8 6612! 6703 j 6799 6892 6986 7733 7826 7920 7079; 7 '73: 7266, 736a 94 465 1453 8386 7346 7640 8oi3! 8106 8199 8293 93 466 8479 8572 8665. 8759 8852 1 89451 9o38 9i3i 9224! 93 467 468 9317 9410 95o3 9596! 9689 : 9782 9875! 99671 ••60' •i53t 93 070246 0339 043 1 o524! 0617 0710 1 0802 0895; 0988: io8oi 91 i 469 1173 1265 i358 i45ii i54'j i636: 1728 1821 1913 2836 2oo5 9^ 1 470 672098 2190 2283 2375, 2467 256o 2652 2744 2929 92 4-1 3o2i 3ii3 32o5 3297J 3390 3482 3574 3666 3758 38Jo, 92 47 i 3942 4861 4034 4126 42i8l 43io 4402 ; 4494 4586 4677 4769 92 473 4953 5870 5o45 5i37 5228 5320 5412 55o3 5595 5687 92 474 5778 6694 5962 6876 7789 8700 t)o53| 6145 6236 6328 64«9 65u 6602 92 475 6785 6968! 7059 7i5i 7242 7333 7424 7516 8427 9' 476 2^"2 7698 7881 8791 8882 8o63 81 54 8245i 8336 91 477 478 8oi8 8609 8973 9882 9064 9i55j 9246 9337 9' 94?8 9519 9610 9700 979' 9973 ••63 •i54 •245 91 480 68o336 0426 o5i7 0607 0698 0789 0879 0970 1060 ii5i 91 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 3i37 3227 3317 3407 3497 3587 3677 3767 3857 90 483 3947 4037 4127 4217 43o7 4396 4486 4576 4666 4756 90 484 4845 4935 5o25 5ii4 5204 5294 6189 7o83 5383 5473 55fe3 5652 ?9 485 5742 583 1 ?i 6010 6100 6279 6368 6458 6547 486 6636 6726 6904 ^t 8064 7261 81 53 7351 7440 89 487 7329 8420 7618 6dq8 7796 m 8242 833 1 89 488 85o9 8687 8776 8953 9042 9i3i 9220 89 489 9309 93n8 0283 94S6 9575 9664 9753 Q841 9930 ••19 •107 ^'^ 490 690106 1081 0373 0462 o55o 0639 0728 0816 0900 -V093 89 491 1 1 70 1258 i347 1435 i524 1612 1700 • 780 1877 88 492 1965 2o53 2142 223o 23i8 2406 2494 2583 2671 2759 88 493 1847 2935 38i5 3o23 3iii 3'99 3287 4166 3375 3463 355i 3639 83 494 3727 3903 3991 4078 4254 4342 443o 4517 88 495 4605 4693 4781 4868 4956 5832 5o44 5i3i 5219 5307 5394 88 496 5482 5569 5657 5744 5919 6007 6004! 6182 6269 ?7 497 6356 6444 653 1 6618 6706 6793 6880 6968! 7055 7142 8014 l^ 498 7229 tsl 7404 7491 7578 8449 7665 7752 7839 7926 8709 8796 t^ 499 8101 8275 8362 8535 8622 8883 P 5oo 698970 9057 9144 923. 9317 9404 9491 9578 9664 975i 87 5oi 9838 9924 ••11 ••98 •184 •271 •358 •444 •53 1 •617 87 502 700704 0790 0877 0963 io5o ii36 1222 1309I 13^5 2172I 2238 1482 86 5o3 1 568 i6d4 1741 1827 1913 1999 2086 2344 86 5o4 243 1 25i7 26o3 2689! 2775 2861 2917 3o33 3 1 19 32o5 86 5o5 3291 3377 3463 3549! 3635 3721 3807 3893 3979 4o65 86 5o6 4i5i 4236 4322 44o8i 4494 4579 4665 4731 4837 4922 86 1 507 5008 5094 5179 5265| 5350 5436 5522 56o7 5693 5778 86 5o8 5864 'r, 6o3d 6120 6206 6291 6376 6462, 6547! 6632 85 1 509 6718 6888 6074 7059 7144 7229 73 1 5 7400I 7485 85 5io 707570 7655 7740 7996 8081 8166 825i 8336 85 5ii 8421 85o6 8591 8846 8931 90 1 5 9100; 9185 85 , 5ia 9270 9355 9440 9524' 9609 9694 9779 9863 9948| ••33 8'> 1 5i3 710117 0202 0287 037 1 j 0456 o54o 0625 0710 0794: 0879 85 1 5i4 0963 1S07I 1048 Il32 1217 i3oi i385| 1470' i554' 1639' 1723 84 1 2566 84 1 5i5 1892 2734 1976 2060' 2144 2229 23i3 2397 2481 1 5i6 265o' 2§.8 2902' 2986 3070: 3i54 3238 3323 3407 84 ^'1 5i6 3491! 433o' 3575; 3659 3742; 3826 3910 3994 4749! 4833 4078 4162 4246 84 44i4i 4407 458 1 4665 525i| 5335 5418 55o2 4916 5ooo 5oS4 84 5i9 5167I 55861 5669 5753! 5836 5q20 84 D. i.^'J ! I I 2 ! 3 1 4 ;5 i JLJ 7 i 8 ! 9 TAIiLK OF LOGAIUTI1M8 FROM 1 TO 10,000. ~N.* 1 I 1 2 3 4 5 6421 6" 7 8 I , D. 1 520 716003 0087 6170 6254 6337 6304 6588 6671! 6754 S3 7504! 7^87! «3 521 6838, 6921 7004 7088 Ti7il 7254 7338 800 3 8086' 8169 8834 891 71 9000 7421 523 523 83o2 mjm 1% 8253 83361 8419: 83 9o83] 9i65j 9248, 83 524 933. 9414' 9497 9580 9663, 9745i 9S28 99111 9994! ••77, 83 0738: 082 i| 0903I 83 1563 164&' 1728, h 525 720159 0242, 0323 0407 1233 0490' 0373! o655 'J26 0986 1068 n5i i3i6: 1398 1481 527 iBi. 1893 .975 2o58 2i4o| 2222 23o5 2387 2469 2552 d? i 52a 2634 2716 2> 2881 29O3, 3o45 3127 3948 3209 329. 3374 82 529 3456 3538 3620 3702 3i84' 3866 4o3o 4112 4194 8j 53o 724276 4358 4440 4322 4604! 4685 m 4849 4931 5oi3 82 53 1 5095 5176 5258 5340 5422' 5303 56^7 6483 5748 6564 583o 82 532 5912 5993 6075 61 56 6238, 632o' 6401 6646 82 533 6727 6809 6890 6972 7033 7134 7216 7866 7948 8029 8678: 8739 8841 7297 8110 8922 7379 819. 9003 7460 8273 9084 81 534 5iD 7041 8354 7623 6435 7704 85i6 nil 81 8i 536 9165 9246 9327 •i36 9408 9480 9370 .2981 ^378 IIOD, 1186 9651 9732 9813 9893 81 537 9974 ••55 •217 1266 •340 •621 •702 i5oS 81 538 73o«2 i589 oS63 0944 1024 i347 1428 81 539 1669 1750 1 830 1911 1991 2072 2l52 2233 23.3 81 540 732394 2474! 2555 2635 2715 llfs 2876 2956 3o37 3117 80 541 3197 3278; 3358 3438 35.8 3679 3759 4560 3839 3919 80 542 3999 4079, 4 1 60 4240 4320 4400 4480 4640 4720 55.9 80 D43 4800 48S0! 4960 5o4o 5.20 5200 5279 5359 5439 80 544 5599 5679; 5739 5838 5918 5998 6078 6i57 6237 63i7 80 545 6397 6476 6556 6635 67.5 73.1 63o5 8384 6874 6954 7o34 71.3 80 545 7193 6067 7352 6146 7431 S3 IJ^'^ tv. 7908 79 54-' 9672 6225 8543 8701 79 548 8860 8939 9018 90971 9177 9256 9335 9414 nil 79 549 965 1 9731 9810 9889 9968 ••47 •126 •2o5 79 55o 740363 0442 0321 0600 0678 0757 o836 09.5 0994 1782 2568 1073 79 1 55 1 Il52 I230 288a 1 388 .467 1546 1624 1703 i860 79 552 1939 2018 2175 2254 2332 2411 2489 2647 l^ 553 2723 2804 2961 3o3o 3823 3n8 3196 39S0 3270 3353 343. 554 35io 3588 3667 3745 3902 4684 4o58 4.36 42i5 78 555 4293 4371 4449 4328 4606 4762 4840 4919 4997 78 556 5075 5.5] 5231 5309 5387 5465 5543 5621 5599 im 7« 557 5855 59331 601 1 6089 6167 6245 6323 6401 ^479 .7« 55a 6634 6712; 6790 6S68 6945 7023 710. i 8o33 8808 7334 8.10 8885 7? 559 56o 741a 748188 liU m s? IV,1 7800 8376 gl? 78 77 56 1 8963 9040 91 !8 9193 9272 935o 9427 9304 9582 9639 77 562 9736 9'i.4 989I 9968 ••45 •123 •200 •277 •354 •43 1 77 563 7=.o5o8 o586 0663 0740 i5io 0SI7 0894 0971 1048 1.23 1202 77 564 1279 1 356 1433 .587 1664! 1741 1818 1895 2663 1972 77 565 2048 2125 2202 2279 2356 2433 25o9 2386 l^ 77 566 2816 2893 2970 3047 3.23 3 200 3277 3353 3430 77 567 3583 366o 3736 45oi 38i3 3889 3966 4042 4119 4883 4195 4272 5o36 77 568 4348 4425 4578 4654 4730 4807 4960 76 569 5lI2 5189! 5265 5341 5417 5494 6256 5570 5646 5722 5799 76 5/0 755875 5q5i 6027 6io3 6180 6332 6408 6484 6560 76 57' 6636^ 671 2I 6788 6864 6940 7016 70.72 7168 7244 8oo3 8761 7320 8079 8836 76 572 573 'it, m iz 7624 8382 7700 8458 ^533! leJ) Zi 76 -6 574 8912 9668 8988 1 9o63 9139 9214 9290I 9366 9441 9517 9392 7$ 575 9743; 9819 0498J 0573 9894 9970! ••4:11 "ni G?5o •272 •347 7? 575 76.1422 0649 0724' 0799! 0875 .4^7: 1532! .627 22:8: 23o3! 2378 1025 iroi ^^ P,l 1176 i25i 1326 1402 1702 1778, 1853 75 '^t :oo3 2078 2153 2453 25291 2604' 75 3278' 3353i 75 579 2754 2829 2904 29781 3o53! 3128 32o3 , N. I 2 3 4 J 5 i 6 7 8 1 9_1 D.J lO A TABLE OP LOGARITHMS FROM 1 TO 10,000. N. 3 3 4 5 6 1 7 1 8 9 I), 58o 763428 35o3 "35^ 3653 3727 38o2 3877i 3952! 4027 4101 58i 4176 425i 4326 4400 4475 455o; 4624I 4699' 5296 5870' 5443 4774 4848 582 641 3 4998 5072 5U7 5221 5520 5594 583 5743 58i8 6636 5966 6041 6ii5| 6190 6264 6888 584 6487 6562 6710 6785 6859I 698 J' 7007 7082 585 7i56 7280 7304 7379 7453 7527 7601 1 7675, 7749 7823 586 7898 7972 8046 8120 %t 8268: 8342J 8416! 8490 8564 587 588 8638 8712 8786 8860 9008I 9082 91 56 9746] 98201 9894; 0484 o557| 068 1 j 1220 1298' 1867 9280 9808 I'i 9377 770115 945 1 9525; 9599 96,3 9968 ••42 589 0189! 0263 o336 0410 0703 0778 5oo 770852 09261 0999 1073 1146 1440 i5i4 591 1087 1661 1784 1808 1881 1955 2028 2102; 2175 2248 592 2822 2895 2468 2542 26i5 2688J 2762 2835: 2908 2n8l 5^3 3o55 8128 3201 3274 3848 8421 3494 3567! 8640 37.3 594 3786 886oi 8988 4006 4079 4i52 4225 4298I 4371 4444 595 4517 4390 4663 4786 4809 5588 4882 49551 5028; 5,00 5,73 596 5246 58.9 5392 5465 56io 5688 5736' 5829 5902 597 5974 6047 6120 6193 6265 6338 641. 6488 6556 6629 5^8 6701 6774 6846 6919 6992 7064 7187 7209' 7282 7354 599 7427 7499 7572 7644 8368 7717 7789I 7862I 7984' 8006 8079 600 778151 8224 8296 8441 85i3 8585 8658, 8780 8802 601 8874 8947 9019 9091 9168 9286 9808 9380; 9452 9324 602 9596 9669 9741 9813 9885 9957 ••29! •lOlj •.78 •245 6o3 780817 0889 0461 o538 o6o5 0677 0749! 0821 1468J .540' 0893 0965 604 1087 1109 1181 1258 1824 1896 .6.2 1684 6o5 n55 .827 1899 1971 2688 2042 2114 2186 2258: 2829 2401 606 2473 8189 2544 2616 2759 8473 2881 2902 2974' 8046 3.17 607 8260 8882 3408 3546 86.8 36So 440.3! 876. 8882 608 8904 8975 4046 4118 4189 4261 4332 4475 4546 609 4617 4689 4760 4881 4902 4974 5o45 5. .61 5.87 5259 610 785830 5401 5472 5548 56.5 5686 5757 582S1 5899 5970 611 6041 6112 6i83 6254 6825 6896 6467 6538: 6609 6680 612 6751 6822 6893 6964 7o35 7106 7'77 7248| 73,9 7890 6i3 7460 758i 7602 7678 7744 7815 7885 7956 8027 8098 614 8168 8289 8810 838 1 8451 8522 8598 86631 8734 8804 6i5 8875 8946 9016 9087 9157 9228 9299 9869 9440 95.0 616 958 1 965 1 9722 9792 9868 9933 •••4 ••74' •144 •2.5 70 617 790285 08 56 0426 0496 o567 0687 0707 0778: 0848 0918 70 618 0988 1059 1 1 29 1199 1269 1840 .4.0 1480 i55o 1620 70 619 1691 1761 i83i 1901 1971 2041 2111 2.8l| 2252 2822 70 620 792892 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 8092 8162 8281 83oi 8871 8441 85.1 858i| 865i 372. 70 622 3790 386o 8980 4000 4070 4. 39' 4209 4279 4849 5045 44.8 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5ii5 70 624 5i85 5254 5324 58o3 6088 5463 5532I 5602 56721 5741 58ii ^ 625 5880 5949 6019 6i58 6227 ^297 6366; 6436 65o5 626 6574 6644 6713 6782 6852 6921 6990 7614 7688 83o5 8874 7060' 7.29 7198 69 627 628 7268 7337 7406 7475 7545 7752; 7821 7800 8582 69 7960 8029 8098 87^0 9478 8167 8858 8286 8443, 85.3 69 629 865 1 8720 8927 8996 1 9065 9134' 9208 9272 ^ 63o 799341 9400 0098 9547 9616 9685! 9754 9828: 9892 9961 69 63i fi 30029 0167 0236 o8o5 0873 0442 o5 1 1 1 o58o 0648 69 63a 0717 0786 o854 0923 0992 1061 1129 i8i5 1 198! 1884^ 1266 1335 69 633 1404 1472 i54i 1609 1678 1747 .952 2021 69 634 2089 2i58 2226 2295 2363 2482 25oo 2568, 2687 2705 ^ 635 2774 2842 2910 2979 3o47 3.16' 8184 3252, 3321 3389 636 3457 3525 3594 3562 8780 8798, 8867 8935 4008 4071 68 637 638 4189 4208 4276 4957 4344 4412 4480 4548 4616 4685 4753 68 4821 4889 5o25 5093 5i6ij 5229 5297 5365 5433 68 639 35oi 556^ 5637 5705 5773 5841 1 5908 5976 6044 611J ^\ N. ■ - 1 ^^ 3 4 5 1 6 7 1 8 ^1— 1). A TABLE OP LOOAKITHSdG FROM 1 TO 10,000. n N. 1 , j 2 3 4 5 6 7 1 8 1 9 D. j 640 806180' 6348 63i6 6384 645i 65i9 6587 66551 6723 6790 68 1 641 6858 6926! 6994 7061 7129 7«97 7264 7332 7400 I^^J 68 642 7535 8211 7603 1 jt>io 8279! 8346 7738 8414 7806 7873 794, 848, 85491 86,6 8008 8076 8,43 68 643 8684 8751 88,8 67 644 88S6 8953 9021 9088 9i56 92231 9290 9358; 9425 9492 67 643 9360 9627 1 9694 9762 &i\ tt\ t^i ••3,1 ••98 •,65 67 646 810233 o3ooi o367 0434 0703 1 0770 fdl J*^ 647 0904- 0971 j io39 1106 ii-ji ,240 ,3o7 1374! 1441 67 648 1575 1642 1709 1776 1843 19101 ,977 2044^ 21,1 2178 67 649 2245 23l2 2379 2445 25,2 9379 2646' 2713 2780 2847 ^7 6DO 812913 2980 3o47 3114 3i8, 3247 33i4 338, 3448 35i4 67 65i 3d8i 3648 3714 3781 3848 3914 3981 4048 4114 4181 67 65i 4248 43i4 438i 4447 43,41 438, I 4647 4714 4780 4847 67 653 4913 4980 5046 5m3 f/79! 5246: 53,2 5378 5445 55,, 66 654 5578 5644 5711 5777 58431 59,0! 5976 6042 6,09 6,75 66 655 6241 63o8 6374 644o 65o6l 6373 1 6639 6705 677, 6838 66 656 6904 6970 7o36 7102 7169! 7235i 73o, l^^l 7433 8820 66 tu 7365 8226 7631 ^3?8 7764 8424 7830 8490 m 7962 8622 8028 8094 8688 8734 66 66 659 8885 9017 9083 9,49 "^vi 9281 9346 9412 '^ll 66 660 819344 9610 9676 ;S 9807 9873 9939 •••4 ••70 66 661 820201 0267 0333 0464 033o 03^3 066, 0727 0792 66 662 o858 0924 1D79 0989 ,,20 ,,86 123, i3i7 ,382 1448 66 663 i5i4 i64D 1710 1775 ,84. 1906 1972 2037 2io3 65 664 2168 2233 2299 2QD2 2364 243o 2493 2 560 2626 2691 2756 65 665 2822 2H87 3oi8 3o83 3,48 32,3 3279 3344 3409 65 666 3474 3539 36o5 1 3670 3735 38oo 3865 3930 3996 406, 65 ^ 4126 4i9« 4256 432, 4386 445, 45,6 458, 4646 471, 65 4776 4B41 nt 4971 5o36 5,0, 5,66| 523, 5296 536, 65 669 5426 5491 562, 5686 6^9^ 58,5 5880 5945 6o,o 65 670 826075 6140 6204 6269 6334 6464 i 6528 6393 6658 65 671 6723 6787 6852 6917 698, 7046 7,,, 1 7«73 7240 73o5 65 672 7434 7499 ! 7563 7628 1^^ 7757 7821 7886 853, 795, 65 673 8080 8144 ' 8200 ' 8853 8273 8338 84021 8467 8395 64 674 8660 8724 8789 X 8982 9046 911, 9,75 9239 64 67D 9304 9368 9432 1 9497 9625 9600 9754 9818 9882 64 676 9947 **i I ••75 : •,39 •204 •268 •3321 •3q6| •460 •525 64 678 830D89 0653 0717 1 0781 0845 0909 09731 IO37J ,,02 1,66 64 I230 1294 1934 i358 1 1422 ,486 ,330 ,6,41 16781 ,742 ,8o6j 64 679 1870 1998 2637 2062 2,26 2,80 2828 2253! 23i7 238, 2445| 64 680 832509 2673 j 2700 2764 2892I 2956 3020 3o83 64 681 3i47 3211 3275 1 3338 3402 3466 353ol 35931 3657 4,66' 423o 4294 372, 64 682 3784 3848 3912 1 3975 4o39 4io3 ' 4357 64 683 4421 4484 A-US 1 461, 4673^ 4739 53, o| 5373 4802 4866 4929 5437 55oo! 5564 4993 64 684 5o56 5l20 5i83 i 5247 5627 63 685 6691 5754 58i7 588 i 5944I 6007 6071I 6,34' 6,97 6704! 6767 j 6830 , 6261 63 680 6324 6387 6451 1 65,4 63771 6641 7210 7273 ' 68941 63 687 688 tA 7020 7o83 7146 7336 7399 7462 1 7525I 63 8,561 63 7652 l]\l 7778; 7841 7904 8408 8471 8334 7967 8o3o, 8093 8597 8660 8723 689 8219 8282 8786; 63 691 638840 9478 8912 9341 8975 9604 1 9o38j 910, 9164 ; 9667, 9729' 9792 9227 9280 o352 9855 9918 9981 94i5i 63 ••431 63 692 840106 0169 0232 I 0294 0357 0420 0482 o;)45i 0608 06711 63 693 0733 0796 0850 1422 1485 ' 092, 0984 1046 ,,09 ,172! ,234 13971 63 1735, 1797 i860 ,0221 63 236o 2422 1 2484 2347 i 62 694 \lsl ; 1347 16,0: 1672 695 2047! 2110 2,72 2235 2297 696 260c 26721 2734 3233 3295I 3357 2796 2859 292, 2o83 3046 3,08 3,70! 62 ^ 3420 3482: 3544 36o6 3660! 3731! 37931 62 3855! 3918: 3g8G 4042 4104 4166 4229 42911 4353 441 5 6a ! 699 N. 44771 4539 4601 1 4664 4726 4788', 485o 4912' 4974 5o36| 62 . 3 1 3 1 4 ! 5 1 6 ! 7 1 8 1 9 1 r). ) 12 A TABLE OF LOGARITHMS FROM 1 TO 10,00U. N. I 2 3 4 5 6 5470 7 8 9 D. 62 7GO 845098 5i6o 5222 5284 5346 5408 5532 5394 5656 701 5718 5780 5842 5904 5966 602S 6090 6.5. 62.3 6273 6894 62 -!02 6337 6399 6461 6523 6385 6646 6708 6770 6832 62 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 75.1 62 704 7573 7634 7696 7758 7819 7881 8497 ifi> 8oo4 8066; 8128 U 1 7o5 8189 8231 83i2 8374 843! 8620 86821 8743 61 706 88o5 8866 8928 9542 8989 905 1 9112 9'74 9235 9297 9358 61 ^S o-94'9 9481 9604 9665 9726 o34o 9788 9849 99" 9972 61 830033 0095 oi56 0217 0279 0401 0462 0324 o585 61 709 0646 0707 0769 o83o 089; 0932 .014 1075 ii36 "97 61 710 851258 l320 i38i 1442 i5o3 1 564 1625 1686 '747 .809 6i 711 1870 1931 1992 2o53 2.14 2175 2236 2297 2353 2419 61 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 3090 3i5o 321. 3272 3333 3394 3455 35.6 3577 3637 61 714 3698 3759 3820 388i 3941 4002 4063 4124 4 1 85 4245 61 715 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 716 4913 4974 5o34 5095 5i56 5216 5277 5337 5398 5459 61 7n 5319 5580 5640 5701 5761 5822 5882 5943 6oo3 6064 61 718 6124 6i85 6245 63o6 6366 6427 6487 6548 6608 6668 60 7>9 6729 6789 685o 6910 6970 7574 703 1 7091 7i52 7212 7272 60 720 857332 7393 7453 75i3 7634 7694 7755 78.5 8417 7875 60 721 7935 7993 8o56 8116 8176 8236 8297 8357 8477 60 722 ^Hl 8397 8657 8718 8778 8838 8898 8958 90.8 9078 60 723 9 1 38 9198 9258 9318 9370 9i39 9499 9559 9619 9679 •278 60 724 0.9739 0398 9859 0458 99,8 9978 ••38 ••98 •i58 •218 60 723 86o338 03l8 0578 0637 0697 0757 08.7 0817 60 726 0937 0996 io56 1II6 1176 1236 1293 i355 I4i5 1475 60 727 1 534 1394 1 654 I7I4 1773 i8'33 1893 2489 1932 2012 2072 60 728 2l3l 2191 225l 23lO 2370 243o 2549 2608 2668, 60 729 2728 2787 2847 2906 2966 3o25 3o85 3.44 3204 3263i 60 730 803323 3382 3442 3301 356i 3620 3680 3739 3799 3858 59 73 I 3917 3977 4570 4o36 4096 4i55 4214 42-:4 4333 4392 4452 59 732 45ii 463 4689 4748 4808 4867 4926 49S5 5o45 59 733 5io4 5i63 5222 5282 5341 5400 5459 56:9 5578 5637 59 734 5696 6287 5755 58i4 5874 5933 6524 5992 6o5i 6110 6169 6228 59 735 6346 64o5 6465 6383 6642 6701 6760 6819 59 736 6878 6937 '^.tt 7o55 7114 7173 7232 729. 7350 7409 7998 59 737 7467 7526 8ii5 7644 7703 8292 7762 835o 7821 7880 7939 59 738 8o56 B174 8233 8409 8468 8327 8586 59 739 8644 8703 8762 8821 8879 8938 8997 9036 9114 9173 59 740 869232 9290 9349 9408 9466 9525 95«4 9642 9701 9760 59 741 9818 9877 9935 9994 ••53 •ill •170 •228 •287 •345 742 870404 0462 052l 0570 0638 0696 0755 08.3 0872 oq3o 58 743 0989 1047 1 106 1 164 1223 1281 1339 IQ23 1398 I4561 13.5 58 744 1673 i63i l6qo 1748 1806 1 865 1981 2564 2040 2098 2681 58 745 2 1 56 22l5 2273 2331 2389 2448 25o6 2622 58 746 2739 2797 2855 2913 2972 3o3o' 3o88 3146 3204 3262 58 747 3321 3379 3437 3495 3553 36ii' 3669 2727 3785i 3844 58 748 3902 3960 4018 4076 4i34 4192' 4250 43o8 4366, 4424: 58 749 4482 4540 4398 4656 4714 4772' 4830 4888 4945 5oo3 58 750 875061 5i 19 5698 5,77 5235 5293 535 1 1 5409 5466 5324 5582 58 75i 5640 5756 58i3 5871 5929' 5987 65o7 6564 6045 6102 6160 58 702 6218 6276 6333 6391 6449 6622 6680 6737 58 7-53 6795 6853 6910 6968 7026 7083, 7141 7199 7256 7314 58 754 7371 7429 7487 7544 7602 7659' 7717 7774 8349 7832I 7889 58 755 7947 8322 8004 8062 8119 8177 8234I 8292 8407 8464 57 756 8579 8637 8694 8752 8809 8866 9383 9440 8924 8981 9039 57 7Dd 9096 9153 9211 926S 9325 9497 9555 9612! 57 9669 9726 9734 9841 9898 9936 ••i3 ••70 •127 •i85 57 759 N. 880242 0299 0356 o4i3 0471 o528' o585 0642 0699 0756 9 57 D. I l_i_. 3 4 A±L 7 8 A lAULK UF LOGARITHMS FROM 1 TO 10,000 ib N. 760 761 76a 763 764 765 766 768 769 77c 77« 772 773 774 775 776 77^ I 779 78c 781 782 783 7«4 785 786 7«7 788 789 790 79" 792 793 794 795 796 797 798 799 800 801 803 8o3 804 8o5 806 807 808 809 810 811 I 812 8i3 I 8.4 8i:i 81S 8n I 8id I 819 I "n. i 880814 i3S5 1953 2533 3093 366 1 4229 4793 536 1 5926 886491 7034 7617 8179 8741 9302 9862 89042 1 0980 !d37 892095 265 1 3207 087 1 j .442; 2012 23Si 3i5o' 3718 4285 4852i 5418 5983! 6347' 7iir, 7674 8236 99i8j 04771 .o33^ .593' 2 .do' 2707 „.„,, 3262, 3762! 38.7, 43i6j 4371 4870 4923 5423I 5478 5975J 6o3o' 6526; 658.^ 7i32 76H2 0928' 0985 1499! i556 2069! 2.26 2638J 2695 3207; 3264 3775 3832 4342I 4399 49091 4960 5474 5d3i 6039 6096 6604! 6660 7.67; 7223 773o' 7786 8292; 8348 88531 8909 941. 9974 9470 ••3o 7077 S97627 8.76 8-125 9273 9821 900367 09.3 .458 2003 2547 903090 3633 8231 8780 9328 9875, 0422' 0968 ID|3 2057 2601 3.44 3687 4229 o533i o589| .09. 1.47 .649 .703 2206! 2262 2762I 2818 33i8l 3373 38731 3928 4427 4482 49S0! 5o36 5533| 5588 6oS5 6.40 6636 6692 7187 7242 7737I 7792 8286I 8341 8835, 8890 4770 33.0 585o! 6389I 6927 4174 47'6 5256 5796 6335 6874 , 74.. I 7465 7949' 8002 908483' 8539 9021 9074 9556I 96.0 910091I 0.44 06241 0678 1.58 1690 2222 2753 3284 .21. 1 .743 2275 2806 333 r 9383 94^7 99J0 9983 0476 od3. 1022 .077 .567 1622 21.2 2166 2655 2710 3199 3253 3741 '4 4878 4283 4824 5364 5418 5904 5958 6443 6497 7o3D 698. p.9 8o56 7573 8..0 8592 8646 9.28 9.8. 9663 97.6 0197 073. 02D. o]84 .3.7 1264 1797 2328 i85o 238. 28591 29.3 3390} 3443 .042 .6.3 2.83 2752 332.1 3888: 4455' 5022| 5587 6i52| 67161 7280' 7842I 64041 89651 9.')26 ••86' 0645; .203, .760; 23.7; 2873' 3429 39S4 4D38 509.1 5644 i 6.95 6747! 7297| ^96! 8944! 94921 ••39; o586| ..3. .676' 222.1 2764 3307! 3849! 43q«| 49^2 5472; 60I2| 655. 7089; 76261 6.63, 8609! 92331 9770 o3o4! o838| .37.1 .903 1 2435 2q66 3496 1099 1670 2240 2809 3377 3945 43.2 8 1.56 1727 22971 28661 3434I 40021 4369! 5078; 5.33, 5644: 5700 6209! 6265; 67731 6829I 7i36| 7302I 7808 7955; 8460; 85.6 902. i 9077 1 9582 j 9b38: •14. 1 •.97I 0700, 0736; .259 13.4I 1816: 1872! 2373| 24:9' 2929 2985' 3484| 3310, 4039 4094 1 45931 4648 5i46| 5201 5699' 5734 623.1 63o6 6802 6837 ; 7352 7902 843. 8999 9547 '94 7407; 83o6 906 4: 9602 •.49' 0640 1 0693 ..86' 1240 173.1 .783, 2275i 2320 28.8, 2873, 336. 3416' 3904 4445 4986 5326 6066 6604 7143 7680 8217 8753 3958, 4499' 5o4o 558o 6. 19 6658 7196; 7734 8270 880-: 1 9289 9342 i 9823: 9877J o338 04.. I 0891 0944 1424' 1477! 19561 2009 2488 2541 j 30.9 3072 3549 36o2 I2l3{ 1784I 2354' 2923 1 349. 4039' 4623, 5.92 5p7 6321 6885 7449 80.. 8573 9.34 9694 •233 0812 1370 I928I 2484| 3o4o 3595 4130 4704I 5237 3S09' 636. 1 6912. .462' 6012 856 1 9109 9636 •203 0749 .293 1840: 2384 2927 3470 4012 4553 5094 5634 6173 67.2: 725o 7787^ 8324 8860 9396 9930 0464 2o63: 2594: 3.25! 3655 1>. 127. 1841 24.1 2980 3348 4.13 4682 5248 t8.3| 6378 6942 73o5 8067 8629 9190 9730 •309I 0868! 1426 .983 1 2340' 3096 365. 42o5 4759 53.2 5864 64.6 6967 73.7 6067 86.5 i328 1898 2468 3o37 3603 4172 tilt 5870! 6434! 8.23 8685 9246 9806 •365 0924 1482 2039 2393 3i5. 3706 426. 4814 5367 5920 6471 7022 7572 8.22 8670I 9164! 9218 •258: 0804 .349 .8o.i 24i8 298.; 3324 4066; 4607 1 5.48; 5688 6227' 6766' 73o4j 784. 8378; 8q.4' 9449 03.8 io5. 15841 2..6j 26471 3.78, 9766 •3.2 0859 1404 .948 2492 3o36 3578 4120' 466. 5202 5742 628. 6820! 73581 m 8967! 93o3: ••37! 057. .Jo4 .637 2.69 2700 323. 3761 I u 56 56 56 56 56 56 56 56 56 56 56 56 56 55 55 55 55 55 55 55 55 55 55 55 55 55 55 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 53 53 53 53 53 53 53 53 A 1ABL& OF LOGARITHMS FROM 1 TO lO,OOU. 820 . ! , 1 J \ 4 1 5 6 1 1 4184 8 1 9 I 9i38i4 3867; 39201 39731 4026; 4079 4 1 32 4237] 4290 821 4343 4396' 4449' 45o2| 4555' 4608 4660! 4713 4766; 48.9 822 4872 4925 4977 5o3o 5o83j 5i36 5189' 5241 52941 5347 823 5400 5453 55o5 5558 56ii| 5664 5716, 5769 5822 5875 824 5927 5980 6o33 65o7 6559 6o85 6i38: 6191 6243' 6296 6349! 6401 82c 6454 6612 6664: 6717 6770'; 6822 6875, 6927 74001 7453 826 6980 7033 7083 7i38 7190, 72431 7295^ 7348 828 7D06 7558: 761 1 7663 7716; 7768I 7820: 7873 8240' 8293 8345: 8397 7925j 7978 8c3o 8o83 81 35 8188 845o! 8302 829 8555 8607' 8659 8712 8764 8S16, 8869I 8921 8973; 9026 83o 919078 9i'Jo' 9183 9235 92871 9340 9392! 9444 9496, 9549 83 1 9601 9653 9706 9758 9810 9S62 9914' 99^^7 ••19 ••71 832 920123 0176 0228 0280 o332i o384 0436 0489 o54il 0593 833 0645 0697 0749 0801 o853; 0906 0938' 1010 1062 1114 834 1166 1218 1270 l322 i374 1426 1478^ i53o i582, i634 835 1686 1738 1790 1842 1894' 1946 1998J 2o5o 25i8j 2570 2102' 2 1 54 836 2206 2258, 23l0 2362 2414 2466 2^2 2674 837 838 2725 2777 2829 2881 2933 2q85 3037: 3089 3i4ol 3192 3244 3296, 3348 3399 345 1 i 35o3 3555| 3607 3658, 3710 839 3762 3Si4 3865 39.7 3969' 4021 40721 4124 4176! 4228 840 924279 4331! 4383 4434 4486: 4538 4589 i 4641 46931 4744 841 4796 4848 48r,9 49^1 5oo3 5o54 5io6 5i57 5209' 5261 842 53i2 5364 54 J 3 5467 55i8 5570 562 1 i 5673 3723; 5776 843 5828 5879 5931 5982 6o34 6o85 6i37 6188 6240 6291 844 6342 6394 6445 6497 6548 6600 665ii 6702 6754' 68o5 845 6857 6908, 6950 7422j 7473 7011 7062. 71 i4 7i65! 7216 7268I 7319 846 7370 8o3^ 7576 7627 7678, 7730 9191: 8242 77811 7832 8293 8345 847 848 7883 79351 7986 80S8 8140 8396 8447 i 8498 8549 86oi| 8652 8703. 8754 88o5 8857 849 8908 8959 9010 9061 9112! 9i63 92i5! 9266 9317! 9368 85o 929419 9470 9321 9572 9623 9674 9725, 9776 9827: 9879 85i 9930 oqSi. ••32 ••83 •i34' •iSS •236| •287 •338! ^389 852 930440 6491 0542 0592 0643 0694 0745^ 0796 08471 0898 853 0949 1000 io5i 1102 n53 1204 1254' i3o5 i356j 1407 854 1458 1 509' i56o 1610 i66i 1712 17631 1814 1865 1915 855 1966 2017: 2068 2118 2169 2220 227IJ 2322 2372 2423 856 2474 2524: 2575 2626 26771 2727 27781 2829 2879! 2930 858 2981 3o3ii 3o82 3i33 3i83 3234 3285' 3333 3386 3437 3487 3538 3589 3639 3690 3740 3791! 3841 3892 3943 859 ^3993 4044 4094 4145 4195! 4246 4296; 4347 4397 4448 860 934498 4549 4599 46 5o 4700I 4751 4801! 4852 4902 4953 861 5oo3 5o54 5 1 04 5i54 52o5 3255 53o6: 5356 5406 5457 862 5507 5558 56o8 5658 5709I 5759 58091 586o 5910 5960 863 601 1 6061 i 61 11 6162 6212 6262 63 13! 6363 6413 6463 864 6514! 6564: 66i4 6665 6715, 6765 68j5 6865 6916 6966 865 ''Vit 7066; 71 17 7167 7217I 7267 7317 7367 7418 7468 866 8019 7568^ 7618 8069^ 81 19 7668 7718, 7769 7819 7869 7919 7969 868 8169 8219' 8269 8320! 8370 8420I 8470 8520 8570 8620 8670 8720 8770 8820! 8870 8920 8970 869 9020 9070 9120 9170 9220 9270 9320J 9369 94iq 9918 746Q 870 939519 9369 9619 94001 8j 0068 0118 9719 9769 0218; 0267 9819 9869 ^8 871 0168 o3i7 0816 0367 j 0417 0467 872 o5i6j o566 0616 0666 07161 0765 o865j 0915 0964 873 1014 1064 1114 1 1 63 i2i3 1263 i3i3 i362i 1412 1462 874 i5u i56i 1611 1660 1710! 1760 1809 1859' 1909 ,958 875 2008 2o58 2107 2157 2653 2207 2256 23o6^ 2355; 24o5| 2455 876 25o4I 2554 26o3 2702! 2752 2801 i 285i: 2901 3297 3346 3396 2950 t^l 3ooo 3o49 3oo9 3148 3198; 3247 3445 3495 3544 3593 3643 3692, 3742 4186: 4236 3791' 3841 3890 4285' 4335 4384 3939 4433 879 N. 3989 4o38| 4o88| 4137 , 1 2 3 4 i 5 (3 7 1 8 . ^1 A TABLE OF LOGARITOMS FROM 1 TO 10,000, 16 N. 1 1 1 2 3 ; 4 1 5 1 6 1 7 8 ] 9 1 T). 49 88o 944483 4532 458 1 463 1, 4680' 4729I 4779' 4828 4877^ 4927 68i 4976: 5o25j 5o74| 5i24 5173 5222: 5272 5321 5370 5419 49 882 5469I 55 18 5567 56i6 5665 57i5; 5764 58i3 5862^ 5912 49 883 59611 60101 6059 6108 6157 6207 6256 63o5| 6354' 64o3 49 884 6452 65oi 655i 6600 6649' 6698 6747' 6796! 6845 6894 7090 7'4o 7>89' 7238, 7287I 7336 7385 49 885 6g43, 6992 7434 74H3 7041 49 886 7532 -i5Si^ 763o 7679 7728, 7777i 7826 7875 49 88^ 7924 7Q73 841 3, 8462 8022 6070 81 19' 816S 8217 8266 83i5 8364 49 85ii 856o 8609 8657; 8706I 87551 8804 88531 49 889 8902! 8951 9 49390 1 9439 8999 9048 9097 9146 9195 9244' 9292 9341 49 9536 9585 9634' 9683: 97311 9780; 9829! 49 890 94«8 891 9S78 9926 9975 ••24 ••73 •i2ii •170; •219 •267, •3i6' 49 o5ir o56o' 0608 o657| 0706 0754: o8o3 49 892 9D0365 0414 0462 ^^ o85i 0900 i386 0949 0907 1046 1095 1143! 1 192; 1240' 1289 49 1483 i532; i58o 1629 1677 1726 1776 4q 196c' 2017 2066 2114! 2i63; 221 1 ! 2260 48 2453 25o2 255o 2599 2647I 2696 27441 48 ^i 1 338 I43DI 895 1823 1872 1920J 896 23o8. 2356 24o5 897 2792 1 2841 28i-;9! 33731 2938 2986 3o34 3o83 34211 3470 35 1 81 3566 3i3i| 3i8o 3228 43 898 1 3276; 3325 36i5| 3663: 37 1 1 48 899 376c 38o8 3856 39051 3953 4001 4049 4098 4146 4194 48 900 954243 4291 4339! 4387 4435 4484 4532 458o 4628 4677 5062' 5iio 5i58 48 901 4725 4773 482 1 1 4S69' 49J8 4966 5oi4l 48 902 5207 5688 5255 53o3, 535 1 i 5399 5832 588o 5447 5495 5543 5S92' 5640 48 903 5736 5784' 5928 5976; 6024 6072I 6120 48 904 61681 6216 6265' 63 1 3 6361 6400 6457 1 65o5, 6553 6601 43 905 6649 712S 6697 6745 6793 6840 6888; 6936' 6984 7o32 7080 48 906 7176 7224 72721 7320 73681 7416 7464I 7512' 7559 48 907 7607 7655 8i34 7703 775i! 7799 7847 7894 83251 8373 7942 8421 7990 8o38 48 908 8086 8181 8229' 8277 8468 85 1 6 48 909 8564 8612 8659 8707 8755 88o3; 8850 8898 8946 8994 48 910 9V4I 9089 9.37 9185 9232 9280. 9328 9375 9423 9471 48 yll . 9518 9566 9614 9661 9709 VM 9804 9852 9900 •328 •376 9947 48 912 9995 ••42 ••90 •i38 •i8d •280 •423 48 913 960471 o5i8 o566 061 3 0661 0709 0756 0804 o85i 0899 48 94 0946 0994 I04I 1089! n36 i563 161 1 1184' I23l 1270 1753 i326 .374 47 9i5 1421 1460 1943 i5i6 16581 1706 1801 1848 47 916 1895 2369 1990 2o38 2o85 2l32 2180 2227 2275 2322 47 918 2417 2464 25ll 2559 2606 2653 2701 2748 2795 47 284i 2890 2937 2985 3o32 3079 3i26 3.74 3221 3268 47 919 33i6 3363 3410 3457 35o4 3552 3599 3646 3693 3741 47 920 i 963788 3835 3882 3929 3977 4024 4071 4118 4i65 4212 47 921 4260 4307 4354 4401 4448 449^ 4542 4590 5o6i 4637 4684 47 922 473 1 4778 4825 4872 i%l 4966 5oi3 5io8 5i55 47 923 5202 5249 5296 5343 5437 5484 553 1 5578 5625 47 924 5672 5719 5766 58i3 586o 5oo7 6376 5954 6001 6048 6095 47 923 6142 6189 6658 6236 6283 6329 6423 6470 65i7 6564 47 926 661 I 6705 6752 6799 6845 6892 6939 7408 6986 -033 47 928 7080 7127 7173 7595 7642 8062 8109 7220 7267 73i4 7361 7454 7501 47 7548 8016 768S 778* 8249 7829 7875 7922 7969 47 929 8i56 8296 83431 8390 8810! 88d6 8436 47 930 968483 853o 8576 8623 8670; 8716 8763 8903 .,369 47 931 8950 8996 9043 9090 9i36 9i83 9229 9276 9323 47 931 9416 9463 9509 ! 99-8, 9973 95561 9602 9649 9742 9789! 9B35 47 933 9S82 1 M,,! MfeS •ii4 •207 •254 •3or 47 934 970347 ; o3q3i 0440 1 0486 o533 o579 0626 0672 0719 0765 n83 122s 46 93D 0812 o85J 5 0904 1369 1 0951 0997 1044 1461 i5o8 \z 1137 46 936 1276 i32: I4i5 1601 1647 16931 46 :iIo' 2157 46 $1 1740 i78< ) i832 1879 2388; 2434 2018 2064 2203 224t ) 2293 2342 2481 25271 2573 2619! 46 ! 2989I 3o35 3082; 46 j 939 2666 1 27121 2738 2804I 285 1 ! 2897 2943 IT. 1 L'_[.' 3 1 4 1 ^ 1 6 r 1 8 1 9 1 i>:i 16 K. 940 941 942 943 944 946 9=i7 948 949 900 93, q5i 953 934 955 906 957 938 959 960 961 962 963 964 965 966 967 968 969 970 97' 972 973 974 976 977 978 979 980 981 982 983 984 983 986 987 9S8 989 990 991 992 993 994 995 996 997 998 999_ A TAHLR OF LOQARrrUMS FROM 1 TO ib,UUU. I 2 1 3 1 4 1 5 6 7 8 j 9 D. i 46 973128 3174 3220 3266' 33i3: 3359 34o5 34' I 3497 3543 3390 3636 3682: 3728! 3774 3820 3866 391 3i 3959 4oo5 46 4031 4097 4143 4189 4235' 4281 4327 4374! 4420, 4466 46 45i2 4538 4604' 465o 4696 4742 5o64: 5no 5i56 5202 4788 4834 4880 4926 46 4972 5oi8 5248 5294 5340 5386 46 5432 5478 5524' 5570I 56i6 5662 5707 5753 5799 6258 5843 46 5891 68?4 5983' 6029 6075 6121 6167 6212 63o4 46 6330 6442' 6488 6533 6579 6625 6671 6717 7175 6763 46 6808 6900 6946 6992 7037 74q5 7083 7129 7220 46 7266 73.2 7358 74o3; 7449 7541 7586 7632 8089 1678 ^135 46 977724 7769 7815 7861 IZ & It 8043 46 8181 8226; 8272' 83i7 85oo 8546 8591 46 8637 86831 8728 8774 8819 8865] 8911 8956 9002 9047 46 9093 9.38 9184 923o! 9273. 932ii 9366 9412 9437 95o3 46 9348 9394 9639 9685j 9730 9776' 9821 9867 ^6? 9953 46 980003 0049 o5o3 0094 0140 oiB5 023 1 ' 0276 o322 0412 43 0458 0349 0394' 0640 o685 0730 077^ 0821 0867 43 0912 1 366 0957 ioo3 1048 1093 1 1 39' 1184 1229 1 683 I3S5 1 3 20 43 1411 1456 i5oi i547 1592' 1637 1728 1773 45 1819 1864 1909 1954 2000 2045 j 2090 2 1 35 2I8I 2226 45 982271 23i6 2 362 2407 2432 2497| 2543 2588 2633 2678 45 27231 2769! 2814 2859 2904 2949' 2994 3356 340 1 i 3446 3 040 3o85 3i3o 45 3175 3220J 3265 33io 3491 3536 358i 45 3626 3671! 3716 3762 3807 3852] 3B97 3o42 3987 4o32 45 4077! 4122 4167 4212 4257 43o2 4347 4^92 4437 4482 45 4527 4372 4617 466: 4707 4732, 4797 4842 4887 4932 45 4977 5o22! 5067 5ii2 5i57 5202' 5247 6292 5337 5382 45 5426 5471! 55i6 556i 56o6 565i! 56q6 574. 5786 5830 45 5873; 5920, 5965 6010 6o55 6100 6144 6189 6234 6279 45 6324 6369' 64 1 3 645s 65o3 6548; 6593 6637 6682 6727 45 98677J; 6817! 6861' 6906 6951 6996 7040 7o85 7i3o 7175 43 7219 7264 7309: 7353 7398 7443] 7488 7532 7577 7622 40 7666I 77 u 7756; 7800 8n3| 8r57 8202' 8247 7845 7890 7934 8336 838i 7979 8425 8024 8068 43 8291 8737 8470 85i4 45 8559' 8604: 8648; 8693 9003, 9049 9°94: 9i38 87S2 8826 8871 8916 9361 8960 45 9i83 9227! 9272 93i6 940 5 43 9450; 9494 9339 9583 9805; 9939 9983| ••28 990339 o383 o428| 0472 0783; 0827] 0871 0916 991226 1270 i3i5| i359 9628 9672 9717 9761 9806 9850 44 ••72 •1.7 •161 •206 •25o •294 44 o5i6 o56i o6o5 o65o :fi^' 0738 44 0960 1004 1049 \t. 1182 44 i4o3 I448| 1492 1890 1935 2333 2377 i58o 1625 44 1669! 1713: 1758 1802 1846 1979 2023 2067 44 211li 2l56 2200 2244 22S8 2421 2465 25o9 44 2554 2598 2642 2995: 3039; 3o83 2686 2730 2774 2819 2863 2907 2931 44 3127 3172 32i6i 3260 33o4 3348 3392 3833 44 3436 3480 3524 3568 36i3 3657 3701 3745 3789 44 38771 3921! 3965 4317I 4361! 44o5 4009 4o53 4097 4141 4i85 4229 4273 44 4449 4493 4537] 458i 4625 4669 47 1 3 44 4737' 4801! 4845 4889 if. 49771 5o2i 5o65 5 1 08 5i52 44 5196' 5240' 5284! 5328 54.6 1 5460 55o4 5547 5591 6o3o 44 595635; 5679' 5723 5767 58ii 5854 1 5898 5942 5986 44 6074' 6117 6161 62o5 6249 6293 6337 638o] 6424 6468 44 65i2: 6555 6599 6949 6993 7037 7386 743o! 7474 6643 6687 6731 6774 6818 6862 % 44 7080 7124 7168 7212 7255 7299 7692 7736 8129 8172 44 7517 7561 7605; 7648 8041 8o85 7779 8216 44 7823 7867 7910 8239! 83o3 8347 7934 1% 44 8390 8477! 8521 8564! 8608 8652 44 8693 8739' 8782! 8826 8869' 8913] 8956 93o3, 9348; 9392 9000 9043 9087 44 9i3r 9174 9218; 9261 9435 9479 ' 9522 ■44 9565 9609 9652 9696! 9739' 9783j 9826 9870 9913 1 9957 1 43 i • 1 2 .' 1 4 1 5 i 6 7 1 _«. 1 9 ! D.__ A TABLE OF . LOGARITHMIC SINES AND TANGENIS FOK EVEKT DEGREE AND MINUTE OF THE QUADRANT. Kemark. The minutes in the left-hand column of each 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 DPIGREES.) A TABLF OF LOGARITHMIC M. Sine 0- 000000 1). 1 Cosine ! I), i Tang. 1). Cotang. 60 10.000000 • oooooo Infinite. I 6.468726 5017-17, 000000; "00 6-468776 5017.17 13-53627 i a 764756 2934-85 000000 -00 764756 2984 83 285244 3 940847 2082-31 000000, -00 940847 2082 81 0591 58 57 4 7.065786 1615-17I i3i9-68 ni5-75 0000001 -00 7-065786 i6i5 •7 12^984214 56 5 162696 000000 : -00; i6?696 1819 iii5 H 887804 758122 55 6 241877 9.999999' -01 241878 54 I 308824 066-53 852-54' 999999 -oi 308825 ^r 591175 638 1 83 53 366816 999999 .01 866817 54 52 9 417968 762-63; 999999 999998 -01 417970 762 63 582o3o 5i 10 463725 689-88 -01 468727 689 88 586278 12.494880 457091 5o II 7.5o5ii8 629-81! 9.999998 •01 7.5o5i2o 629 81 4q 48 12 542906 579-36 999997 •CI 542909 577672 579 33 i3 577668 536-41 999997 •01 586 42 422828 47 14 609853 499-38 999996 •01 609857 499 ',t 890143 46 i5 639816 467-14 438-81 999996 .01 689820 43J 860180 45 i6 667845 999995 .01 667849 82 382i5i 44 \l 694173 4l3-72 999995 .01 694179 4i3 73 3o582i 43 718997 391.35 999994 •01 719004 891 36 280997 257516 42 ^9 742477 764754 371-27 999993 -01 742484 371 28 41 20 353-15 999993 -01 764761 35i 86 235289 40 21 '■& 336-72 9-999992 -01 7-785951 836 73 12.214049 198845 89 38 22 321-75 999991 -01 806155 321 76 23 825451 3o8-o5 999990 -01 825460 3o8 06 174540 37 24 843934 295.47 999989 •02 848944 861674 295 49 i56o56 36 25 861662 283.88 999988 999988 •02 288 90 188826 85 26 878695 895085 273-17 -02 878708 278 18 121292 34 11 263-23 999987 -02 895099 263 25 104901 38 910879 Ifl 999986 .02 & 254 01 089106 32 29 926119 999985 .02 245 40 078866 3i 3o 940842 237-33 999983 .02 940858 287 35 059142 3o 3i 7.955082 229-80 9-999982 .02 7.955100 229 81 1 2 - 044900 ll 32 968870 222-73 999981 .02 968889 222 75 08 1 1 1 1 33 982233 216-08 999980 .02 982253 216 10 017747 27 34 995iq8 8-007787 209-81 999979 .02 995219 lit 88 004781 26 35 2o3.qo 999977 •02 8-00780^ ?3 11-992191 25 36 020021 198-31 999976 .02 020043 198 24 ll 081919 193.02 999975 .02 081945 188 o5 23 043 DO I 188.01 999973 .02 048527 08 956473 22 39 054781 183-25 999972 .02 054809 i83 27 945I9I 21 40 065776 178-72 999971 .02 o658o6 178 74 984194 20 41 8-076500 174-41 9-999969 999968 .02 8-076581 174 44 11-928469 9 1 8008 :? 42 086965 170-31 -02 086997 170 34 43 097183 166-39 999966 •02 097217 166 42 002788 n 44 107167 162-65 999964 -03 107202 162 68 802797 888037 16 45 I 16926 159.08 Q99963 • 03 1 16968 ',U 10 i5 46 I 2647 1 153.66 999961 .03 126D10 68 873490 4 47 i358io 152.38 999959 •03 i3585i l52 41 864149 i3 48 144953 149-24 999958 ■o3 I44Q96 149 27 855004 12 49 153907 146-22 999956 .03 1589^2 146 27 846048 II 5o 162681 143.33 999954 .03 162727 143 86 887273 10 5i 8-171280 I797'3 187985 196102 140-54 9.999952 .o3 8.171828 140 57 11.828672 I 52 53 54 137-86 1 135-29 1 i32-8o 999950 999948 Q99946 .o3 • 03 .o3 188086 196156 182 84 820287 811964 808844 55 204070 i3o.4i 999944 • 03 204126 180 -44 ffi^ 5 56 21 1895 219581 j 128.10 999942 -04 211953 128 -14 4 u 1 125.87 999940 -04 219641 125 • 90 780359 772805 3 227134 123.72 999988, -04 227195 123 -76 2 59 234557 ; 1 2 1 . 64 999986, -04 284621 121 -68 765379 I 60 241855 i 119-63 9999841 -04 241921 119-67 758079 Cosine 1). Sine j89°| Cotang. D. _;rang. M. SINES AND TANGENTS (1 DEGllEE.) 19 }L. Sine 1 D. 1 CoBine 1 D. 1 Tang. D. 1 Cotang. 1 8.241855 119.63 9.999934 •04 8-241921 119-67 11.758079! 60 I 249033 117.68 999932 .041 249102 117-72 7 50898 1 59 743833 58 a 256094 ii5-8o 999929 -04 256i65 113-84 3 363o42 113.98 999927 -041 263ii5 114-02 736885 U 4 269881 112.21 999923 •04' 269936 112-25 730044 5 276614 263243 iio-So 999922 •04! 276691 110-54 723309 55 6 108-83 999920 •04I 283323 io8-87 716677 54 I 289773 1 07 . 2 1 999918 •04 289836 107-26 710144 53 290207 103-65 999913 .04 296292 103-70 703708 52 9 302346 104- i3 999913 •04: 3026J4 104-18 697366 5i iO 308794 102-66 999910 •041 308884 102-70 691 116 5o i: 8.314904 101-22 9.999907 999905 •04 8.3i5o46 101-26 11-684954 4Q 13 321027 90.82 98-47 •04i 321122 99-87 678878 48 a 327016 332924 999902 •04 327114 .o5 333025 98-51 672886 47 1 / 97-14 999899 97-19 95-90 666973 46 15 338753 93-86 999897 •05 338856 661144 45 i6 344D04 94-60 999894 •o5 344610 94-65 655390 44 \l 35oi8i 93-38 999891 .o5l 330289 •o5; 333895 93-43 6497 ' ' 43 355783 999888 92^24 644 io5 42 >9 36i3i5 999885 •oSj 36i43o 91-08 82-85 638510 41 20 366777 999882 .05! 366895 633 1 o5 40 31 8.372171 9-999879 999876 -o5 8-372292 11.627708 622378 ^ 33 377499 87-72 -o5 377622 87-77 86-72 33 382762 86-67 999873 .05 382889 617111 37 34 387962 393 101 85-64 999870 ■ o5 388092 393234 83-70 611908 36 35 84-64 999867 .05 84.70 606766 35 36 398179 83-66 999''^64 .o5 398315 83.71 601685 34 11 4o3i99 82-71 999861 .o5 403338 82.76 81-82 596662 33 408161 tu 999838 • 05 4o83o4 5867^7 32 39 4i3o68 999854 .05 4i32i3 80-91 3i 3o 417919 79.96 999831 .06 418068 80 -02 581932 3o 3i 8-422717 ]U 9.999848 .06 8.422869 427616 70-14 78.30 11-377131 11 33 427462 999844 .06 572382 567685 33 432156 77-40 76-37 999841 .06 4323i5 77.45 27 34 436800 999838 .06 436962 44i56o 563c38 26 35 441394 75-77 999834 •06 75.83 558440 23 36 445941 74-99 999831 -06 446110 75-05 553890 24 549387 23 U 450440 74-22 999827 -06 45o6i3 74-28 454893 73-46 999823 .06 455070 73-52 544o3o 22 540319! 21 39 459301 72-73 999820 .06 439481 463849 72-79 40 463665 72-00 9998161 .06 72-06 536i5i| 20 41 8.467985 71-29 9.999812 .06 8-468172 71.35 11-531828 :? 43 472263 70-60 999801 .06 472434 70-66 527546 523307 519108 43 44 '^ 69.91 60-24 68-59 .06 .06 485o5o 69.^8 6?-65 17 16 45 484848 999797 999793 •07 514950 5io83o i5 46 488963 67-04 67.31 66.69 66.08 .07 489170 49J25o 68-01 14 % 493040 999790 999786 -07 67.38 506750 i3 TI •07 497293 66.76 502707 \2 49 999782 •07 501298 505267 66-15 498702 11 5o 5o5o45 65.48 999778 •07 65-55 494733 11-490800 486902 10 5i 53 516726 52o55i 64-89 64-3i 9-999774 999769 •07 •07 8.509200 5 1 3098 64-Q6 64-39 § 53 63-75 999763 •07 63-82 483o39 7 54 63-19 999761 •07 5243^6 63-26 479210 47^414 6 55 524343 62-64 9997^7 •07 62-72 5 56 528102 62 - ! 1 999753 •07 528349 62.18 4]i65i 401920 4 U 531818 61.58 999748 •07 532080 61 -65 3 535523 6i.o6 999744 •07 535779 6i-i3 464221! 2 59 539186 60.55 999740 •07 a^I 60-62 ;6o533 1 6o 543819 60. 04 999735 -07 60 -13 456916 1 Cosine D. 1 Sine 88° Cotang. D. Tang 1 749 3o 95 •14 3i 10 168252^ 7 54 332607 3o 82 99^lh •14 8336.3 3o t 166387: 6 55 834456 3o tl .14 835471 3o 83 1645Z9I 5 56 836297 83Siio 3o 998976 ^8958 .,4 837321 3o 70 57 162679' 4 ll 3o 43 .i5 839.63 3o 160837 3 839956 3o 3o •151 840998 .,5] 842825 3o 45 i5ooo2! 3 59 841774 8435^5 3o 17 998950 3o 32 157.75! I 155356! 60 3o 00 998941 -15! 844644 30-19 ; Cosino E ^Sine 8'60 Cotang. D J Tang. >lj r^ [1 1 DEOKEKf .) A TABLE OF LOOAUITHMIC TJ Sine 1 D. ! Cosine j D. Tang. 1 D. 1 Cctang. 1 ""1 8.843585 3o-o5 9-9q894i| .15 8-844644! 30-19 II -155356: 60 i I 845387 847183 29-02 29-80 998932! • i5 846455; 30-07 1 53545; So i5i74o! 58 a 998923 ..5 848260 2q-95 3 848971 2Q-67 99^914 .i5 850057I 29-82 ;g?gi u 4 85oi5i 29-55 998905 .i5 85 1846' 29-70 5 85n25 29-43 9988^7 .i5 853628] 29-58 146372 55 1 6 8542QI 29-3i • i5 855403 i 29-46 144597 54 I 856o49 29-19 998878 ..5 857171 29-35 :42S'39 141CO8 53 857801 29-07 998869 • 15 858932 29-23 52 9 85g546 28-96 998860 .i5 29-11 139314 5i 10 861283 28-84 998851 .i5 862433 29 • cc 28-88 13-567 5o II 8-863014 28-73 9-998841 .i5 8-864173 11 135827 ii 13 864738 28-61 998832 • 15 865906 28.77 134094 i3 866455 28-50 998823 .16 867632 28-66 132368 47 14 868i65 28-39 28-28 998813 .16 869351 28-54 1 30649 46 i5 869868 998804 .16 871064 28.43 128936 45 i6 871565 28-17 998795 .16 872770 28.32 127230 44 17 873255 28-06 998785 .16 874469 28-21 1 2 553 1 43 i8 874938 27-95 998776 .16 876162 28-11 123838 42 19 876615 27-^6 998766 .16 877849 28-00 I22l5l 41 20 878285 37-73 998757 .16 879529 27-89 120471 40 ai 8-879949 27-63 9-998747 • 16 8-881202 27-79 II 118798 l^ 32 881607 883258 27-52 998738 .16 882869 27-68 ii7i3i 23 27-42 998728 .16 884530 27-58 115470 ll 24 884903 886542 27-31 998718 .16 886 1 85 27-47 Ii38i5 25 27-21 998708 •16 887833 27-37 1 12167 35 26 888174 27-11 9986^9 •16 889476 27.27 1 io524 34 27 889801 27-00 •16 89.112 27-17 108888 33 28 891421 26-90 26-80 998679 • 16 892742 27-07 107258 32 29 893035 99S669 •17 894366 26.97 26.87 105634 3i 3o 894643 26-70 998659 •17 895984 8-897596 io.ioi6 3o 3i 8-896246 26-60 9-998649 •17 26.77 11-102404 ll 32 8Q7842 26-51 998639 •17 899203 36.67 100797 33 899432 26-41 998629 •17 900803 26-58 099197 27 34 901017 902596 26-31 998619 •n 902398 903987 905570 26-48 097602 26 35 26-22 998609 •17 26-38 096013 25 36 904169 26-12 998599 •17 26-29 094430 24 ll 905736 26 -03 998580 99857^ •17 907147 26-20 092853 33 907297 25.93 •>7 908719 910285 26-10 Oni28l 089715 088 1 54 22 09 908803 25-84 998568 •17 26-01 21 40 910404 25.75 998558 •17 911846 25-92 25-83 30 41 8-911949 913488 25-66 9 -998548 •17 8-913401 11-086599 \l 42 25-56 998537 •17 914951 25-74 o85o49 o835o5 43 9l5022 25-47 998527 •17 916495 918034 25-65 17 44 9i655o 25-38 998516 .18 25-56 081966 16 45 918073 25-29 998506 .18 919568 25-47 25-38 0S0432 i5 46 919591 25-20 998495 • 18 921096 078904 077381 14 % 921103 25-12 998485 • 18 932619 25-30 i3 922610 25-o3 998474 .18 924 1 36 25-21 075864 la 49 924112 24-94 24-86 998464 .18 925649 25-13 074351 u 5o 925609 998453 .18 1 927156 25-03 072844 10 5i 8-927100 24-77 9-998442 .18 : 8-928658 24-95 24-86 11-071342 I 52 928587 93oofS 24-69 998431 .18 93oi55 069845 53 24-60 998421 •i8i 931647 24-78 068353 7 54 931 544 24-52 998410 .181 933i34 24-70 066866 6 55 933oi5 24-43 998399 .18 934616 24-6i 065384 5 56 934481 24-35 998388 .18 936093 34-53 o6390T 4 U 935942 24-27 998377 998366 998355 .18 937565 24-45 062435 3 937398 938850 24-19 .18 939032 24-37 060068 2 5g 24-11 .18' 940494 .18! 941962 24 -30 , o595o6 I 6o_ 940296 24-c3 998344 24-21 D, • o59o48' .Cosine I>. 1 Sin«_ 85 °i CoUmg. i_Tang.__; :«. SINKS AND TANGKXTS (5 DKGREKS.) ^: Sine 1). Cosine 3 Tanc:. D. Cotanj?. o 8 940296 9417J8 24 o3 9-998344 '9 8-94.952 24 21 ii.o58o48' 60 I 23 t^ 998333 »9 943404 24 »3 056596 i? a q43i74 23 998323 19 944852 24 03 o55i48 3 9i46o6 23 79 9983.1 '9 946295 947734 23 97 o537o5 57 4 946034 23 7> 998300 »9 23 r. o52266 56 5 947456 23 63 9982S9 »9 949.68 23 o5o832 55 6 7 948814 23 23 ^i 998277 998266 »9 '9 950397 952021 23 23 -4 66 049403 047979 54 53 » 95.69^ 23 40 998255 •9 953441 23 60 046359 52 9 953100 23 32 998243 '9 954856 23 5i 045.44 5i 10 954499 23 25 998232 j 19 956267 23 44 043733 5o II 8-9''38o4 937284 958670 23 17 9.998220, 19 8-957674 23 37 .1-042326 ^2 'I 23 .0 9982091 19 939073 23 29 040925 48 i3 23 02 llV^ 19 960473 23 23 039^27 47 14 960032 22 9^^ '9 961866 23 14 036.34 46 i5 961429 22 h 998.74 »9 963255 ?3 07 036745 45 i6 962801 22 80 998163. '9 964639 23 00 03536. 44 \l 964 1 7c 22 73 99815. >9 966019 22 & 03398. 43 965534 22 66 998.39' 998128 20 967394 968766 2 2 032606 42 '9 966893 22 59 20 22 79 o3i234 41 20 968249 22 52 998116' 20 970.33 22 71 020867 ...028504 40 2l 8.969600 22 44 9.998104 20 8-07.496 22 65 ?2 22 970947 22 38 998002 i 998080 20 972835 ' 22 V 027145 38 23 972289 22 3. 20 974209 22 5i 025791 ?2 34 973628 22 24 99806S1 20 973560 22 44 024440 36 25 974962 22 '7 998056 20 976906 22 37 023oq4 35 26 976293 22 10 998044! 20 978248 22 3o 021752 34 \l 977619 978941 22 o3 998032 j 20 979586 22 23 0204.4 33 2. 97 998020 20 980921 22 n 019079 32 ?9 9H0259 98,573 2. ?? 998008; 20 982251 22 10 017749 016423 3. 3o 2. 997996, 20 983577 22 04 3o 3i 8-982883 2. 77 9-997985) 20 8-984899 21 97 ii-oi5ioi 29 32 984.89 2. 2? 997972; 20 9862.7 21 l\ 013783; 28 33 98549. 2. 997939! 20 987532 2. 012468 ll 25 34 35 988083 21 21 57 5o 997947' 997933 20 988842 990149 2. 2. 78 7' 011.58 ooo85i 36 989374 2. 44 997922 99145. 21 65 008549 24 \l 990660 2. 38 997010 997897 997883, 992750 21 56 007250 003955 23 99'9i3 2. 3. 994045 21 32 22 39 993222 2. 25 995337 21 46 004663 2. 4o 994497 21 »9 997872 996624 21 40 003376 20 41 8.995768 2. 12 9-997860 8-997908 21 34 11-002092 ;i 42 997036 21 06 997847 999188 21 27 000812 43 998299 2. 00 997835 9 • 000465 21 21 20-999535 99S262 17 44 99<)56o 20 l^ 9978221 00.738 2. i5 16 45 9 • 0008 1 6 20 997809! oo3oo7 21 S 996993 i5 46 002069 0033 18 20 82 997797! 997784 004272 21 995728 .4 % 20 .76 005534 20 97 994466 i3 004563 20 .-0 99777 «i 006792 20 tl 993208 12 i^ oo58o5 20 • 64 9977581 008047 009298 20 99.953 II 5o 007044 9-008278 20 -58 9977451 20 80 990702 10.989454 988210 10 5i 20 -52 9.997732 9.010546 20 74 I 52 0095.0 20 -46 9977'9; 997706 01 1700 oi3o3i 20 68 53 010737 20 -40 20 62 986969 7 54 01.962 20 •34 9976031 997680 014268 20 56 985732 6 55 0.3.82 20 -.1^ oi55o2 20 5i 984498 983268 5 56 014400 20 997667 016732 20 45 i ll 0.56.3 20 ■17 997654! 017959 019183 20 40 982041 016824 20 -.2 99764 1 ' 20 33 980817 2 59 oi8o3i 20 -06 997628 020403 20 28 978380 I 6c 019235 Coeine 20 -00 9976141 021620 20 23 __ 1). Sine [Sr-^ _Corang. P _ Tar4r._ m7 Hi (^ DEGREES.) A TABLE OF LOGAKITiiMlt it. Sine D. Cosine ). Tang. J;, n Cotang. 60 9.019235 20-00 9-997614 I2 9-021620 20-23 10-978380 I 020435 •9 o5 997601 22 02834 20 17 9771661 5*1 975q56 58 a 021632 19 B9 997588 22 024044 20 3 022825 19 84 997374 22 025231 20 06 974749' 57 4 024016 19 78 9973611 22 026455 20 00 973543! 56 5 025203 19 73 997547 22 021655 '9 93 972345 55 6 026386 J9 67 997534 23 028852 '9 90 971148; ^4 I 027367 19 62 997320 23 o3oo46 '9 85 969954 53 028744 19 57 997307 23 o3i237 '9 79 968763 52 9 029918 '9 5i 997493 997480 23 032425 '9 74 967575 c' 10 031089 19 47 23 o336o9 '9 69 966391 5o II 9-032257 19 4i 9-997466 23 9-034791 19 64 10-965209 ^ 12 033421 19 36 997452 23 035969 19 58 964031 i3 034582 '9 3o 997439 23 037144 19 53 962856 47 14 035741 >9 25 9974231 23 o3S3i6 >9 48 961684 46 i5 036896 '9 20 9974 HI 23 039485 '9 43 96051 5 45 i6 o38o48 '9 i5 9973971 23 040631 '9 38 95q349 938187 44 n 039197 •9 10 997383 23 041813 19 33 43 i8 040342 ;? o5 997369 23 042973 '9 28 957027 42 19 041485 99 99733-) 23 044 1 3o 19 23 955870 41 20 042625 18 94 997341 23 045284 '9 18 954716 40 21 9-043762 18 89 9-997327 24 9-046434 19 i3 10-953566 ? 22 044895 18 ^4 9973 1 3 24 047582 '9 08 952418 23 046026 18 79 697299 9972S3 24 048727 '9 o3 951273 37 24 047154 18 73 24 049869 18 98 95oi3i 36 25 048279 18 70 997271 24 o5ioo8 18 t lt?,l 35 26 049400 18 65 997257 24 032144 18 34 11 o5o5i9 o5i63! 18 60 997242 24 053277 18 84 946723 33 18 55 997228; 24 034407 18 79 9/45593 ?' ^9 032749 053859 18 5o 997214; 24 035535 18 74 944465 3i 3o 18 45 997 '99 24 036639 18 ^? 943341 3o 3i ■9 - 054966 iS 41 9.9971831 24 9-057781 18 65 10-942219 It 32 056071 18 36 997570 24 038900 .8 ^ 94 1 1 00 33 058271 18 3i 997156 24 060016 .8 930984 938870 n 34 18 27 997'4i| 24 061 i3o .8 5i 26 35 059367 18 22 997' 27 i 24 062240 18 46 937760 25 36 060460 iS 17 997112 24 063348 18 42 936652 24 ll o6i55i 18 i3 997008, 997083, 24 064453 18 37 935547 23 062639 18 08 25 o65556 18 33 9344 a 22 59 063724 064806 18 04 997068; 25 066655 18 28 933345 21 40 17 99 997053 25 067752 18 24 932248 20 41 9-065885 17 q4 9-997.)39J 25 9-068846 18 '9 io-93ii54 >9 42 06696: 17 00 9970241 25 069938 18 i5 930062 18 43 o68o36 I / ^6 9.;7009: 25 071027 18 10 928973 927S87 '7 44 069 1 07 17 81 906994 25 072113 18 06 16 45 070176 J7 77 9'<76| -20 141340' 15.48 54 i5.i6 99^^59, •29 142269' 143196 15.45 55 139037 l5.|2 99>'^4i; -29 15.42 856804 5 56 139944 i4o«5o 13.09 995S23 •29 1441 21' 15.39 855879 4 854956 3 u l5.o6 995806 1 •29 143044 15.35 141754 142655 l5-o3 995788 .29 143966 146885 13-32 834034 i 59 I5.00 99^77' -29 15-29 15.26 833ii5 I 6o^ 143555 14-96 9957531 -29 147803 852197 1 Ci^irie _J>:_i Siiie 1^30 ! C0t!Ulg._| _l^-._.i Tiiiig. J_il. J ZG {^ DEGREES.) A TABLE OF LOGARITHMIC TIT" Sine D. Cosino D. Tan-. B. Cctang. 9-143555 14-96 9.995753 ".3o 9-147803 13.26 IO-852I07 851282 6o" I 144453 14 93 995735 .30 148718 i5 23 u 5 145349 14 90 995717 .30 149632 i5 20 85o368 3 146243 14 87 995699 -3o 1 5o544 i5 17 849456 u 4 i47i36 14 84 995681 -3o 1 5 1454 i5 14 848546 5 148026 14 81 995664 .30 152363 i5 II 847637 55 6 1489 1 5 14 7S 995646 -3o 153269 i5 08 846731 54 I 140802 14 75 995628 •3c 154174 i5 o5 845826 53 1 50686 14 72 995610 -3o 1 55077 i5 02 844923 5a 9 i5i569 14 69 995591 •3o 155978 14 99 844022 5i 10 1 5245 1 14 66 995573 -30 156877 14 96 843123 5o II 5-i5333o 14 63 9-995355 -3o 9-137773 14 93 10-842225 ^^ 12 i542o8 14 60 995537 -3o 158671 14 ?7 841329 840435 i3 i55o83 14 57 995019 -30 1 59565 14 ii 14 i55n57 1 56830 14 54 995501 -3i 160457 14 84 839543 i5 14 5i 995482 • 31 161347 14 81 838653 i6 157700 14 48 995464 -31 162236 14 79 837764 44 \l 158569 14 45 995446 .3i 1 63 1 23 14 76 836877 43 159435 14 42 995427 -31 164008 14 73 835992 42 '9 i6o3oi 14 39 995409 -3i 164892 14 70 835io8 41 20 161 164 14 36 995390 • 31 165774 14 67 834226 40 21 9-162025 14 33 9-995372 • 31 9-166654 14 64 10-833346 U 3i 162885 14 3o 995355 • 3i 167532 14 61 832468 -.3 163743 14 27 995334 -3i 168409 14 58 83 159 1 37 24 164600 U 24 995316 -3i 169284 14 55 830716 36 25 165454 14 22 995297 -3i 170157 14 53 829843 35 26 1 66307 U '9 995278 -3i 171029 14 5o 828971 34 1 27 1 67 09 14 16 995260 • 31 171899 14 47 828101 33 28 168008 14 i3 995241 -32 172767 14 44 827233 32 -jg 168856 14 10 995222 -32 173634 14 42 826366 3i 3o 169702 14 07 995203 -32 174499 14 39 825501 1 3o 3i 9-170547 14 o5 9-995184 .32 9-175362 14 36 io.824638i 20 823776 28 32 171389 14 02 9951 65 -32 176224 14 33 33 172230 i3 99 995146 •32 177084 14 3i 8229161 27 34 173070 i3 96 995127 •32 177942 178799 14 28 8220581 36 35 173908 i3 94 995108 .32 14 25 821201! 25 36 174744 i3 11 995089 .32 179653 14 23 820345 24 u 175578 i3 995010 995o5i .32 i8o5o8 14 20 819492 23 17641 1 i3 86 .32 i8i36o 14 n 818640 22 39 177242 i3 83 995o32 -32 182211 14 i5 817789 21 40 178072 i3 80 9950 1 3 •32 iS3o59 14 12 816941 20 41 9-178900 i3 77 9.99499^ -32 9.183907 14 09 10-816093; 19 81 5248; 18 42 179726 i3 74 994974 -32 184732 14 07 43 i8o55i i3 74 9949^^ -32 185597 186439 1 1 0', 814 '.o3, 17 44 181374 i3 9949^3 -32 M 02 81 3561 1 16 43 182196 i3 66 994916 994896 • 33 187280 i3 99 812720J i5 46 i83oi6 i3 64 •33 188120 i3 96 811880 14 47 183834 i3 61 994877 994857 -33 188938 i3 93 811042I i3 4^ 18465: i3 59 -33 189794 1 3 % 810206: 12 4n 185466 i3 56 994838 -33 190629 i3 809371; II 8oS538, 10 5o 186280 i3 53 994818 -33 191462 i3 86 5i 9.187092 i3 5i 9-99479"^ .33 9- 192294 i3 84 10.8077061 9 806876 9 52 187903 i3 48 99 '•779 994759 .33 193124 i3 81 53 188712 i3 46 .33 193953 i3 79 806047 7 54 189519 190325 i3 43 994739 .33 194780 i3 76 8o5220 6 55 i3 41 994719 .33 195606 i3 74 804394 5 56 191130 i3 38 994700 .33 196430 i3 7' 803570] 4 U 191933 i3 36 994680 .33 197253 i3 69 802747; 3 19273 i i3 33 994660 .33 198074 i3 66 801926 2 bq ; 193534 1 3 3o 994640 -33 198894 i3 64 801106! I 60 194332 i3 58 994620 .33 199713 i3.6i 800287 1 Cosine ~P _ Sine 81=^ Cotang. D . Tan£.__ M 1 SINKS AND TAN OK NTS . (9 DEGREES.] 27 ii. Sino _^!_ Conine 1 P. , Tang. D. Cotang. "o- 800287 "60" o 9 (94332 i3. 28 9-994620"" .33 9.199713 i3 61 I 1951 iO i3. 26 994600: •3{ 200520 i3 59 7994-1 7986)5 5I 2 195923 i3 23 994580 .33; 201343 i3 56 3 196719 io83o2 i3 21 994360, •34' 202139 i3 54 797841 ^I 4 i3 18 994340^ .34! 202971 34' 203782 i3 52 797c 20 56 5 i3 • 6 994519 i3 49 796118 55 793408 54 6 199091 li i3 99i499, •34 204392 i3 ii 42 40 2 "•ouSi i3 1 1 08 06 994479, 99443Q 99443^1 •34 •34 2o54oo 206207 207013 i3 i3 i3 794600 793793 792987 53 52 5i 10 202234 04 994418: •34 207817 1 3 38 792183 5o II ; 2o3oi7 01 9.994397 •34' 9-208619 i3 35 IO-79I38I ^? 12 2o37'/7 99 994377 •34' 209420 i3 33 7Qo58o i3 204577 96 994336 .34 2IO?20 i3 3i 780780 ii U 2o53d4 94 .34 2tioi8 i3 28 78M2 i5 2o6i3i t^ 9943 1 6 •34 2ii8i5 i3 26 788185 45 i6 206906 994295 •34 212611 i3 24 785802 44 \l 201679 208452 87 85 994274 994254 .35 •35 2i34o5 214198 i3 i3 21 «9 43 42 »9 209222 82 994233 .35 2149H9 13 \l 785011 41 20 209992 80 994212 .35 215780 9-216568 i3 784220 40 31 9-210760 7S 9-994191 .35 i3 12 !€• 783432 39 22 2in26 75 9941 7 1 .35 217356 21S142 i3 li 782644 38 23 212291 73 994-50 .35 i3 cS 781858 37 24 2i3od5 71 994120 994108 .35 218926 i3 c5 781074 780290 36 25 2i38i8 68 .35 219710 i3 c3 35 26 214579 1 2 66 994087 994066 .35 220492 i3 ti 779508 778728 34 »7 215338 64 .35 221272 222052 12 C9 33 28 216097 61 994045 •35 12 ^7 777948 32 ?9 216S34 59 994024 .35 222830 12 H 777170 3t 3o 217600 9-2i836i 12 5] 99ioo3 .35 2236o6 12 ?2 -76394 3o 3i 5t 9.9939S1 • 35 9-224382 12 88 10.775618 11 32 2191 16 53 993960 .35 225i56 12 774844 33 219S68 So 99393q! • 35 225929 12 86 774071 2 34 220618 48 .35 226700 12 84 773300 35 221367 222115 12 46 •36 111% 12 81 772029 25 36 44 993875 • 36 12-79 771761 24 3t 222861 >> 42 993854 • 36 23o539 12.77 11-73 770993 23 3« 2236o6 12 39 993832 • 36 770227 22 39 224349 37 99381 1 • 36 II.73 769461 21 4o 223002 9-225833 33 ,•&"? •36 23l302 1271 760698 20 4i 33 • 36 9-232065 12.69 10-767935 \t 42 226573 3i 99^746 •36 232826 12.67 U.65 767174 43 227311 28 993725 • 36 233586 766414 \l 44 328048 26 993703 •36 234345 12.62 765655 45 228184 24 993681 • 36 235io3 12. 60 764897 i5 46 229518 22 993660 .36 235859 12-58 764141 14 % 2302 02 20 993638 • 36 236614 (2^56 763386 i3 2J09S4 18 993616 •36 237368 238120 12^54 762632 12 49 231714 16 993391 12. 52 761880 11 ^o 232444 14 993572 238872 12. 5o 761128 10 5i 0'233i72 12 9-9^3550 9.23962a 12.48 10-760378 ? 52 233899 09 993528 240371 |2^46 ??C 5! 234623 07 993506 241 1 18 I2^44 I fii a)534Q 236073 o5 993^84 241865 12^42 758i35 55 o3 993462 242610 12.40 757390 5 56 236795 12 01 993440 243354 12-38 756646 u 237515 238235 M 99 993418 244097 244839 12-36 755903 97 993396 12.34 755161 J9 238953 239670 95 993374 245579 1232 754421 6o 93 993351 246319 CoUng. 1 ^•3o 753681 Ocaiiie -_? Siiie i JOo u K Tang. ij 28 __J^ DKGUEEB.} A lABLE OF LOGARITHMIC rs~ Sine D. Cosine D. Tang. ]). Cotang. 9 • 239670 11-93 9-993351 .37 ~9^2"46379 12-30 10 •753681 60' I 240386 11.91 11-89 993329 .3t 247057 12-28 752943 5o 2 241 lOI 993307 993285 .37 247794 248530 12-26 752206 58 3 241814 11-87 •37 12-24 751470: 57 4 242526 11-85 993262 •37 249264 12.22 7507 36i 56 5 243237 11-83 993240 -.11 249998 25o73o 12-20 750002 1 55 b 243947 ii-Si 993217 993195 12-18 7492-'oi 54 748539: 53 I 244656 11-79 • 38 231461 12-17 245363 11-77 993172 • 38 252191 I2-l5 747809 52 9 246069 246773 ,1.75 993149 .38 252920 233648 12.13 747080; 5i 10 11.73 993127 • 38 1211 746352 5o 11 0-247478 248181 n.71 9 -993 '04 .38 9-254374 12 09 10.745626 49 13 11-69 993081 • 38 255IOO \lll 744900, 48 i3 248883 wii ^^31 .38 255824 744176 47 14 249583 • 38 256547 i2.o3 743453 46 i5 230282 11-63 ^3oi3 • 38 257269 12-01 742731 45 i6 230980 11-61 992990 .38 237990 258710 1200 742010 44 17 231677 232373 ii.5o 11.58 992967 •38 11.98 741290 43 18 992044 .38 259429 11.96 740571 42 •9 233067 11-56 992Q2I .38 260146 n-94 739854 41 ao 253761 11.54 992898 •^^ 260863 11-92 739137 10.738422 40 21 9-234453 11-52 9-992873 •38 9.261578 11-89 l^ 22 235144 H 5o 992852 •?8 262292 737708 23 235834 11.48 9^^2829 .39 263oo5 11-87 736995 37 24 256523 n-46 992806 .39 263717 11-85 73/3283 36 25 257211 11-44 992783 •39 264428 11-83 735572 35 26 2585?3 11-42 992739 •39 263138 11-81 734862 34 ^5 11.41 992736 .39 263847 11-70 11-78 734153 33 259268 11.39 992713 •39 266555 733445 32 29 259951 11-37 992690 •?9 267261 11-76 732739 732033 3i 3o 260633 11.35 992666 .39 267967 11-74 3o 3i 9-26i3i4 11.33 0-99ib43 •39 9-268671 11-72 10-731329 11 32 261994 11-31 992619 •^9 269375 11-70 730625 33 262673 11 -3o 992596 •39 270077 11.'^^ 729923 27 34 263351 11-28 992572 •39 270779 \::p. 729221 26 35 264077 11-26 992549 •39 271479 272178 728321 25 36 264703 11.24 992525 •59 n-64 727822 24 II 265377 11.22 9925oi •39 272876 11-62 727124 23 266031 11.20 992478 • 40 273573 11-60 726427 22 39 266723 II. 19 992454 • 40 274269 11-58 725731 21 40 267395 II-IT ii-i5 992430 • 40 274964 11-57 11-55 725o36 20 41 9 -268065 9-992406 • 40 9-275658 10-724342 ;i 42 268734 ii-i3 992382 • 40 276351 11-53 7236.19 43 269402 ii-ii 992359 .40 277043 11-51 722957 17 44 270069 11-10 992335 •40 277734 11-50 722266 16 43 270735 11-08 9923 1 1 .40 278424 11-48 721576 i5 46 271400 11-06 992287 .40 279113 11-47 720887 720199 14 S 272064 II. o5 992263 .40 2&88 11-45 i3 272726 11-03 992239 .40 11-43 719312 12 49 2 /3388 11-01 992214 •40 281174 28i858 11-41 718826 11 5o 274049 Q 274708 10-90 10-98 992190 •40 11-40 718142 lO 5i 9-992166 .40 9-282542 11.38 10 717438 I 52 275367 10-96 992143 .40 283225 11.36 716775 53 276024 10-94 992117 992093 •41 283907 984588 11-35 716093 I 54 276681 1092 •41 11-33 7i54i:- 55 777337 10-91 10-89 992069 •41 285268 ii-3i 714732 5 56 277991 992044 •41 285947 11^30 714053 4 U 278644 10-87 992020 •41 286624 11-28 713375 3 279297 10-86 991996 991971 •41 287301 11-26 712695 2 59 279948 10-84 •41 287977 11^25 7 1 202 J 1 60 260399 10-82 991947 -41 193 288652 11-23 711348 ..Tun^. ^ LT. _Cosiue_ ! D. Sine Cotung. D. jd. SIKE8 AND TAN0EXT3. (11 DBGnetcs. ) 29 ir 1 Sine 9380599 381248 D. 1 Cosine : D. 1 Taiijr. 1 D. ' Cotaiijr. 1 1 o 10.83 9-9Q«9'«7. •41 9-388652 I 11-23 .0-711348 60 I 10-81 991922 •41 1 289326 1 11-23 7106/4 5q 710001 58 5 38.897 10-79 99.897 •41 289999 1 1 1 • 20 3 282344 I0-7J 10-76 991873 •41 1 29067. 11 -18 ]Xl^ u 4 283190 991848 •4: j 29.342 1 11-17 ! ii-i5 5 383836 10-74 991823 •41 1 2920.3 707987 55 6 384480 10-72 991799 •41 i 299682 I 11-14 7073 1 8 54 I 285124 10-71 99' 774 •43 i 293350 1 11-12 7o665o 53 285766 1069 991749 •43 29 '.017 II 11 703983 53 9 286408 1067 991724 •42 29^4684 . 1 -09 7033i6 , 5i 10 287048 10-66 991699 •42 295349 I. -07 70465 1 5o 11 z 287687 10-64 9-99.6741 •42 9-2960.3 11-06 10-703987 703323 2 It ;88326 10-63 99 '649! •42 296677 11-04 li 288964 10-61 9916241 •42 297339 II -o3 702661 47 14 289600 lo-So 10-58 99.599' •42 1 298001 11-01 ^^•^ 46 l5 290236 99.574! •42 298662 11-00 45 i6 290870 10-56 99.549 •42 299322 10-98 700678 44 \l 291304 10-54 99'524 •42 2999^^0 10-96 700020 43 29213-7 292768 10-53 99.4981 42 3oo638 10-95 69^705 1 42 »9 io-5i 991473: 42 301395 10-93 1 4' 20 293399 io-5o 991448 42 30.93. 10-92 698049 10-697393 4c 31 9-294029 294658 10.48 9-991422 42 9-302607 10-90 10-89 l^ 33 10-46 991397 42 3o326i 696739 33 295286 10-45 99.372 43 3039.4 304567 10-87 696086 37 34 295913 296539 10-43 991346 43 10-86 693433 36 35 10-42 99.321! 43 3o32i8 10-84 694782 35 36 297 '64 10-40 99.295 43 3o5869 10-83 694131 34 37 297788 10-39 99.2701 43 3o65.9 10-81 693481 33 38 298412 10-37 991244 43 307.68 10-80 692832 32 2Q 299034 10-36 991218! 43 3078.5 10-78 692185 3i 3o 299655 10-34 99«i93i 9-991.67 43 3o8463 10-77 69.537 3c 3i 9-300276 IO-32 43 9-309.09 1075 10-69089. 11 33 300895 io-3i 99n4i! 43 309734 10-74 600246 33 3oi5i4 10-29 10-28 9911.5 43 3.0398 10-73 689602 27 34 302l32 991090 43 3llo42 10-71 688938 6883.5 26 35 302748 3o3364 10-26 991064 43 31.685 10-70 35 36 10-25 99.038 43 3.2327 10-68 687673 24 ll 303979 304393 10-23 99.012 43 3.2967 10-67 687033 23 10-22 990986 43 3.36o8 10-65 686392 685753 22 39 303207 10-20 990960 43 3.4247 10-64 21 40 3o58i9 10-19 990934 44 3.4885 10-62 685ii5 20 41 9 -306430 10-17 9-990908, 990882 44 9^3.5523 10-61 10-684477 ;? 43 307041 10-16 44 3.6.59 3.6795 10-60 68384. 43 3o765o 10-14 990855 44 10-53 683205 \i 44 308259 10-13 ^zi . 44 3 17430 10-57 682570 45 308867 10-11 44 3.8064 10-55 681936 i5 46 309474 3 1 0080 10-10 990777 • 44 3.8697 10-54 6S.3o3 14 s 10-08 990750 • 44 3.9329 10-53 6806711 i3 3 1 0685 10-07 io-o5 990724 • 44 3.9961 10-5! 68oo3q 13 679ioA 11 J^ 311289 311893 990697, • 990671; • 44 320392 10- 5o 5c 10-04 44 32.222 10-48 678778 10 5i C-3i2i95 10-03 9-990644 44 9-32i85i 10-47 10-45 IC-678.49 9 53 f^ 10-01 9906.8, . 44 322479 323.06 677521 8 53 lO-OO 99o5oil . 990565 . 44 10-44 676894I 7 54 314297 9-98 44 323733 10-43 676267 6 55 314897 3 1 5493 9-97 990538 1 . 44 324358 10-41 675643 5 56 9-96 99o5iil . 45 3249S3 10-40 675017 4 i 316092 9-94 990485 . 45 325607 10-39 tl^ 3 316689 9 93 990458, . 45 32623. .0.37 ■2 S 317284 9-91 99043 1 • 45 326853 10-36 673 1 /,7 672535 I 6o 317879 9.90 990404 • 45 327475 10-35 Cosine D. Sine 7 8° CotaiiS'. - D. Tang. 30 (12 DEGREES.) A TABLE OP LOGARITHMIC M. j Sine 1). j Cosine | T>. i Tang. D. Cotanar. 60 9.377879" 9.90 9-99o4oi -45 ' 9-327474 10.35 10-672526 I 318473 990378 -451 3:8095 10.33 671905 U 2 319066 9.87 99o35ii .45 328715 10-32 671285 3 319658 ?-86 990324I -45 329334 10-30 670666 u 4 320249 9.84 9902071 .45 329953 10-29 670047 5 320840 9-83 990270' '451 33o57o 10.28 669430 6688 1 3 55 6 321430 9-82 9902 i3 .45 ; 331187 10-56 54 I 322019 9-8o 990215; -45 ! 33i8o3 IO-25 668197 667 58 2 53 322607 9-79 900 1 88, .451 332418 10-24 52 9 323194 ! 523780 9-77 990 1 6 1 1 .451 333o33 10-23 666967 666354 5i ic 9.76 990134 i -45 333646 10-21 DO ii 9-324366 9.75 9-990107 •46 ; 9-334259 10-20 10-665741 it 13 324950 9.73 990079 i -46 334871 10- 19 665129 6645.8 i3 325334 9-72 990002 1 -46 335482 10-17 10-16 il 14 326117 9-70 990025 989997 .46 336093 663907 i5 326700 li^ .46 J36702 .o-i5 663298 45 i6 327281 989970 .46 337311 io-i3 662689 44 11 327862 9-66 989942 •46 337919 10-12 662081 43 328442 9-65 989015 989887 .46 338527 10-11 661473 42 19 329021 9.64 .46 339.33 10-10 660867 41 20 329699 9-62 989860 .46 339739 io-o8 660261 40 21 9-330176 9-6i 9-989832 .46 9-340344 10-07 10-659656 It 22 33oi53 9-60 989804 .46 340948 34i552 10 -06 659052 658448 23 33i3:9 Q-58 989777 .46 10-04 11 24 331903 9.57 989749 •47 342155 io-o3 657845 25 332478 9-56 989721 •47 342757 10-02 657243 35 26 333o5i 9-54 989693 •47 343358 10-00 656642 34 11 333624 9-53 989665 •47 343958 344558 9-99 9-98 656o42 33 334195 9.52 989637 •47 655442 32 - 29 334^66 9 -So 989609 •47 345i57 9-97 654843 3 1 3o 33533- 9-49 989582 •47 345755 9-96 654245 3o 3i 9-335906 9-48 9.989553 •47 9-346353 9-94 .0-653647 It 32 336475 9-46 989525 •47 346949 347545 9-93 653o5. 33 337043 9-45 989497 •47 9-92 652455 U 34 337610 9.44 989469 •47 348141 9-91 65i859 65.265 35 338176 9-43 989441 •47 348735 9-90 25 36 338742 9-41 9894.3 •47 349329 9-88 650671 24 il 339306 9-40 9S9384 •47 349922 35o5i4 lii 650078 23 38 339871 9-39 989356 •47 649486 648894; 22 39 340434 9.37 989328 .47 35iio6 9-85 21 40 340996 9-341558 9-36 ^8^300 .47 351697 9-352287 352876 353465 9-83 6483o3 20 41 9-35 989271 •47 9-82 io-6477'3| ',t 42 342119 9-34 ' 989243 •47 9-81 6471241 43 342679 9-32 989214 •47 9-80 646535; 17 645047 i 16 44 343239 9-3i 989186 •47 354053 Q-79 45 343797 9 -30 9S9157 •47 354640 9-77 645360' .5 46 344355 9-29 989128 .48 355227 3558i3 356398 356982 357566 9-76 644773; 14 % 344912 345469 9.27 9-26 989100 989071 :|- 9-75 9-74 644187: 1 3 643602 .2 i9 346024 9-25 989042 •48 9-73 643-18 :i 5o 346579 9-24 989014 •48 9-71 642434 10 5i 9 347 '34 9-22 9-988985 .48: 9.358149 9-70 io.64i85i, 9 641269 § 5i 347^87 9-2' 988956 .48: 35873. 9-6o 1 9.68 ; 53 348240 9-20 988927' -48 359313; 640687 7 640107 6 54 ' 348792 9-19 988898. .48: 359893 9.67 55 1 349343 9-17 988869; .48: 360474 9-66 639326 5 56 ; 349893 9.16 988840 .48; 36io53 9-65 63^947 4 i1 350443 9-15 9888.11 .49' 36.632 Q-63 638368 3 5b ' 350992 9-14 988782 .49' 3622 10 9-62 637790 J 59 35i34o Q.l3 988753 .49' 362787, 9-61 647213 1 60 352088 1 9-II 988724 .49' 363364 9-60 636636 u 1 Cosine ' D. Sine 770 Cotnng. T). __TaLiS^l M.| 8INES AND TANGENTS. (13 DKORKKS. ) 3 '.^ Sine 1). Cosine D. Tan?. 1). Cotanj;, 9.352088 9-11 "^988724 —49 9-363364 9-60 10 036636 "6^ I 352635 9-10 988695 .49 363940 9 S 636o6o 5? 2 353i8i 9. 09 908 988666 .49 364^15 9 635485 3 353726 988636 .49 365otio 9 U 6049.0 57 4 354271 9.07 9-03 988607 .49 365664 9 634J36 56 5 354815 9H8578 .49 366237 9 54 633763 55 6 355358 904 988548 .49 366810 9 53 633.901 54 2 355901 903 988519 .49 367382 9 52 6326i8' 53 356443 9-02 988489 .49 367953 368524 9 5i 632047 52 9 356oH4 357524 9.358064 901 h 988460 .49 9 5o 631476 5i 10 II 988430 9.988401 .49 •49 9.369663 9 Q tt 630906 io.63o337 5o it 12 3586o3 8-97 988371 •49 370232 9 46 629768 i3 35914 1 8-96 988342 •49 370799 9 45 629201 628633 47 14 359678 8.95 988312 .50 371367 9 44 46 i5 36021 5 8.93 988282 .50 371933 9 43 628067 45 i6 360752 8-92 988252 .50 372499 9 42 627501 44 \l 361287 8-91 988223 .50 373064 9 41 626936 43 361822 8-90 988193 .50 ^?^^P 9 40 626.171 42 »9 362356 8-80 988163 .50 9 ^ 625807 41 20 362889 8-88 988133 .50 3747^6 9 623244 40 21 9-363422 8-87 9.988103 .50 9.375319 9 ll 10-624681 ll 22 363954 8-85 988073 .50 375881 9 624. .9 623558 23 364485 8.84 988043 •50 376442 9 34 37 24 3650 16 8-83 988013 .50 377003 9 33 622997 622437 36 25 365546 8.82 987983 .50 377563 378122 9 32 35 26 366075 8-81 987953 .50 9 3i 621878 34 11 366604 8.80 tt^ .50 378681 9 3o 62.3.9 33 367i3i 8-79 .50 379239 9 ll 620761 32 29 367659 368 1 85 8-77 987862 .50 '^Itl 9 620203 3i 3o 8-76 987832 .51 9 ll 619646 3o 3i 9-36871 1 8.75 9.987801 .51 9-380910 9 10-619090 618534 ll 32 369236 8-74 987771 .51 381466 9 25 33 369761 8.73 987740 .51 382020 9 24 617980 27 34 370285 8.72 . 987710 • 51 382575 9 23 617425 26 35 370808 8.71 987679 •51 383129 9 22 616871 25 36 37i33o 8-70 987640 987618 .51 383682 9 21 6i63i8 24 ll 371852 .51 384234 9 20 615766 23 372373 8.67 8-66 987588 .51 384786 9 \l 6l32l4 22 39 372894 987557 •51 385337 385888 9 614663 21 4o 3734.4 8-65 987526 •51 9 \l 614112 20 41 9.373933 8-64 9.987496 .51 9.386438 9 io.6i3562 \l 42 374452 8-63 987465 • 51 388084 9 14 6i3oi3 43 374970 8-62 9H7434 .51 9 i3 612464 \l 44 375487 8-61 987403 .52 9 12 61 1916 6ii369 45 376003 8-60 987372 .52 388631 9 II i5 46 376519 8-5? 987341 .52 3891-78 9 10 610822 14 49 37703d 987310 .52 389724 9 ^ U 610276* i3 IS 8.57 987270 987248 .52 .52 390270 390815 I 609730 609185 12 II 5o 9.379089 8-54 987217 .52 391360 9.391903 9 608640 ry 5i 8.53 9.987186 .52 9 o5 10-608097 I 5a 379601 38aM3 852 987,55 .52 392447 9 04 607533 53 8.5i 9«7I24 .52 392989 9 o3 607011 6.36469 I 54 380624 8-5o 987092 •52 393331 9 02 55 38n34 8.45 8-48 987061 .52 394073 9 01 605927 6o5386 5 56 38i643 987030 .52 394614 t 00 4 U 382152 8.47 8-46 986998 .52 395.54 90 604846 3 382661 986467 .52 3^233 8 98 6o43o6 2 59 383i68 8-45 986930 .52 8 97 603767 I fe^ 383675 8.44 gHbtfo^ •52 396771 8 96 6o3239 « , Cofine J). ISijie 763 Cotang. 1>. T.-^i^ J^:. 2r. 52 (14 DBGREES.) A TABLE OF LOGARITHMIC o Sine D. CoHiiie D. 1 Taii^. 5. Cotang. 60 9-383675 8.44 9-9869041 -52 9.396771 8.96 10.603229 I 384182 S.43 986873 -53 3973og 8.96 602691 It a 384687 8.42 98684. -53 397846J 8.95 602154 3 385192 8-41 986809 -53 398383 8.94 601617 57 4 385697 8.40 986778: -53 3989.9 1 8-93 60 1 08 1 56 5 386201 8^3^ 986746 -53 3994551 8.92 6oo545 55 6 386704 9867 14I -53 399990 8-91 600010 54 1 387207 8.37 986683, -53; 4oo524 S.90 1 tk 599476 1 598942 53 £ 387709 8-36 98665 1 1 -53 4o.o58 52 9 ! 388210 8-35 986619! -53 401591 598409 597876 5i 10 ' 3887.1 ' 8-34 986587; -53 402.24 8-87 5o II 9-389211 8-33 9.9865551 -53 9-402656 8.86 10.597344 8 12 3897.1 8-32 986523 -53 403.87 8.85 596813 i3 3902.0 8.3i 986491 -53 4037.8 8.84 596282 tl 14 390708 8.3o 986459 .53 404249 404778 8.83 595751 i5 39.206 8.28 986427 .53 8.82 595222 45 i6 39.703 8.27 986395 •53; 4o53o8 8.81 594692 44 n 392199 8.26 986363 ■54 405836 8.80 594164 43 iS 392695 8-25 , 986331 •54 406364 8.70 8.78 593636 42 19 39819. 8.24 986299: -54 406892 593.08 41 20 393685 8-23 986266 •54 407419 8-77 8.76 592581 ^° 21 9-394.79 394673 8-22 9.986234 •54] 9.407945 10-592055 t^ 22 8-21 986202 •54 408471 8.75 Hfj,^ 23 395.66 8.20 986169 .54 408997 40952 1 8.74 37 24 395658 8.19 8.i8 986.37 If 8-74 590479 589953 36 25 396.50 986104 •54 4.0045 8.73 35 i6 396641 8.17 986072 •54 410569 8.72 589431 34 ^J 397.32 8.17 8.16 986039 •54 41.092 8.71 588908 33 397621 986007 •54 4ii6i5 8.70 588385 32 ?9 398 m 8.i5 985974 •54 412.37 8.69 8-68 587863 3i 3o 398600 8.14 985942 .54 4.2658 587342 3o 3i 9-399088 8.i3 9.985909 • 55 9-413.79 8.67 10.586821 11 32 399575 8-12 985876 .55 413699 8.66 586301 33 400062 8-11 985843 .55 414219 414738 8-65 585781 27 34 400549 401035 8.10 9858.1 .55 8-64 585262 26 35 8.09 8.08 985778 .55 415257 415775 8-64 584743 25 36 40.520 985745 .55 8.63 584225 24 37 4o2oo5 8-07 985712 .55 416293 8.62 583707 23 38 402489 8-06 985679 .55 4168.0 8.61 583.90 22 39 402972 8-05 985646 .55 417326 8.60 582674 21 40 403455 8-04 9856.3 .55 417842 8.59 8.58 582158 20 41 9-403938 8-03 9.985580 .55 9.418358 10.581642 \t 42 404420 8-02 985547 .55 418873 8.57 581.27 43 404901 4o5382 8-01 9855.4 .55 419387 8.56 5806.3 n 44 8.00 985480 .55 419901 8.55 580099 16 579585 1 1 5 45 4o5862 7-9^ 985447 .55 420415 8.55 46 406341 985414 .56 420927 8.54 570073: 14 578560I 1 3 ii 406820 7-97 7.96 985380 • 56 421440 8.53 407299 985347 • 56 421952 8-52 578048 12 49 407777 7-95 9853.4 -S^ 42 2463 8.5i 577537 " 5o 408254 7-94 985280: .561 422974 8.5o 577026 TO ! ;)i 9 408731 7-94 9.9852471 -56 9.423484 8.49 8.48 10.576516 I 55 409207 7-93 9852.3 -56 424503 576007 53 409682 7.92 985.80: -56 8.48 575497 1 6 54 410.57 7.91 985.46: -56 425oii 8.47 574989 55 4io632 985.13 -56 4255.9 8.46 574481 5 56 4 1 II 06 985079! -56 98504D -56 426027 8.45 573973 4 U 411579 426534 8-44 573466 3 4I2052 7-87 98001 1 -56 427041 8.43 572950 572433 a 59 412524 •^.86 984978 -56 427547 8.43 I _6o_ 412996 7-85 984944' 56i 428052 8.42 571948 1 L Cosine 1 D Sine 1 T50| Cotang. D. Tnne. M. | sixes AND TAXGEMS. (15 DEGREES. ) 33 [M. Siiio D. Cosine 1>. Tailg. D. CotAng o 7-65 9-984944 Q -428052 8-42 10-571948 60 1 7-84 984910 •371 ' 428557 8-41 571443 i? 3 7-83 984876 •37 429062 8-40 570938 3 414408 7.33 984842 •57 429566 8^3? 570434 u 4 ^l^H-jS 7-82 984H08 •57 430070 569930 5 41334] 4i58iD 7.81 984774 •57 430573 8-38 5^^^? 55 6 7-8o 984740 •57 431075 •57 43.577 fM 54 I 9 4i6a83 984706 563423 53 416751 417217 984637 9846o3 •57 4320-79 •57i 432580 8.35 8-34 567921 567420 l\ 10 417684 <;.4iBi5o •57I 433o8o 8.33 566920 5o II 7-75 9.984569 •57! 9-433580 8-32 .0-566420 % la 4186.5 7-74 98433d •^7 434080 8.32 565920 i3 419079 7-73 984500 'V f^m 8.3, 565421 47 14 419544 7-73 984466 :tl 8-30 564922 ^i |5 420007 7-72 984432 435576 8-29 8-28 564424 45 i6 420470 7-7» 984397 • 58 436073 563927 44 \l 42qq33 421395 984363 • 58 436570 8-28 563430 43 984328 • 58 iP^li 8-27 562933 42 '9 421857 984294 • 58 8-26 562437 41 20 4223id 7-67 984239 • 58 438o59 8-25 56.941 40 21 9-422778 7.67 9-984224 •58! Q-438554 8-24 10-56.446 It 32 423238 7.66 984 • 90 • 58 439048 8-23 560932 23 42^697 7-65 984133 • 58 439543 8-23 560457 u 24 424106 7-64 984120 • 58 440036 8-22 559964 25 424615 7-63 984083 • 58 440529 8-21 5%78 558486 35 26 425073 7.62 984050 • 58 441022 8-20 34 U 425530 7-6i 984015 • 58 44i5i4 8..9 33 425987 7.60 983981 .58 442006 8-19 8-. 8 557994 32 29 426443 7-60 983946 • 58 442497 442988 557303 3i 3o 9-427354 7.59 98391 1 • 58 8..7 5570.2 3o 3i 7-58 9-983873 • 58 Q-44347q 8-16 10-556521 ll 32 428263 7-57 983840 •59| ■ 443968 8.16 556o32 33 7-56 9838o5 •59| 444458 8-15 555542 ll 25 34 35 428717 429170 7.55 7-54 $l]lt •591 444947 •59 445435 8.14 8..3 555o53 554565 36 429623 7-53 983700 •59 445923 •59 446411 8-. 2 554077 5535B9 24 ll 430075 7-52 983664 8-. 2 23 43o527 7-52 983629 •59 446898 8-II 553.02 22 39 430978 7-5i 9«^?9i •39 447384 8-.0 5526.6 21 40 431429 7-5o 983338 •59 44.870 8^09 552 1 3o 20 41 9431879 7-49 9-983523 •59 9-448356 8-09 8-08 10-55.644 :? 4: 432329 7-49 983487 •^ 448841 55.. 59 43 432778 7-48 983452 •59 449326 8-07 550674 n 44 433226 7-47 983416 •59 449S10 806 550.90 .6 45 433675 7-46 983381 •59 450294 8-06 §49706 .5 46 434122 7-45 983345 •39: 450777 8-o5 549223 548740 14 % 434569 7-44 9832i8 •59, 45.260 8-04 i3 435016 7-44 •60, 45.743 8-03 548937 12 49 435462 7-43 •60 452225 8-02 547775 II 1 5c 435908 9.436353 7-42 983202 •60! 452706 8-02 547294 10 1 5i 7-41 9-983166 •60; 9.453.87 8-01 10-5468.3 t 5a 436798 7 40 983 i3o •6oj 453668 8-00 54633: 53 437242 7-40 983094 •<>0| 454148 7-99 545852 54 437686 ?:^ 983038 •60 454628 ?:^ 545373 55 4.^8129 983022 •60 455.07 54489.' 56 43'5572 7-37 982986 •60 455586 ]'-U 544414 U 439014 7-36 983950 •601 456064 543936 439456 7-36 982914 •60 456542 7-96 543458 ^ '^l 7.35 7-34 982878 982842 •6o| 457019 •60 457496 7-95 7-94 D. 542981 54a5o4 1 Cosiue 1). _ S^iue _ 74''l Cotaug. 84 (16 DEGRKE8.) A TABLE OP LOGARITHMIO M. Sine L\ Cosine 1). Taii^. D. Cotansr. 1 9-440338 7-34 9-982842 .60 9.457496 7-94 10- 542504' 60 I 440778 -33 982805 .60 457973 7-93 5420271 59 54i55r' 58 3 441218 -32 982769 .61 458449 7.93 3 441658 -31 982733 .61 458923 7.92 541075 57 4 442096 .31 982696 •61 459400 7.91 540600 56 5 442535 7 •3o 982660 .61 459875 7-90 540. -.5 55 6 442973 7 -.11 982624 •61 460349 460823 539651 34 I 443410 - / 982587 •61 539177 53 538703 53 443847 7 •27 9825511 -61 461297 9 444284 - / •27 982514 -61 461770 j'ss 538230 5i 10 444720 •26 982477 •61 462242 7-87 537758, 5o II 9 -445 1 55 7 •25 0- 982441 .61 9-462714 7.86 .'3-537286: 49 536814' 48 12 445590 •24 982404 .61 463 186 7-85 i3 446025 7 •23 982367 •61 463658 7.85 5:6342 47 14 446459 •23 982331 -61 464129 7-84 535871' 46 i5 446893 22 9822941 •61 464599 7-83 535401! 45 i6 . 447326 21 982257 •61 465069 7-83 53493 r 44 *l 447759 ^ 20 982220 -62 465539 7.82 534461 i 43 i8 448191 1 20 982183 .62 466008 7.8. 533992! 42 ^9 448623 ' Is 982146 .62 466476 7.80 533524 41 20 449054 982109 •62 466945 7.80 533o55i 40 21 9-449485 7 17 9-982072 .62 9-467413 7-79 10-5325871 39 532120! 38 22 449915 16 982035 -62! 467880 7.78 23 45o345 16 981998 .621 468347 7.78 53 1 653 i 37 24 450775 i5 981961 -62 468814 I'll 53 r 1 861 36 25 45 1204 14 981924 -62 469280 7.76 530720 35 26 45i632 i3 981886 •62 469746 7-75 530254! 34 11 452060 7 i3 981849 •62 4702 1 1 7-75 529780; 33 452488 _ 12 981812 -62 470676 7-74 529324I 32 29 452915 453342 1 1 981774 •62 471141 7-73 528859! 3 1 528395 3o 3o 7 10 981737 -62 471605 7-73 3i 9-453768 10 9-981699 •63 9-472068 7-72 10.527932 2q 32 454194 :i 981662 -63 472532 7-71 527468| 28 33 454619 981625 •63 472995 473457 7.71 527005I 27 34 455o44 07 981587 -63 7.70 526543 26 35 455469 455893 07 981549 •63 473919 7.69 526081 25 36 06 981512 -63 474381 -tl 5256 I Q I 24 j ll 4563 16 o5 98' 474 •63 474842 525i58 23 456739 04 981436 •63 4753o3 7-67 524697 22 39 457162 04 981399 •63 475763 7-W 524237 21 40 457584 o3 981361! •63 476223 7.66 523777 20 41 9 -458006 02 9-981323 •63 9-476683 7-65 10.523317 522858 \t 42 458427 01 981285 •63 477142 7.65 43 458848 01 981247 -63 477601 7-64 52 2399 ]l 44 459268 CO 981209 •63 478059 7-63 521941 45 459688 9^^ 981171 • 63 478517 478975 7-63 521483 i5 46 460108 981 i33 -64 7.62 52I025 14 il 460527 98 981095 981057 -64 479432 7.61 520568 i3 46?364 97 .64 i& 7.61 520I11 12 ^9 96 981019 .64 7.60 519655 II 5o 461782 95 980981 • 64 480801 7.59 519199 10 10.518743' 9 518288; 8 5i 5-462199 95 9-980942 .64 9-481257 ]i^ 52 462616 94 980904 980866 • 64 481712 33 463o32 93 • 64 482167 I'^i fi7833 7 5,7379 6 516925I 3 54 463448 93 980827 .64 482621 1-^1 55 463864 92 980789 •64 483075 7-56 56 464279 91 980750 • 64 483529 7.55 516471J 4 ll 464694 90 980712 • 64 483982 7.55 5i6oi8i 3 1 465 I 08 To 980673 -64 484435 7-54 5i5565' 2 , ^ 465522 980635 -64 484887 7.53 5i5ii3i I ; 6o 4b5935 6- 88 980596 -64 485339 7.53 5i466i " ; Coeino D. J Sine -J 30 (Jotang. D. Taiisr. JLJ SIVK8 AND TANGENTS. (17 DEORKE8. ) 35 BT. 1 Sino 1 D. Cosiiio D. Tan?. D. Cotancr. 10.514661 ! 9-465935 466348 6-88 Q. 980596 .64 9.485339 7.55 60 I 6-88 ^ 98o5d8 .64 485791 7.5a 514209 5i3758 u a 466761 6.87 6-86 980519 • 65 486242 7.51 3 467173 467585 9S0480 .65 486693 7.51 5i33o7 u 4 6-85 980442 .65 487143 7.50 512857 56 5 467996 b-St 980403 • 65 487593 7-49 5 12407 ^^ 6 468407 6-84 980364 .65 488043 lit iwtu li I 468817 6-83 980325 .65 488492 53 4S9227 6-83 980286 .65 488941 489390 489838 7-47 5iio59 32 9 10 469637 470046 6-82 6.81 980247 98020S • 65 • 65 ]■■% 5io6io 31 510162 30 11 9-470455 6.80 9-980169 .65 9-490286 7.46 10.509714! 40 509267 48 u 470863 6.80 980180 .65 490733 7.45 i3 47127' Ui 980091 • 65 491180 7-44 508820 47 14 471679 472086 980002 • 65 491627 7-44 5o8373 46 i5 6-78 980012 •65 492073 7-43 507927 45 i6 472492 6-77 979973 •65 492019 7-43 507481 44 \l 472898 6-76 979934 979893 .66 492963 7-42 507035 43 473304 6.76 .66 493410 7-41 5o65qo 42 '9 473710 6.75 979835 .66 493854 7.40 5o6i46 41 20 4741 1 5 6-74 979816 .66 494299 9-49474i 7.40 503701 40 21 9-474510 474923 475327 6.74 9.979776 .66 7.40 10-503237 It 22 6.73 979737 .66 493186 vu 504814 38 23 6-72 979697 .66 495630 504370 37 24 475730 6.72 979638 .66 496073 7-37 503927 36 25 476133 6.71 979618 .66 4965 1 5 7-37 5o3485 35 26 '2 476536 476938 477340 6-70 6-69 979579 979539 .66 .66 ti^ 7.36 7.36 5o3o43 5o26oi 34 33 2S 6-69 6-68 979499 • 66 497»4i 7.35 5o2 1 39 501718 32 ?9 477741 478142 979439 .66 498282 7.34 3i 3o 6-67 979420 .66 498722 7-34 501278 3o 3i 9-478542 6.67 6-66 6-65 9-979380 .66 9-499'63 7.33 io.5oo837 11 27 32 33 478942 479^42 979340 979300 .66 -67 499603 5ooo42 7.33 7-32 499938 34 479741 4^0 I 40 480539 6-65 979260 -67 5oo48i 7-31 4993 '9 26 35 36 6-64 6.63 979220 979180 -67 .67 500920 5oi359 7-31 7-3o 499080 498641 25 24 u 480037 481334 6-63 979 « 40 -67 mm 7.30 498203 23 6-62 979100 •67 ]:lt 497765 22 39 481731 661 979059 .67 502672 497328 21 40 482128 6.61 9790 I Q 9 •978979 -67 5o3io9 7.28 496891 20 41 9-482525 6.60 .67 9 503546 7-27 10-496434 '5> 42 482921 4833 1 6 6-59 978o3o 978898 .67 503982 7.27 496018 la 43 6-5? .671 5o44iB 7.26 493582 17 493146 16 44 483712 978838 .67, 504854 7-25 45 484107 6.57 978817 .67! 505289 7-25 49471 1 '5 46 484501 484895 485289 6.57 6-56 978777 978736 .67I 5o5724 .67! 5o6i59 •68! 506593 7-24 7-24 494276 14 493841 «3 655 978696 7-23 493407' 12 f^ 485682 6. 55 978635 •68 507027 7-22 492973 " 5o 486075 6.54 978615 .68 507460 7.72 492340 10 Si 9-486467 6-53 9-978574 •68; 9.507893 7-21 ic 492107: 9 491674 " 53 486860 6-53 978533 .68 5o8326 7-21 53 487251 6.5a 978493 • 68 508759 7.20 491241; 7 54 487643 488034 6.5i 978432 .68 509191 7. 19 490809 6 400378, 5 55 6.5i 978411 .68 509622 ?::? 56 488424 6.5o 978370 .68 5ioo54 1] 5q 6o 488814 489204 6.5o 649 6.48 6.48 978247 978206 .68 .68 .68 .68 5 1 0485 510916 5ii346 511776 7.18 ,..6 48^5 ■ 489C84 488654 488234 3 2 1 Conine ! D. Sine 7 2- Cotaiig. D. Taiij^r. .?• i 30 (18 DEGREES.) A TABLE OF LOCiARlTHMlC Ti: Sine D. Cosine 1 1). Taiicr. 1 ^• Cotaug. 9-489082 6-48 9.978206' .68 9.511776 1 7-i6 10.488224 60 I 490071 1 6-48 978165; -68 5l2206 7-16 4877941 59 7 4907^9 ! 6-47 978124' -68 512635 7.15 487363 58 3 49n47 ! 6-46 978083 -69 5 1 3064 7-14 486936 57 4 491535 ' 6.46 978042; -69 513493 7-14 486307 56 5 491922 6.45 978001 -69 51392, 7-i3 486079 55 6 492308 i 6-44 977950 -69 9770,8 .69 977077: -69 514349 7-i3 485651 54 7 492695 493081 6-44 514777 7-12 485223 53 8 ^.43 5i52oi 7-12 484796 5i 9 493466 6.42 977835 .69 5i563, 7-II 484369 483943 5, IC 49385 I 6-4J2 977794 -69 5i6o57 7-10 5d 11 9 -454:35 6.4! 9-9777321 -69 9.5,6484 7.10 10.48J316 1? 12 494621 6.41 977711 -69 5169I0 517335 7-09 483090 i3 495co5 6.40 977660' -69 977628^ -69 7-0? 482665 47 14 495388 6.39 51776, 5i§i85 482239 4818,5 46 i5 493772 tl^ 977586 : .69 7-o8 45 i6 496154 977544 1 -70 5i86io ]:2 48,390 44 n 496537 6.37 9775o3 ! .70 5,9034 480966 480342 43 i8 496919 497 JO I 6-37 977461 i -70 519458 7-o6 42 19 6-36 977419 . .70 5,9882 7-05 480118 41 20 497682 6-36 977377 .70 52o3o5 7-o5 479695 40 21 9-498064 6-35 9.977335 •70 9.520728 7-04 10.479272 39 38 22 498444 6.34 977293 977231 •70 521, 5t 7-03 478849 23 498825 6.34 •70 52,573 7-03 478427 ll 14 499204 6.33 977209 •70 521995 7.03 478003 i5 499584 6.32 977167 •70 522417 7.02 477583 35 i6 499963 6-32 977125 •70 522838 7-02 477162 34 n 5oo342 6.3i 977083 •70 52325g 7-01 476741 33 i8 500721 6-31 977041 •70 52368o 7-0, 47^)320 32 1 i*^ 501099 6.3o 976999 •70 524100 7-00 475900 3i 1 io 501476 6-29 976937 •70 524520 6.99 475480 3o ti 9-5oi854 6-29 6.28 9.976014 976872 •70 9-524939 6-99 ,0.475061 29 I2 50223l •71 525359 6.9^ 474641 28 53 502607 6.28 97683o •71 525778 6.98 474222 27 H 502984 6.27 976787 976745 •71 526197 6-97 473803' 26 )5 5o336o 6-26 •71 526615 6-97 473385 25 16 503735 6.26 976702 •71 527033 6.96 472967 24 !^ 5o4iio 6-25 976660 •7' 52745, 6-96 472349 23 504485 6-25 976617 •71 527868 6.95 472132 i2 ^9 504860 6-24 976574 •71 528285 6-95 47i7'5, 2, 40 5o5234 6-23 976532 •71 528702 6.94 471298 20 4i 9 -505608 6-23 9.976489 •71 9-5291,9 6.93 ,0.47088, IQ i2 505981 5o6354 6-22 976446 •71 529535 6.93 470465 18 43 6-22 976404 •71 529930 6.93 47oo5o| 17 44 506727 6-21 976361 •71 53o366 6.92 469634' 16 45 507099 6-20 676318 •7' 530781 1 6.9, 469219' ,5 468804; 4 46 507471 6-20 976275 •71 53,196 6.91 47 507843 508214 6-19 976232 •72 53i6ii; 6.90 468389 i3 i 48 6-19 6-i8 976189 •72 532025 6-90 6.89 467973 12 I 49 5o8585 976146 •72 532439 467361 " 1 5o 508956 9.509326 6.18 1 976103 •72 532853 tli 467147 ID 1 5i 6-17 : 9 976060 .72 9 '533266 10.466734 9 I 5s 5,w^; 6.16 1 976017 •72 533679 6.88 466321 8 1 53 :jioo65 6.16 i 975974 • 72 534092 6.87 465908 7 465496 6 465o84i 5 54 510434! 6-15 975930 97D887 •72 534504 6.87 55 5io8o3 6.i5 •72I 534916 6.86 56 511172 6-14 1 975844 -72 535328 6-86 464(372 4 U 5ii54o 6.i3 ! 975800! .72 535739' 6-85 464261 3 511907 6-13 975757I .72 536i5o 6.85 463850 2 59 512275 6-12 973714! -72 536561 ; 6-84 463439' I (jo 512642 6-12 9736701 .72 536972 6-84 463028, Coaiiio __.I^-.__ Siijo 171° Cotaiig.^l _i>. _Ttiiig^l M^ SINES AND TANQENTfi. (19 L»EGUEEri. 1 8^ o Sine D. Cosiiio D. _Tuug^ 1 ^' Cotang. ._ 0-512642 6-12 9.975670 •73 9-536972 6-84 10-463028; 60 I 5i3oo9 61. i^m •73 537382 6-83 462618 !? 2 5i3373 6 I. •73 537792 538102 53861 1 6.83 462208 3 4 51374. 514107 6-10 609 975539 975496 973432 •73 •73 6.82 6.82 461798 46.389 U 5 514472 6.09 6.08 .73, 539020 6.8( 460980 460571 55 6 5.4837 975408 •73 539429 6.81 54 I 5.5202 6.08 975365 975321 •73 539837 6-80 460163 53 5i5566 6-07 •73 540243 6-8o« 459755 52 9 10 5.5930 5.6204 9.516657 tS Q. 975189 .73; 540653 .73' 54.061 6-79 6.70 6. 78 45q347 438939 10-458532 5i 5a if 6.o5 •73 9-54.468 tt 13 5.7020 6-05 975.43 •73 54.875 6-78 458.25 i3 5.7382 5-04 975.0. •73 542281 6-77 457719 ii 45 14 i5 5.2745 518107 5.8468 604 6 03 975057 975oi3 •73 •73 542688 543094 6.76 4573 m 456906 456501 i6 6-o3 974960 974925 •74 543499 6.76 44 '.I 5.8829 6-02 •74 54390! 6.75 456095 43 519190 6-0. 974880 •74 5443.0 6.75 455690 455285 42 '9 5195DI 6-0. 974836 •74 544715 6.74 4. 20 519911 600 974792 •74 545119 6-74 454881 40 21 9-52027. 600 9-974748 •74 9-545524 6.73 10-454476 ^ 22 ^ 52063. 5-99 974703 •74 545928 546331 6.73 454072 23 520990 5-99 974659 •74 6.72 453669 453265 u 24 521349 5-98 974614 •74 546735 6-72 25 521707 5-98 974570 •74 547.38 6.7. 452S62 35 26 522066 5.97 974525 •74 547540 6-71 452460 34 11 5224^4 5.96 974481 •74 liit$ 6.70 452057 33 5227S1 5.96 974436 •74 6-70 431655 32 29 523i3S 5-95 97439. •74 548747 6-69 451253 3i 3o 523495 5.95 974347 •75 549149 6-69 6-68 45o85i 3o 3i 9-523832 5.94 9.974302 •75 9.'j4955o io-45o45o It 32 524208 5.94 974257 •75 P^[ 6-68 450049 449648 33 524564 5.93 9742.2 •75 6-67 '\ 34 524920 5.93 974.67 •75 550752 6-67 449248 448848 35 523275 5.92 974122 •75 55ii52 6-66 25 36 525630 5-9. 974077 •75 55i552 6-66 448448 24 ^2 525584 5.91 974032 .75 55.952 552351 6-65 448048 23 526339 526693 5.90 973987 •75 6-65 447649 22 39 5% 973942 •75 552750 6-65 447250 21 40 527046 973^^7 •75 553149 9-553548 6-64 44685. 20 41 9.527400 5-8q 9-973832 •75 6-64 10.446452 \t 43 522753 528 1 o5 5.88 973307 •75 553946 534344 6-63 446054 43 5-88 973761 •75 6-63 445656 \l 44 528458 5.87 973671 •76 55474. 6.62 445259 45 528810 It •76 555 1 39 555536 6-62 444861 .5 46 529161 973625 .76 6-61 444464 .4 s 529513 5-86 9735SO •76 555933 556J29 6-6. 444067 i3 529864 5-85 973535 •76 6-60 443671; n 1 49 53021 5 5.85 ?73489 •76 55672! 660 4432751 I 5o 53o565 5.84 973444 •76 55712. 6.59 442870 1.) .0-442483: 442087' 8 5i o.53o9i5 5.84 9.973398 .76 9.557517 6.59 5a 53.265 5-83 973352 .76 557913 5583o8 tu 53 53i6.4 5.82 973307 .76 441692J 7 441298I 6 54 531963 532312 5.83 973261 .76 558702 6.58 55 5.81 9732.5 .76 559097 6.57 4409031 5 , 56 532661 5.81 973169 .76 5598^5 tu 440509 4401.5 4 59 533009 5.80 973.24 •76 3 533357 533704 5. So 973078 973o32 .76 •77 56io66 6 56 6. 55 439721 2 I 60 534052 972986 •77 6.55 43?934 Coaiud I). Sine TOO Cotang. 1> TaiM?. 88 (20 DEGREES.) A TABIE OIT LOGARITHMIC M. Sine D. Cosine |I>. 1 Tang. D. 1 Cotancr. n o 9.534052 5.78 9.972986 •77 9.56106b 6.55 10.438934' 60 I 534399 5-77 972940 •77 561459 6.54 438341! 59 438.49 58 2 D34743 5-77 972894 •77 56i85i 6.54 3 535092 53543H t^ 972848 •77 562244 6.53 ^ u 4 972802 •77 562636 6.53 5 535783 5.76 972755 •77 563028 6.53 436972 55 6 536129 5.75 972709 972663 •77 563419 6.52 436381 54 I 536474 5-74 •77 563811 6.52 436189 53 435798 52 5368 1 8 5-74 972617 •77 564202 6.5i 9 537.63 5.73 972570 •77 564592 6.5i 435408 5i 10 537307 5.73 972524 •77 564983 9-565373 6.5o 435017 5o 11 9-537851 5.72 9.972478 •77 6.5o 10.434627 49 434237 i 48 12 538194 5.72 972431 •78 565763 6-49 i3 538538 5.7, 972385 .78 566 1 53 6-49 43JS47 47 433438 46 14 538880 5.71 972338 .78 566542 6-49 6.48 i5 539223 5.70 972291 •78 566932 567320 433068 45 i6 539565 5.70 972245 .78 6.48 432680; 44 \l 539907 5.69 972198 .78 568486 6-47 432291 j 43 540249 t^ 972IDI .78 6-47 6-46 431002; 42 43i5.4 41 »9 540590 972!o5 .78 20 540931 5.68 972o58 .78 568873 6-46 43 1.271 40 21 Q. 541272 5.67 9-972011 .78 9.569261 6-45 10.430739' 39 43o352! 3§ 32 54i6i3 5.67 97 '964 .78 569648 6-45 23 541953 5-66 971017 971870 .78 570035 6-45 429965: 37 429578; 36 24 542293 542632 5-66 •78 570422 6.44 25 5-65 971823 •78 57158. 6-44 429.9. 35 26 542971 5-65 97 J 776 .78 6.43 4288o5i 34 27 543310 5-64 971729 •79 6.43 428419' 33 428033 32 28 543649 5-64 97 » 682 •79 571967 572352 6-42 29 544325 5-63 971635 •79 6-42 4276481 3i 3o 5-63 971588 •79 572738 6-42 427262 3o 3i Q. 544663 5-62 9.971540 •79 9-573.23 6-41 10.426877: 2Q 426493 28 32 545000 5.62 971493 •79 573507 6.41 33 545338 5.61 971446 •79 573892 6.40 426.08 27 425724: 26 34 545674 5-6i 971398 •79 574276 6.40 35 54601 1 5.60 9713DI •79 574660 6.39 425340 25 36 546347 546683 5.60 97i3o3 •79 575044 6.39 4249561 24 424573; 23 37 5.59 971256 •79 575427 6.39 6.38 38 547019 y? 971208 •79 5738.0 4241901 22 39 547354 971161 •79 576.93 6.38 423807 1 21 40 547689 5.58 971113 •79 576576 6.37 423424! 20 41 9.548024 5.57 9-971066 .80 9-576958 577341 tu 10.423041! 19 422639! 18 42 548359 548693 5.57 971018 .80 43 5.56 970970 .80 577723 6-36 4222771 17 44 549027 5.56 970022 • 80 578104 6.36 421896! 16 45 549360 5.55 970874 .80 578486 6.35 42i5i4! i5 46 549693 5-55 970827 .80 578867 579248 6.35 421.33 14 ii 550026 5.54 970779 970731 .80 6.34 420732' 1.3 55o359 5.54 .80 579629 6-34 42037 II 12 49 550692 5.53 970683 .80 580009! 6.34 41999.1 II 1 5o 55i024 5-53 970635 .80 53o389| 6-33 419611 10 5i 9.55i356 5.52 9-970586 .80 9-5807691 6.33 10-419231 4i885i ? 52 551687 552018 5.52 97o538| .80 58.. 491 58.5281 6.32 53 5-52 970490 • 80 6-32 418472 7 54 552349 5.5i 970442 .80 58.907! 6-32 4.B093; 6 1 55 552680 5.5i 970394 .80 5822861 6.3. 417714 5 56 553010 5.5o 970345 • 81 582665' 6-3. 4.7335 4 U 553341 5.50 970297 .81 583043! 6-30 416378 3 553670 5.49 970249 .81 583422 6-30 2 59 554000 5.45 5.48 970200 •81 583800 6.29 416200! I 1 60 554329 970152 .81 584177 6- 29 415823 1 Oofeiiie b. Sine 69°, Cotang. 1). 1 Tang. M. I SINKS AND TANOENT8. (21 DKUREKS/ 3 M. o Si no D. Cosine J>^ Tang. D. Cotanjf. 9-554329 5 48 9970152 .81 9-5a4i77 6-29 10 •415823 6c I 554658 5 48 970103 .81 58455D 6 \% 415445 5o 4i5o68 58 2 554987 5553 ID 5 47 970055 .81 584932 6 3 5 47 970006 .81 585309 6 28 414691 57 4 555643 5 46 969957 .81 585686 6 27 414314 56 5 555971 5 46 969909 • 81 586062 6 27 4139381 55 6 556299 5 45 969860 .81 586439 6 27 4i356i| 54 I 556626 5 45 969811 .81 586813 6 26 4i3i85 53 556953 5 44 969762 .81 587190 6 26 412810 53 a 557280 557606 5 44 9697" 4 .81 587566 6 25 412434 5i r 10 5 43 969665 .81 587941 6 25 412059 5o II 9-557932 5 43 9-969616 .82 9-5883i6 6 25 10.411684 49 12 558258 5 43 969567 .82 588691 6 24 411309 48 i3 558583 5 42 969518 .82 589066 6 24 410Q34 410360 47 14 558909 5 42 9^^469 .82 589440 6 23 46 i5 559234 5 41 969420 82 589814 6 23 410186 45 i6 559558 5 41 969370 .82 590188 6 23 4098 1 2 44 12 559883 5 40 969321 .82 590062 6 22 409438 43 560207 5 40 969272 •82 590935 6 22 409065 408692 42 »9 56o53i 5 ^9 969223 .82 59«3o8 6 22 41 20 56o855 5 39 969173 .82 591681 6 21 408319 40 21 9-561178 5 38 9-09>24 .82 9-592054 6 21 10-407946 ^9 22 56i5oi 5 38 969075 •82 592426 6 20 407374 33 23 561824 5 V 969025 •82 592798 6 20 407202 ^"^ U 562146 5 V. 968976 •82 593170 6 •9 406829 36 25 562468 5 36 9O8926 •83 593542 6 \l 406458 35 26 562790 5 36 968877 •83 593914 6 406086 34 'i 563112 5 36 968827 •83 594285 6 18 405715 33 563433 5 35 968777 •83 594656 6 18 405344 32 29 563755 5 35 968728 •83 595027 6 n 404973 3i 3o 564075 5 34 968678 •83 595398 9-595768 6 17 404602 3o 3i 9-564396 5 34 9-968628 •83 6 \i 10.404232 29 32 564716 5 33 968578 •83 596138 6 403862 28 33 565o36 5 ZZ 96S528 •83 5q65o8 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 565095 5663 1 4 5 3i 968379 •83 597616 6 i5 402384 24 !2 5 3i 968329 968278 •83 m 6 i5 40201 5 23 566632 5 3i •83 6 14 401646 22 39 56695. 5 3o 968228 •84 598722 6 14 401278 21 z 567369 5 3o 96S17S .84 599091 6 i3 10 -40054 I 20 41 9-567587 5 29 9-968128 .84 9-599459 6 i3 \l 42 567904 568222 5 % 968078 •84 599827 6 i3 400173 43 5 96S027 .84 600194 6oo562 6 12 399806 •7 44 568539 568856 5 28 967977 •84 6 12 399438! 16 45 5 28 967027 967876 • 84 600929 6 II 398704 1 «4 46 569172 5 27 .84 601296 601662 6 II % 569488 5 5? 9678:6 • 84 6 II 398338, 1 3 397971! 12 569804 5 967775 • 84 602029 602393 6 10 i9 570120 5 26 9677:«5 • 84 6 10 397605! II 5o 570435 5 25 967674 • 84 602761 6 10 397239; 10 io^396873, 3965071 8 5i 9-570751 5 25 9-967614 .84 9-6o3i27 6o34q3 6 09 53 571066 5 24 967573 • 84 6 09 53 57 1 38c 5 24 9675'..2 .85 6o38d8 6 S 396142 7 54 57i6q5 5 23 967471 • 85 604223 6 395777 6 S5 V^ 5 23 967421 •85 604588 6 08 395412 5 56 5 23 967370 .85 604953 6o53i7 6 07 3^68] 4 % 572636 5 22 967310 .85 6 07 3 572950 5 22 96726^1 •85 6o5682 6 -1 394318 2 59 573263 5 21 967217 •85 606046 6 "^ 1 60 573575 5-21 967166 •85 606410 6 06 Cosine D. Sine «8o Colang. D. Taiig. 40 (22 DEGKEES.) A TABLK OF LOGARITHMIO u7 Sine D. Coaiiie | D. | Tuntr. D. 1 Coi&ng. 9-573575 573888 5.21 9-967166, .85 9-6o64ic 6-06 10-393590 60 I 5.20 967115, .85 606773 6-06 3928681 58 2 574200 5.20 967064! -85 607137 1 6-o5 3 574512 5.19 967013 .85 607500 6-05 392500I 57 1 4 574824 5.19 966961 .85 607863 6-04 392187 56 5 573136 5.19 5.18 066859 .85 608223 6-04 891770 55 6 575447 .85 6o8588 6-04 391412; 54 I 575758 5.18 966808 .85 6o8q5o 6093 1 2 6-o3 391050I 53 576069 5-17 966756 .86 6-o3 390688I 52 9 576379 5.17 966705 .86 609674 6-o3 890826 389964 5i . IC 576689 5-16 966653 .86 6ioo36 6-02 5o 11 9-576999 5-i6 9-966602 .861 9-610397 6-02 10.889608 il 12 t]fa 5-16 966550 • 86 610739 6-02 389241 388880 i3 5.i5 966499 .86 611120 6-01 47 14 577927 5-15 966447 .86 611480 6-01 388520 46 i5 578236 5.14 966895 .86 611841 6-01 888159 45 i6 578545 5.14 966344 .86 612201 6-00 887799 887489 44 \l 578853 5.i3 966292' .86 6i256i 6-00 43 579162 5-i3 966240I .86 612921 6-00 387079 42 19 579470 5-13 966188! .86 613281 5-99 886719! 41 20 579777 5-12 966136 .86 618641 til 386359I 40 21 V58oo85 5-12 9-966085 .87 9-614000 10.386000 89 385641 38 22 580392 5-11 966033 .87 614359 5.98 23 580699 5.11 960981 .87 614718 5.98 385282 37 884928 36 384365! 85 24 25 58I00D 58i3i2 5. II 5.10 96'')928 965876 .87 .87 6i5o77 613435 5-97 5.97 26 58i6i8 5.10 960824 .87 615798 5.97 884207! 34 11 581924 5.09 965772 .87 6i6i5i 5-96 888849I 33 582229 5.09 963720 .87 6i65o9 5.96 388491! 32 388i88j 3i ^9 582533 t:i 963668 .87 616867 5.96 3o 582840 965613 .87 617224 5.95 382776 3o 3i 9-583145 5.08 9-963363 .87 9-617582 5.95 10-882418 11 32 583449 5.07 9655 1 1 .87 617989 5.95 382061 33 583754 5.07 965458 .87 618293 6i8652 5.94 381705 27 34 584058 5.06 965406 il 5-94 381848! 26 35 584361 5.06 965353 619008 5.94 880992 25 380686! 24 36 584665 5.06 965301 .88 619864 5-93 ll 584968 5.o5 965248 .88 619721 5-98 380279! 23 585272 5.o5 965195 .88 620076 5.93 879924! 22 39 585574 5-04 965143 .88 620482 5.92 379368! 21 40 585877 5-o4 905090 9-965087 .88 620787 5.92 879218 20 4i 9-586179 5.o3 .88 9-621142 5.92 10-378858 ;? 42 586482 5.o3 964984 .88 621497 621852 5.91 878508 43 586783 5-o3 96493 1 .88 5.91 878148 17 44 587085 5.02 964879 .88 622207 5.90 3777981 16 877489! i5 45 587386 5-02 964826 .88 622561 5.90 46 587688 5.01 964773 .88 622915 IX 877085; 14 s 587989 5.01 964719 .88 623269 628628 376781! i3 588289 5-01 964666 .89 s'o^ 876877; 12 jg 588590 5.00 964613 .89 628076 376024, 11 1)0 588890 5.00 964560 .89 624880 875670, 10 5i 9-589190 589489 4-99 9-964507 .89! 9-624683 5.88 10.375317! 9 374964! 8 52 4.99 964454 .89! 625o36 5.88 53 '^l^ 4-99 4.98 964400 .89I 625388 5.87 3746121 7 54 964347 .89! 625741 5.87 374259! 6 55 590387 4-98 964294 .891 626093 .89 626445 5.87 3789071 5 56 590686 4-97 964240 5.86 373355! 4 u 590984 4-97 964187 .89! 626797 .89' 627149 5.86 378203' 3 591282 4-97 4-96 964133 5.86 37285i[ 2 59 59080 964080 .891 627501 5.85 372499 I 372148: ^ 59187b- 4-96 964026 .89! 627852 5.85 Cosuie D. 1 Si 110 G7°l Cotiui • 02 684001 5 37 315999 i3 638730 4-35 954396 I -02 6^4324 5 37 3.5676 12 1 49 638981 4-35 9543351 • 02 684646 5 37 3.5354 > 56 639242 4-35 954274 ' • 02 6H4968 5 U 3i5o37 •0 5i 9 -639503 4-34 9-954213 I • 02 g 685290 5 r/%.3147101 Q J 1 4388 8 52 639764 4.34 954152 I •02 ' 6S5612 5 36 53 640024 4-34 954090 I •02! 6S5934 5 36 3.4066 7 54 640284 4-33 954029 I 953968 I -021 6S6255 5 36 3i3745i 6 55 640344 4-33 -02; 6H6577 •02 6H6893 5 35 3i3423! 5 56 640804 4-33 953906 I 953845 I 5 35 3i3io2: 4 U 641064 4-32 •02 681219 5 35 312781J 3 641324 432 953783 I •02I 687340 5 35 3i2l6o 2 59 641584 4-32 953722 I •o3 (,HiHb\ 5 -34 312.39 1 ~- 641842 4-3i 9536601 •o3 688182 , Cotan^. 5 .34 3ii8i8 1. Cosine D. Sine t 140 □ D.' ~ 1 Tang. a (26 DEGUEES.) A I'ABLB OF LOUARITnMIC '^r Siiio D. Cosine D. Tang. D. Cotang. 9 •641842 4-31 9 ■ 953660 r^3 9.688182 5.34 io.3ii8i8 60 1 642101 4.31 im o3 688502 5 34 311498 5q 2 642360 4-31 o3 688823 5 34 311177 58 3 642618 4-3o 953475 953413 o3 689143 5 33 3 108571 57 i 642877 4-3o o3 689463 5 33 310537; 56 5 6431 35 4 -30 953352 o3 689783 5 33 310217; 55 6 643393 4-3o 953290 I o3 690103 5 33 309897; 54 I 643630 4-29 o53228 o3 690423 5 33 30^258 52 643908 4-29 q33i66 o3 690742 5 32 Q 644165 4-29 q53io4 o3 691062 5 32 308938 5 1 lo 644423 4-28 953042 o3 691381 5 32 308619 5o II 9.644680 4 28 g. 952980 04 9.691700 5 3i io.3o83oo; 49 307981 48 307662 47 12 6X4936 4 28 952918 04 692019 5 3i i3 645193 4 27 952855 04 692338 692656 692975 5 3i 14 6454D0 4 27 952793 932731 04 5 3i 807344 46 i5 645706 4 27 I 04 5 3i 307025 45 1 i6 645962 4 26 932669' I 04 693293 5 3o 306707 3o6388 44 11 646218 4 26 952606 I 04 693612 5 3o 43 646474 4 26 952544 I 04 693930 5 3o 306070 3o5752 42 •9 646729 4 25 952481 04 694248 5 3o 41 20 646984 4 25 952419 04 694566 5 29 3o543ii 40 21 9-647240 4 25 9.952356 04 9-694883 i 29 io.3o5n7! 3q 304799! 38 304482 1 37 22 647494 4 24 952294 952231 04 695201 5 29 23 647749 4 24 04 693318 5 29 24 648004 4 24 952168 o5 695836 5 11 304164! 36 25 648208 4 24 952106 o5 696153 5 3o3847i 35 26 648012 4 23 952043 o5 696470 5 28 3o353o^ 34 27 648766 4 23 951980 o5 696787 5 28 3o32i3: 33 2d 649020 4 23 931917 o5 697103 5 28 3028971 52 29 649274 4 22 951 854 o5 697420 5 27 3j258oI 3i 3o 649327 4 22 951791 1 o5 697736 5 27 302264 3o 3i 9-649781 4 22 Q. 93 1728 o5 9.698033 5 27 10.3019471 29 3oi63ii 28 32 65oo34 4 22 ^ 931665 o5 698369 5 27 33 650287 4 21 901602 1 o5 698683 5 26 3oi3i5j 27 300999! 26 3oo6G4i 25 34 65o539 4 21 95i539 o5 69900 1 5 26 35 650792 4 21 951476 o5 699316 5 26 36 65 1 044 4 20 951412 1 o5 699632 5 26 3oo368; 24 37 651297 4 20 951 349 06 699947 5 26 3ooo53i 23 38 39 65 1 549 65 1 800 4 4 20 19 951286 951222 06 06 700263 700578 5 5 25 25 2997371 23 29942 2 1 21 40 652052 4 19 95,159 06 70c 893 5 25 299107 20 41 «.6523o4 4 \t Q. 951096 1 06 g. 701208 5 24 10.298792 \l 42 652555 4 95io32 I 06 701523 5 24 298477 43 652806 4 18 950968 06 701837 5 24 298163 n 44 653o57 4 18 950905 I 06 702152 5 24 297848 16 45 6533o8 4 18 950841 1 06 702466 5 24 297534 i5 46 653558 4 17 930778 06 702780 5 23 297220 14 47 6538o8 4 17 95o7i4;i 06 703095 5 23 296^05! 1 3 48 6540O9 4 \l g5o65o 1 06 ]:l^ 5 23 296391 12 49 634309 4 95o586:i 06 5 23 296277 11 5o 654558 4 16 95o522:i 07 704036 5 22 295g64i 10 5i 9.654808 4 16 9.9504581 1 07 9.704350 5 22 10.295650; 9 295337I 8 295o23j 7 52 655o58 4 .16 950394 I 95o33o I 07 704663 5 22 53 655307 4 i5 07 704977 5 22 54 655556 4 i5 950266 1 07 705290 5 22 294710- 6 55 6558o5 4 i5 950202 I 07 703603 5 21 294307; 5 294084 4 56 656o54 4 14 950 1 38,1 07 705916 5 21 u 6563o2 4 14 930074' I 07 706228 5 21 293772' 3 65655: 4 14 950010 I 07 706541 706854 5 21 293439' 2 59 6567CX' 1 4 i3 949045 1 07 5 21 298146; I 2928341 66 1 65704 ; Cosin'j 1 4 •13 949881 1-07 707166 5-20 1 ~; D. Sijje !63=^ Cotang. D. Taiig. M-J BINES ANli TANGENTS (27 DEGREES. ) 60 MT Sino D. Cosine 1 D. 1 Tang. D. Cotang. •>fp 4-i3 9-949881 1-07 j 9-707166 5-20 10-292834 I 4-i3 949816 1-07 1 707478 5-20 2925i2| 5o a 657542 4-12 949732 1-07 707790 708102 5-20 292210 5b 3 6577O0 ' 4-12 949688 1-08 5-20 t^, u i. 658o37 4-12 9I9623 1-0^3 1 70S414 5-19 t 658284 ! 4-12 949558 I -08 1 708726 5-19 291274 55 6 658531 1 4-II 949494 1- 08 709037 5-19 2909631 54 I 658778 4-II 949429 \-oi 709349 5-19 29065 1 53 659025 4-11 949364 I -08 709660 t\l 290340 5a 9 659271 4-10 949300 1-08 1 709971 2?97i8' 5a 10 659517 ' 4-10 949235|i-oS 1 710282 5-18 11 9-659763 4-10 9-949170 I-O^ j 9-710593 5.18 288785 % 47 46 la i3 660009 660255 4-09 4-09 949105 I- oi 949040 I -oS 7 '0904 1 711215 5.18 5.18 14 66o5oi 4-09 948975 I -08 948010 I -08 i 711525 5.17 288475 i5 660746 4-09 711836 5-17 288164 45 i6 660991 4-o8 94HS45 i-o8 712146 5-17 287854 44 \l 661236 4-oS 948780; I -09 712456 5-17 287544 43 661481 4-o8 948715 1-09 712766 5.16 287234 42 '9 661726 4-07 948650 1-09 713076 5-16 286924 41 30 661970 4-07 948584 1-09 713386 5-16 286614 40 ai 9-662214 4-07 9-948519,1 -09 9.713696 5-16 10-286J04 'i 27 662459 662703 4-07 9484541-09 714005 5.16 285995 23 4-o6 948388,1-09 714314 5.i5 285686 u 24 662946 4-06 948323,1-09 714624 5.i5 285376 25 663190 6634i3 4-o6 948257 1-09 714933 5.i5 285067 35 26 4-o5 948192 1-09 715242 5-15 284758 34 J2 663677 4-o5 948126 1-09 7i555i 5.14 284449 33 663920 4-o5 948060 1-09 7i586o 5-14 284140 32 ?9 664163 4-o5 947995 >-'o 716168 5.14 283S32 3i 3o 664406 4-04 947929 I -10 7 '6477 5-14 2H3523 3o 3i 9-664648 4-04 9-947863 l-IO 9-716785 5-14 10.283215 '& 32 664891 4-04 947797 947731 1 -10 717093 5-13 282907 33 665 ill 4o3 I-IO 717401 5-13 282D99 27 34 665375 4-o3 947665 I -10 717709 5-13 282291 281983 26 35 6('>56i7 4 -03 947600:1-10 718017 5.i3 25 36 665859 4-02 947533 I -10 718325 5.i3 281670 24 'A 666100 4-02 947467! I -10 718633 5-12 281367 23 666342 4-02 947401 |i- 10 718940 5-12 281060 22 39 666583 4-02 9473351-10 719248 5-12 280752 21 4o 666824 4-01 9472601-10 9-947203,1-10 719555 5-12 280445 20 4i 9-667065 4-01 9.719862 5-12 io.28oi38 \l 42 667305 4-01 947i36ji-ii 720169 720476 5-11 279831 43 667546 4-01 947070 I -II 5-11 279524 17 44 667786 4-00 947004 I -11 720783 5-II '^W 16 45 668027 4-00 946037 III 946871 I -11 721089 5-11 i5 46 668267 4-00 721396 5-11 278604 14 % 6685o6 3.99 946804 I'll 721702 5-10 278298 i3 668746 3-99 946738 I - 1 1 722009 7223lD 5-10 277901 12 49 66S986 3-99 946671 I -11 5-10 277685; II 5o 669225 \^ 946604 1 • 1 1 722621 5-10 277379 10 10. 277073 9 276768 8 \' 9.660464 9 946538,1-11 9-722927 5-10 52 669703 3.98 94647i|i-ii 723232 5.09 53 669942 3.98 946404|i-n 723538 5.09 276462 ] 2-6 n6l 6 ?^ 670181 3.97 946337 I- M 723844 5.09 55 670419 670658 3.97 946270I1-12 724149 5-09 275851 5 56 3.97 9462031- 12 724454 'a 275546 4 i2 670806 3-97 946i36!i-i2 724759 725o6d 275241 3 671 i34 3.96 946069 1-12 5.08 274935 2 J? 671372! 3-96 94600211-12 725369 5 08 274631 I 6o 671609' 3-96 945935J1-12 725674 5.08 274326 Coeiiie D. Sme i62o Cbtaiw?. D 1 Taiig. [ Tl 46 (28 DKORKES.) A TABLE OP LOG A-RnfiMIfJ M. 9 10 II 13 i3 14 i5 i6 \l 19 20 31 32 23 24 35 36 aS 29 3o 3i 33 33 34 35 36 3^^ 39 40 41 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 u 5q 6o Sine I D. 9*671609 671847 672084 672321 672553 672795 673032 673268 673505 67374 67397 9-674213 6744/18 674684 674919 675155 675390 675624 675859 676094 676328 9-676562 676796 6770J0 677264 677498 677731 677964 678197 678430 678663 9-678895 679128 679360 679592 679824 68oo56 680288 68o5i9 680750 680982 9-68i2i3 681443! 681674 681905 682135 682365 682595 682825 683o55 683284 9-683514 683743 683972 684201 684430 684658 684887 685ii5i 6853431 685571 3.96 3.95 3.95 3.95 3.95 3.94 3.94 3.94 94 93 93 93 92 92 92 92 91 91 91 91 3' go 3 90 3.90 3.Q0 CoBino 1 D. Coftine I D. 9-945o35 1.12 945868 1. 1 2 945800 I • 1 2 945733 1-12 945666 I 945598 I 945531 I 945464 I 945396 I 945328 I 945261 I 9-945193 I 945125 I 945o58 I 944990 I 944922 I 944854 I 944786 I 944718 I 944650 I 944582 I 9-944514 I 944446 I 944377 I 944309 I 944241,1 9441721 944104 I . 944o36 1-14 94396711-14 943899!! .14 9-943830 I - 14 943761 1 1- 14 943693 I - 1 5 943624 I- 15 943555,1-15 943486 1-15 943417 I- 15 943348 i-i5 943279 I -15 943210 1-15 9-943141 i-i5 943072 I -15 943oo3 I -15 942934 I- 1 5 942864 I- 1 5 942795 I -16 942726 I- 16 942656 i-i6 942587 1-16 942517 i.i6 9-942448 I 942378 I 942308; I 942239 I 942169 I 942099 I 942029 I 94i9'^9 I 941889 1-17 941819 1-17 Tang. Sine 9-725674; 725979' 726284! 726588. 726892! 727197: 727501 727805 728109' 7284121 728716I 9.729020I 729323; 729626I 729929' 73o233i 73o535: 73o838 731141 731444' 731746' 9-732048, 73235il 732653; 732955J 733257I 733558| 7338601 734162! 7344631 734764! 9- 735066 I 735367 735668! 7 3 5969 I 736269! 736570 736871 737171 737471 737771 9-738071 738371 738671 738971 739271 739570 739870 740169 740468 740761 9.741066 741365 741664 741962 742:61 742559 742858; 743i56j 743454! 743753] Cctang . _[ D. Cotang. 1 5-08 10-274336 '6^ 5.08 274021 It 5.07 273716 5.07 273412 u 5-07 273108 5.07 272803 55 5-07 5.06 27249Q 272195 54 53 5-06 271891 52 5.06 271588 5i 5-06 271284 5o 5-06 10-270980 1? 5-o5 270677 5-o5 270374 47 5-o5 270071 46 5-o5 269767 45 5-05 269465 44 5-04 269162 43 5-04 268859 41 5.04 268556 41 5-04 268254 40 5-04 10-267952 It 5-o3 267649 5-o3 267347 37 5-o3 267045 56 5-o3 266743 35 5-o3 266442 34 5-02 266 1 40 33 5-02 265M38 32 5-02 265537 3i 5-02 265236 3o 5.02 10-264934 It 5-02 264633 5-01 264332 27 5-01 26403 1 26 5-01 263731 25 5-01 26343o 24 5-01 263129 23 5-00 262829 22 5-00 262529 21 5 -00 262229 20 5-00 10-261929 \l 5.00 261629 4-99 261329 17 4-99 261029 16 4-99 260729 i5 4.99 260430! 14 4.99 260 i3o i3 4.90 4-98 2598311 12 259532 11 4-98 1 259233, 10.258934! 10 4-98 ? 4-98 258635! 4-98 2583 36i I 4-97 268o38: 4-97 257739: 4-97 257441 4-97 257142 4-97 256844 4-97 356546; 4-96 256248. D. 1 __Tan^ J^ J^!L 8INE8 AND TAN0KNT8. (29 DKOIIBES.^ > 47 M.' Sine 9-685571 D. Cosine 1 D. 1 Tang. D. CoUing. I 3.80 9-9418191.171 9-743752 496 10-256248, 60 I 685799 3.79 9417491-17! 744o5o 4.96 355950! 59 2556521 58 3 686027 3-79 94.679 .•.71 744348 4-96 3 686254 3.79 94.6091-171 744645 4-96 2553551 57 4 686482 i^ 94.539 I -17I 744943 4.96 255o57 56 5 ^5? 94.4601-171 743240 94.3981-17! 745538 4.96 254760J 55 6 3-78 4-95 254462 54 I 687163 3.78 94.3281-171 745835 4-95 254.65 53 687389 3.78 94.2581-171 746132 4-95 253868 52 9 687616 3-77 941187 '-n: 746429 4-95 253571 5i 10 687843 3-77 941 1 17 1 -ni 746726 4-95 253274 5o II "■^ 3-77 9.941046 1-18 9-747023 4.94 10-252977 25268. il la 3.77 9409751-18: 7473.9 4-94 id 688521 3-76 940005.1-18 747616 4.94 2523S4 ^J 14 688747 3-76 940834 I -181 7479>3 4.94 252087 46 15 688972 3-76 Q40763 1-18 748209 7485o5 4-94 25.79. 45 i6 689198 3.76 940693 1-18 4-93 23.495 44 \l 689423 3-75 940622,1-18! 748801 493 25o9o3 43 689648 3.75 94055 1 1 1-18 749097 94o48oji.i8| 749303 9404001-181 749689 9-94o338ji-i8, 9-749983 940267; I -18 750281 4-93 42 19 689873 3.75 4-93 2 50607 4. 20 21 690098 9 690323 3.75 3-74 4-93 4-93 25o3.. io-i5oo.5 40 ^2 22 690548 3.74 4-92 249719 38 23 690772 3.74 9401961-18 750576 4-92 249424 37 24 690996 3 74 940.25:1-19 750872 4-92 249.28 248833 36 25 691220 3-73 94oo34{i-i9i 751.67 4-92 35 26 691444 3.73 939982 1.19 751462 4-92 248538 34 27 691668 3-73 939911 I -19 751757 4-92 248243 33 28 691892 3-73 939840 1-19 752032 4-91 247948 32 29 692115 3-72 939768,1-19 752347 4-9' 247653! 3i 3o 692339 3.72 939697,1-19 752642 4.91 247358, 3o 3. 9-692562 3-72 9.939625,1-19 9.752937 4-9' 10-2470631 2n 246769 :8 32 692785 3-7. 939354 1 - 19 753231 4-9' 33 693008 3-71 939482 1-19 753526 4-9» 2464-4! 27 34 693231 3-71 9394.0 1-19 753820 4-90 246180 20 35 693453 3.7. 939339 1 - 19 7541.5 4.90 245885 25 36 693676 3-70 939267 1-20 939195 1 -20 ti^ 4.90 24559. 24 37 693898 3-70 4.90 243297 23 38 694120 3-70 939123 l-2o[ 754997 4.90 245oo3 22 39 4o 694342 694564 3-70 3.69 93903211 -20! 733291 9389S0J1-20J 755585 tx 244709 2444.5 21 20 41 9-694786 3.69 9-938908 1-20, 9.753S78 4-89 10-244.22 ;§ 42 695007 3-69 938836 1-20 756.72 4.89 243828 43 690229 3.69 3-68 938763 1-20 756465 4.89 243535 17 44 695450 938691 1 -20 756759 4.89 24324. 16 45 69567. 3-68 9386i9Ji-2o 757052 4-88 242948 i5 46 695892 3-68 9385471 « -20 938475|i-2o 757345 242655 14 il 696113 3-68 757638 4-88 242362 i3 696334 3.67 938402 75793. 4-88 24:069 12 49 696554 3.67 938330 758224 4-88 24.776 1. 5o 696775 3-67 938258 7585.7 4-88 241483 10 5i 9-696993 3-67 9-938i85 9-758S10 4-88 10-241190 « 52 6972.5 3-66 938 n 3 759102 4-87 240898 8 53 697435 3-66 938040 ]m; 4-87 24o6o5 I 54 697654 3-66 ttl 4-87 24o3i3 55 697874 698094 3-66 759979 4-87 240021 5 56 3-65 937822 760272 4-87 239728 4 tl 6983.3 3-65 937749 760564 4-87 4-86 239436 3 693532 3-65 937676 1-21 760856 23QI44 238852 2 59 69875. 3-65 937604 I -21 761 148 4-86 I 60 698970 3.64 937531 1-21 761439 4-86 238561 CoRine D. Sine 60° Cotang. D. Tmig. M. 18 (30 DEGREES.) A FABLE OF LOGARITHMIC ld7 Smo 1). Cosine | D. 1 Tang. D. Cotang. \~~^ 9-698970 3-64 9-937531 1-21 9-761439 4-86 io.23856i! 60 I 699189 3-64 937458 1-22 761731 4-86 288269 50 287077 58 2 699407 3-64 937385 1-22 762023 4-86 3 699626 3-64 937312 1-22 762814 4-86 287685! 57 4 699844 3-6: 937238 1-22 762606 4-85 287894! 56 5 700062 3-oJ 937165 1-22 768188 4-85 287103 55 6 700280 3-63 937092 1-22 4-85 286812' 54 I 700498 3-63 937019 1 -22 768479 4-S5 236521! 53 700716 3-63 936046 1-22 936872 1-22 768770 4-85 286230! 52 9 700933 3-62 764061 4-85 235989' 5i 10 70i:5i 3-62 936799 1-22 9-936725 1-22 764352 4-84 235648 5o II 9-701368 3-62 9-764648 4-84 13-235357 4^ 47 12 i3 701585 701802 3-62 3.61 936632 1-23 936578 1-23 764933 765224 4-84 4-84 235067 284776 14 702019 3-6i 9365o5i-23 936431 !i -23 765514 4-84 284486 46 i5 702235 3.61 7658o5 4-84 284195 45 i6 702452 3-6i 936357ii-23 766095 4-84 288905 44 ]l 702669 702885 3-6o 9362841-23 766385 4-83 2336i5 43 3-6o 936210 1-23 766675 4-83 233325 42 19 7o3ioi 3-6o 936i36 1-23 766965 4-83 233o35 41 20 703317 3-6o 986062 1-23 767255 4-83 282745 40 21 9-703533 3-59 9-935988;! -23 9-767545 4-83 10-282455 39 22 703749 3.59 9359i4!i-23 935840 1-23 767884 4-83 282166 38 23 703964 3.59 768124 4-82 281876 37 24 704179 704395 3.59 935766^1-24 768418 4-82 23 1 587 36 25 Ifs 035692 1-24 768708 4-82 281297 35 26 704610 9356i8;i-24 768992 4-82 281008 34 11 704825 3-58 935543,1-24 769281 4-82 230719 33 7o5o4o 3-58 935469:1-24 769570 4-82 280480 32 29 705204 3-58 935393 1-24 769860 4-81 280140 3i 3o 705469 3.57 935320 1-24 770148 4-81 220852 3o 3i 9-705683 3-57 9-935246 I -24 9.770487 4-8i 10-229563 11 32 705898 3.57 93517111-24 770726 4-8i 229274 228985 33 706112 3-57 935097! 1-24 771015 4-8i 27 34 706326 3-56 935022 1-24 771808 4-8i 228697 26 35 706539 3-56 934948 934873 1-24 77i5o2 4-8i 228408 25 36 706753 3-56 1-24 77.8^0 4-80 228120 24 11 706967 3-56 934798 1-25 772168 4-80 227882 23 707180 3-55 934723 1-25 772457 4-8o 227543 22 39 707393 3-55 934649! 1-25 772745 4-80 227255 21 40 707606 3-55 934574 1-25 778088 4-8o 226967 20 41 9-707819 3-55 9-934499 1-25 9.778821 4-8o 10-226679 '9 18 42 708032 3-54 9344241-25 778608 4-79 226892 43 708245 3-54 934349 1-25 773806 4-79 226104 17 44 708458 3-54 934274,1-25 774184 4-79 2258i6 16 45 708670 3-54 934199'! -25 774471 4-79 225529 i5 46 708S82 3-53 934123 1-25 774759 4-79 225241 14 47 709094 3-53 9340481 1-25 775046 4-79 224954 i3 48 709806 3-53 933973: 1-25 775333 4-79 4-78 224667 12 49 709518 3-53 933898,1-26 775621 224879 11 5o 709730 3-53 933822 1-26 773908 4-78 224092 10 5i 9-709941 3-52 9-933747ii-26 9-776105 4-78 10 2238o5 i 53 710153 3-52 933671 1-26 776482 4-78 2 235 1 8 53 7io364 3-52 933596 1-26 776769 4-78 228281 7 54 710375 3-52 93-332o'i.26 777o5d 4-78 222945 6 55 710786; 3-5i 933445 1-26 777342 4 78 222658 5 Go 710997: 71120S' 3-5i 933369' 1. 26 777628 4-77 22J372 4 u 3-5i 933293 1 .26 777915 4-77 222085 3 711419 3-5i 9332 1 7>- 26 778201! 4-77 221799 2 59 71 1629 3-5o 9331 41 ;i- 26 778487! 4-77 22l5l2j I 6o 71 1839 3-30 933066; I -26! 778774I 4-77 221226! Cosine 1 Ih^ 1 Si]ie_ J590]_Cotang. [ D. _ Tang. J _M.J WNES AND TAN'GEN'TtJ. (31 DEGREKt5.) 41) W. 9 in II 13 |3 14 |5 i6 \l 19 20 21 22 23 24 25 26 II 3i 32 33 34 35 36 i ll 3q 40 41 42 43 44 45 46 % 5o 5i 52 53 54 55 56 U Sine 9-711839 7i2o5o 712260 712469 712880 713098 7i33o8 713517 713726 713935 9-7I4I44 714352 71456! 714760 714978 7i5t86 7 1 5391 7 1 56o2 7 1 5809 7 1 60 1 7 ! 9-716224 716432 716639! 716846! 7170531 717259' 717466. 717673. 717879 7i8o85 9-718291' 718497 718703 718909; 7i9"4i 719320 719525 719730 719935 72or4o 9-720345 750549' 7207J4 720953 721162' 721366' 72i57c| 721774' 721978 722181I 9-722385 722588 722791! 722994 723197, 7 23400 I 7236o3| 7238o5 724007 724210 D. Cosine I D. | Taii^. i D. | Cotaiig. f~ 3-50 3-50 3-50 3-49 3-49 3-49 3.49 3.4$ 3.4b 3.48 3-48 3-45 3-45 3.45 3-45 3-45 3-44 3-44 3-44 3-44 3.43 3-43 3.43 3-43 3.43 3-42 3-42 3.42 3.42 3.41 3.41 3 3 3 3 3- 41 40 4c 40 3-40 3.40 3 3 3 3 3 3. CJosine 39 39 ll It 3-38 3-38 3-38 3.37 3.37 3.37 3.37 9-933o66'i-26| 982990 I -27! 932oi4;i-27| 932$;j8 1-271 932]62 i-27| 932685; I -271 93260Q 1-27! 932534 1-27I 932457 I -271 932380 1-27! 982304 I -27, 9932228 1-27' 932i5i 1-27' 982075 I -28! 981998 1-28, 981 921' I -281 981845 I-28| 981768 1-28^ 981691 1-28' 981614 1-28, 981587I1.28 9>93i46o I -28 981883 1.28j 981806 1-28 981229 1 -29' 981 1 52 I -29' 981075 1-29: 980998 1-29' 93oo2i[i.29l 980843, 1 -29 980766 1-29' 9-980688 1-29' 98061 1 I -291 98o538 1-29^ 980456 I -291 9808781 1.29I 980800 i.3oj 980228 I -3oi 980145 1 -30] 980067 1 .3o; 92998Q 1 .3o| 9-92091 1 i-3oi 929888, 1. 3oj 929755 I -80 929677 i.3o 929599 1 .30 929531 I -801 929442 i.3o 929 ?6^ i-8ij 9292% i-3i' 929207 1 .31 ; 9-929129 I -31 92905c 1-81 9:8972 I -311 928K98 I 81 928815 i.3r 928786 1.31 1 928657 I • 3 1 92S5781.81 Q28499 I -31 I ^284201 -31 i D. I680 9-778774; 779060 779346 779682I 7799'8{ 780203 780489 780773; 7810^)0. 781846I 781681 9-781916 782301, 7S2486 782771 7«8o56 7888 J I 788626 788910 784195 784479 9-784764 78,^50 ',8 785882 7856 1 6 785900 786181 786468 ■786752 787086 787810 9-787608 7878S6 78S170 788453 788786 789019 789I02 789585 789868 790i5i 9-790488 790716 790999 7912S1 791568 791846 792128 792iio 792692 79297 i ?• 798256 79^538 7938 1 9 79iioi 794883 79^045 795327 795508 i 795789 1^ Cot&n^, 4-77 4- 4-77 4-76 76 76 76 76 76 76 75 75 75 75 75 75 4 4 4 4 4 4 4 4 4 4 4 4 4-7-' 4-75 4-74 74 74 74 74 74 73 73 73 73 73 73 73 72 72 72 72 72 72 72 7> 7' 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4' 4 4-7« 4-71 469 469 4 69 4 69 4-69 4-69 4-69 4-68 4-68 I|0-22I226 220940 220634 220368 220082 219797 2I95I I 219225: 2 I K940 218654) 2IS869I 10 218084 2i779<; 217314 217229 2 ! 69 216659 216874 2 1 6090 2i58o5 2 1 552 1 10-215286 214952 214668 214884 2 1 4 1 00 218816 218582 218248 2 1 2964 212681 10-212897 212114 211880 21 1 547 2 1 1 264 2 1 098 1 2 1 0698 2 1 041 5 210182 209H49 10-209567 209384 209001 208719 208487 208 1 541 207872 207590 2,t78o8 207026] 10-206744 206462 206181 205899 205617I 2o5336l 2o5o55' 204773 204492 2042 1 1 60 u 55 54 53 52 5i 5o 47 46 45 44 43 42 41 40 ll 35 34 33 32 3i 3o It ll 25 24 23 22 21 20 I Taiiff. 50 (32 DEGREES.) A TABLE OP LOGARITHMIC M. Sine 1 ^• Cosine | D. Tanof. 1 ^' Cotaiig. 1 9-724210 3-37 9-928420 1-82 9.795789! 4-68 10' 20421 1 ; 60 I 724412 1 3 •37 928842 1-32 796010 4.68 2089301 5o 2 724614 1 3 •36 928268 I -82 79633! 4-68 2086491 58 203368! 57 208087! 56 3 724816 3 -86 928188 1-82 796682 4-68 4 725017 3 -86 928104 1-32 796913 4-68 5 725219 3 • 36 928025 1-82 797194 4-68 202806! 55 6 725420 3 35 927946 1-82 927867 1-82 797475 4-68 20x525; 54 2 725622 3 85 797755 4-68 202245! 53 725823 3 35 927781 1-82 798086 4-67 20.964; 52 9 726024 3 35 927708 1-32 798816 4-67 2oi684j 5! 10 726225 3 35 927629 1-82 798596 4-67 201 404 j 5o II 9.726426 3 34 9.927549 1-32 9.798877 4-67 IO-20I128l 49 2008431 48 la 726626 3 34 927470 1-38 799157 4-67 i3 726827 3 34 927890 1.38 799437 4-67 2oo563 47 U 727027 3 34 927810 1.83 799717 4-67 200283 46 i5 727228 3 34 927281 !.33 799997 800277 4-66 200008 45 i6 727428 3 33 927.51 1.83 4-66 199723' 44 \l 727628 3 33 927071 1.88 800557 800886 4-66 199448! 43 727828 3 38 926991 1.88 4-66 199164 42 ^9 728027 3 33 9269TI 926S81 !.33 801 1 16 4-66 19S884! 41 20 728227 3 33 1.33 801896 4-66 198604! 40 21 9-728427 728626 3 32 9-926-51 1.38 9.801675 801955 4-66 10-198825; 39 198045; 38 22 3 82 026671 1.38 4-66 23 728825 3 32 926591 !-33 802284 4-65 197766' 37 24 729024 3 82 9265 11 1.34 8o25i3 4-65 197487; 36 25 729223 3 81 926481 1-84 802792 4-65 197208 35 26 729422 3 81 926851 1-34 808072 4-65 196928 34 11 -720621 3 81 926270 1-34 8o885i 4-65 196649 33 729820 3 3i 926190 1-34 808680 4-65 196870; 32 29 780018 3 3o 926110 1-84 808908 4-65 196092 3 1 3o 780216 3 3o 926029 1-34 804187 9-804466 4-65 195813, 3o 3i 9-73o4i5 3 80 '■fM^ 1-34 4-64 10- 195534 29 1952551 28 32 7806 1 3 3 3o 1-34 804745 4-64 33 780811 3 3o 925788 1-34 8o5o23 4-64 I 949771 27 1946981 26 34 781009 3 29 925707 1-34 8o53o2 4-64 35 73 1 206 3- 29 925626 1-34 8o558o 4-64 194420 25 36 781404 3. 29 925545 1-35 8o5859 4-64 I94i4!l 24 ll 781602 3- 29 925465 1-35 806187 4-64 198868! 23 781799 3- U 925884' 1-35 8064 1 5 4-63 193585, 22 39 781996 3- 925308: 1-35 806693 4-63 198807 31 40 782.98 3. 28 925222I !-85 806971 4-63 198029 20 41 9-732390 3. 28 9-925i4i 1.35 9-807249 4-68 .0-192751 \t 42 782587 3- 28 925o60| 1.35 807527 4-63 192473 43 782784 3- 28 924979, 1.35 807805 4-63 , 192.95 ll 44 782980 3- 27 924897 !.85 80S088 4-68 191917 45 788.77 3- 27 924816 1-85 808861 4-63 .9.689 :5 46 783378 3. 27 924735 1-86 808688 4-62 191862! 14 ii 783569 788765 3- 27 924654; 1-86 808916 4-62 .91084 18 3. 27 924572J1-86 809.98 4-62 190S07J 12 49 7330. 3. 26 924491; I -36 809471 4-62 190329; 11 5c 784157 8. 26 924409' 1 -86 9-924828, 1-36 809748 4-62 190252! !0 5i 9-784553 8 26 9.810025 4-62 10-1899751 Q 1896981 8 §2 784549 3- 26 924246 1-36 810802 4-62 53 734744 3- 2: 924i64:i-36 8io58o 4-62 189420 7 54 734989 8- 2) 924088 I -361 810857 4-62 189.43 188866 6 55 785.85 3- 25 924001;! '36 811184 4-6i 5 56 78588o 3- 25 9289 1 9' 1-36.' 811410 4.61 188590 4 ll 785525 3- 25 928887, 1 -36; 811687 4-6i i883i3 3 735719 -i- 24 928755;!. 87 81 1964 4-61 i88o36 a 59 735914 3. 24 928678 1-37 92359IJI-37 812241 4-61 »?775q I 1 6o 786109 3-24 812517 4-61 187483 Cosine D. Sine J5TO! Cotang. i D. 1 Tang. M. BINBG A NT) TANGENTS. (3.3 DKORRKS. ) b\ \^ bine D. Cosine | D. | Tamr. D. Cot&ng. 1 o '•]& 3.24 9-92359i|i.37i 9.812517 61 10-187482 "60 I 3 24 923509' I •371 8!2794 61 187206 59 a 736498 3 24 923427;! 923J45!i •37 813070 61 186930! 58 1 3 ?a? 3 23 •37 813347 4 .60 186653 57 4 3 23 923263 I •37 81 3623 4 60 186377 5o 5 737080 3 23 923i8i]i •37 813899 4 60 186101 55 6 737374 3 23 923098' I .37' 814175 .37! 8!4452 60 185825 54 I 737467 3 23 923oi6|i 60 185548; 53 737661 3 22 922q33i 92285i 1 .37 814728 60 .85272 5j g 737855 3 22 •37 8i5oo4 60 184996 5i 1 IC 738048 3 22 9227681 9 .922686 1 1 .38 8i5279 60 18472 1 5o ^ II 9-738241 3 22 .381 o.8i5555 59 10-184445 49 .84169 4!^ 183893' i7 19 738434 3 22 922603 I -38! 8i583i 59 !3 738627 8 21 92252o'l -381 8r6i07 59 t/i 738820 3 21 922438,1 -38i 8i6382 59 1 836 18] 46 183342 45 i5 739013 3 21 922355 I -38 8 16658 59 |6 739206 3 21 922272 I 922189 I • 38 816933 59 1 83067 1 44 ]l 739398 3 21 .38 817209 ^ 182791! 43 739500 739783 3 20 922106 I -38 817484 59 182516 42 «9 3 20 922023 I .38 8.7759 8i8o35 U 182241! 41 2o 739975 3 20 921940 I 9-921857 I -38 181965 10-181690 40 21 9-740167 3 20 .39 9-8i«3io 58 ^ 22 740359 3 20 931774 I .39 8.8585 58 181415 23 74o55o 3 19 92169I I .39 818860 53 181140J 37 24 740742 3 19 921607 I -39 819135 58 .8o865i 36 25 740934 3 19 921524 I -39 819410 58 .R0590! 35 26 741125 3 »9 921441 1 -39 819684 58 1803 i6j 34 27 74i3i6 3 \t 921357 I .39 8.0959 58 18004 1 1 33 28 74i5o8 3 9312741 -39! 820234 58 1797661 32 2y 741699 3 18 921 190 I .39 82o5o8 4 57 1794931 3i 3o 741889 3 18 921 107 I .39 820783 / 57 170217' 3o 3i 9.742080 3 18 9-921023 I -39 9-821057 57 io-ii>^,,iZ 29 178668; 38 32 743271 3 18 Pl?! •40 82.332 57 33 742462 3 17 .40 821606 4 57 178394; 37 34 742652 3 17 920772'! .40 821880 , 57 1 78 120, 26 35 743842 3 17 920688! ! .40 822154 57 , 177846 25 36 743o33 3 17 920604!! .40 822429 57 ^ 177571; 34 37 743223 3 '7 92o52o!i .40 822703 ll 177297! 23 38 743413 3 16 920436|i .40 822977 177023i 22 39 743602 3 16 920352I1 .40 823250 56 17675OJ 2. 40 9-743982 3 16 920268,1 -40! 823524 56 .76476, 20 . ii 3 16 9-930i84li -40; 9-823798 56 10.176202 \l 42 744171 3 16 920099 I .40 824072 56 175928 175655 43 744361 3 i5 9300l5|I •40! 824345 56 17 44 744550 3 i5 919931 I 919846 I •41 824610 .41 82489J 4 56 175381 16 45 744730 74493S 3 i5 56 175.07 15 46 3 i5 919762 I .41 825166 56 174834 14 S 745117 3 i5 9J9677 I -41 825i3Q ■41 825713 55 174561 13 745306 3 14 919593 I 55 174287 !2 49 745404 3 14 919508,1 •41 825986 55 174014 «I 5u 745683 3 14 919434 1 •41 826259 55 1737411 10 5i 9-745871 746o5o 74624B 3 14 9919339 I •41 9-826532 55 10-173468 9 92 3 14 9192541 •41 826805 55 173195 8 « 3 i3 919169!! 919085:1 -41 827078 55 172922 7 1726491 6 54 746436 4 i3 .41 8273DI 55 55 746624 3 i3 919000'! •41 827624 55 172376] 5 56 746812 3 i3 9i89!5:! •42 827897 828170 54 172103 4 ■u 746909 747 '87 3 i3 9i883o ! •43 54 171830 3 3 12 918745 I 918659'. -42 828442 54 I7I558 2 ^) 747374 3 12 -42 828715 54 171285 I 6o 747562 3 12 918574 ! •42 828987 54 171013 _Coamo_ D. Sine 6O0I Cotang^ ~D.~I . Tanff. ^J 52 (34 DEOREEB.) A TABLE OF LOGARITHMIC M. Bine D. Cosine 1 D. 1 Tang. D. Cotang. 1 ' o 9 -747562 3.12 9.9i8574;i-42 918489' 1. 42 9-828987 4-54 10-171013 1 60 I 747749 3.13 829260 4-54 170740 a 747936 3-13 918404 1-42 829532 4-54 170468 58 3 748123 3. II 9i83i8|i.42 829805 4-54 170195 57 4 748310 3.11 918233I1.42 830077 4-54 169923 56 5 748497 7486$3 3. II 918147 1-42 83o349 4.53 I 6965 I 55 6 3-11 918062 1.42 83o62i 4-53 169879 54 I 748870 3.11 917976 917891 1-43 1 830893 4.53 16^835 53 749056 3.10 1.43 83ii65 4-53 5a 9 749243 3.10 917805 1.43 831437 4-53 168563 5i 10 749429 9-749615 3.10 917719 |i-43 881709 4-53 168291 5o II 3.10 9-917634 |i-43 9.831981 4-53 10.168019 ^ la 749801 3-10 917548 1.43 832253 4-53 167747 i3 749987 3.09 917462 1.43 832525 .4-53 167475 2 a 750172 3.09 917376 1.43 832796 4-53 167204 i5 75ol')8 3.09 91729011.43 833o68 4-52 i6'j932 45 i6 75o543 3-09 917204 1.43 833339 4-52 166661 44 \l 700729 3':^^ 917118 1.44 833611 4-52 166889 166118 43 7D0914 917032 1.44 833882 4-52 43 J9 Villi 9.751469 3-08 916046 1-44 834 1 54 4-52 165846 41 20 3-08 916859 1.44 9.91677311.44 834425 4-52 165375 40 31 3.08 9.834696 4-52 io-i658o4 ^ 23 75i654 3.08 916687 1.44 834967 4-52 i65o83 23 75i83q 732023 3.08 916600 1-44 833238 4-52 164762 37 24 3.07 9i65i4 1.44 835509 4-52 164491 36 23 732208 3-07 9 -'642 7 1-44 835780 4-5i 164220 35 26 732392 3.07 916341 1-44 836o5i 4-5i 168949 34 u 752576 3.07 916254 1-44 836322 4-5i 168678 33 752760 3.07 916167 1.45 836593 4-5i 168407 32 o9 752944 3.06 916081 1.45 836864 4-5i i63i36 3i So 753128 3.06 9159941-45 837134 4-5i 162866 3o 3i 9.753312 3.06 9-913907 1.45 9-837405 4-5i 10.162395 It 33 753495 3.06 915820 1.45 837675 4-5i 162825 33 753679 3.06 915733 1.45 83-946 4-5i 162034 27 34 733862 3.o5 915646 1.45 8382x6 4-5i 161784 26 35 754046 3.o5 915559 1.45 838487 4-5o i6i5i3 25 36 754229 3.o5 915472 1-45 838757 4-5o 161243 24 ll 754412 3-05 9i5385|i.45 889027 4-5o 160973 23 754595 3.o5 913297 1-45 839297 4-5o 160708 22 39 754778 754960 3-04 915210 1.45 889368 4-5o 160482 21 4o 3.04 9i5i23 1.46 889888 4-5o 160162 20 41 9.755143 3-04 9.915035 1.46 9-840108 4-5o 10-159892 1? 42 755326 3-04 vx 1.46 840878 4-5o 159622 43 755508 3-04 1.46 840647 4-5o 139353 17 16 44 755690 3.04 914773 1.46 840917 4-49 159088 45 753872 3.o3 914685 1.46 841187 4-49 i588i3 i5 46 756o54 3.o3 914598 1.46 841457 4-49 158543 14 ^I 756?36 3.o3 914510 1.46 841726 4-49 158274 i3 48 756418 3.o3 914422 ..46' 841996 4-49 i58oo4 12 ^9 736(>oo 3.o3 914334!! -46 842266 4-49 157734 II 5o 756782 3.02 91424611 -47 842385 4-49 157465 10 5i 9.756963 3.02 9-914158^1.47 9-842805 4.49 10.157195 I 53 757144 3.02 914070,1.47 913982,1.47 9138941-47! 843074 4-49 156926 53 757326 3.02 843343 4-49 156657 7 54 757688 3.02 843612 V^ 156388 6 55 3.01 9138061.47! 848882 i56ii8 5 56 757869 3.01 9137181.47 8441 5 I 4-48 155849 4 ll 758o5o 3.01 9i363o 1-47 844420 4-48 1 55580 3 758230 3.01 913541 1-47 844689 844958 4-48 i553ii 2 ^ 758411 3.01 9i3453|i.47 4-48 i55o42 I 6o 75359. 3.01 913365 1-47 845227 4.48 154773 1 Coeino D. 8ine 55" Cotimg. D. Tang. IT' HINES AND TANOKNTb. (35 DEGREES.' 58 ¥7 Sine 1 D. Cosine D. Tancr. D. Cotanj?. 60 9-758591 3.01 9-913365 1.47 9-845327 4-48 io-:54773 1 ]fsll\ 3-00 3oo 913276 I 913187 I .47 845496 -48 843764 4-48 4-48 1 54504 154236 ^ 3 759133 3oo 913099 I .48 846033 4.48 [5396^ 57 1 4 75931a 3.00 9i3oio^i .48' 846302 4-48 1 53430 56 5 759492 3oo 912922 I 913833I1 912744 I 913655 I •48; 846570 4-47 55 6 I 759H53 2-99 2-99 .48' 846839 •48 847107 .48 847376 4-47 4-47 i53i6i 152893 54 53 76003 I 2-99 4-47 152624 52 9 7602 II 2-99 912566 I •48 847644 4-47 152356 ?' lo 9.7605?^ V.^ 912477 » 9.9123881I .48 847913 4-47 152087 5o 11 .48 9-848181 4-47 io.i5i8i9 it la 76074S 3.98 912299 I •4o 848449 4-47 i5i55i i3 760937 3.98 912210 I •49 848717 4-47 1 51383 47 14 761106 2.98 912121 I .49 848986 4-47 i5ioi4 46 i5 761285 2.98 9i2o3i 1 .49 849254 4-47 1 50746 45 i6 761464 3-98 911942I1 -49 849322 4-47 1 50478 44 •7 761642 2-97 9ii853|i -49; 849790 .49' 85oo58 4-46 l5o2IO 43 i8 761821 2-97 9n763|i 4-46 149942 42 «9 761999 2-97 91167411 •49 85o325 4-46 149675 41 30 9.763336 2-97 911 584' I .49 85o593 4-46 149407! 40 31 3.97 9-911495 I •49 9 -850861 4-46 '"■'X^^llt 33 762534 3.96 911405 I •49 851129 85i396 4-46 33 7637:3 763889 3.96 9u3i5:i .50 4.46 1486041 37 148336! 36 24 3.^ 911336 I -5o' 85i664 4-46 25 763067 763245 3.46 911136 1 -So 85i93i 4.46 148069 33 36 3.^ 911046 1 -5o 832199 •So! 852466 4.46 147801 34 27 763422 3.^ 9ioo56 I 910866; I 4-46 147534 33 23 763600 3.^ • So 852733 4-45 147267 33 29 763777 3.95 910776^1 010686,1 • 5o 853001 4-45 146909I 3 1 1 3o 763934 3.95 .50 853268 4-45 1467 J 2 3o 3i 9-764i3i 395 oqio5q6'i .50 9-853535 4-45 10 -146465 11 33 764308 3.95 9io5o6ii •50 853802 4-45 146198 1459J1 145664 33 764485 3-94 91041511 .5o 854336 4-45 27 34 764662 3-94 9io325ii .5i 4-45 26 35 764838 2-94 9102351 9JOi44|i .5i 8546o3 4-45 145397 i45i3o 25 36 76301 5 2-94 .51 854870 4-45 24 ■?, 765191 3-94 910054' I • 51 855i37 4-45 144863 23 3^ 765367 3.94 909783 1 •5i 855404 4-45 144596 32 -9 765544 3.93 .5i 855671 4-44 144329 21 40 765730 9.765S96 3.93 .5i 855938 4-44 144062 30 4i 3.93 9-909691 I .5i 9-856204 4-44 10.143796 \t 43 766073 a.?.3 909601 I .5i 856471 4-44 143529 143263 43 766347 3.93 909510 I .5i 856737 4-44 \l 44 766433 3.93 909419 I 909328.1 • 5i 857004 4-44 142906 142730 45 766598 3.93 .52 857270 4-44 i5 46 766774 3-93 909237 I .52 857537 4-44 142463 U 47 766949 2-92 909146 I .52 857803 4-44 142197 1419J1 141664 i3 48 49 767124 767300 2.93 3-93 900055^1 •52 •52 858069 858336 4-44 4-44 13 11 S 767475 3.91 •52 858602 4-43 i4i3q8 IO-l4ll32 10 5i 9-767649 3.91 Q. 90878 I I 908690 I .52 9-858868 4-43 i 5i 767824 3-91 •52 859134 4-43 140866 53 'Xi 3 91 9085991 .52 859400 4-43 140600 I U 3-9! 908507 I •52 839666 4-43 140334 55 768348 390 908416 I .53 859932 4.43 140068 5 56 768523 3.9c 9083241 .53 860198 4-43 139803 4 U 768697 3-90 I 90S233 I .53 860464 4-43 139536' i 768S71 3-90 908141,1 .53 860730 4.43 139370; 3 1 S 769043 769319 3.90 a 90 D. 908049 I 907958,1 ill K 4.43 4-43 1 39005 13^739 1 Cosine Sine la ^40 Cotang D. Tanjf. WJl 54 (30 DEGREES.) A TABLE OF LOGARITHMIC 'm:~ Sine D. Cosine D. Tang. D. CoTIUlg. '■'Zi^ 2-90 9-907958 1 90786611 -'5i 9-86126! 4-43 !0- 138739 138473 GO I 89 .53 861 527 4 43 ^ 2 769566 89 907774 1 -53 86.792 862058 4 42 138208 3 769740 89 90768211 .53 4 42 137942 U 4 769913 89 907590 1 -53 862323 4 42 137677 5 770087 2 ^ 907498:1 -53 862589 4 42 .37411 55 6 770260 907406: 1 -53 862854 4 42 .37.46 54 ^ 6 770433 * 88 907314 I 907222|I •54 863119 863385 4 42 .3688! 53 770606 88 -54 4 42 .366.5 5j 9 770779 88 907129I •54 863650 4 42 .36350 5i 10 770952 88 90703711 •54 863915 4 42 .36o85 5o II 9-77"25 88 9-90694511 906852 '1 •54 9^864.80 4 42 I3-. 35820 it 12 771298 87 •54 864445 4 42 .35555 i3 771470 87 906760I1 •54 864710 4 42 135290 47 14 771643 87 906667:1 •54 864975 4 41 .35025 46 i5 771815 87 906575 I -54 865240 4 41 .34760 45 i6 77 '9^7 87 90648211 •54 8655o5 4 41 .34495 !3423o 44 n 772139 11 906389!! 9062961 •55 865770 4 4! 43 i8 772331 2 •55 866o35 4 4! 133965 42 19 7725o3 86 9062041 •55 866300 4 41 133700 4! 20 772675 86 906 1 1 1 i I .55 866564 4 4. .33436 40 21 9-772847 773018 86 9-906018 I .55 9-866829 4 4! .o-i33i7! It 22 86 905925! I • 55 867094 4 4! .32906 23 773.90 86 9o5832'i .55 867358 4 4! .32642 11 24 773361 85 905739 I 905645, 1 • 55 867623 4 4i .32377 25 773533 85 •55 867887 4 41 .32113 35 26 773704 85 9055521 .55 8681 52 4 40 131848 34 11 773875 85 9054591 905366! I .55 868416 4 40 i3i584 33 774046 85 •56 868680 4 40 i3i32o 32 29 774217 85 90527211 • 56 868945 4 40 i3io55 3i 3o 774388 84 905.70 1 9-9o5o8^. • 56 869200 4 40 i 30794 3o 3i 9-774558 84 • 56 9-869473 4 40 io-!3o527. It 32 774729 774899 84 904992 I 904898 I • 56 869737 4 40 i3o263 33 84 • 56 870001 4 40 \l^t 27 34 775070 - 84 904804 I • 56 870265 4 40 26 35 775240 84 9047 1 1 I 904617 I • 56 870529 S-jo-iqi 871057 4 40 .2947. 25 36 775410 _ 83 • 56 4 40 . .79207 24 u 775580 83 904523!! .56 4 40 .'.8043 23 775750 „ ^ 83 9044291. •57 871321 4 40 12C679 12841 5 22 39 775920 2 83 904335 I •57 871585 4 40 2! 40 776000 83 90424! I •57 871849 4 39 I28i5i 20 4t 9-776259 83 9-904147 I •57 9-872112 4 39 10.127888 \t 42 776420 77659§ 2 &2 904053 I • 57 872376 4 39 127624 43 82 9039591! •57 872640 4 39 127360 17 44 776768 82 903864 I •57 872903 4 39 .27097 16 45 776937 82 903770 ! •57 873167 4 39 .26833 i5 46 777106 82 903676 I -57 873430 4 39 .26570 14 tl 777275 81 90358! I •57 873694 873957 4 39 !:63o6 :3 777444 81 903487 ! •57 4 39 1 26043 I? 49 777613 81 903392 I • 58 874220 4 39 125780 11 So 777781 81 903298 ! • 58 874484 4 39 !255i6 ID 5i 9 777950 •} •* 81 9 -903203 ! • 58 9.874747 4 39 10.125253 t 52 778119 81 903 1 08 I • 58 875010 4 l^ 1 24990 53 778287 778455 80 9o3o!4 I • 58 875273 4 124727 7 54 80 902919 I 902824 ! • 58 875536 4 38 .24464 6 55 778624 2 80 • 58 875800 -4 38 .24200 5 56 778792 80 902729 ! • 58 876063 4 38 123937 4 U 778960 80 9026341. • 58 876326 4 33 .23674 3 779128 80 902539 I .59 876589 4 38 1234!! a 59 779295 79 902444 I • 59 876851 4 38 I23i49 I 60 779463 2.79 902349 I .59 877114 4-38 122886 u Cosine _ JX _ Sine . 53<3i Cotan?. D^ Tang. rsINEE \ AND TANGENTS. (37 DEGREES. ) 5fi ^ Sine D. Cosine 1 I). Tang. 1 ^■ Coteng. 9.779463 a •79 ^-902849^59 902253 1-59 9-877114 1 4-38 10-122886 "60" I 779631 •79 877377 4-38 122623 ^ 2 779708 779966 •79 902158 1.59 877640I 4-38 122860 3 79 902068 I - 59 877908! 4-38 122007 121885 U 4 780133 :?§ 901967 i-So 901872 1-59 878165 1 4-38 5 780800 878428 4-38 121572 55 6 780467 .78 901776 1.59 9016811.59 878691 878953 4-38 i2i3o9 54 I 780634 78 4-87 121047 53 780801 •78 901 585 1.59 879216 4-37 120784 5a 9 780968 •78 901490 1-59 879478 4-37 i:o522 3i 10 781 i34 •78 901894 1-60 879741 4.37 120259 5c II o-78i3oi •77 9-901298 i-6o 9 -880008 4-37 0- 1 19997 S n 781468 •77 901202 1-60 880265 4-87 119735 i3 -»8;634 77 901106 1-60 880528 4-37 119472 ii 14 781800 77 901010 1.60 880790 88io52 4-37 IIC210 118948 i5 781966 77 900914 900818 1.60 4-87 45 i6 782182 77 1-60 881814 1-37 1 1 8686 44 ]l 782298 76 900626 1-60 881576 4.37 118424 43 782464 76 1-60 88i8?,9 4-37 118161 42 »9 782680 76 900529 1-60 882101 4.37 ',\]l?, 41 30 782796 9-782961 76 900488 1-61 882868 4-36 40 21 76 9-900887 1-61 9-882625 4-36 10-117875 It 22 788127 76 900240 1-61 882887 888148 4-36 117118 33 788292 75 900144 1-61 4-36 116852 ii 24 7334:)8 75 s 1. 61 883410 4-36 1 16590 25 788623 75 1-61 888672 4-36 116828 35 26 788788 75 1-61 888984 4-86 1 16066 34 U 788953 75 899757 1-61 884196 4-36 ii58o4 38 784118 75 899660 1-61 884457 4-36 115548 32 29 784282 74 899564 1.61 884719 4-86 115281 3i 3o 784447 74 899467 1-62 884980 4-36 Il5020 80 3i 9-784612 74 9-899370 1.62 9-885242 4-36 10.114758 ll 32 784776 74 899273 1-62 8855o8 4-36 1 14497 1 14235 33 784941 74 899176 1-62 885765 4-36 27 34 7S5io5 74 1-62 886026 4-36 118974 26 35 785269 785488 73 1.62 886288 4-36 118712 25 36 73 898084 1-62 886549 4.35 118451 24 ll 785597 73 898787 1-62 886810 4-35 118190 23 785761 73 898689 1.62 887072 4-35 1 1 2928 1 1 2667 22 39 785925 73 898592 1-62 887888 4-35 21 40 7860H9 73 898494 1-63 887594 4-35 112406 20 41 9-7S6252 72 9-898897 1-68 '•XI 4-35 10-112145 \t 42 786416 72 898299 1-63 4.35 1 11884 43 786579 72 898202 1-63 888877 4-35 111628 \l 44 786742 72 898104 1-63 888689 4.35 Iii36i 45 786906 72 898006 1-63 888900 4-35 lillOO i5 46 787069 72 897008 897810 1-68 889160 4-35 110840 14 s 787282 71 1-68 880421 4-35 110818 i3 787895 787557 71 897712 1-68 889682 4-35 12 49 7« 891614 1-63 889948 4.35 110057 II 5o 787720 71 897516 i-68| 890204 4-34 109796 10 5i 9-787888 V 9-897418 1-64' 9 890465 4.34 10-109535 I 52 788045 V 897820 i-64i 890725 4-34 109275 53 788208 71 897222 1-64! 890986 4-34 109014 10S753 7 54 ]mi 70 gi 1-64; 891247 4-34 6 55 70 1.64 891507 4.34 108233 5 56 7S 70 1-64! 891768 4-34 4 ll 70 i-64i 892028 4-34 10-.972 3 789018 70 896729 I -641 892289 4-34 107711 3 59 789180 70 896681 1-64 892549 4-34 107451 I 60 789342 69 896582 I -641 892810 4-34 107190 Coeiro D. Sine e20| Cot!in£r. D. Tiinj?. M. 27' 56 (38 DEGREES.) A TABLE OF LOGARITHMIC pr Sine D. Cosine I). Tang. D. Cctang. ' ~ 9-789342 2.69 "9^896532 1-64 "^8928lc 4-34 10-107190 106930 60 I 789504 2.69 806433! I -65 89307c 4.34 5? 3 789665 2.69 896335 11-65 893331 4-84 106669 3 789827 2-69 896286 1-65 893591 898851 4-34 106409J 57 4 •789988 2-69 896187 896088 1-65 4.34 106 149; 56 5 790 '49 1-65 8941 u 4-34 109889 55 6 790810 895089 1-65 894871 4.34 ioj62c 105368 54 I 790471 2-68 895840 'i-65 894682 4-38 53 790682 2-68 895741 1-65 894892 4-33 io5io8 52 9 790798 2-68 895641 1-65 895152 4-83 104848 5i 10 790934 2-68 895542 1-65 895412 4-33 104588 5o II 9-791II5 2-68 9-895448 1-66 9-895672 4-38 10-104328 i^ 12 791275 2.67 895848 1-66 890982 4-33 104068 i3 791486 2-67 895244 1-66 896192 4-83 io38o8 47 46 14 791596 2.67 895145 1-66 896402 4-33 103548 i5 791737 2.67 895045 1-66 896712 8^6971 4-83 108288 45 i6 791917 2.67 894945 894846 1-66 4-83 108029 44 \l 792077 l:tl 1-66 897281 4-83 102769 43 792287 894746 1-66 897491 897751 4-83 102509 42 19 792897 2-66 894646 1-66 4-33 102249 41 20 792537 2-66 894546 1-66 898010 4-33 101900 10-101780 40 21 9-792716 2-66 9-894446 1.67 9-898270 4-33 ^? 22 792876 2-66 894846 1.67 898530 4-88 101470 38 23 798085 2-66 894246 1.67 898789 4-38 101211 37 24 ]llt 2-65 894146 1-67 899049 899808 4-32 100951 36 25 2-65 894046 1.67 4-32 100692 35 26 79^514 2-65 898046 898846 1-67 899568 4-82 100482 34 11 798673 2-65 1-67 899827 4-32 100178 33 798882 2-65 898745 1-67 900086 4-32 099914 32 ^9 798991 2-65 898645 1-67 900846 4-82 099654 3i 3o 794 I 5o 2-64 898544 1.67 900605 4-32 099805 3o 3i 9-794808 2-64 9-898444 1-68 9-900864 4-32 10-099186 29 28 32 794467 2-64 898848 1-68 901124 4-32 09S876 33 794626 2-64 898243 1-68 901888 4-32 098617 11 34 794784 2-64 898142 1-68 901642 4-82 098858 35 794942 2-64 898041 1-68 901901 4-82 098099 25 36 795101 2-64 892940 1.68 902160 4-82 097840 24 1 11 795259 2-63 892889 1-68 902419 4-32 097581 23 795417 795575 795733 9-795891 2-63 892739:1.68 89268811-68 902679 902988 4-32 097821 22 39 2-63 4-82 097062 21 40 2-63 892586{i.68 9-908455 4-3i 096808 20 41 2-63 9-892435:1.69 4-3i 10-096545 13 42 796049 796206 2-63 8923341-69 908714 4-3i 096286 43 2-63 892233! I- 69 908978 4.31 096027 17 44 796864 2-62 892182 1.69 904282 4-3i 095768 16 45 796521 2-62 892080! I -69 904491 904750 4-3i 095509 i5 46 796679 2-62 891929' 1-69 4-3i 095250 14 % 49 796886 2-62 8918271-69 9o5oc8 4-3i ! 094992 OV74733 094474 i3 796993 797 1 5o 2-62 2-61 891726I1-69 891624' I -69 905267 905526 4-8i 1 4-3i 12 5o 797807 2-61 891528 1.70 905784 4-3i 094216 10 5i 9-797464 2-61 9-891421 1-70 9-906048 4-3i 10-093957 t 52 797621 2-61 891819 1-70 90680a 4-3i 098698 53 797777 797984 2-6l 891217 1-70 9o656o 4-3i 098440 I 54 2-61 891 1 15 1-70 906819 4-3i 098 181 55 798091 2.61 891018 1-70 907077 4-3i 092Q23 5 56 798247 2.61 89X9 y-70 907886 4 3i 092664 4 U 798403 2-60 1-70 907594 907852 9081 1 1 4-3i 092406 3 798560 2.60 890707:1.70 4-31 092148 a 59 « 2.60 890605^1-70 4-30 091889 I 6o 2.60 890508 1.70 908869 4-3o 091681 CoBine D. Sine la 10 Cotang, D. Tang. SINES AND TANGENTS. (39 DEGREES. ) 61 IT. Sine D. Cosine | D. Tauor. D. Cotang. 1 1 9.798872 2-60 9.890503:1.70 9-9o83tQ 4.30 10-091631 60 I 799028 60 890400 1. 7 1 90862S 3o 091372 ^ a 799184 60 890298 1. 7 1 908886 3o 091114 58 3 799339 59 890195 1.71 909144 3o 090856 57 4 799493 59 Sl;:^; 909402 3o 090598 56 1 5 799631 5? 009660 3o 090340 55 6 799806 59 9099 1 8 3o 090082 089823 54 I •799962 59 889785 I.? I 910177 910435 3o 53 8001 17 59 889682:1.71 3o 089565 52 9 800272 58 889579>-7» 910693 3o 089307! 5l 10 800427 38 88g477ii-7i 9i09-)i 3o 089049, 5o IX 9.800582 58 9-889374 1.72 9.911209 3o 10-088791 088533 it la 800737 58 8892711.72 91 146- 3o 1 '3 800892 2 58 8891681.72 911724 3o 088276 47 14 801047 58 iEPi 9119S2 3o 088018 46 i5 801201 58 912240 4 3o 087760 45 i6 80 1 356 2 57 912498 4 3o 087502 44 \l 8oi5ii 57 88875511.72 912756 4 3o 087244 43 80 1 665 57 88S651I1.72 9i3oi4 4 29 086086 42 «9 801819 57 888548! 1. 72 913271 4 29 086729 41 20 801973 57 888444I1.73 913529 4 29 086471 40 31 9-802128 57 9-88834i;i.73 9-9I3787 4 29 10-086213 It 72 802282 56 8882371.73 914044 4 29 085956 23 802436 56 8881341.73 914302 4 29 085698 u 24 802589 802743 56 8880301.73 914560 4 29 085440 25 56 8879261.73 8878221.73 914817 4 29 o85i83 35 26 802897 56 9i5o75 4 29 084925 34 U 8o3oDo 56 887718I1.73 915332 4 29 084668 33 803204 56 8876141.73 915590 4 29 084410 32 29 803357 55 8875io'i.73 9'5847 4 29 084 1 53 i 3 1 3o 8o35ii 55 887406' 1-74 916104 4 29 083896 3o 3i 9.803664 55 9-887302,1-74 9-916362 4 29 10-083638 It 32 8o38i7 55 887198! I-. 74 916619 4 29 08338 1 33 803970 55 886885! 1 .74 916877 4 29 o83i23 27 34 804123 55 9<7>34 4 29 082866 26 35 804276 34 917391 4 29 082609 25 36 804428 54 886780 1-74 917648 4 29 082352 24 ll 804581 54 886676 1-74 917905 4 It 082095 081837 23 804734 804886 54 886571 1-74 918163 4 22 39 54 8864661.74 918420 4 28 o8i58o 21 4o 8o5o39 54 8863621 1. 75 918677 4 28 08 1 323 20 4i Q-8o5i9i 54 9. 886257 '1.75 9-918934 4 28 10-081066 \t 42 805343 53 886152:1.75 919191 4 28 080809 43 805495 53 886047! 1. 75 919448 4 28 o8o552 ]l 44 8o5647 53 885042^1-75 885837 1.75 919705 4 28 080295 45 805709 80595 1 53 919962 4 28 o8oo38 1 5 1 46 53 8857321.75 920210 920476 4 28 079781 14 % 806 1 o3 53 885627 1.75 4 28 079524 i3 806254 53 885522 1.75 920733 4 28 079267 12 49 806406 52 8854161.75 920990 4 28 0790 10 II 078753 10 5o 806557 52 88531111.76 921247 9-92i5o3 4 28 5i '■»o 52 9-8852o5li.76 4 28 10-078497: 9 078240I s 52 52 885 100' 1.76 921760 4 28 53 807JII 52 884004:1-76 8848ao'i.76 884783:1.76 Q22017 4 28 0779831 7 077726. 6 54 807163 52 922274 4 28 55 807314 52 922530 4 28 077470 5 56 807465 5i 8846771.76 922187 4 28 C72956 U 807615 5i 8845721.76 923044 4 28 3 807766 5i 8844661.76 923300 4 28 076700 i te? 5i 5i 8843601.76 884254 1-77 9238i3 4 4 27 27 076443 076187 Twiff.- -M-. Cosino D. Sine IfiQo Cotem?. 'Z? >. 1 5tt (40 DEGREES.) A TABLE OF LOGARITHMIC M. Sine D. Cosine D. Tung. D. Cotan^. 0-808067 2.5l 9.884354 9 -92381 3 4 •27 10-076187 60 I 808318 2 5i 884148 924070 4 .27 075980 5? 3 8o8368 2 5i 884042 924327 4 27 075673 3 8o85i9 2 5o 883936 924583 4 27 075417 U 4 808669 2 5o 883829 883723 924840 4 27 075160 5 808819 2 5o 925096 4 27 074904 55 6 808969 2 5o 8836 1 7| I 925332 4 27 074648 54 I 8091 19 809269 2 2 5o 5o 883510 883404 ]] 92586? 4 4 27 27 074891 074135 53 52 9 809419 2 49 883297 78 926122 4 27 078878 5i 10 809569 9.809718 2 49 883191 9.883084 78 926378 4 27 078622 5o II 2 49 78 9.926634 4 27 ic- 078866 % 13 809868 2 49 882977 78 926890 4 27 078110 i3 810017 2 49 882871 78 927147 4 27 072853 .% 14 810167 2 'S 882764 78 927403 4 27 072597 i5 8io3i6 2 882657 78 927659 9271915 4 27 072841 45 i6 810465 2 48 882550 78 4 27 072085 44 \l 810614 2 48 882443 78 928171 4 27 071829 071573 43 810763 2 48 882336 79 928427 4 27 42 J9 810912 2 48 882229 79 928683 4 27 071817 41 20 811061 2 48 882121 79 928940 4 27 071060 40 21 9-8ii2io 2 48 9-882014 79 9.929196 929452 4 27 10 070804 39 38 23 8ii358 2 47 881907 79 4 27 070548 23 8ii5o7 2 47 881799 79 929708 4 27 070292 37 24 8ii655 2 47 881692 79 929964 4 26 070086 36 25 81 1804 2 47 88 1 584 79 980220 4 26 069780 35 26 811952 2 47 881477 79 930475 4 26 069525 34 27 812100 2 47 881369 79 930781 4 26 069269 33 28 812248 2 47 881261 80 980987 4 26 069013 32 29 812896 2 46 881 1 53 80 981248 4 26 068757 3i 3o 812544 2 46 881046 80 981499 9.931755 4 26 o685oi 3o 3i 9-812692 2 46 9-880938 80 4 26 10-068245 ll 33 812840 2 46 88o83o 80 982010 4 26 067990 33 812988 2 46 880722 80 982266 4 26 067784 27 34 8i3i35 2 46 880613 80 982522 4 26 067478 26 35 8i3283 2 46 88o5o5ii 80 982778 4 26 067222 25 36 8i343o 2 45 880397 I 80 988088 4 26 066967 24 ll 813578 2 45 880289 I 81 988289 988545 4 26 066711 23 813725 2 45 880180 I 81 4 26 066455 22 39 813872 2 45 880072! I 81 988800 4 26 066200 31 40 814019 2 45 879Q63I 9-879855!! 81 984056 4 26 063944 ao 41 9'8i4r66 2 45 81 9-984811 4 26 10-065689 \i 42 8i43i3 2 45 879746:1 81 984567 4 26 065433 43 814460 2 44 879637' I 81 984828 4 26 o65i77 u 44 814607 2 44 879529 I 81 935078 4 26 064922 45 814753 2 44 879420 I 81 935383 4 26 064667 i5 46 814900 2 44 879311 I 81 985589 4 26 064411 14 % 8 1 5046 2 44 879202 I 82 935844 4 26 064 1 56 i3 815193 815339 2 44 870003! i 8780841 1 878§75|i 82 9861 .■>o 4 26 068900 12 49 2 44 82 936355 4 26 068645 II 5o 8 1 5485 3 43 82 986610 4 26 068800 io-o63i34 10 5i 9-8i563i 3 43 9.8787661 82 9-986866 4 25 9 52 815778 3 43 878656ii 82 987121 4 25 062879J 8 53 815924 3 43 878547! I 82 987876 4 35 0626241 7 062868 6 54 816069 816215 3 43 878438 I 82 987682 4 25 55 3 43 87832811 82 987887 4 25 0621 i3 ' 5 56 8i636i 3 43 878219 I 83 988142 4 35 06 1 858 4 U 8i65o7 3 43 878109 I 83 938898 938653 4 25 061602 3 8 16652 3 43 877999 I 83 4 25 061847 2 59 816798 3 43 877850 I 83 988908 4 25 061092 060837 I 66 816943 2 42 8777go I 83 989168 4 35 M. Cosine D. Sine 49^ Cotang. D. Tang. HlSRi AND TANGKNTb. (41 DEGREES. 1 60 M.'l c r-SineH T). CoBUie 1 D. Tang. D. "Oomng. "6^" 9.816943 2.42 9.8777801.83 9-939163 10-060837 1 817088 42 877670 I •83! 939418 o6o582 U a 817233 42 877560 1 .83! 939673 060327 3 817379 42 877450 I -83l 939928 060072 57 4 817524 41 877340 1 •83 94018^ 059817 56 5 817668 41 877230 1 -84 940438 359562 55 6 817813 41 877120 I •84 940694 059306 54 I 817958 41 877010I1 •84 940949 039001 058796 53 8i8io3 41 876899 I .84 941204 32 9 818247 41 8766781 .84 941458 058342 5i lO 818392 9.818536 41 .84 941714 o5Sj86 5o 11 40 9.876068 I .84 9^941968 10 -05803 2 '^ la 818681 40 8764571 •84 942223 tl]l]l i3 818825 40 876347; I ■84 942478 % i4 818960 81911J 40 876236,1 .85 942733 057267 i5 40 876125 1 •85 942988 057012 45 i6 819257 40 876014 1 .85 943243 056757 o565o2 44 \l 819401 40 875904 1 .85j 94349^ .85! 943752 43 819545 39 875703 1 875682 I 056248 42 19 819689 39 •85 944007 n^l 41 20 819832 39 875571'! -85 944262 40 21 9.819976 39 Q.87545Q 1 •85 9-944517 10.055483 ^ 22 820120 39 875348,1 •85 94i77» 055229 23 820263 39 875237:1 -85 945026 054974 U 24 820406 It 875126 I •86 945281 054719 25 82o55o 875014 I •86 945535 054463 35 26 820693 820836 38 874903 I -86 945790 054210 34 u 3 38 874791 1 874680 I .86 946045 053955 33 820979 _ 38 -86 946299 053701 32 29 821122 38 874568 I •86 9465D4 053446 3i 3o 821265 38 874456 I •86 946808 053192 3o 3i 9-821407 38 9.87/J44 1 •86 9-947063 10-052937 052682 29 32 82i55o 38 874232 1 •87 947318 28 33 821693 37 87412111 •87 94757a 052428 '^ 34 821835 37 874009 I •87 947826 052174 35 821977 37 8738^ I •87 94^°^' 051919 25 36 822120 37 8737841 87367211 •87 948336 o5i664 24 U 822262 37 •87 948590 o5i4io 23 822404 37 873560! 1 • 87 948844 o5ii56 22 39 822546 11 873448! I •87 949099 9493d3 030901 21 40 8226S8 873335 I •87 o5o647 20 41 9.822880 36 9.873223,1 •87 9 949607 10 -050393 o5oi38 \l 42 822972 36 8731101 -88 949862 43 823114 36 Z^t;. • 88 950116 049884 \l 44 823255 36 • 88 950370 049630 45 823307 823539 36 8727721 8726591 ^88 950625 049375 i5 46 36 • 88 &^ 049121 048867 14 a 823680 35 872547:1 •88 i3 823821 35 8724341 -88! 931388 4 048612 12 49 823963 35 872321 1 •881 951642 048358' 11 1 56 824104 35 872208;! •88 951896 •80' 9-952i5o 048104 10 5i 9-824245 35 9.8720951 10-047850 I 52 824386 35 l]\& •8^f 952405 047595 53 824527 35 • 8? 95265o 95291J 047341 54 824668 34 871755;! .8^ 047087 046833 55 824808 34 871641I1 •8^ 953167 56 824949 34 871528,1 •8? 953421 046579 046325 ll 825090 825230 34 871414 1 ■si 953675 34 871301 I •89 953929 954183 046071 59 825371 34 871187,1 •89 045817 045563 6o 8255II 34 871073,1 •90 954437 (V.sine D. 8iiie |48c Cotaiig. I). Tatl^. tL 1 GO (42 DEOREKS.) A TABLE OF LOOARITHMIC Sine D. Cosine \ D. | Tanof. I). Cotanor. 9-8255ii 2.34 9-871073 I. 90I 9-954487 4.23 !0. 045563 60 I 825651 83 870960:1 870846;! •90 954691 4-23 oi^3o9 o45o55 3 825791 8a593i 33 -90! 934945 4-23 3 33 87078211 •90 955200 4-23 044800 h 4 816071 83 870618;! .9c 955454 4-23 044546 5 t' 762 1 1 33 870504 I • 90 955707 4-23 044298 044089 048785 55 6 826351 33 870890 ! • 90 955961 .4-23 54 I 826491 33 870276!! .90 956215 4-23 53 826681 83 870161 I -90 956469 4.23 04353! 52 9 826770 32 870047 |l 869983 j I •91 95672J 4-23 048277 048028 5i 10 826910 32 .91 956977 4-23 5o II 9.827049 32 9.869818 I •91 9"9i723i 4-23 10-042769 tl la 827189 827828 82 8697041 •91 957485 4-28 0425lD i3 32 8695891 .9! 957780 957998 4-23 042261 47 14 827467 32 869474I1 .9! 4-28 042007 46 i5 827606 32 869860 1 1 .9! 958246 4-23 041754 45 i6 827745 82 86924511 •91 958500 4-23 o4i5oo 44 \l 827884 81 8691801 -91 958754 4.28 041246 43 828023 81 8690151 8689001 •92 959008 4-23 0.10992 42 19 828 1 62 3i -92 959262 4-23 010788 41 20 828801 81 868785] I .92 959516 4-28 040484 40 21 9-828489 828578 81 9-868670 I -92 9-959769 960023 4-23 10-040281 It 22 3i 868555 I .92 4.28 089977 23 828716 81 868440 I .92 960277 4-28 089728 37 24 828855 3o 868324 I .92 960531 4-23 089469 36 25 828998 80 868209 1 868098 ! .92 060784 4-23 089216 35 26 829181 3o .92 961088 4-23 088962 34 11 829269 3o 867978 I 867862 1 •93 961291 4.28 088709 038455 33 829407 80 .98 961545 4-23 32 29 829545 3o 867747 I ■93 961799 962052 4.28 088201 8! 3o 829688 80 867681 I .98 4-23 087948 3o 3i 9.829821 29 9-8675i5 I .98 9-962806 4-23 10-037694 It 32 829959 29 867809 I 867283 I -98 962560 4-23 087440 33 880097 880284 29 .98 962818 4-23 087187 11 34 29 867167 I .93 968067 4-23 086933 35 880872 29 867051 I -93 968820 4-23 086680 25 36 83o5o9 29 866935 I 866819 I 866708 1 .94 968574 4.28 086426 24 U 880646 29 .94 968827 4-23 086178 23 880784 It •94 964081 4-23 035919 22 39 880921 866586 I •94 964335 4-23 035665 21 40 88io58 28 866470 I •94 964588 4-22 085412 20 41 9.881195 881882 28 9-866353 i .94 9-964842 4-22 io.o35!58 \t 42 28 866287 1 .94 965095 4-22 084905 43 881469 28 866120 I .94 965849 4-22 08465! 17 !6 44 881606 28 866004 1 .95 965602 4-22 084898 45 881742 28 865887 1 -95 965855 4-22 084145 i5 46 881879 832015 28 865770 I -95 966105 4-22 08889! 14 47 27 8656531! -95 966862 4-22 088688 i3 48 882152 27 865586 I .95 966616 4-22 038884 12 49 832288 27 865419 I -95 Z^i 4-22 o83i3i II 5o 832425 27 86530? I .95 4-22 082877 ID 5i ^ 882561 27 9-865i85|r .95 9-96737^ 4-22 IP 082624 53 882697 • 27 865o68|i -95 967629 4-23 082371 8 53 882888 11 864950 j! 864833! .95 967888 968186 4-22 032!I7 I 54 882969 .96 4-22 081864 55 838ioD 26 8647161 .96 968889 4-22 08161! 5 56 «3824i 26 864598 I .96 968643 4.22 o3i857 4 U 833377 26 864481 ! -96' 968896 4-22 o3iio4 3 833512 26 864868 I -96 969140 .96 969408 4-22 o3o85i 2 ; 59 833648 26 864245 I 4-22 o3o597 I 6o 833783 26 864127 I -96 969656 4-22 o3o344 Tarur. CI<«iae D. Sine kto! Cotane. 1 D. SINES AKD TANGENTS. (43 DEOIiEES.' 01 Sine D. Cosine | D, Tansr. D. Cotane. ~6r 9.833733 2.26 9-864127 1.96 9.969656 4-22 ioo3o3i4 I 833019 • 25 664010 1 .96 969909 4 22 03009! j 5q 029838 5& 1 834054 25 863892 1 -97 970162 4 22 3 4 834189 834325 2 25 25 863774 I 863656 I •97 -97 970416 970669 i 22 22 l^^\ u 5 834460 2 25 863538 I ■97 970922 4 22 0290781 55 028825 5i 6 834595 834730 834»65 25' 863419 I -97 •97 971175 4 2? I 25 863301 I 971429 4 22 028571 53 25 863 1 83 1 •97 9716R2 4 22 0283 18 52 9 834999 24 863o64 I •97 971935 4 22 0280651 5 1 10 835i34 24 862946 I 9-862827,1 .98 972188 4 2: 027fii2! 5o II 9.835269 ^ 835403 24 .98 9972441 4 22 10^027559 4q 027306; 48 12 24 862709 1 .98 972694 4 22 i3 835538 24 8625901 .98 972948 4 22 027052, 47 14 835672 24 862471 I .98 973201 4 22 026799' 46 15 835807 24 862353 I .98 973454 4 22 026546 45 |6 835941 24 862234 I .98 973707 4 22 026293, 44 ',1 836075 23 862115 1 .98 973960 4 22 026040! 43 836209 836343 23 861996 I 861877 I .98 974213 4 22 0257871 42 '9 23 .98 974466 4 22 025534; 41 20 836477 23 861758 I .99 974719 9-974973 4 22 025281 40 31 9.83661 1 23 9-86i63S I .99 4 22 10-025027 39 22 836745 23 86i5i9 1 .99 975226 4 22 024774; 38 23 836S78 23 861400 I .99 975479 4 22 024521' 37 24 837012 22 861280 1 •99 975732 4 22 024268 36 25 837140 22 861 161 I -99 9759S5 4 22 024015 35 26 837279 22 861041 1 .99 976238 4 22 023762; 34 11 83i4i2 22 860022' I 860^02 I .99 976491 4 22 023509 33 ■ 837546 * 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 g. 837945 - 22 9-8604422 •00 9 -977503 4 22 10 022497 20 32 83S078 21 866322 2 •00 977756 4 22 022244 28 33 838211 21 860202 2 •00 978009 4 22 0219911 27 021738 26 34 838344 21 860082:2 .00 978262 4 22 35 838477 21 859^4212 .00 9785 1 5 4 22 021485 25 36 838610 21 .00 978768 4 22 021232 24 ll 838742 21 859721 2 8596012 •01 979021 4 22 020979! 23 020726} 22 838875 21 •01 979274 4 22 39 839007 21 8594802 •01 979527 4 22 020473 i 21 4c 839140 20 8593602 •01 979780 4 22 020220, 20 41 9.839272 20 9-8592392 •01 9-980033 4 22 10-019967 10 019714, 18 42 839404 20 8591 19'2 •01 980286 4 22 43 83q536 20 858998.2 858877 2 -01 980538 4 22 019462; 17 01920-)! 16 018956! 1 5 44 839668 20 •01 980791 4 21 45 839800 20 8587562 •02 981044 4 21 46 839932 20 858635 2 •02 981297 4 21 018703, 14 s 840064 '9 8585i4'2 •02 981550 4 21 018450 i3 840196 "9 858393 2 •02 981803 4 21 018197 12 49 84032S »9 858272,2 •02 982056 4 21 01 7044 II 5o 84045Q '9 858i5i2 •02 982309 4 21 017601 r. 5i 9.840591 2 19 9-858029 2 857908,2 •02 9-9^2562 4 21 io^oi74J8 9 017186 r 52 840722 19 .02 982814 4 21 53 840854 19 857786*2 •02 983067 4 21 016933 7 016680 6 54 8^0985 It 85766512 • 03 983320 4 21 55 841116 85754312 • o3 9S3573 4 21 oi64i7i 5 56 841247 841378 18 857422;2 .o3 983826 4 21 016174 4 u 18 8573oo'2 .o3 984079 4 21 01 592 1 3 015669 2 841509 18 857178,2 03 984331 4 21 59 841640 18 857o56l2 .03 984584 4 21 oi54i6| I ^ 84177' 18 85693412 •03 984837 4 21 oifi63 CoAino D. Sii- Ueo Cotanfr. D. Tail?. M^ 62 (44 [ DEGREES.) A TABLE OF LOGARITHMIC k. Sine 1 D. - - Cosine T). Tang. ] D. ) Cotang. "9^841771" 1 2-18 9-856934 2-03, 9-9S4837 4 •21 io.oi5i63 60 I 841902 2 -18 8568 1 2 2-03; 985090 4 -21 OI49IO ^ 2 842033 2 -18 856690! 2 -04 985343 4 •21 014657 3 842163 2 •«7 856568:2-04 985596 4 -21 014404 u 4 842294 2 •17 85644612 -04 985848 4 -21 0l4l52 5 842424 2 •17 856323!2-o4 986 1 01 4 -21 013899 55 6 842555 2 •n 856201 2 -04 986354 4 -21 0136461 54 1 I 842685 2 •17 856078 2-04 986607 4 -21 013393 53 842815 2 •17 855956 2-04 986860 4 -21 oi3i4o 52 9 842946 2 •17 855833 2-o4i 987112 4 •21 012888 5i 10 843076 2 •17 8557II 2-05 9-^7365 4 -21 012635 5o II 9-843206 2 .16 9-85558Bi2-o5! 9-987618 4 -21 10-012382 it 12 843336 2 .16 855465 2-o5l 9B7871 4 -21 O12129 i3 843466 2 .16 855? <2 2-o5 988123 4 21 011877 47 14 843595 2 .16 855219 2-o5 988376 4 21 011624 46 i5 843725 2 .16 855096 2 -05 988629 4 -21 011371 45 i6 843855 2 • 16 854973 854850 2-05 988882 4 21 OIII18 44 \l 843984 2 -16 2.05 989134 4 21 010866 43 8441 14 2 -15 854727 2.06 989387 4 21 oio6i3 42 «9 844243 2 -15 8546o3 2.06 989640 4 •21 oio36o 41 20 844372 2 i5 854480 2.06 989893 4 21 010107 40 21 9-844502 2 i5 9-854356 2.06 9-990145 4 21 10-009855 ll 22 84463 1 2 i5 854233 2-o6 990398 9906a I 4 21 009602 23 844760 2 i5 854109 2-o6 4 21 009349 37 24 844889 845oi8 2 i5 853986|2-o6 990903 4 2 1 009097 21 008844 36 25 2 i5 853862 2-o6 99II56 4 35 26 845147 2 i5 853738 2 -06 9^)1409 4 21 008591 34 11 845276 2 14 853614 2-07 991662 4 21 008338 33 845405 2 i4 853490 2-07 99I9I4 4 2 1 008086 32 ?9 845533 2 i4 853366 2-07 992167 4 21 007833 3i 3o 845662 2 14 853242 2-07 992420 4 21 007580 3o 3i 9-845790 2 14 9-853ii8 2-07 9-992672 4 21 10 • 007328 It 32 845919 2 i4 852994 852869 2-07 992925 4 21 007075 33 846047 2 14 2-07 993178 4 21 006822 27 34 846175 2 14 852745 2-07 993430 4 21 006570 26 35 846304 2 14 852620 2-07 993683 4 21 oo63i7 25 36 846432 2 i3 852496 2-o8 993936 4 2 1 006064 24 U 846560 2 i3 852371 2-08 994189 4 21 oo58ii 23 846688 „ i3 852247 2-o8 994441 4 21 005559 21 oo53o6 22 39 846816 2 i3 852122 2-o8 994694 4 21 4o 846944 2 i3 85i997 2-o8 994947 4 21 oo5o53 20 41 9-847071 2 i3 9.85i872!2.o8 9-995I99 995452 4 21 10-004801 \t 42 847199 2 i3 85.747 2.0S 4 21 004548 43 8473?7 2 i3 85i622 2-08 995705 4 21 004295 'I 44 847454 2 12 851497 2-09 995957 4 2 1 004043 45 847582 2 IS 85i372 2-09 996210 4 21 003790 21 003537 i5 46 847709 847836 2 12 85i246 2-09 996463 4 14 ii 2 12 85 II 21 2-00 996715 4 21 oo3285j i3 1 847964 2 12 850996:2-09 996968 4 21 oo3o32 12 1 49 848091 2 12 850870:2-09 997221 4 21 002779 II 5o 848218 2 12 85o745j2-09 997473 4 21 002527 10 5i 9.848345 12 9-85o6i9 2-09 85o49Ji2-io 9-997726 4 21 10-002274 t 52 848472 II 997979 4 21 002021 53 848599 11 85o368 2-10 998231 4 21 001760 21 ooi5i6 I 54 848726 11 850242|2.I0 998484 4 55 848852 II 85oii6 2-10 998737 4 21 001263 5 56 848979 II 849990 2-10 998989 4 21 OOIOII 4 ^ 849106 II 849864! 2- 10 999242 4 21 000755 3 849232 II 8497382.10 999495 4- 21 ooo5o5 s 59 849359 849485 II 84961 1 2-10 999748 4- 21 000253 I 60 II 849485 2-10 10-000000 4- 21 10-000000 Cosine D. Sine 450,' Cotftng. _ 1 ). Tang. ._ M- • L-» %^KJ JfU^J 924173 J] 3 THE UNIVERSITY OF CALIFORNIA LIBRARY