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Tappan, Eli Todd, Treatise on geometry and trigonometry : Stanford University Libraries
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DEPARTMENT OF EDUCATION ■
JAN 2 9 1909 LELA N D 3 1 A N ,'fOR D J JUNIOR UNIVERSITY.
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SCHOOL OF EDUCATION
STANFORD N^p/ UNIVERSITY LIBRARIES
ECLECTIC EDUCATIONAL SHRIE8.
TEEATISE
GEOMETRY TRIGONOMETRY: COLLEGES, SCHOOLS AND PRIVATE STUDENTS.
WRITTEN F0B THE MATHEMATICAL COUR8E 0F
JOSEPH RAY, M. D.,
ELI T. TAPPAN, M. A., Professor of Mathematics, Ohio University.
CINCINNATI:
WILSON, HINKLE & CO. PHIL'A: CLAXTON, REMSEN & HAFFELFINGER. NEW YORK: CLAEK & MAYNAED.
623170 THE BEST J.NQ CHEAPEST.
MATHEMATICAL SERIES. Ray's Primary Arithmetic : Simple Mental Lessons and Tables. For little Learners. Ray's Intellectual Arithmetic: the most interesting and valuable Arithmetic extant. Ray's Rudiments of Arithmetic : combining mental and practical exercises. For beginners. I Ray's Practical Arithmetic: a full and practical treatise on the inductive and analytic methods of instruction. Ray's Higher Arithmetic: the principles of Arithmetic analyzed and practically applied. Ray's Test Examples: three thousand practical problems for the slate or blackboard. For drill exercises and review. Ray's New Elementary Algebra: a simple, thorough, and progressive elementary treatise. For Schools and Academies. Ray's New Higher Algebra : a progressive, lucid, and comprehensive work. For advanced Students and for Colleges. Ray's Elements of Geometry : a comprehensive work on Plane and Solid Geometry, with numerous practical exercises. Ray's Geometry and Trigonometry : Plane and Spher ical Trigonometry, with their applications; also a complete set of Logarithmic tables, carefully corrected. Ray's Differential and Integral Calculus: in course of preparation, and to be published during the present year. To be followed, at an early day, by other works, forming a com plete Mathematical Course for Schools and Colleges. Entered according to Act of Congress, in the year 1868, by 8ARGENT, WIL8ON & HINKLE, In the Clerk's Office of the District Court of the United 8tates, for the Southern District of Ohio. ELBCTRoTYPED AT THE FRANKL1N TYPE FoUNDRY, C1NC1NNAT1.
PREFACE. The science of Elementary Geometry, after remaining nearly stationary for two thousand years, has, for a century past, been making decided progress. This is owing, mainly, to two causes: discoveries in the higher mathematics have thrown new light upon the elements of the science ; and the demands of schools, in all enlightened nations, have called out many works by able mathematicians and skillful teachers. Professor Hayward, of Harvard University, as early as 1825, defined parallel lines as lines having the same direc tion. Euclid's definitions of a straight line, of an angle, and of a plane, were based on the idea of direction, which is, indeed, the essence of form.
This thought, employed in
all these leading definitions, adds clearness to the science and simplicity to the study. In the present work, it is sought to combine these ideas with the best methods and latest discoveries in the science. By careful arrangement of topics, the theory of each class of figures is given in uninterrupted connection. No attempt is made to exclude any method of demonstration, but rather to present examples of all. The books most freely used are, "Cours de g6ometrie elementaire, par A. J. H. Vincent et M. Bourdon;" " G6' ometrie thfiorique et pratique, etc., par H. Sonnet;" "Die (iii)
iv
PREFACE.
reine elemental'' ma thematik, von Dr. Martin Ohm;" and " Treatise on Geometry and its application to the Arts, by Rev. D. Lardner." The subject is divided into chapters, and the articles are numbered continuously through the entire work. The con venience of this arrangement for purposes of reference, has caused it to be adopted by a large majority of writers upon Geometry, as it had been by writers on other scien tific subjects. In the chapters on Trigonometry, this science is treated as a branch of Algebra applied to Geometry, and the trig onometrical functions are defined as ratios. This method has the advantages of being more simple and more brief, yet more comprehensive than the ancient geometrical method. For many things in these chapters, credit is due to the works of Mr. I. Todhunter, M. A., St. John's College, Cam bridge. The tables of logarithms of numbers and of sines and tangents have been carefully read with the corrected edi tion of Callet, with the tables of Dr. Schron, and with those of Babbage. ELI T. TAPPAN. Ohio University, Jan. 1, 1868.
CONTENTS. PAET FIRST.—INTRODUCTORY. CHAPTER I. PRELIMINARY. PAGE.
Logical Terms, General Axioms, Ratio and Proportion,
9 .11 12
CHAPTER II. the subject stated. Definitions, Postulates of Extent and of Form, Classification of Lines, Axioms of Direction and of Distance, Classification of Surfaces, Division of the Subject,
....
17 19 22 23 24 26
PART SECOND.—PLANE GEOMETRY. CHAPTER III. STRAIGHT LINES.
Problems, Broken Lines, Angles,
' .
28 31 32
vi
CONTENTS. PAGE.
Perpendicular and Oblique Lines
38
Parallel Lines,
43 CHAPTER IV. CIRCUMFERENCES.
,
General Properties of Circumferences,
...
52
Arcs and Radii,
53
Tangents,
58
Secants,
59
Chords, Angles at the Center,
60 64
Intercepted Arcs, Positions of Two Circumferences,
72 78
CHAPTER V. TRIANGLES.
General Properties of Triangles,
85
Equality of Triangles, Similar Triangles,
93 101
CHAPTER VI. QUADRILATERALS.
General Prpperties of Quadrilaterals,
.
.
.
119
Trapezoids,
122
Parallelograms,
123
Measure of Area, Equivalent Surfaces,
128 135
CHAPTER VII. POLYGONS.
General Properties of Polygons,
143
Similar Polygons,
147
CONTENTS.
Til PAOI.
Regular Polygons, isoperimetry,
151 159
CHAPTER VIII. CIRCLES.
Limit of Inscribed Polygons, Rectification of the Circumference, Quadrature of the Circle
164 166 172
PART THIRD.—GEOMETRY OF SPACE. CHAPTER IX. STRAIGHT LINES AND PLANES.
Lines and Planes in Space, Diedral Angles, Parallel Planes, Triedrals, polyedrals,
177 185 190 195 209
CHAPTER X. POLYEDRONS. Tetraedrons, Pyramids, Prisms, Measure of Volume, Similar Polyedrons, Regular Polyedrons,
213 222 226 232 239 241
CHAPTER XI. SOLIDS OF REVOLUTION.
Cones, Cylinders,
247 249
vm
CONTENTS. PAGE.
Spheres,
250
Spherical Areas, Spherical Volumes,
261 270
Mensuration,
276
PART FOURTH.—TRIGONOMETRY. CHAPTER XII. PLANE TRIGONOMETRY. Measure op Angles,
277
Functions of Angles,
279
Construction and Use of Tables, Right angled Triangles,
296 302
Solution of Plane Triangles,
304
CHAPTER XIII. SPHERICAL TRIGONOMETRY. Spherical Arcs and Angles, Right angled Spherical Triangles,
314 324
Solution of Spherical Triangles,
329
CHAPTER XIV. LOGARITHMS. Use of Common Logarithms,
334
TABLES. Logarithmic and Trigonometric Tables,
.
.
.
345
ELEMENTS
GEOMETRY
CHAPTER I.—PRELIMINARY. Article 1. Before the student begins the study of geometry, he should know certain principles and defini tions, which are of frequent use, though they are not peculiar to this science. They are very briefly pre sented in this chapter. LOGICAL TERMS. 3. Every statement of a principle is called a Propo sition. Every proposition contains the subject of which the assertion is made, and the property or circumstance asserted. When the subject has some condition attached to it, the proposition is said to be conditional. The subject, with its condition, if it have any, is the Hypothesis of the proposition, and the thing asserted is the Conclusion. Each of two propositions is the Converse of the other, when the two are such that the hypothesis of either is the conclusion of the other. (0)
10
ELEMENTS OF GEOMETRY.
3. A proposition is either theoretical, that is, it de clares that a certain property belongs to a certain thing ; or it is practical, that is, it declares that something can be done. Propositions are either demonstrable, that is, they may be established by the aid of reason ; or they are indemon strable, that is, so simple and evident that they can not be made more so by any course of reasoning. A Theorem is a demonstrable, theoretical proposition. A Problem is a demonstrable, practical proposition. An Axiom is an indemonstrable, theoretical propo sition. A Postulate is an indemonstrable, practical propo sition. A proposition which flows, without additional reason ing, from previous principles, is called a Corollary. This term is also frequently applied to propositions, the demonstration of which is very brief and simple. 4. The reasoning by which a proposition is proved is called the Demonstration. The explanation how a thing is done constitutes the Solution of a problem. A Direct Demonstration proceeds from the premises by a regular deduction. An Indirect Demonstration attains its object by showing that any other hypothesis or supposition than the one advanced would involve a contradiction, or lead to an impossible conclusion. Such a conclusion may be called absurd, and hence the Latin name of this method of reasoning—reduetio ad absurdum. A work on Geometry consists of definitions, proposi tions, demonstrations, and solutions, with introductory or explanatory remarks. Such remarks sometimes have the name of scholia.
GENERAL AXIOMS.
\\
5. Remark. —The student should learn each proposition, so as to state separately the hypothesis and the conclusion, also the condition, if any. He should also learn, at each demonstration, whether it is direct or indirect ; and if indirect, then what is the false hypothesis and what is the absurd conclusion. It is a good exercise to state the converse of a proposition. In this work the propositions are first enounced in general terms. This general enunciation is usually followed by a particu lar statement of the principle, as a fact, referring to a diagram. Then follows the demonstration or solution. In the latter part of the work these steps are frequently shortened. The student is advised to conclude every demonstration with the general proposition which he has proved. The student meeting a reference, should be certain that he can state and apply the principle referred to.
GENERAL AXIOMS. 6. Quantities which are each equal to the same quan tity, are equal to each other. 7. If the game operation be performed upon equal quantities, the results will be equal. For example, if the same quantity be separately added to two' equal quantities, the sums will be equal. 8. If the same operation be performed upon unequal quantities, the results will be unequal. Thus, if the same quantity be subtracted from two unequal quantities, the remainder of the greater will exceed the remainder of the less. 9. The whole is equal to the sum of all the parts. 10. The whole is greater than a part. EXERCISE. 11. What is the hypothesis of the first axiom ? eral quantities are each equal to the same quantity.
Ans. If sev
12
ELEMENTS OF GEOMETRY.
What is the subject of the first axiom ? Ans. Several quan tities. What is the condition of the first axiom ? Ans. That they are each equal to the same quantity. What is the conclusion of the first axiom? Ans. Such quan tities are equal to each other. Give an example of this axiom.
RATIO AND PROPORTION 12. All mathematical investigations are conducted by comparing quantities, for we can form no conception of any quantity except by comparison. 13. In the comparison of one quantity with another, the relation may be noted in two ways : either, first, how much one exceeds the other; or, second, how many times one contains the other. The result of the first method is the difference be tween the two quantities ; the result of the second is the Ratio of one to the other. Every ratio, as it expresses " how many times " one quantity contains another, is a number. That a ratio and a number are quantities of the same kind, is fur ther shown by comparing them; for we can find their sum, their difference, or the ratio of one to the other. When the division can be exactly performed, the ratio is a whole number ; but it may be a fraction, or a radical, or some other number incommensurable with unity. 14. The symbols of the quantities. from whose com parison a ratio is derived, are frequently retained in its expression. Thus, The ratio of a quantity represented by a to another represented by b, may be written , . A ratio is usually written a : b, and is read, a is to b.
RATIO AND PROPORTION.
13
This retaining of the symbols is merely for conven ience, and to show the derivation of the ratio; for a ratio may be expressed by a single figure, or by any other symbol, as 2, m, j/3, or jr. But since every ratio is a number, therefore, when a ratio is thus expressed by means of two terms, they must be understood to represent two numbers having the same relation as the given quantities. The second term is the standard or unit with which the first is compared. So, when the ratio is expressed in the form of a frac tion, the first term, or Antecedent, becomes the numera tor, and the second, or Consequent, the denominator. 15. A Proportion is the equality of two ratios, and is generally written, a : b : : c : d, and is read,
a is to b as c is to d,
but it is sometimes written, a : b = c : d, , or it may be,
a c b = d'
all of which express the same thing: that a contains b exactly as often as c contains d. The first and last terms are the Extremes, and the second and third are the Means of a proportion. The fourth term is called the Fourth Proportional of the other three. A series of equal ratios is written, a : b : : c : d : : e : f, etc. When a series of quantities is such that the ratio of each to the next following is the same, they are written, a : b : c : d, etc.
14
ELEMENTS OF GEOMETRY.
Here, each term, except the first and last, is both an tecedent and consequent. When such a series consists of three terms, the second is the Mean Proportional of the other two. 16. Proposition.— The product of the extremes of any proportion is equal to the product of the means. For any proportion, as a : b : : c : d, is the equation of two fractions, and may be written, a c_ b~d' Multiplying these equals by the product of the denom inators, we have (7) aXd = bXc, or the product of the extremes equal to the product of the means. IT1. Corollary—The square of a mean proportional is equal to the product of the extremes. A mean pro portional of two quantities is the square root of their product. 18. Proposition. — When the product of two quanti ties is equal to the product of two others, either two may be the extremes and the other two the means of a proportion. Let aXd=bXc represent the equal products. If we divide by b and d, We have b==oV or' a '• ° ''' c : d.
(1st.)
If we divide by c and d, we have c = 5' 0r' a: c '''' ° ' d' If we arrange the equal products thus : bXc = aXd,
(2d0
RATIO AND PROPORTION.
15
and then divide by a and c, we have b : a : : d : c. (3d.) By similar divisions, the student may produce five other arrangements of the same quantities in pro portion. 19. Proposition. — The order of the terms may be changed without destroying the proportion, so long as the extremes remain extremes, or both become means. Let a : b : : c : d represent the given proportion. Then (16), we have aXd = bXc. Therefore (18), a and d may be taken as either the extremes or the means of a new proportion. 20. When we say the first term is to the third as the second is to the fourth, the proportion is taken by alternation, as in the second case, Article 18. When we say the second term is to the first as the fourth is to the third, the proportion is taken inversely, as in the third case. 21. Proposition —Ratios which are equal to the same ratio are equal to each other. This is a case of the first axiom (6). 22. Proposition. — If two quantities have the same multiplier, the multiples will have the same ratio as the given quantities. Let a and b represent any two quantities, and m any multiplier. Then the identical equation, mXaXb= mXbXa, gives the proportion, mXa : mXb : : a : b (18). 23. Proposition.—In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.
16
ELEMENTS OF GEOMETRY.
Let a : b :: e : d :: e :f :: g : h, etc., represent the equal ratios. Therefore (16), aXd = bXc aXf=bXe aXh = bXg To these add aXb=bXa aX (b+d+f+h) = bX(a+c+e+g). Therefore (18), a+e+e.+g : b-\-d-\-f-\-h : : a : b. This is called proportion by Composition. 34. Proposition. — The difference between the first and second terms of a proportion is to the second, as the dif ference between the third and fourth is to the fourth. The given proportion, a : b : : c : d, may be written,
h=d'
Subtract the identical equation, b_d b~d' The remaining equation, a—b c-— d
,
may be written, a — b : b : : c — d : d. This is called proportion by Division. 25. Proposition.—If four quantities are in proportion, their same powers are in proportion, also their same roofs. Thus, if we have a : b c : d, then, a2 : b2 c2: d2; also, ]/a : \/b |/c : \/d. These principles are corollaries of the second gen eral axiom (7), since a proportion is an equation.
THE SUBJECT STATED.
17
CHAPTER II. THE SUBJECT STATED. 26. .We know that every material object occupies a portion of space, and has extent and form. For example, this book occupies a certain space; it has a definite extent, and an exact form. These prop erties may be considered separate, or abstract from all others. If the book be removed, the space which it had occupied remains, and has these properties, extent and form, and none other. 27. Such a limited portion of space is called a solid. Be careful to distinguish the geometrical solid, which is a portion of space, from the solid body which occu pies space. Solids may be of all the varieties of extent and form that are found in nature or art, or that can be imagined. 28. The limit or boundary which separates a solid from the surrounding space is a surface. A surface is like a solid in having only these two properties, extent and form; but a surface differs from a solid in having no thickness or depth, so that a solid has one kind of extent which a surface has not. As solids and surfaces have an abstract existence, without material bodies, so two solids may occupy the same space, entirely or partially. For example, the position which has been occupied by a book, may be now occupied by a block of wood. The solids represented •Geom.—2
18
ELEMENTS OF GEOMETRV.
by the book and block may occupy at once, to some ex tent, the same space. Their surfaces may meet or cut each other. 29. The limits or boundaries of a surface are lines. The intersection of two surfaces, being the limit of the parts into which each divides the other, is a line. A line has these two properties only, extent and form ; but a surface has one kind of extent which a line has not: a line differs from a surface in the same way that a surface does from a solid. A line has neither thick ness nor breadth. 3©. The ends or limits of a line are points. The intersections of lines are also points. A point is unlike either lines, surfaces, or solids, in this, that it has neither extent nor form. 31. As one line may be met by any number of oth ers, and a surface cut by any number of others; so a line may have any number of points, and a surface any number of lines and points. And a solid may have any number of intersecting surfaces, with their lines and points. DEFINITIONS. 32. These considerations have led to the following definitions : A Point has only position, without extent. A Line has length, without breadth or thickness. A Surface has length and breadth, without thick ness. A Solid has length, breadth, and thickness. 33. A line may be measured only in one way, or, it may be said a line has only one dimension. A surface has two, and a solid has three dimensions. We can not
THE POSTULATES.
19
conceive of any thing of more than three dimensions. Therefore, every thing which has extent and form be longs to one of these three classes. The extent of a line is called its Length; of a sur face, its Area ; and of a solid, its Volume. 34. Whatever has only extent and form is called a Magnitude. Geometry is the science of magnitude. Geometry is used whenever the size, shape, or posi tion of any thing is investigated. It establishes the principles upon which all measurements are made. It is the basis of Surveying, Navigation, and Astronomy. In addition to these uses of Geometry, the study is cultivated for the purpose of training the student's pow ers of language, in the use of precise terms ; his reason, in the various analyses and demonstrations; and his inventive faculty, in the making of new solutions and demonstrations. THE POSTULATES. 35. Magnitudes may have any extent. We may conceive lines, surfaces, or solids, which do not extend beyond the limits of the smallest spot which represents a point ; or, we may conceive them of such extent as to reach across the universe. The astronomer knows that his lines reach to the stars, and his planes extend be yond the sun. These ideas are expressed in the fol lowing Postulate of Extent—A magnitude may be made to have any extent whatever. 36. Magnitudes may, in our minds, have any form, from the most simple, such as a straight line, to that of the most complicated piece of machinery. We may
20
ELEMENTS OP GEOMETRY.
conceive of surfaces without solids, and of lines without surfaces. It is a useful exercise to imagine lines of various forms, extending not only along the paper or blackboard, but across the room. In the same way, surfaces and solids may be conceived of all possible forms. The form of a magnitude consists in the relative posi tion of the parts, that is, in the relative directions of the points. Every change of form consists in changing the relative directions of the points of the figure. Every geometrical conception, however simple or com plex, is composed of only two kinds of elementary thoughts—directions and distances. The directions de termine its form, and the distances its extent. Postulate of Form.— The points of a magnitude may be made to have from each other any directions whatever, thus giving the magnitude any conceivable form. These two are all the postulates of geometry. They rest in the very ideas of space, form, and magnitude. 37. Magnitudes which have the same form while they differ in extent, are called Similar. Any point, line, or surface in a figure, and the simi larly situated point, line, or surface in a similar figure, are called Homologous. Magnitudes which have the same extent, while they differ in form, are called Equivalent.
MOTION AND SUPERPOSITION. 38. The postulates are of constant use in geomet rical reasoning. Since the parts of a magnitude may have any posi tion, they may change position. By this idea of mo'
FIGURES.
21
tion the mutual derivation of points, lines, surfaces, and solids may be explained. The path of a point is a line, the path of a line may be a surface, and the path of a surface may be a solid. The time or rate of motion is not a subject of geome try, but the path of any thing is itself a magnitude. 89. By the idea of motion, one magnitude may be mentally applied to another, and their form and extent compared. This is called the method of superposition, and is the most simple and useful of all the methods of demon stration used in geometry. The student will meet with many examples. EQUALITY. 40. When two equal magnitudes are compared, it is found that they may coincide; that is, each contains the other. Since they coincide, every part of one will have its corresponding equal and coinciding part in the other, and the parts are arranged the same in both. Conversely, if two magnitudes are composed of parts respectively equal and similarly arranged, one may be applied to the other, part by part, till the wholes coin cide, showing the two magnitudes to be equal. Each of the above convertible propositions has been stated as an axiom, but they appear rather to constitute the definition of equality.
FIGURES. 41. Any magnitude or combination of magnitudes which can be accurately described, is called a geomet rical Figure.
22
ELEMENTS OF GEOMETRY.
Figures are represented by diagrams or drawings, and such representations are, in common language, called figures. A small spot is commonly called a point, and a long mark a line. But these have not only extent and form, but also color, weight, and other proper ties ; and, therefore, they are not geometrical points and lines. It is the more important to remember this distinction, since the point and line made with chalk or ink are constantly used to represent to the eye true mathemat ical points and lines. 42. The figure which is the subject of a proposition, together with all its parts, is said to be Given. The additions to the figure made for the purpose of demon stration or solution, constitute the Construction. 43. In the diagrams in this work, points are desig nated by capital letters. Thus, the points A and B are at the ex A Y, tremities of the line. c D Figures are usually designated \ \ by naming some of their points, as \ \ the line AB, and the figure CDEF, P E or simply the figure DF. a
When it is more convenient to desig nate a figure by a single letter, the small letters are used. Thus, the line a, or the figure b.
LINES. 44. A Straight Line is one which has the same di rection throughout its whole extent.
THE STRAIGHT LINE.
23
A straight line may be regarded as the path of a point moving in one direction, turning neither up nor down, to the right or left. 45. A Curved Line is one which constantly changes its direction. The word curve is used for a curved line. 46. A line composed of straight lines, is called Broken. A line may be composed of curves, or of both curved and straight parts. THE STRAIGHT LINE. 4y. Problem—A straight line may be made to pass through any two points. 48. Problem.— There may be a straight line from any point, in any direction, and of any extent. These two propositions are corollaries of the post ulates. 49. From a point, straight lines may extend in all directions. But we can not conceive that two separate straight lines can have the same direction from a common point. This impossibility is expressed by the following Axiom of Direction.—In one direction from a point, there can be only one straight line. 5©. Corollary—From one point to another, there can be only one straight line 51. Theorem—If a straight line have two of its points common with another straight line, the two lines must coin cide throughout their mutual extent. For, if they could separate, there would be from the point of separation two straight lines having the same direction, which is impossible (49).
24
ELEMENTS OF GEOMETRY.
52. Corollary.—Two fixed points, or one point and a certain direction, determine the position of a straight line. . 53. If a straight line were turned upon two of its points as fixed pivots, no part of the line would change place. So any figure may revolve about a straight line, while the position of the line remains unchanged. This property is peculiar to the straight line. If the curve BC were to revolve upon the two points B and C as piv ots, then the straight line con necting these points would remain at rest, and the curve would revolve about it.' A straight line about which any thing revolves, is called its Axis. 54. Axiom of Distance. — The stra Now, the triangles DAF t— // \vv and DAE are equivalent, for / 1 / / \ \\ r they have the same base DA, / y / \ \ A and equal altitudes, since / /\ / \ f\ '' their vertices are in the line // \/ \/ \\ EF parallel to the base (264). £ f g-^ To each of these equals, add the figure ABCD, and we have the quadrilateral FBCD equiva lent to the polygon ABCDE. In this manner, the number of sides may be diminished till a triangle is formed equivalent to the given polygon. In this diagram it is the triangle FDG.
401. Problem— To draw a square equivalent to a given triangle. Find a mean proportional between the altitude and half the base of the triangle. This will be the side of the required square. EQUIVALENT SQUARES.
402. Having shown (379) how an area is expressed by the product of two lengths, it follows that an equa
EQUIVALENT SURFACES.
137
tion will represent equivalent surfaces, if each of its terms is composed of two factors which represent lengths. For example, let a and b represent the lengths of two straight lines. Now we know, from algebra, that what ever be the value of a and b,
This formula, therefore, includes the following geomet rical 403. Theorem.—The square described upon the sum of two lines is equivalent to the sum of the squares described on the two lines, increased by twice the rectangle contained by these two lines. Since the truths of algebra are universal in their application, this theorem is demon h> i strated by the truth of the above ah i equation. Such a proof is called algebraic. a1 ah It is also called analytical, but with doubtful propriety. Let the student demonstrate the theorem geometrically, by the aid of this diagram. 404. Theorem.— The square descr2n — 4. Adding to both members of this inequality, v -\' 4, and subtracting 2n, we have 4 > v. That is, the sum of the angles at the vertex is less than four right angles. This demonstration is a generalization of that of Article 587. The student should make a diagram and special demonstration for a polyedral of five or six faces.
DESCRIPTIVE GEOMETRY.
211
613. Theorem. — In any convex polyedral, the sum of the diedrals is greater than the sum of the angles of a polygon having the same number of sides that the poly edral has faces. Let the given polyedral be divided by diagonal planes into triedrals. Then this theorem may be demonstrated like the analogous proposition on polygons (423). The remark made in Article 346 is also applicable here. DESCRIPTIVE GEOMETRY. 614. In the former part of this 'work, we have found problems in drawing to be the best exercises on the principles of Plane Geometry. At first it appears im possible to adapt such problems to the Geometry of Space ; for a drawing is made on a plane surface, while the figures here investigated are not plane figures. This object, however, has been accomplished by the most ingenious methods, invented, in great part, by Monge, one of the founders of the Polytechnic School at Paris, the first who reduced to a system the elements of this science, called Descriptive Geometry. Descriptive Geometry is that branch of mathemat ics which teaches how to represent and determine, by means of drawings on a plane surface, the absolute or relative position of points or magnitudes in space. It is beyond the design of the present work to do more than allude to this interesting and very useful science. 0 EXERCISES. 615.—1. What is the locus of those points in space, each ot which is equally distant from three given points? 2. What is the locus of those points in space, each of which is equally distant from two given planes?
212
ELEMENTS OP GEOMETRY.
3. What is the locus of those points in space, each of which is equally distant from three given planes? 4. What is the locus of those points in space, each of which is equally distant from two given straight lines which lie in the same plane ? 5. What is the locus of those points in space, each of which is equally distant from three given straight lines which lie in tlie same plane? 6. What is the locus of those points in space, such that the sum of the distances of each from two given planes is equal to a given straight line ? 7. If each diedral of a triedral be bisected, the three planes have one common intersection. 8. If a straight line is perpendicular to a plane, every plane parallel to the given line is perpendicular to the given plane. 9. Given any two straight lines in space ; either one plane may pass through both, or two parallel planes may pass through them respectively. 10. In the second case of the preceding exercise, a line which is perpendicular to both the given lines is also perpendicular to the two planes. 11. If one face of a triedral is rectangular, then an adjacent diedral angle and its opposite face are either both acute, both right, or both obtuse. 12. Apply to planes, diedrals, and triedrals, 'respectively, such properties of straight lines, angles, and triangles, as have not already been stated in this chapter, determining, in each case, whether the principle is true when so applied.
TETRAEDRONS.
213
CHAPTER X. POLYEDRONS. 616. A Polyedron is a solid, or portion of space, bounded by plane surfaces. Each of these surfaces is a face, their several intersections are edges, and the points of meeting of the edges are vertices of the poly edron. 617. Corollary The edges being intersections of planes, must be straight lines. It follows that the faces of a polyedron are polygons. 618. A Diagonal of a polyedron is a straight line joining two vertices which are not in the same face. A Diagonal Plane is a plane passing through three vertices which are not in the same face.
TETRAEDRONS. 619. We have seen that three planes can not inclose a space (581). But if any point be taken on each edge of a triedral, a plane passing through these three points would, with the three faces of / the triedral, cut oif a portion / of space, which would be in closed by four triangular faces. A Tetraedron is a polyedron having four faces.
214
ELEMENTS OF GEOMETRY.
620. Problem—Any four points whatever, which do not all lie in one plane, may be taken as the four vertices of a tetraedron. For they may be joined two and two, by straight lines, thus forming the six edges ; and these bound the four triangular faces of the figure. 621. Either face of the tetraedron may be taken as the base. Then the other faces are called the sides, the vertex opposite the base is called the vertex of the tetraedron, and the altitude is the perpendicular distance from the vertex to the plane of the base. In some cases, the perpendicular falls on the plane of the base produced, as in triangles. 622. Corollary—If a plane parallel to the base of a tetraedron pass through the vertex, the distance between this plane and the base is the altitude of the tetrae dron (574). 623. Theorem— There is a point equally distant from the four vertices of any tetraedron. In the plane of the face BCF, suppose a circle whose circumference passes through the three points B, C, and F. At the center of this circle, erect a line perpendicular to the plane of BCF. Every point of this per pendicular is equally distant from the three points B, C, and F (531). In the same manner, let a line perpendicular to the plane of BDF be erected, so that every point shall be equally distant from the points B, D, and F. These two perpendiculars both lie in one plane, the plane which bisects the edge BF perpendicularly at its
TETRAEDRONS.
215
center (520). These two perpendiculars to two oblique planes, being therefore oblique to each other, will meet at some point. This point is equally distant from the four vertices B, C, D, and F. 624. Corollary.—The six planes which bisect perpen dicularly the several edges of a tetraedron all meet in one point. But this point is not necessarily within the tetraedron. 625. Theorem.— There is a point within every tetrae dron which is equally distant from the several faces. Let AEIO be any tetraedron, and let OB be the straight line formed by the intersection of two planes, A one of which bisects the .y/\ diedral angle whose edge is ^s / \ AO, and the other the die' >^g.-/.C \ dral whose edge is EO. \~ ~T~~^^' Now, every point of the i first bisecting plane is equally distant from the faces IAO and EAO (560) ; and every point of the second bisecting plane is equally distant from the faces EAO and EIO. Therefore, every point of the line BO, which is the intersection of those bisect ing planes, is equally distant from those three faces. Then let a plane bisect the diedral whose edge is EI, and let C be the point where this plane cuts the line BO. Since every point of this last bisecting plane is equally distant from the faces EAI and EOI, it follows that the point C is equally distant from the four faces of the tet raedron. Since all the bisecting planes are interior, the point found is within the tetraedron. 626. Corollary—The six planes which bisect the several diedral angles of a tetraedron all meet at one point.
216
ELEMENTS OF GEOMETRY. EQUALITY OF TETRAEDRONS.
637. Theorem— Two tetraedrons are equal when three faces of the one arc respectively equal to three faces of the other, and they are similarly arranged. For the three sides of the fourth face, in one, must be equal to the same lines in the other. Hence, the fourth faces are equal. Then each diedral angle in the one is equal to its corresponding diedral angle in the other (599). In a word, every part of the one figure is equal to the corresponding part of the other, and the equal parts are similarly arranged. Therefore, the two tetraedrons are equal. 628. Corollary.—Two tetraedrons are equal when the six edges of the one are respectively equal to those of the other, and they are similarly arranged. 629. Corollary—Two tetraedrons are equal when two faces and the included diedral of the one are respect ively equal to those parts of the other, and they are similarly arranged. 630. Corollary—Two tetraedrons are equal when one face and the adjacent diedrals of the one are respect ively equal to those parts of the other, and they are similarly arranged. 631. When tetraedrons are composed of equal parts in reverse order, they are symmetrical.
MODEL TETRAEDRON. 632. The student may easily construct a model of a tetrae' dron when the six edges are given. First, with three of the edges which are sides of one face, draw the triangle, as ABC. Then, on each side of this first triangle, as a base, draw a triangle equal to the corresponding face; all of which can be done, for the
TETRAEDB.ONS.
217
edges, that is, the sides of these triangles, are given. Then, cut out the whole figure from the pa per and carefully fold it at the lines AB, BC, and CA. Since BF is equal to BD, CF to CE, and AD to AE, the points F, D, and E may he united to form a vertex. In this way models of various forms may be made with more accuracy than. in wood, and the student may derive much help from the work. But he must never forget that the geometrical figure exists only as an intellectual conception. To assist him in this, he should strive to generalize every demonstration, stating the argu ment without either model or diagram, as in the demonstration last given. To construct models of symmetrical tetraedrons, the drawings may be equal, but the folding should, in the one case, be up, and in the other, down.
SIMILAR TETRAEDRONS.
633. Since similarity consists in having the same form, so that every difference of direction in one of two similar figures has its corresponding equal differ ence of direction in the other, it follows that when two polyedrons are similar, their homologous faces are simi lar polygons, their homologous edges are of equal die' dral angles, and their homologous vertices are of equal polyedrals. 634. Theorem.— When two tetraedrons are similar, any edge or other line in the one is to the homologous line in the second, as any other line in the first is to its homolo gous line in the second. If the proportion to be proved is between sides of homologous triangles, it follows at once from the simi larity of the triangles. Geom.—19
218
ELEMENTS OF GEOMETRY.
When the edges taken in one of the tetraedrons are not sides of one face; as, AE : BC : : I0 : DF, A
then, CD, as just proved, and CD. Therefore, DF. Again, suppose it is to be proved that the altitudes AK and BH have the same ratios as two homologous edges. AK and BH are perpendicular lines let fall from the homologous points A and B on the opposite faces. From K let the perpendicular KN fall upon the edge I0. Join AN, and from H let the perpendicular HO fall upon DF, which is homologous to I0. Join BG. Now, the planes AKN and EIO are perpendicular to each other (556), and the line IN in one of them is, by construction, perpendicular to their intersection KN. Hence,#IN is perpendicular to the plane AKN (557). Therefore, the line AN is perpendicular to IN, and the diedral whose edge is I0 is measured by the angle ANK. In the same way, it is proved that the diedral whose edge is DF, is measured by the angle BGH. But these two diedrals, being homologous, are equal, the angles ANK and BGH are equal, and the right an gled triangles AKN and BHG are similar. Therefore, AK : BH : : AN : BG.
TETRAEDRONS.
219
Also, the right angled triangles ANI and BGD are similar, since, by hypothesis, the angles AIN and BDG are equal. Hence, Therefore,
AI : BD : : AN : BG. AK : BH : : AI : BD.
Thus, by the aid of similar triangles, it may be proved that any two homologous lines, in two similar tetraedrons, have the same ratio as two homologous edges. 635. Theorem. — Two tetraedrons are similar when their faces are respectively similar triangles, and are simi larly arranged. For we know, from the similarity of the triangles, that every line made on the surface of one may have its homologous line in the second, making angles equal to those made by the first line. If lines be made through the figure, it may be shown, by the aid of auxiliary lines, as in the corresponding proposition of similar triangles, that every possible an gle in the one figure has its homologous equal angle in the other. The student may draw the diagrams, and go through the details of the demonstration. 636. If the similar faces were not arranged similarly, but in reverse order, the tetraedrons would be symmet rically similar. 637. Corollary.—Two tetraedrons are similar when three faces of the one are respectively similar to those of the other, and they are similarly arranged. For the fourth faces, having their sides proportional, are simi lar also. 638. Corollary.—Two tetraedrons are similar when two triedral vertices of the one are respectively equal to two of the other, and they are similarly arranged.
220
ELEMENTS OF GEOMETRY.
639. Corollary.—Two tetraedrons are similar when the edges of one are respectively proportional to those of the other, and they are similarly arranged. (>IO. Theorem— The areas of homologous faces of similar tetraedrons are to each other as the squares of their edges. This is only a corollary of the theorem that the areas of similar triangles are to each other as the squares of their sides. 641. Corollary.—The areas of homologous faces of similar tetraedrons are to each other as the squares of any homologous lines. 642. Corollary.—The area of any face of one tetrae' dron is to the area of a homologous face of a similar . tetraedron, as the area of any other face of the first is to the area of the homologous face of the second. 643. Corollary. — The area of the entire surface of one tetraedron is to that of a similar tetraedron as the squares of homologous lines.
TETRAEDRONS CUT BY A PLAN.E. 644. Theorem.—If a plane cut a tetraedron parallel to the base, the tetraedron cut off is similar to the whole. For each triangular side is cut by a line parallel to its base (572), thus making all the edges of the two tetraedrons respectively proportional. 649. Theorem—If two tetraedrons, having the same altitude and their bases on the same plane, are cut by a plane parallel to their bases, the areas of the sections will have the same ratio as the areas of the bases. If a plane parallel to the bases pass through the ver tex A, it will also pass through the vertex B (622). But
TETRAEDRONS.
221
such a plane is parallel to the cutting plane GHP (566). A
b
Therefore, the tetraedrons AGHK and BLNP have equal altitudes. The tetraedrons AEIO and AGHK are similar (644). Therefore, EIO, the base of the first, is to GHK, the base of the second, as the square of the altitude of the first is to the square of the altitude of the second (641). For a like reason, the base CDF is to the base LNP as the square of the greater altitude is to the square of the less. Therefore, EIO : GHK : : CDF : LNP. By alternation, EIO : CDF : : GHK : LNP. 646. Corollary—When the bases are equivalent the sections are equivalent. 647. Corollary—When the bases are equal the sec tions are equal. For they are similar and equivalent. REGULAR TETRAEDRON. 648. There is one form of the tetraedron which de serves particular notice. It has all its faces equilateral. This is called a regular tetraedron. 649. Corollary—It follows, from the definition, that
222
ELEMENTS OF GEOMETRY.
the faces are equal triangles, the vertices are of equal triedrals, and the edges are of equal diedral angles. 650. The area of the surface of a tetraedron is found by taking the sum of the areas of the four faces. When two or more of them are equal, the process is shortened by multiplication. But the discussion of this matter will be included in the subject of the areas of pyra mids. The investigation of the measures of volumes will be given, in another connection. EXERCISES. 651.—1. State other cases, when two tetraedrons are similar, in addition to those given, Articles 635 to 639. 2. In any tetraedron, the lines which join the centers of the opposite edges bisect each other. 3. If one of the vertices of a tetraedron is a trirectangular tri' edral, the square of the area of the opposite face is equal to the sum of the squares of the areas of the other three faces.
PYRAMIDS. 653. If a polyedral is cut by a plane which cuts its several edges, the section is a polygon, and a portion of space is cut off, which is called a pyramid. A Pyramid is a polyedron having for one face any
polygon, and for its other faces, triangles whose vertices meet at one point.
PYRAMIDS.
223
The polygon is the base of the pyramid, the triangles are its sides, and their intersections are the lateral edges of the pyramid. The vertex of the polyedral is the vertex of the pyramid, and the perpendicular distance from that point to the plane of the base is its altitude. Pyramids are called triangular, quadrangular, pentag onal, etc., according to the polygon which forms the base. The tetraedron is a triangular pyramid. 653. Problem—Every pyramid can be divided into the same number of tetraedrons as its base can be into triangles. Let a diagonal plane pass through the vertex of the pyramid and each diagonal of the base, and the solu tion is evident.
EQUAL PYRAMIDS. 654. Theorem.— Two pyramids are equal when the base and two adjacent sides of the one are respectively equal to the corresponding parts of the other, and they are simi larly arranged. For the triedrals formed by the given faces in the two must be equal, and may therefore coincide; and the given faces will also coincide, being equal. But now the vertices and bases of the two pyramids coin cide. These include the extremities of every edge. Therefore, the edges coincide; also the faces, and the figures throughout.
SIMILAR PYRAMIDS. 655. Theorem.— Two similar pyramids are composed of tetraedrons respectively similar, and similarly arranged ; and, conversely, two pyramids are similar when com posed of similar tetraedrons, similarly arranged.
224
ELEMENTS OF GEOMETRY.
656. Theorem.— When a pyramid is cut by a plane parallel to the base, the pyramid cut off is similar to the whole. These theorems may be demonstrated by the student. Their demonstration is like that of analogous proposi tions in triangles and tetraedrons.
REGULAR PYRAMIDS. 637. A Regular Pyramid is one whose base is a regular polygon, and whose vertex is in the line perpen dicular to the base at its center. 658. Corollary.—The lateral edges of a regular pyra mid are all equal (529), and the sides are equal isosce les triangles. 659. The Slant Hight of a regular pyramid is the perpendicular let fall from the vertex upon one side of the base. It is therefore the altitude of one of the sides of the pyramid. 660. Theorem— The area of the lateral surface of a regular pyramid is equal to half the product of the pe rimeter of the base by the slant hight. The area of each side is equal to half the product of its base by its altitude (386). But the altitude of each of the sides is the slant hight of the pyramid, and the sum of all the bases of the sides is the perimeter of the base of the pyramid. Therefore, the area of the lateral surface of the pyr amid, which is the sum of all the sides, is equal to half the product of the perimeter of the base by the slant hight. 661. When a pyramid is cut by a plane parallel to the base, that part of the figure between this plane and
PYRAMIDS.
225
the base is called a frustum of a pyramid, or a trunc ated pyramid. 663. Corollary—The sides of a frustum of a pyra mid are trapezoids (572); and the sides of the frustum of a regular pyramid are equal trapezoids. 663. The section made by the cutting plane is called the upper base of the frustum. The slant hight of the frustum of a regular pyramid is that part of the slant hight of the original pyramid which lies between the bases of the frustum. It is therefore the altitude of one of the lateral sides. 664. Theorem— The area of the lateral surface of the frustum of a regular pyramid is equal to half the prod uct of the sum of the perimeters of the bases by the slant hight. The area of each trapezoidal side is equal to half the product of the sum of its parallel bases by its altitude (392), which is the slant hight of the frustum. There fore, the area of the lateral surface, which is the sum of all these equal trapezoids, is equal to the product of half the sum of the perimeters of the bases of the frustum, multiplied by the slant hight. 665. Corollary—The area of the lateral surface of a frustum of a regular pyramid is equal to the product of the perimeter of a section midway between the two bases, multiplied by the slant hight. For the perimeter of a section, midway between the two bases, is equal to half the sum of the perimeters of the bases. 666. Corollary.—The area of the lateral surface of a regular pyramid is equal to the product of the slant hight by the perimeter of a section, midway between the vertex and the base. For the perimeter of the middle section is one'half the perimeter of the base.
226
ELEMENTS OP GEOMETRY. MODEL PYRAMIDS.
667. The student may construct a model of a regular pyra mid. First, draw a regular polygon of any number of sides. Upon these sides, as bases, draw equal isosceles triangles, taking care that their altitude be greater than the apothem of the base. The figure may then be cut out and folded.
EXERCISES. 668.—1. Find the area of the surface of a regular octagonal pyramid whose slant hight is 5 inches, and a side of whose base is 2 inches. 2. What is the area in square inches of the entire surface of a regular tetraedron, the edge being one inch ? Ans. j/3. 3. A pyramid is regular when its sides are equal isosceles triangles, whose bases form the perimeter of the base of the pyramid. 4. State other cases of equal pyramids, in addition to those given, Article 654. 5. When two pyramids of equal altitude have their bases in the same plane, and are cut by a plane parallel to their bases, the areas of the sections are proportional to the areas of the bases. PRISMS. 669. A Prism is a polyedron which has two of its faces equal polygons lying in par allel planes, and the other faces parallelograms. Its possibility is shown by supposing two equal and parallel polygons lying in two par allel planes (569). The equal sides being parallel, let planes unite them. The figure thus formed on each plane is a parallelogram, for it has two opposite sides equal and parallel.
PRISMS.
227
The parallel polygons are called the bases, the paral lelograms the sides of the prism, and the intersections of the sides are its lateral edges. The altitude of a prism is the perpendicular distance between the planes of its bases. 670. Corollary.—The lateral edges of a prism are all parallel to each other, and therefore equal to each other (573). 671. A Right Prism is one whose lateral edges are perpendicular to the bases. A Regular Prism is a right prism whose base is a regular polygon. 672. Corollary—The altitude of a right prism is equal to one of its lateral edges ; and the sides of a right prism are rectangles. The sides of a regular prism are equal. 673. Theorem—If two parallel planes pass through a prism, so that each plane cuts every lateral edge, the sections made by the two planes are equal polygons. Each side of one of the sections is parallel to the corresponding side of the other section, since they are the intersections of two parallel planes by a third. Hence, that portion of each side of the prism which is between the secant planes, is a parallelogram. Since the sections have their sides respectively equal and parallel, their angles are respectively equal. There fore, the polygons are equal. 674. Corollary.—The section of a prism made by a plane parallel to the base is equal to the base, and the given prism is divided into two prisms. If two paral lel planes cut a prism, as stated in the above theorem, that part of the solid between the two secant planes is also a prism.
***i
228
ELEMENTS OF GEOMETRY. HOW DIVISIBLE.
675. Problem.—Every prism can be divided into the same number of triangular prisms as its base can be into triangles. If homologous diagonals be made in the two bases, as EO and CF, they will lie in one plane. For CE and OF being parallel to each other (670), lie in one plane. There fore, through each pair of these homologous diagonals a plane may pass, and these diagonal planes divide the prisms into triangular prisms. 6*76. Problem.—A triangular prism may be divided into three tetraedrons, which, taken two and two, have equal bases and equal altitudes. Let a diagonal plane pass through the points B, C, and H, making the intersections BH and CH, in the sides DF and DG. This plane cuts off the tetraedron BCDH, which has for one of its faces the base BCD of the prism; for a second face, the triangle BCH, being the sec tion made by the diagonal plane ; and for its other two faces, the triangles BDH and CDH, each being half of one of the sides of the prism. The remainder of the prism is a quadrangular pyra mid, having the parallelogram BCGF for its base, and H for its vertex. Let i± be cut by a diagonal plane through the points H, G, and B.
PRISMS.
229
This plane separates two tetraedrons, HBCG and HBFG. The two faces, HBC and HBG, of the tetraedron HBCG, are sections made by the diagonal planes; and the two faces, HCG and BCG, are each half of one side of the prism. The tetraedron HBFG has for one of its faces the base HFG of the prism ; for a second face, the triangle HBG, being the section made by the diagonal plane; and, for the other two, the triangles HBF and GBF, each being half of one of the sides of the prism. Now, consider these two tetraedrons as having their bases BCG and BFG. These are equal triangles lying in one plane. The point H is the common vertex, and therefore they have the same altitude ; that is, a perpen dicular from H to the plane BCGF. Next, consider the first and last tetraedrons described, HBCD and BFGH, the former as having BCD for its base, and H for its vertex; the latter as having FGH for its base, and B for its vertex. These bases are equal, being the bases of the given prism. The vertex of each is in the plane of the base of the other. Therefore, the altitudes are equal, being the distance between these two planes. Lastly, consider the tetraedrons BCDH and BCGH as having their bases CDH and CGH. These are equal triangles lying in one plane. The tetraedrons have the common vertex B, and hence have the same altitude. 6T7. Corollary. — Any prism may be divided into tetraedrons in several ways ; but the methods above ex plained are the simplest. 678. Remark.—On account of the importance of the above problem in future demonstrations, the student is advised to make a model triangular prism, and divide it into tetraedrons. A po tato may be used for this purpose. The student will derive most benefit from those models and diagrams which he makes himself.
230
ELEMENTS' OF GEOMETRY. EQUAL PRISMS.
679. Theorem.— Two prisms are equal, when a base and two adjacent sides of the one are respectively equal to the corresponding parts of the other, and they are simi larly arranged. For the triedrals formed by the given faces in the two prisms must be equal (599), and may therefore be made to coincide. Then the given faces will also coin cide, being equal. These coincident points include all of one base, and several points in the second. But the second bases have their sides respectively equal, and parallel to those of the first. Therefore, they also coin cide, and the two prisms having both bases coincident, must coincide throughout. 680. Corollary—Two right prisms are equal when they have equal bases and the same altitude. 6S1. The theory of similar prisms presents nothing difficult or peculiar. The same is true of symmetrical prisms, and of symmetrically similar prisms.
AREA OF THE SURFACE. 683. Theorem.— The area of ihe lateral surface of a prism is equal to the product of one of the lateral edges by the perimeter of a section, made by a plane perpen dicular to those edges. Since the lateral edges are parallel, the plane HN, perpendicular to one, is perpendicular to all of them. Therefore, the sides of the polygon, HK, KL, etc., are severally perpendicular to the edges of the prism which they unite (519). Then, in order to measure the area of each face of the prism, we take one edge of the prism as the base
PRISMS.
231
of the parallelogram, and one side of the polygon HN as its altitude. Thus, area AG = AB X HP, area EB = EC X HK, etc. By addition, the sum of the areas of these parallelograms is the lateral surface of the prism, and the sum of the altitudes of the parallelograms is the perim eter of the polygon HN. Then, since the edges are equal, the area of all the sides is equal to the product of one edge, multi plied by the perimeter of the polygon. 683. Corollary—The area of the lateral surface of a right prism is equal to the product of the altitude by the perimeter of the base. 684. Corollary—The area of the entire surface of a regular prism is equal to the product of the perime ter of the base by the sum of the altitude of the prism and the apothem of the base.
EXERCISES. 685.—1. A right prism has less surface than any other prism of equal base and equal altitude ; and a regular prism has less surface than any other right prism of equivalent base and equal altitude. 2. A regular pyramid and a regular prism have equal hexag onal bases, and altitudes equal to three times the radius of the base; required the ratio of the areas of their lateral surfaces. 3. Demonstrate the principle stated in Article 683, without the aid of Article 682.
232
ELEMENTS OF GEOMETRY.
MEASURE OF VOLUME. 686. A Parallelopiped is a prism whose bases are parallelograms. Hence, a parallelopiped is a solid in closed by six parallelograms. 687. Theorem.— The opposite sides of a parallelopiped are equal. For example, the faces AI and BD are equal. For I0 and DF are equal, being opposite sides of the parallelogram IF. For a like reason, EI is. equal to CD. But, since these equal sides are also par allel, the included angles EIO and CDF are equal. Hence, the parallelograms are equal. 688. Corollary.—Any two opposite faces of a paral lelopiped may be assumed as the bases of the figure. 689. A parallelopiped is called right in the, same case as any other prism. When the bases also are rectangles, it is called rectangular. Then, all the faces are rectangles. 690. A Cube is a rectangular parallelopiped whose length, breadth, and altitude are equal. Then a cube is a solid, bounded by six equal squares. All its verti ces, being trirectangular triedrals, are equal (602). All its edges are of right diedral angles, and therefore equal (555). The cube has the simplest form of all geometrical solids. It holds the same rank among them that the square does among plane figures, and the straight line among lines.
MEASURE OF VOLUME.
233
The cube is taken, therefore, as the unit of measure of volume. That is, whatever straight line is taken as the unit of length, the cube whose edge is of that length is the unit of volume, as the square whose side is of that length is the measure of area.
VOLUME OF PAKALLELOPIPEDS. 691. Theorem— The volume of a rectangular paral lelepiped is equal to the product of its length, breadth, and altitude. In the measure of the rectangle, the product of one line by another was ex plained. Here we have three lines used with a similar meaning. That is, the number of cu bical units contained in a rectangular parallelo' piped is equal to the product of the numbers of linear units in the length, the breadth, and the alti tude. If the altitude AE, the length EI, and the breadth 10, have a common measure, let each be divided by it ; and let planes, parallel to the faces of the prism, pass through all the points of division, B, C, D, etc. By this construction, all the angles formed by these planes and their intersections are right angles, and each of the intercepted lines is equal to the linear unit used in dividing the edges of the prism. Therefore, the prism is divided into equal cubes. The number of these at the base is equal to the number of rows, mul tiplied by the number in each row; that is, the product Geom.—20
234
ELEMENTS OF GEOMETRY.
of the length by the breadth. There are as many layers of cubes as there are linear units of altitude. Therefore, the whole number is equal to the product of the length, breadth, and altitude. In the diagram, the dimensions being four, three, and two, the volume is twenty'four. But if the length, breadth, and altitude have no com mon measure, a linear unit may be taken, successively smaller and smaller. In this, we would not take the whole of the linear dimensions, nor would we measure the whole of the prism. But the remainder of both would grow less and less. The part of the prism meas ured at each step, would be measured exactly by the principle just demonstrated. By these successive diminutions of the unit, we can make the part measured approach to the whole prism as nearly as we please. In a word, the whole is the limit of the parts measured ; and since the principle demon strated is true up to the limit, it must be true at the limit. Therefore, the rectangular parallelopiped is meas ured by the product of its length, breadth, and altitude. 693. Theorem— The volume of any parallelopiped is epial to the product of its length, breadth, and altitude. Inasmuch as this has just been demonstrated for the rectangular parallelopiped, it will be sufficient to show that any parallelopiped is equivalent to a rectangular one having the same linear dimensions. Suppose the lower bases of the two prisms to be placed on the same plane. Then their upper bases must also be in one plane, since they have the same altitude. Let the altitude AE be divided into an infinite number of equal parts, and through each point of division pass a plane parallel to the base AI. Now, every section in either prism is equal to the
MEASURE OF VOLUME.
235
base ; but the bases of the two prisms, having the same length and breadth, are equivalent. The several par tial infinitesimal prisms are reduced to equivalent fig
ures. Although they are not, strictly speaking, paral lelograms, yet their altitudes being infinitesimal, there can be no error in considering them as plane figures ; which, being equal to their respective bases, are equiva lent. Then, the number of these is the same in each prism. Therefore, the sum of the whole, in one, is equivalent to the sum of the whole, in the other ; that is, the two parallelopipeds are equivalent. Besides the above demonstration by the method of infinites, the theorem may be demonstrated by the or dinary method of reasoning, which is deduced from principles that depend upon the superposition and co incidence of equal figures, as follows . Let AF be any oblique 'parallelopiped. It may be shown to be equivalent to the parallclopiped AL, which has a rectangular base, AH, since the prism LIIEO is equal to the prism DGAI. But the parallelopipeds AF and AL have the same length, breadth, and altitude.
236
ELEMENTS OF GEOMETRY.
By similar reasoning, the prism AL may be shown to be equivalent to a prism of the same base and alti tude, but with two of its opposite sides rectangular. This third prism may then be shown to be equivalent to a fourth, which is rectangular, and has the same dimen sions as the others. 693. Corollary.—The volume of a cube is equal to the third power of its edge. Thence comes the name of cube, to designate the third power of a number.
MODEL CUBES.
694. Draw six equal squares, as in the diagram. Cut out the figure, fold at the dividing lines, and glue the edges. It is well to have at least eight of one size.
695. Corollary. —The volume of any is equal to the product of its base by its 696. Corollary—The volumes of any pipeds are to each other as the products dimensions.
parallelopiped altitude. two parallelo' of their three
VOLUME OF PRISMS. 697. Theorem.— The volume of any triangvlar prism is equal to the product of its base by its altitude. The base of any right triangular prism may be con sidered as one-half of the base of a right parallelopiped. Then the whole parallelopiped is double the given prism, for it is composed of two right prisms having equal bases and the same altitude, of which the given prism
MEASURE OF VOLUME.
237
is one. Therefore, the given prism is measured by half the product of its altitude by the base of the parallel' opiped ; that is, by the product of its own base and altitude. If the given prism be oblique, it may be shown, by demonstrations similar to the first of those in Article 692, to be equivalent to a right prism having the same base and altitude. 698. Corollary.—The volume of any prism is equal to the product of its base by its altitude. For any prism is composed of triangular prisms, having the com mon altitude of the given prism, and the sum of their bases forming the given base. 699. Corollary The volume of a triangular prism is equal to the product of one of its lateral edges mul tiplied by the area of a section perpendicular to that edge. VOLUME OF TETEAEDRONS. TOO. Theorem.— Two tetraedrons of equivalent bases and of the same altitude are equivalent. Suppose the bases of the two tetraedrons to be in the
same plane. Then their vertices lie in a plane parallel to the bases, since the altitudes are equal. Let the edge AE be divided into an infinite number of parts,
238
ELEMENTS OF GEOMETRY.
and through each point of division pass a plane parallel to the base AIO. Now, the several infinitesimal frustums into which the two figures are divided may, without error, be consid ered as plane figures, since their altitudes are infinitesi mal. But each section of one tetraedron is equivalent to the section made by the same plane in the other tet raedron. Therefore, the sum of all the infinitesimal frustums in the one figure is equivalent to the sum of all in the other; that is, the two tetraedrons are equiv alent. 701. Theorem.— The volume of a tetraedron is equal to one'third of the product of the base by the altitude. Upon the base of any given tetraedron, a triangular prism may be erected, which shall have the same alti tude, and one edge coincident with an edge of the tet raedron. This prism may be divided into three tetrae drons, the given one and two others, which, taken two and two, have equal bases and altitudes (676). Then, these three tetraedrons are equivalent (700); and the volume of the given tetraedron is one'third of the volume of the prism ; that is, one'third of the prod uct of its base by its altitude. VOLUME OF PYRAMIDS. 702. Corollary.—The volume of any pyramid is equal to one'third of the product of its base by its altitude. For any pyramid is composed of triangular pyramids; that is, of tetraedrons having the common altitude of the given pyramid, and the sum of their bases forming the given base (653). 703. Corollary.—The volumes of two prisms of equiv alent bases are to each other as their altitudes, and the
SIMILAR POLYEDRONS.
239
volumes of two prisms of equal altitudes are to each other as their bases. The same is true of pyramids. 704. Corollary—Symmetrical prisms are equivalent. The same is true of symmetrical pyramids. *705. The volume of a frustum of a pyramid is found by subtracting the volume of the pyramid cut off from the volume of the whole. When the altitude of the whole is not given, it may be found by this proportion : the area of the lower base of the frustum is to the area of its upper base, which is the base of the part cut off, as the square of the whole altitude is to the square of the altitude of the part cut off.
EXERCISES. 706.—1. What is the ratio of the volumes of a pyramid and prism having the same base and altitude ? 2. If two tetraedrons have a triedral vertex in each equal, .their volumes are in the ratio of the products of the edges which contain the equal vertices. 3. The plane which bisects a diedral angle of a tetraedron, divides the opposite edge in the ratio of the areas of the adjacent faces.
SIMILAR POLYEDRONS. 70T. The propositions (640 to 643) upon the ratios of the areas of the surfaces of similar tetraedrons, may be applied by the student to any similar polyedrons. These propositions and the following are equally appli cable to polyedrons that are symmetrically similar. TOS. Problem—Any two similar polyedrons may be divided into the same number of similar tetraedrons, which shall be respectively similar, and similarly arranged. For, after dividing one into tetraedrons, the construc'
240
ELEMENTS OF GEOMETRY.
tion of the homologous lines in the other will divide it in the same manner. Then the similarity of the re spective tetraedrons follows from the proportionality of the lines. 709. Theorem.— The volumes of similar polyedrons are proportional to the cubes of homologous lines. First, suppose the figures to be tetraedrons. Let AH and BG be the altitudes.
Then (641), EIO : CDF : : EI2 : CF2 . : AH2 : BG2. By the proportionality of homologous lines, (634), i AH : i BG : : EI : CF : : AH : BG. Multiplying these proportions (701), we have AEIO : BCFD : : EP : CF3 : : AH3 : BG3, or, as the cubes of any other homologous lines. Next, let any two similar polyedrons be divided into the same number of tetraedrons. Then, as just proved, the volumes of the homologous parts are proportional to the cubes of the homologous lines. By arranging these in a continued proportion, as in Article 436, we may show that the volume of either polyedron is to the vol ume of the other as the cube of any line of the first is to the cube of the homologous line of the second.
REGULAR. POLYEDRONS.
241
•710. Notice that in the measure of every area there are two linear dimensions ; and in the measure of every volume, three linear, or one linear and one superficial. EXERCISE. VU.. What is the ratio between the edges of two cubes, one of which has twice the volume of the other? This problem of the duplication of the cube was one of the celebrated problems of ancient times. It is said that the oracle of Apollo at Delphos, demanded of the Athenians a new altar, of the same shape, but of twice the volume of the old one. The efforts of the Greek geometers were chiefly aimed at a graphic so lution ; that is, the edge of one cube being given, to draw a line equal to the edge of the other, using no instruments but the rule and compasses. In this they failed. The student will find no difficulty in making an arithmetical solution, within any desired degree of approximation.
REGULAR POLYEDRONS. .712. A Regular Polyedron is one whose faces are equal and regular polygons, and whose vertices are equal polyedrals.
The regular tetraedron and the cube, or regular hexa' edron, have been described. The regular octaedron has eight, the dodecaedron twelve, and the icosaedron twenty faces. Geom.—21
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ELEMENTS OF GEOMETRY.
The class of figures here defined must not be con founded with regular pyramids or prisms. 713. Problem.—It is not possible to make more than five regular polyedrons. First, consider thoso whose faces are triangles. Each angle of a regular triangle is one-third of two right angles. Either three, four, or five of these may bo joined to form one polyedral vertex, the sum being, in each case, less than four right angles (612). But the sum of six such angles is not less than four right angles. Therefore, there can not be more than three kinds of regular polyedrons whose faces are triangles, viz. : the tetraedron, where three plane angles form a vertex ; the octaedron, where four, and the icosaedron, where five angles form a vertex. The same kind of reasoning shows that only one regular polyedron is possible with square faces, the cube ; and only one with pentagonal faces, the dode' caedron. Regular hexagons can not form the faces of a regular polyedron, for three of the angles of a regular hexagon are together not less than four right angles ; and there fore they can not form a vertex. So much the more, if the polygon has a greater num ber of sides, it will be impossible for its angles to be the faces of a polyedral. Therefore, no polyedron is possible, except the five that have been described.
MODEL REGULAR POLYEDRONS. 714. The possibility of regular polyedrons of eight, of twelve., and of twenty sides is here assumed, as the demonstration would occupy more space than the principle is worth. However, the student may construct models of these as follows. Plans for the regular tetraedron and the cube have already been given.
REGULAR POLYEDRONS.
243
For the octaedron, draw eight equal regular trian gles, as in the diagram.
For the dodecaedron, draw twelve equal regular penta gons, as in the diagram.
For the icosaedron, draw twenty equal regular trian gles, as in the diagram.
There are many crystals, which, though not regular, in the geometrical rigor of the word, yet present a certain regularity of shape. EXERCISES. 715.—1. How many edges and how many vertices has each of the regular polyedrons? 2. Calling that point the center of a triangle which is the inter section of straight lines from each vertex to the center of the opposite side; then, demonstrate that the four lines which join the vertices of a tetraedron to the centers of the opposite faces, inter sect each other in one point 3. In what ratio do the lines just described in the tetraedron divide each other? 4. The opposite vertices of a parallelopiped are symmetrical triedrals. 5 The diagonals of a parallelopiped bisect each other; the lines which join the centers of the opposite edges bisect each other; the lines which join the centers of the opposite faces bi.
244
ELEMENTS OF GEOMETRY.
sect each other; and the point of intersection is the same for all these lines. 6. The diagonals of a rectangular parallelopiped are equal. 7. The square of the diagonal of a rectangular parallelopiped is equivalent to the sum of the squares of its length, breadth, and altitude. 8. A cube is the largest parallelopiped of the same extent of surface. 9. If a right prism is symmetrical to another, they are equal. 10. Within any regular polyedron there is a point equally distant from all the faces, and also from all the vertices. 11. Two regular polyedrons of the same number of faces are similar. 12. Any regular polyedron may be divided into as many regu lar and equal pyramids as it has faces. 13. Two different tetraedrons, and only two, may be formed with the same four triangular faces; and these two tetraedrons are symmetrical. 14. The area of the lower base of a frustum of a pyramid is five square feet, of the upper base one and four'fifths square feet, and the altitude ie two feet; required the volume.
SOLIDS OF REVOLUTION.
245
CHAPTER XI. SOLIDS OF REVOLUTION. 716. Of the infinite variety of forms there remain but three to be considered in this elementary work. These are formed or generated by the revolution of a plane figure about one of its lines as an axis. Figures formed in this way are called solids of revolution. 717. A Cone is a solid formed by the revolution of a right angled triangle about one of its legs as an axis. The other leg revolv ing describes a plane surface (521). This surface is also a circle, having for its radius the leg by which it is de scribed. The hypotenuse describes a curved surface. The plane surface of a cone is called its base. The opposite extremity of the axis is the vertex. The alti tude is the distance from the vertex to the base, and the slant hight is the distance from the vertex to the cir cumference of the base. 718. A Cylinder is a solid described by the revolution of a rectangle about one of its sides as an axis. As in the cone, the sides adjacent to the axis de scribe circles, while the opposite side describes a curved surface. The plane surfaces of a cylinder are called its bases,
246
ELEMENTS OF GEOMETRY.
and the perpendicular distance between them is its altitude. These figures are strictly a regular cone and a regular cylinder, yet but one word is used to denote the figures defined, since other cones and cylinders are not usually discussed in Elementary Geometry. The sphere, which is described by the revolution of a semicircle about the diameter, will be considered separately. "719. As the curved surfaces of the cone and of the cylinder are generated by the motion of a straight line, it follows that each of these surfaces is straight in one direction. A straight line from the vertex of the cone to the circumference of the base, must lie wholly in the sur face. So a straight line, perpendicular to the base of a cylinder at its circumference, must lie wholly in the surface. For, in each case, these positions had been occupied by the generating lines. One surface is tangent to another when it meets, but being produced does not cut it. The place of contact of a plane with a conical or cylindrical surface, must be a straight line ; since, from any point of one of those surfaces, it is straight in one direction.
CONIC SECTIONS. 720. Every point of. the line which describes the curved surface of a cone, or of a cylinder, moves in a plane parallel to the base (565). Therefore, if a cone or a cylinder be cut by a plane parallel to the base, the section is a circle. If we conceive a cone to be cut by a plane, the curve formed by the intersection will be different according to the position of the cutting plane. There are three dif
CONES.
247
ferent modes in which it is possible for the intersection to take place. The curves thus formed are the ellipse, parabola, and hyperbola. These Conic Sections are not usually considered in Elementary Geometry, as their properties can be better investigated by the application of algebra. CONES. 731. A cone is said to be inscribed in a pyramid, when their bases lie in one plane, and the sides of the pyramid are tangent to the curved surface of the cone. The pyramid is said to be circumscribed about the cone. A cone is said to be circumscribed about a pyramid, when their bases lie in one plane, and the lateral edges of the pyramid lie in the curved surface of the cone. Then the pyramid is inscribed in the cone. "722. Theorem.—A cone is the limit of the pyramids which can be circumscribed about it; also of the pyramids which can be inscribed in it. Let ABODE be any pyramid circumscribed about a cone. The base of the cone is a circle inscribed in the base of the pyramid. The sides of the pyramid are tangent to the surface of the cone. Now, about the base of the cone there may be described a polygon of double the num ber of sides of the first, each alternate side of the second polygon coinciding with a side of the first. This second polygon may be the base of a pyramid, having its vertex at A. Since the sides of its bases are tangent to the base of the cone, every
248
ELEMENTS OF GEOMETRY.
side of the pyramid is tangent to the curved surface of the cone. Thus the second pyramid is circumscribed about the cone, but is itself within the first pyramid. By increasing the number of sides of the pyramid, it can be made to approximate to the cone within less than any appreciable difference. Then, as the base of the cone is the limit of the bases of the pyramids, the cone itself is abo the limit of the pyramids. Again, let a polygon be inscribed in the base of the cone. Then, straight lines joining its vertices with the vertex of the cone form the lateral edges of an inscribed pyramid. The number of sides of the base of the pyr amid, and of the pyramid also, may be increased at will. It is evident, therefore, that the cone is the limit of pyramids, either circumscribed or inscribed. 723. Corollary.—The area of the curved surface of a cone is equal to one-half the product of the slant hight by the circumference of the base (660). Also, it is equal to the product of the slant hight by the circumfer ence of a section midway between the vertex and the base (666). '724. Corollary.—The area of the entire surface of a cone is equal to half of the product of the circumfer ence of the base by the sum of the slant hight and the radius of the base (499). 725. Corollary—The volume of a cone is equal to one'third of the product of the base by the altitude. 726. The frustum of a cone is defined in the same way as the frustum of a pyramid. 727. Corollary—The area of the curved surface of the frustum of a cone is equal to half the product of its slant hight by the sum of the circumferences of its bases (664). Also, it is equal to the product of its slant
CYLINDERS.
249
hight by the circumference of a section midway between the two bases (665). "7S8. Corollary.—If a cone be cut by a plane paral lel to the base, the cone cut off is similar to the whole (656). EXERCISES. '729.—1. Two cones are similar when they are generated by similar triangles, homologous sides being used for the axes. 2. A section of a cone by a plane passing through the vertex, is an isosceles triangle.
CYLINDERS. 730. A cylinder is said to be in scribed in a prism, when their bases lie in the same planes, and the sides of the prism are tangent to the curved surface of the cylinder. The prism is then said to be circumscribed about the cylinder.
A cylinder is said to be circum scribed about a prism, when their bases lie in the same planes, and the lat eral edges of the prism lie in the curved surface of the cylinder ; and the prism is then said to be inscribed in the cylinder. 731. Theorem.—A cylinder is the limit of the prisms which can be circumscribed about it; also of those which can be inscribed in it. The demonstration of this theorem is so similar to that of the last, that it need not be repeated.
250
ELEMENTS OF GEOMETRY.
732. Corollary—The area of the curved surface of a cylinder is equal to the product of the altitude by the circumference of the base (683). 733. Corollary—The area of the entire surface of a cylinder is equal to the product of the circumference of the base by the sum of the altitude and the radius of the base (684). 734. Corollary.—The volume of a cylinder is equal to the product of the base by the altitude (698).
MODEL CONES AND CYLINDERS. 735. Models of cones and cylinders may be made from paper, taking a sector of a circle for the curved surface of a cone, and a rectangle for the curved surface of a cylinder. Make the bases separately. EXERCISES. 736.—1. Apply to cones and cylinders the principles demon strated of similar polyedrons. 2. A section of a cylinder made by a plane perpendicular to the base is a rectangle. 3. The axis of a cone or of a cylinder is equal to its altitude.
SPHERES.
737. A Sphere is a solid de scribed by the revolution of a semicircle about its diameter as an axis. The center, radius, and diame ter of the sphere are the same as those of the generating circle. The spherical surface is described by the circumference.
SPHERES.
251
738. Corollary.—Every point on the surface of the sphere is equally distant from the center. This property of the sphere is frequently given as its definition. 739. Corollary.—All radii of the same sphere are equal. The same is true of the diameters. 740. Corollary.—Spheres having equal radii are equal. 741. Corollary.—A plane passing through the center of a sphere divides it into equal parts. The halves of a sphere are called hemispheres. 743. Theorem—A plane which is perpendicular to a radius of a sphere at its extremity is tangent to the sphere. For if straight lines extend from * the center of the sphere to any other point of the plane, they are oblique and longer than the radius, which is perpendicular (530). There fore, every point of the plane except one is beyond the surface of the sphere, and the plane is tangent. 743. Corollary.—The spherical surface is curved in every direction. Unlike those surfaces which are gen erated by the motion of a straight line, every possible section of it is a curve.
SECANT PLANES. 744. Theorem— Every section of a sphere made by a plane is a circle. If the plane pass through the center of the sphere, every point in the perimeter of the section is equally distant from the center, and therefore the section is a circle.
252
ELEMENTS OF GEOMETRY
But if the section do not pass through the center, as DGF, then from the center C let CI fall perpendicu larly on the cutting plane. Let radii of the sphere, as CD and CG, extend to differ ent points of the boundary of the section, and join ID and IG. Now the oblique lines CD and CG being equal, the points D and G must be equally distant from I, the foot of the perpendicular (529). The same is true of all the points of the pe rimeter DGF. Therefore, DGF is the circumference of a circle of which I is the center. 745. Corollary—The circle formed by the section through the center is larger than one formed by any plane not through the center. For the radius BC is equal to GO, and longer than GI (104). 746. When the plane passes through the center of a sphere, the section is called a great circle; otherwise it is called a small circle. 747. Corollary.—All great circles of the same sphere are equal. 748. Corollary—Two great circles bisect each other, and their intersection is a diameter of the sphere. 749. Corollary.—If a perpendicular be let fall from the center of a sphere on the plane of a small circle, the foot of the perpendicular is the center of the cir cle ; and conversely, the axis of any circle is a diame ter of the sphere. The two points where the axis of a circle pierces the spherical surface, are the poles of the circle. Thus,
SPHERES.
253
N and S are the poles of both the sections in the last diagram. 750. Corollary.—Circles whose planes are parallel to each other have the same axis and the same poles.
ARC OF A GREAT CIRCLE. 751. Theorem The shortest line which can extend from one point to another along the surface of a sphere, is the arc of a great circle, passing through the two points. Only one great circle can pass through two given points on the surface of a sphere ; for these two points and the center determine the position of the plane of the circle. Let ABCDEFG be any curve whatever on the sur face of a sphere from G to A. Let AKG be the arc of a great circle joining these points, and also AD and DG arcs of great cir cles joining those points with the point D of the given curve. Then the sum of AD and DG is greater than AKG. For the planes of these arcs form a triedral whose vertex is at the center of the sphere. These arcs have the same ratios to each other as the plane angles which compose this triedral, for the arcs are intercepted by the sides of the angles, and they have the same radius. But any one of these angles is less than the sum of the other two (586). Therefore, any one of the arcs is less than the sum of the other two. Again, let AH and HD be arcs of great circles join ing A and D with some point H of the given curve ; also let DI and IG be arcs of great circles. In the
254
ELExMENTS OF GEOMETRY.
same manner as above, it may be shown that AH and HD are greater than AD, and that the sum of DI and IG is greater than DG. Therefore, the sum of AH, HD, DI, and IG is still greater than AKG. By continuing to take intermediate points and join ing them to the preceding, a series of lines is formed, each greater than the preceding, and each approaching nearer to the given curve. Evidently, this approach can be made as nearly as we choose. Therefore, the curve is the limit of these lines, and partakes of their common character, in being greater than the arc of a great circle which joins its extremities. 752. Theorem—Every plane passing through the axis of a circle is perpendicular to the plane of that circle, and its section is a great circle. The first part of this theorem is a corollary of Arti cle 556. The second part is proved by the fact that every axis pass«s through the center of a sphere (749). 753. Corollary—The distances on the spherical sur face from any points of a circumference to its pole, are the same. For the arcs of great circles which mark these distances are equal, since all their chords are equal oblique lines (529). 754. Corollary—The distance of the pole of a great circle from any point of the circumference is a quad rant. APPLICATIONS. Y55. The student of geography will recognize the equator as a great circle of the earth, which is nearly a sphere. The paral lels of latitude are small circles, all having the same poles as the equator. The meridians are great circles perpendicular to the equator. The application of the principle of Article 751 to navigation
\
SPHERES.
255
has been one of the greatest reforms in that art. A vessel cross ing the ocean from a port in a certain latitude to a port in the same latitude, should not sail along a parallel of latitude, for that is the arc of a small circle. 756. The curvature of the sphere in every direction, renders it impossible to construct an exact model with plane paper. But the student is advised to procure or make a globe, upon which he can draw the diagrams of all the figures. This is the more im portant on account of the difficulty of clearly representing these figures by diagrams on a plane surface.
SPHERICAL ANGLES.
757. A Spherical Angle is the difference in the directions of two arcs of great cir cles at their point of meeting. To obtain a more exact idea of this angle, notice that the direction of an arc at a given point is the same as the direction of a straight line tangent to the arc at that point. Thus, the direction of the arc BDF at the point B, is the same as the direction of the tangent BH. 758. Corollary.—A spherical angle is the same as the plane angle formed by lines tangent to the given arcs at their point of meeting. Thus, the spherical angle DBG is the same as the plane angle HBK, the lines HB and BK being severally tangent to the arcs BD and BG. 759. Corollary—A spherical angle is the same as the diedral angle formed by the planes of the two arcs. For, since the intersection BF of the planes of the arcs is a diameter (748), the tangents HB and KB are both perpendicular to it, and their angle measures the diedral.
256
ELEMENTS OF GEOMETRY.
760. Corollary A spherical an gle is measured by the arc of a cir cle included between the sides of the angle, the pole of the arc being at the vertex. Thus, if DG is an arc of a great circle whose pole is at B, then the spherical angle DBG is measured by the arc DG. 76I0 A Lune is that portion of the surface of a sphere included between two halves of great circles. That portion of the sphere included between the two planes is called a spherical wedge. Hence, two great circles divide the surface into four lunes, and the sphere into four wedges.
SPHERICAL POLYGONS. 762. A Sphekical Polygon is that portion of the surface of a sphere included between three or more arcs of great circles. Let C be the center of a sphere, and also the vertex of a convex polyedral. Then, the planes of the faces of this polyedral will cut the surface of the sphere in arcs of great circles, which form the poly gon BDFGH. We say con vex, for only those polygons which have all the angles convex are considered among spherical polygons. Conversely, if a spherical polygon have the planes of its several sides produced, they form a polyedral whose vertex is at the center of the sphere.
SPHERES.
257
The angles of the polygon are the same as the die' dral angles of the polyedral (759). 763. Theorem.— The sum of all the sides of a spher ical polygon is less .than a circumference of a great circle. The arcs which form the sides of the polygon measure the angles which form the faces of the corresponding polyedral, for all the arcs have the same radius. But the sum of all the faces of the polyedral being less than four right angles, the sum of the sides must be less than a circumference. 764. Theorem.—A spherical polygon is always within the surface of a hemisphere. For a plane may pass through the vertex of the cor responding polyedral, having all of the polyedral on one side of it (609). The section formed by this plane produced is a great circle, as KLM. But since the polyedral is on one side of this plane, the corres ponding polygon must be con tained within the surface on one side of it. 765. That portion of a sphere which is included be tween a spherical polygon and its corresponding polye dral is called a spherical pyramid, the polygon being its base. SPHERICAL TRIANGLES. 766. If the three planes which form a triedral at the center of a sphere be produced, they divide the sphere into eight parts or spherical pyramids, each hav ing its triedral at the center, and its spherical triangle Geom.—22
258
ELEMENTS OF GEOMETRY.
at the surface. Thus, for every spherical triangle, there are seven others whose sides are respectively either equal or supplementary to those of the given triangle. /^- ^.-S? Of these seven spherical tri' /FV' / j\ angles, that which lies vertically /.''.. /\ 7\.; \ opposite the given triangle, as \ / ';';.--./ >. I GKH to FDB, has its sides V !'~;>C.\/ "~"~JD respectively equal to the sides \ I / /° y of the given triangle, but they u*' quarter of its revolution, form an acute angle with XX', equal to the angle BAC. Now, the numer ical value of the function depends upon the acute angle which the revolving line makes with the fixed line (817). Hence, there is an
286
PLANE TRIGONOMETRY.
angle for each quarter whose functions are numerically equal to those of the angle BAC. 829. Corollary.—Any simple function of an angle i.$ numerically equal to the same function of 1st. The supplement of the angle; 2nd. The given angle increased by two right angles ; 3rd. The given angle taken negatively. The sine and cosecant of supplementary angles have the same signs, while the other simple functions of sup plementary angles have opposite signs (825). The cosine and secant of an angle and of its negative have the same signs, while the other simple functions of such angles have opposite signs. The tangent and cotangent of an angle, and of the same angle increased by two right angles, have the same signs, while the other simple functions of such angles have opposite signs. These conclusions as to the sine may be expressed thus: sin. A=sin. (180°-A)=-sin. a80°-f-A) = -sin. (-A). The following more general expressions are easily de duced from the above corollary. If n is 0, or any integer positive or negative, and A is any angle, then The formula n'180°+(— l)nA includes all angles which have the same sine as A ; The formula w360°±A includes all the angles which have the same cosine as A; and The formula wl80°+A includes all angles which have the same tangent as A. 830. Corollary.—Any simple function of any angle may be expressed in terms of the same function of an acute angle.
FUNCTIONS OF ANGLES.
287
EXERCISES. 831.—1. Make a formula analogous to the above for each of the other simple functions. 2. Demonstrate cosec. 600° = — f/3 ; cot 405° = 1. 3. Write a formula containing all the values of A when tan. A=l. LIMITS OP FUNCTIONS.
832. Theorem— The sine of any angle can not be greater than 1, nor less than — 1 ; and the cosine has the same limits. For the leg of a right angled triangle can not be greater than the hypotenuse; and, therefore, the sine and cosine are fractions having the numerator less than the denominator. 833. Theorem.— The secant and cosecant can not have any values between 1 and — 1 ; and the tangent and cotan gent have no limits. These principles also follow immediately from the defi nitions and the nature of a right angled triangle. 834. As the revolving line passes through the first quarter of its revolution, the sine increases from 0 to 1. The sine of a right angle is unity, for in that case the perpendicular coincides with the hypotenuse. Then the sine decreases till the angle is equal to two right angles, when the sine becomes 0. It continues to decrease till the angle becomes three right angles, when the sine is — 1. Then again it increases to the end of the revolu tion, where the sine is 0. The cosine of 0° is 1, which decreases as the angle increases till the cosine of 90° is 0, and the cosine of
288
PLANE TRIGONOMETRY.
180° is — 1. Then it increases through the remaining half of the revolution. The tangent of 0° is 0. As the angle increases the tangent increases without limit, and the tangent of a right angle is infinite. The tangent of an obtuse angle is negative, and as the angle increases the tangent varies from minus infinity to zero. In the third quarter the tangent varies as in the first quarter through all possible positive values ; and the variations of the fourth quarter are like those of the second. The variations of the cotangent, secant, and cosecant may be traced in the same way. These values of the functions at particular points may be expressed as follows : 81N.
0° 90° 180° 270° 360°
. . . . .
. . . . .
. . . . .
. 0 . 1 . 0 .-1 . 0
C08.
1 0 -1 0 1
TAN.
C0T.
8EC.
C08EC
0
GO
1
CO
QO
0
CO
0
CO
—1
GO
0
CO
0
CO
1
1 CO
—1 GO
The versed sine increases from 0 to 2 as the angle increases from 0° to 180°, and decreases from 2 to 0 through the other two quarters.
EXERCISES. 835.—1. Trace the value of this expression: cos. A — sin. A, as A varies from 0° to 3(3j°. 2. What are the sihe and the tangent of 810°? 3. What are the cosine and secant of — 450°? 4. What are the cosecant and cotangent of 150°? 5. Construct an angle greater than 90°, whose sine is ^; one whose tangent is £; one whose cosine is ^.
FUNCTIONS OF ANGLES.
289
RELATIONS BETWEEN THE FUNCTIONS.
836. A simple function of an angle, being a ratio, may be expressed as a fraction. Let a be the perpendicular, b the base, and e the hy potenuse of the triangle used in defining the functions of an angle. In order to include all possible angles, let it be understood that a and b are either positive or nega tive. Then, a sin. A = a tan. A =
T5
6 0
sec. A =
b c' b cot. A =: a' e cosec. A = a' COS.
0
V
A
837. Corollary—The sine and cosecant of an angle are reciprocals ; also, the tangent and cotangent are re ciprocals; and the cosine and secant are reciprocals. That is, sin. A cosec. A = 1, tan. A cot. A = 1, cos. A sec. A = 1. A practical result of these equations is, that the cose cant, secant, and cotangent are less used than the other simple functions. For, if one has occasion to multiply or divide by the cosecant, the object is accomplished by dividing or multiplying by the sine; and similarly of the secant and cotangent. 838. By means of the Pythagorean Theorem and the fractions just stated, any function of an angle may be expressed in terms of any other function of the same angle. For example, let it be required to find the value Tria—25.
200
PLANE TRIGONOMETRY.
of each of the other simple functions in terms of the sine of the same angle. Beginning with the equation, a + b' ' = c\ and dividing both members by c\ a?
bl = 1.
That is, the sum of the squares of the sine and cosine of any angle is equal to unity. Hence, sin. A = Vl — cos.2 A;
also, cos. A = >/l — sin.2 A.
The exponent is given to sin. and to cos., because it is the function that is involved and not the angle. 839. The sine of an angle is equal to the product of the tangent by the cosine. For, a
a
b
- = t X-c b o
That is,
sin. A = tan. A cos. A.
Hence,
tan. A =
sin. A cos. A
sin. A vl — sin.* A
Since the tangent and cotangent are reciprocals, cos. A VI — sin.2 A cot. A = — r- = ; 7 sin. A sin. A Since the secant and cosine are reciprocals,
.
.
r
sin. A = \ 1 '
T~
Vsec.'A — 1
—r- = sec.- A
sec. A
FUNCTIONS OF ANGLES.
291
EXERCISES. 840.—1. By similar methods, find expressions for the cosine and tangent in terms of each of the other functions. 2. Render each formula into ordinary language. exercise should be continued throughout the work.
This valuable
3. Given 2 sin. A = tan. A, to find A. Ans. 0°, 60°, 120°, 180°, 240°, or 300°. 4. If sin. A = f, what is the value of cos. A? 5. If sin. A = J, what is the value of tan. A? 6. Demonstrate sin. 18° = K/5 — lY Notice that 18° is the angle made by the apothegm and radius of a regular decagon.
FUNCTIONS OF (90° ± A).
841. Theorem— The cosine of an angle is the sine of its complement. That is, cos. A = sin. (90° — A). For, in the right angled triangle of the definitions, the acute angles are complementary; and (818) cos. A = - = sin. B. c This demonstration appears to apply only to the case when the angle A is acute, when the revolving line is in the first quarter. The student may construct a figure for each of the other quarters, and show that the proposition is universally true. 842. Corollary.—Similarly, the cotangent and cose cant are respectively the tangent and secant of the com plementary angle. It is from this property that these functions (cos., cot., cosec.) derive their names. '
292
PLANK TRIGONOMKTRY.
S43. Theorem— Sin. (90° + A) = cos. A, and cos. (90° + A) = — «'n. .4. It has been proved that sin. A = sin. (180° — A), what ever is the value of A (829). It is therefore true for (90°+ A). Substituting, we have sin. (90°+A) =sin. (180°-90°-A) =sin. (90°-A)=cos. A. Again, since cos. A = sin. (90° — A) for all values of A, then for A we may substitute 90° + A. Hence, cos. (90°+A)=sin. (90°- 90°— A)=sin. (— A)=— sin. A. EXERCISES. 844.—1. Find the value of tan. (90° + A). 2. Illustrate with diagrams all the principles of this section. 3. Given sin. A=cos. 2A, to find the value of A. 1/(10 + 21/5) 4. Demonstrate tan. 72° = l/5 — 1. FUNCTIONS OF TWO ANGLES.
843. Let the angle DCF be designated by A and the angle FCG by B; then DCG is A+B. From any point G in the line CG let fall GH and GF respectively perpen dicular to CD and CF. From F let fall FD and FK respect ively perpendicular to CD and GH. Then, the angle FGK is equal to FCD, or A (140). Now DF=CFXsin. A, and CF=CGXcos. B; hence, DF=CGXsin. A cos. B.
FUNCTIONS OF ANGLES.
293
Likewise GK = GFXcos. A, and GF=CGXsin. B; hence, GK=OGXcos. A sin. B. Also, GK-{-DF=GK+KH=GH=CGXsin. (A+B); therefore, sin. (A-(-B)=sin. A cos. B+cos. A sin. B,
(i.)
In the above figure the given angles and their sum are acute. The same demonstration will apply for any given angles, constructing the figure exactly according to the directions, producing when necessary the lines on which the perpendiculars fall. The cosine of the sum of two angles may be found in terms of the sine and cosine of the angles, by the above diagram and similar reasoning. Or, it may be derived from the formula just demonstrated, as follows : Regarding 90° + A. as one angle, we have sin.(90°+A+B)=sin.(90o+A)cos.B+cos.(90°+A)sin.B. Substituting for the functions of 90°+ A and 90°+ A+B, their equivalents (843), cos. (A + B) = cos. A cos. B — sin. A sin. B,
(ii.)
In these two formulas for the sine and cosine of the sum of two angles, if — B is substituted for B, then the sign of sin. B is changed, but not of cos. B (825). Thus, sin. (A — B) = sin. A cos. B — cos. A sin. B, (in.) cos. (A — B) = cos. A cos. B-f- sin. A sin. B, (iv.) . These two formulas may be demonstrated independ ently of the former, in the same manner as the formula for the sine of the sum.
294
PLANE TRIGONOMETRY.
The tangent of the sum of two angles is found thus : .. , „.
sin.(A-)-B) cos. (A+B)
sin. A cos. B -|- cos. A sin.B cos. A cos. B — sin. A sin. B '
Dividing both terms of the fraction by cos. A cos. B,
o- -i i Similarly, J
/ > i t.x tan. A -|- tan. B tan. (A v + B)' = 1 — tan.X A tan. B ,
.
, . (v.) v '
. /a t>\ tim- -A- — tan. B tan. (A v — B) =-— 1+ tan. A tan. -, B
.
/ \ (vi.) v '
EXERCISES. 846.—1. Demonstrate formula n in the same manner as for mula i, and both of them for those cases where the angles are not acute. Observe in what quarters the sine and cosine are negative. 2. Express each formula in ordinary language ; for example : the sine of the sum of two angles is equal to the sum of the products of the sine of each by the cosine of the other. 3. Demonstrate cos. 12° - \{,JZQ + 6 v/5 + /5 — 1.)
FUNCTIONS OP MULTIPLES AND PARTS OF ANGLES. 847. In the formulas of the sine, the cosine, and the tangent of the sum of two angles, suppose B = A ; then,
sin. 2A = 2 sin. A cos. A, .
.
'
'
(I-)
cos. 2A = cos.2 A — sin." A, .
'
'
(II-)
2 tan. A tan. 2A = z —r-r, 1— tan.3 A
.
.
(in.)
.
.
By substituting (n — 1)A for B in the original formulas, sin. nA, cos. wA, and tan. nA may be expressed in func tions of A and of (n — 1)A. Thus, when the functions of
FUNCTIONS OF ANGLES.
295
A are known, the functions of 2A, 3A, etc., may be cal culated. Since cos.2 A '\- sin,2 A = 1 (838), we have cos. 2A = 1 — 2 sin,2 A ;
also, cos. 2A = 2 cos.2 A — 1.
These formulas being true for all angles, £A may be substituted for A. Then, transposing, 2 sin.2 JA = 1 — cos. A,
and 2 cos.2 JA = 1 + cos. A.
Therefore, sin. }2A= .J I (1 — cos. A), cos. >A = Vi(,l + cos. A), ....
(iv.)
By these formulas, from the cosine of an angle, may be calculated the sine and cosine of its half, fourth, eighth, etc. EXERCISES. see. A — 1 848.—1. Demonstrate tan. .=. = 2 tan. A 2. What is the value of sin. 15°; cos. 3°; sin. 1° ZV1
FORMULAS FOR LOGARITHMIC USE.
819. In order to render a formula fit for logarithmic calculation, products and quotients must be substituted for sums and differences. This may frequently be done by means of the formulas which follow. The formulas for the sine and cosine of (A ± B) be come, by adding the third to the first, subtracting the third from the first, adding the second to the fourth, and subtracting the second from the fourth (845),
296
PLANE TRIGONOMETRY.
sin. (A + B) + sin. (A — B) = 2 sin. A cos. B,
(i.)
sin. (A + B) — sin. (A — B) = 2 cos. A sin. B,
(n.(
cos. (A + B) + cos. (A — B) == 2 cos. A cos. B, (in.) cos. (A — B) — cos. (A + B) = 2 sin. A sin. B, (iv.) In the above, let A + B = C, and A — B = D ; whence, A = 1(0 + D), and B = 1(0 -D). Then, sin. C-f sin. D = 2 sin. 1(0 + D) cos. 1(0 — D),
(v.)
sin. C — sin. D =2 cos. 1(0+ D) sin. 1(0 — D), (vi.) cos. C + cos. D = 2 cos. 1(0 + D) cos. 1(0— D), (vn.) cos. D — cos. 0 = 2 sin. 1(0 + D) sin. 1(0 — D), (vm.) By dividing v by VI, sin.C+sin.D sin. 0—sin. D
2K '
'
.(C-D),*"' *jg±g> 2K ' tan. 1(0—D)
Hence, sin. C+sin.D : sin. C—sin. D : : tan. KC+D) : tan. 1(C—D).
(ix.)
EXERCISES. 850.—1. Demonstrate sin. 5A = 5sin. A — 20sin.' A+16sin.5A. 2. Demonstrate sin. (A-fB) sin. (A — J5) = sin.2A — sin.2B.
TKIGONOMETRICAL TABLES. 851. By the application of algebra to the geometrical principles used in the construction of regular polygons, the student has found that the sine of 30° is 1, and the sine of 18° is |(,/o — 1). From these may be found the
TRIGONOMETRICAL TABLES.
297
cosines of these angles ; then (847, iv) the sine and co sine of 15°, and then the sine of 3° (845, in). The sine of 1° may be found as follows : sin. 3A = sin. (A4-2A) = sin. Acos.2A-f- cos. A sin. 2A. Substituting the values of cos. 2A and sin. 2A (847), sin. 3A= 3 cos.2 A sin. A — sin.3 A. Hence (838), sin. 3A = 3 sin. A — 4 sin.3 A. Put 1° for A ; then, knowing the value of sin. 3°, and representing the unknown sin. 1° by x, sin. 3°
Only one of the roots of this equation is less than sin. 3°. It must be sin. 1°, and may be calculated by alge braic methods to any required degree of approximation. Similarly, an equation of the fifth degree, may be formed from the value of sin. 5A; and by its means from the known sin. 1° may be found sin. 12'. Thus, by suc cessive steps, the functions of 1' and of 1" may be found to any required degree of accuracy. Having the sine and cosine of these small angles, the functions of their multiples may be calculated (847). This method, however, is tedious and is not used in practice. It serves -to show the possibility of calculating these func tions by elementary algebra and geometry. The higher analysis teaches briefer methods. These numerical functions are called the natural sines, tangents, etc., to distinguish them from the logarithmic functions which will be defined presently.
298
PLANE TRIGONOMETRY.
852. The Table of Natural Sines and Tangents gives these functions to six places of figures for every 10' from 0 to 90°. It also serves as a table of cosines and cotangents. If the sine or tangent of some intermediate angle is required, it may be found by taking a proportional part of the difference, with as much accuracy as the functions given in the table, except when the angle is nearly a right angle. For example, to find the sine 34° 23' 30", the table gives the sine of 34° 20'=. 564007. Since 3' 30" is .35 of 10', multiply 2399, the difference between this sine and that of 34° 30', by .35, and add the product to the given sine; the sum .564847 is the natural sine of 30° 23' 30". At the beginning of this table, the functions vary with almost perfect uniformity, and in proportion to the angle. Thus, the sine and the tangent of 100' differ only by one' millionth from one hundred times the sine or the tangent of 1'. At the close of the table, the tangent varies rap idly and the sine varies slowly, and both irregularly. Therefore, for the intermediate angles (those not given in the table), the last lines are less to be relied upon than the first. The tangent of a large angle may be found with greater accuracy by finding the cotangent of the same angle and taking its reciprocal (837). LOGARITHMIC FUNCTIONS. 853o Before proceeding to the study of this article, the student should understand the use of the tables of logarithms of numbers. A logarithmic sine, tangent, etc., means the logarithm of the sine, of the tangent, etc. In the tables, the char
TRIGONOMETRICAL TABLES.
299
acteriptic of every logarithmic trigonometric function is increased by 10. For example, sin. 30° = \ ; log. J = 1.698970, which is the true logarithm of the sine of 30°; but the tabular logarithmic sine of 30° is 9.698970. The object of this arrangement is simply to avoid the use of negative characteristics, as would be the case with all the sines and cosines and half of the tangents and co tangents. Therefore, whenever in a calculation, a tabu lar logarithmic function is added, 10 must be subtracted from the result to find the true logarithm; and whenever a tabular logarithmic function is subtracted, 10 must be added to the result. If, however, in place of subtracting a logarithmic function, the arithmetical complement, is added, the result does not need correction, the 10 to be added for one reason, balancing that to be subtracted for the other. 854. The table gives the logarithmic sine, tangent, cosine, and cotangent for every 1' from 0 to 90°. The degrees are marked at the top of each page and the min utes in the left hand column descending, for the sines and tangents ; and the degrees at the bottom of each page and 'the minutes in the right hand column ascending, for the cosines and cotangents. The columns marked P. P. 1" contain the proportional part for one second,. to facilitate the proper addition or subtraction. In using the proportional part for the cosine and co tangent, remember that these functions decrease when the angle increases. 855. To find the logarithmic sine, etc., of a given angle. If the angle is expressed in degrees only, or in degrees and minutes, take the corresponding sine or other function directly from Table IV. If the angle is expressed in degrees, minutes, and sec
300
PLANE TRIGONOMETRY.
onds, then take the logarithmic function corresponding to the given degrees and minutes ; multiply the propor tional part for 1" by the number of seconds ; and add the product to the tabular function, for the sine and tangent, and subtract it for the cosine and cotangent. For example, to find the tabular logarithmic sine of 40° 13' 14" tab. log. sin. 40° 13' = 9.810017, P.P. 1"=2.5, ... 2.5X14 .. = 35, Therefore, .
. tab. log. sin. 40° 13' 14" = 9.810052.
To find the tabular logarithmic cosine of 75° 40' 21",
P. P. 1" = 8.23, . Therefore, .
tab. log. cos. 75° 40' = 9.393685, . 8.23 X 21 . . = 173,
. tab. log. cos. 75° 40' 21" = 9.393512.
This method of using the proportional part given in the tables, gives results that are true to six decimal places, except for the sines, tangents, and cotangents of angles less than three degrees, and for the cosines and cotangents of angles greater than eighty-seven degrees. The sines and tangents of small angles increase almost uniformly. Therefore, the logarithmic sine and tangent of one of these small angles may be found nearly, by adding to the logarithmic sine or tangent of one second the logarithm of the number of seconds in the given angle. This result is subject to the correction in Table V. The cosines and cotangents of large angles are found in the same way, since they are the sines and tangents of the small angles (841 and 842.) Since the tangent and cotangent of an angle are recip rocals, the rule just given for finding the tangents of small
TRIGONOMETRICAL TABLES.
301
angles, may be applied to the cotangents also. For the correction, see Table V. For example, to find the logarithmic sine of 45' 23" = 2723", add to . 4.685575, log. 2723, 3.435048; Subtract as in Table V,
8.120623. 13,
tab. log. sin. 45' 23" = 8.120610. 836. To find the angle when its logarithmic sine, tan gent, cosine, or cotangent is given. If the given function is found in Table IV, take the corresponding angle, expressed in degrees, or in degrees and minutes. If the given function is not in the table, take that which is next less ; subtract it from the given function ; divide the remainder by the proportional part for 1" ; the quotient is the number of seconds, to be added, in case of sine or tangent, to the angle corresponding to the tabular function used; and to be subtracted in case of the cosine or cotangent. For example, to find the angle whose tabular logarith mic tangent is 10.456789, tab. log. tan. 70° 44' = 10.456501, P.P.I" =6.75,
....
288 -=- 6.75 = 43.
Therefore, 70° 44' 43" is the angle sought. To find the angle whose tabular logarithmic cotan gent is . . 9.876543, tab. log. cot. 53° 3' = 9.876326, P.P.I" =4.38,
....
217 h- 4.38 =50.
r
302
PLANE TRIGONOMETRY.
Therefore, 53° 2' 10" is the angle whose logarithmic cotangent is 9.876543. When great accuracy is desired and the angle to be found is less than three degrees or greater than eighty' seven, the corrections in Table V may be used, first using Table IV to determine the angle approximately.
EIGHT ANGLED TRIANGLES. 857. The principles have now been established, by •which, whenever certain parts of a triangle are known, the remaining parts can be calculated. Since the trig onometrical functions are the ratios between the sides of a right angled triangle, the problems concerning such triangles need no other demonstration than is contained in the definitions. The sum of the acute angles being 90°, when one is known, the other is found by subtraction. 858. Problem.— Given the hypotenuse and one angle, to find the other parts. The product of the hypotenuse by the sine of either acute angle, is the side opposite that angle. The prod uct of the hypotenuse by the cosine of either acute angle, is the side adjacent to that angle. 839. Problem.— Given one leg and one angle, to find the other parts. The quotient of one leg divided by the sine of the opposite angle is the hypotenuse. The product of one leg by the tangent of the adjacent angle is the other leg. 860. Problem.— Given one leg and the hypotenuse, to find the other parts. The quotient of one leg divided by the hypotenuse is
RIGHT ANGLED TRIANGLES.
303
, the sine of the angle opposite that leg, and the cosine of the adjacent angle. The other leg may then be found by the previous problem. 861. Problem.— Given the two legs to find the other parts. The quotient of one leg divided by the other is the tangent of the angle opposite the dividend. The hypot enuse may then be found by the second problem. When, as in the last two problems, two sides are given, the third may be found by the Pythagorean Theorem. 862. Only the sine, cosine, and tangent are used in the above solutions. The student may easily propose solutions by means of the other functions. Sinc*e none of the above problems requires addition or subtraction, the operations may all be performed by logarithms. For example : A railroad track, 463 feet 3 inches long, has a uniform grade of 3°. How high is one end above the other ? Here the hypotenuse and one acute angle are given, to find the opposite side. log. 463.25 = 2.665815, tab. log. sin. 3° = 8.718800, Omitting the tabular 10, the sum 1.384615 is the logarithm of 24.2446. Hence, the ascent is nearly 24 feet 3 inches. EXERCISES. 863.—1. Construct a figure to illustrate the above, and each of the following. 2. The hypotenuse is 4321, one angle is 25° 3CK. Find the other angle and the two legs. Solve this both with and without loga rithms.
304
PLANE TRIGONOMETRY.
3. Two posts on the bank of a river are one hundred feet apart ; the line joining them is perpendicular to the line from the first post to a certain point on the opposite bank; and the same line makes an angle of 78° 52' with the line from the second post to the same point on the opposite bank. How wide is the river? 4. The instrument used in measuring the angle in the above statement is imperfect, the observations being liable to an error of V. To what extent does that affect the calculated result ? 5. The hypotenuse being 7093, and one leg 2308.5, find the other leg and the angles. 6. An observer standing 60 feet from a wall measures its angu lar height, and finds it to be 15° 37', his eye being 5 feet from the ground, which is leveL How high is the wall? 7. How much would the last result be affected by an error of 5" in observing the angle ? 8. How much if there had also been an error of 2 inches in measuring the horizontal line ? 9. Find the apothegm and radius of a regular polygon of 7 sides, one side being 10 inches. 10. Find the area of a regular dodecagon, the side being 2 feet. 11. The legs being 42.9 and 47.52, find the angles and the hy potenuse. 12. A tower 103 feet high throws a shadow 51.5 feet long upon the level plane ; what is the angle of elevation of the sun ? 13. How much would the last result be affected by an error of 3 inches in the given height or length ?
SOLUTION OF PLANE TRIANGLES. 864. the sum may be angle is
One angle of a triangle being the supplement of of the other two, when two are known the third found by subtraction. Also, the sine of either equal to the sine of the sum of the other two.
The letters a, b, and c represent the sides of a triangle respectively opposite the angles A, B, and C.
V
PLANE TRIANGLES.
305
865. Theorem The square of one side of a triangle is equal to the sum of the squares of the other two sides, less twice the product of those sides by the cosine of their included angle. For, in the first figure (411), o»=J'+cs — 26'AD, and in the second figure (412),
a2=b'+c2+ 2b'AD; but in the first case, AD = cos. A X AB = c cos. A ; and in the second, AD = — cos. A X AB = — c cos. A. Substituting these values of AD in their respective equations, both become a2 = 6s + c5 — 26e cos. A. By similar reasoning, it may be shown that b3 = a? + c2 — 2ac cos. B, and
c2 = a? -\- 6J — 2ab cos. C.
These three equations suffice for the solution of all problems on plane triangles, but they are not suitable for logarithmic calculations. The following are not liable to this objection: Tris.—26. •-
303
PLANE TRIGONOMETRY.
866. Theorem—Expressing the sum of the sides of any triangle by p, then sin. — = -\y^2 »
-. be
For, by the formula just demonstrated, b= -U c> — acos. A = —— 2be Hence (847, iv),
sin. £ = . Kr^sTA) = J (180° — A + B — C) = sin. (B — £E), etc.
Therefore,
cos. % , Jg°^^ffig°^jg) . 2 * sin. B sin. C
Similarly, from the formula for the cosine of half the angle, sin - = J"n-iEsin.(A — jE) '2 * sin. B sin. C
SPHERICAL TRIGONOMETRY.
320
Hence,
tan. £ = I s.n. -E sin. (A- »E) 2 \sin.(B — «E)8iii.(0 — JE)
Since E must be less than 360° (771), sin. JE is pos itive ; and since sin. Ja is a real quantity, sin. (A — JE) must be positive. Therefore, any angle of a spherical triangle is greater than half the spherical excess.
OPPOSITE SIDES AND ANGLES. 882. Theorem— The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. Let ABC be the spherical triangle, and O the center of the sphere. From any point P in OA, let PD fall perpendicular to the plane BOC; make DE, DF per pendicular respectively to BO, OC ; and join PE, PF, and OD. The plane PED is per pendicular to the plane BOC (556). Therefore, OE is perpendicular to the plane PED, the angle PED is the same as the angle B (759), and PEO is a right angle. Therefore, PE = OP ' sin. POE = OP ' sin. c; and PD = PE ' sin. B = OP ' sin. e sin. B. Similarly, therefore, sin. B sin. C
PD = OP . sin. b sin. C ; OP " sin. c sin. B = OP . sin. 6 sin. C. sin. b or sin. c
sin. B : sin. C : : sin. 6 : sin. c.
SPHERICAL ARCS AND ANGLES.
321
The figure supposes b, c, B, and C to be each less than 90°. When this is not the case, the figure and the dem onstration are slightly modified. For example, when B is greater than a right angle, the point D falls beyond BO, and PED becomes the supplement of B, having the same sine.
FOUR CONTIGUOUS PARTS. 883. Theorem— The product of the cotangent of one side by the sine of another, is equal to the product of the cosine of the included angle by the cosine of the second side, plus the product of the sine of the included angle by the cotangent of the angle opposite the first side. We have (878 and 882), . cos. a = cos. b cos. c'\- sin. b sin. c cos. A, cos. c = cos. a cos. b -\' sin. a sin. b cos. C, sin. c =
sin. a sin. C sin. A
Eliminate c by substituting these values of cos. c and sin. c in the first equation, . , , . . , , , sin.asin.Jcos.Asin.C cos.a = (cos.a cos.o 4' sin.a sin.6 cos.U)' cos.o '\ . A; ; v sin.
transposing and reducing, since 1 — cos.2 b = sin.2 b, cos.a sin.'J =sin.a sin.6 cos.b cos.C-j-sin.a sin.J cot.A sin.C ; dividing by sin. a sin. b, cot. a sin. b = cos. b cos. C 'f. cot. A sin. C.
322
SPHERICAL TRIGONOMETRY.
The demonstration being general, may be applied to other angles and sides, making these five additional formulas : cot. b cot. b cot. c cot. c cot. a
sin. a = cos. a sin. c = cos. c sin. b = cos. b sin. a = co3. a sin. c = cos. c
cos. C + cot. B cos. A + cot. B cos. A+ cot. C cos. B -4- cot. C cos. B + cot. A
sin. C, sin. A, sin. A, sin. B, sin. B.
FORMULAS OF DBLAMBRB. 884. Putting ^A and £B for A and B respectively, in formula I, Art. 845, sin. J(A -j- B) = sin. JA cos. JB + cos. JA sin. £B. Substitute the values of the factors of the second mem ber, as found in Art. 880, sin.A+B = sin-(aP-a)+sin-(zJ?~6) Jsin-hP sin- (hP—c) . 2 sin. c * sin. a sin. b ' but, sin.(^_a)+sin.(J/?_&) =sin.( ^—g- )+sin-( \ + -y )» (849, i), .... =2 sin. £c cos. J(a— 6), and (847, i),
sin. c=2 sin. \c cos. £e.
Substituting these values, also cos. £C for the radi cal (880), . A 4B = cos. \{a — b)- cos. iC, ,~ sin ~— =± Jc 2 cos. J or,
sin. J-(A + B) _ cos. J(a — b) cos. JC cos. \e
SPHERICAL ARCS AND ANGLES.
323
Similarly, by beginning with formulas n, in, and iv of Art. 845, we find, sin. \,(A — B) cos. JC
sin. \{a — b) sin. \c
cos. J(A + B) _ cos. \(a + b) sin. ^C cos. \c cos.£(A — B) sin. \G
sin. \(a + b) sin. \c
These four formulas of Delambre were published by him in 1807. NAPIER'S ANALOGIES. 885. Divide the first of the formulas of Delambre by the third, the second by the fourth, then the fourth by the third, and the second by the first, and these results are obtained: tan. £(A + B) _ cos. \(a — b) cot. \G cos. \{a-\- b) tan. £(A — B) _ sin. J(a — b) cot. £C sin. £(a + &)' tan. \{a + b) _ cos. £(A — B) tan. \c cos. £(A + B) tan.i. \{a— b) _ sin. J (A — B) tan. \c ~~ sin. £(A -f- B) These formulas may be stated as proportions, and are called Napier's Analogies, from their inventor, analogy being formerly used as synonymous with proportion.
324
SPHERICAL TRIGONOMETRY.
886. In the first of the above equations, cos. \{a — o) and cot. JC are necessarily positive; hence, tan. J(A-j-B) and cos. J(«+ b) are of the same sign; thus, J(A-f-B) and \{a-\-b) are either both less or both greater than ninety degrees. In the second of the above equations, sin. \(a -j- b) and cot. \G are positive; hence, tan. J(A — B) and sin. \(a — b) have the same sign ; thus, J(A — B) and %(a — b) are either both positive, both negative, or both zero. Therefore, in any spherical triangle, the greater angle is opposite the greater side, and conversely.
EXERCISES. 887.— 1. Find the formula that results from applying the prin ciple of polar triangles to the first of Napier's Analogies ; also, to the first formula of Art. 883. 2. State a theorem applying the principle of Art. 878 to triedrals. 3. Show, from the third of Napier's Analogies, that the sum of any two sides of a spherical triangle is greater than the third.
RIGHT ANGLED SPHERICAL TRIANGLES. 888. The foregoing formulas may be applied to right angled triangles by supposing one of the angles to be right, for example A. In this manner we have: Art. Art. Art. " Art. "
878, 879, 882, " 883, "
1st formula, 1st formula,
1st formula, 6th formula,
cos. a = cos. b cos. c, . (i.) cos. a = cot. B cot. C, . (n.) sin. 6 = sin. a sin. B \ , '. sin. a = sin. a sin. C > tan. b = tan. a cos. C \ , » tan. a = tan. a cos. B >
RIGHT ANGLED TRIANGLES.
Art. " Art. "
883, " 879, "
3rd formula, 4th formula, 2nd formula, 3rd formula,
tan. b = sin. e tan. B tan. c = sin. b tan. C cos. B = sin. C cos. b cos. C = sin. B cos. c
325
(VI.)
In deducing n, IV, and v, the formulas are reduced somewhat by divisions. These are sufficient for the so lution of every case. These principles may be stated as follows : cos. hyp. = product of cosines of sides, cos. hyp. = product of cotangents of angles, sine side = sine opposite angle X sine hyp., tan. side = tan. hyp. X cosine included angle, tan. side = tan. opposite angle X sine other side, cos. angle = cos. opposite side X sine other angle. 889. Since the cosine of the hypotenuse has the same sign as the product of the cosines of the other two sides, it follows either that two of these three cosines are neg ative, or none. Therefore, in a right angled spherical triangle, either all the sides are less than quadrants, or two are greater and one is less. It appears also (v) that the tangent of an oblique an gle and of its opposite side have the same sign. There fore, these two parts of the triangle are either both less or both greater than 90°. This is expressed by saying they are of the same species.
NAPIER'S RULE OP CIRCULAR PARTS. 890. A mnemonic rule for the formulas of right angled spherical triangles was invented by Napier, and published with his description of logarithms in 1614.
326
SPHERICAL TRIGONOMETRY.
The right angle being omitted, five parts of the triangle remain. The two sides which include the right angle, the complements of the other angles, and the complement of the hypotenuse are called the circular parts of the triangle. These are supposed to be arranged around a circle in the order they occur in the triangle. Any one of the five circular parts may be called the middle part, then the two next to it are the adjacent parts, and the remaining two are the opposite parts. Napier's rule is : The sine of the middle part is equal to the product of the tangents of the adjacent parts, also to the product of the cosines of the opposite parts. The words sine and middle having their first vowel the same, also the words tangent and adjacent, also the words cosine and opposite, renders this rule very easy to remember. For example, if the complement of the hy potenuse be the middle part, then the complements of the angles are the adjacent parts, and the sides are the op posite parts ; this gives formulas I and II.
SOLUTION OF EIGHT ANGLED TRIANGLES. 891. Problem— Given the hypotenuse and an oblique angle, to find the other angle and the sides. Find the other oblique angle by formula n, the side opposite the given angle by in, and the adjacent side tby iv. For example, given the hypotenuse 64° 17' 35", and an angle 70°, to find the opposite side, tab. log. sin. 70° . . = 9.972986, tab. log. sin. 64° 17' 35" = 9.954737, tab. log. sin. 57° 51' 11" = 9.927723.
"
RIGHT ANGLED TRIANGLES.
327
Therefore, the required side is 57° 51' 11". It is known to be acute because its opposite angle is acute (889). 892. Problem.— Given one side and the adjacent ob lique angle, to find the other sides and angle. Find the hypotenuse by IV, the other side by v, and the other angle by VI. 893. Problem—Given the two sides, to find the hy potenuse and angles. Find the hypotenuse by I, and the angles by V. 894. Problem— Given the hypotenuse and one side, to find the angles and the other side. Find the included angle by IV, the other side by I, and the remaining angle by in. 893. Problem—Given the two oblique angles, to find the three sides. Find the hypotenuse by II, and the other sides by vr. In the above solutions there is no ambiguous case. Whenever a part is found by means of its sine, its spe cies is determined by the principle of Art. 889. In the 1st and 4th problems, if the given parts are both of 90°, the triangle is indeterminate. The student may show why. 896. Problem.—Given a side and its opposite angle, to find the other sides and angle. Find the hypotenuse by in, the other side by V, and the other angle by vi.
r
328
SPHERICAL TRIGONOMETRY.
Here the triangle is ambiguous, as all the parts are found by their sines. Sup pose BAC to be a triangle right angled at A, and that C and c are the given parts. Produce CB and CA to meet in C. Then the tri angle CAB has the same conditions as the given triangle, for it has a right angle at A, the given side BA, and C = C, the given angle. 897. The solution of an oblique triangle may be made in some cases to depend immediately upon the solution of a right angled triangle. If a triangle has one of its sides a quadrant, then its polar triangle has its corresponding angle a right angle. The polar triangle can be solved by the preceding methods, and thus the elements of the prim itive triangle become known. If a triangle is isosceles, an arc from the vertex to the middle point of the base divides it into two equal right angled triangles, by the solution of which the elements of the isosceles triangle are found. If a triangle has two sides supplementary, as o and c, the sides a and c may be produced to B', making the isosceles triangle B'. B'AC, which may be solved as above, giving the elements of the orig inal triangle. If a triangle has two of its angles supplementary, then its polar triangle has two of its sides supplemental This may be studied in the manner just stated, and thus the parts of the primitive triangle become known.
SPHERICAL TRIANGLES.
329
EXERCISES. 898.—1. Show that in a right angled spherical triangle, a side is less than its opposite angle when both are acute, and greater when both are obtuse. 2. The sides are 57° 51' 8" and 35° 23' 30"; find the hypotenuse and the angles. 3. The hypotenuse is 71° 39' 37" and one angle 79° 56' 4"; find the Sides and the other angle. 4. One side is 140°, the opposite angle is 138° 14' 14"; find the remaining parts. 5. Show that if the hypotenuse is 90°, one of the sides must be 90°, and conversely. 6. The sides are 90°, 76° 49' 55", 41° 45' 46"; find the angles. 7. A lateral edge of a pyramid whose base is a square, makes angles of 60° and 65° respectively with the two conterminous sides of the base ; find the diedral angle of that edge.
SOLUTION OF SPHERICAL TRIANGLES. 899. Problem,—Given the sides, to find the angles. Either of the angles may be found by the formulas of Art. 880. When all the angles are required, the formula for the tangent is to be preferred. 900. Problem—Given the angles, to find the sides. Either of the sides may be found by the formulas of Art. 881. 901. Problem.— Given two sides and the included angle, to find the other angles and side. The half sum of the other angles may be found by the first of Napier's Analogies, and the half difference by the Trig.—28.
330
SPHER1CAL TRIGONOMETRY.
second; and hence, the angles themselves. Then the third side may be found by the proportion of Art. 882. If the ambiguity attendant upon the use of the sine is not removed by observing that the greater side of a tri angle is always opposite the greater angle (886), then the third side may be found by Art. 881, or by the third or fourth of Napier's Analogies, or by one of the formu las of Delarabre. For example, given the side a = 76° 35' 36", b = 50° 10' 30", and the angle C = 34° 15' 3". By the 1st analogy, tan. « (A + B = cot. i C ?; ,.( ' *v 1 ' * cos. )2{a-\- b) tab. log. cot. iC . . . = tab. log. cos. J(a — I) . = a. c. tab. log. cos. \{a +b)= tab. log. tan. £(A + B) = . ' . i (A + B) =
10.511272 9.988355 0.348717 10.848344 81° 55' 47"
By the 2nd analogy, „sin. %(a — b) tan. JV J(A — B)' = cot. iC - .- —f, , b) ,;' J sin. i(«+ tab. log. cot. JC . . . = tab. log. sin. J(a — b) . = a. c. tab. log. sin. \ (a + b) = tab. log. tan. J(A— B) = .-.J(A— B)
10.511272 9.358899 0.048648 9.918819
=39° 40' 33"
Hence,
A = 121° 36' 20",
and
B = 42° 15' 14".
SPHERICAL TR1ANGLES.
331
Since the remaining side must be less than either of the given sides, it may be found by the proportion, sin. A : sin. C : : sin. a : sin. c; or by the 4th analogy, as follows : , sin. i(A+B) tan. 2ie= tan. Ma -———. =jr 2V — b)'sin. }Z(A — B) tab. log. tan. tab. log. sin. a. c. tab. log. tab. log. tan.
J(a — b) . = £(A + B) . = sin. £(A — B) = \c . . . =
9.370544 9.995677 .194877 9.561098
.-. \c = 20° 0' 5", and c = 40° 0' 10". 90S. Problem.— Given one side and the adjacent angles, to find the other sides and angle. The half sum of the other sides may be found by the 3rd analogy, and the half difference by the 4th; and hence, the sides themselves. Then the third angle may be found by the proportion of Art. 882. If the ambiguity attendant upon the use of the sine is not removed by observing that the greater angle is op posite the greater side, then it may be found by Art. 880, or by the 1st or 2nd analogy, or by one of the formulas of Delambre. 903. Problem.— Given two sides and an angle opposite one of them, to find the oilier angles and side. The angle opposite the other given side may be found by Art. 882, and then the remaining angle and side from Napier's Analogies. Since the sine is used to find the first angle, there may be two solutions. The ambiguity i* sometimes removed
332
SPHERICAL TRIGONOMETRY.
by observing that the greater angle is opposite the greater side. When only one value of the angle found from its sine is consistent with this principle, there is but one solution. When both values of the angle thus found are consist ent with this principle, there are two solutions, that is, there are two distinct spherical triangles which have the given elements. When the angle A and the sides a and b are given, b being greater than a, if both values found for B are greater than A, then there are two triangles, ABC and AB'C, which have the given sides and angle. When the same parts are given, and b is less than a, if both values found for B are less than A, there are two solutions. In this case the given angle must have been obtuse, and in the former case it must have been acute. It may happen that neither value of the angle found from its sine is consistent with the principle stated. This shows that the given conditions are incompatible, and that the triangle is impossible. 904. Problem.— Given two angles and a side opposite one of them, to fend the other sides and angle. The side opposite the other given angle may be found by the proportion of Art. 882, and then the remaining angle and side from Napier's Analogies, as in the pre ceding solution. This case is precisely analogous to the last; it pre sents the same ambiguity, and the ambiguity is resolved in the same manner. *
SPHERICAL TRIANGLES.
333
EXERCISES. 905.—1. The sides are 60° 4' 54", 135° 49' 20", and 146° 37' 15"; find the angles. 2. Find the diedral angle of a regular tetraedron. 3. The sides are 105°, 90°, and 75° ; find the sines of the angles without the use of the tables. 4. The angles are 32° 26' 7", 36° 45' 28", and 130° 5' 23"; find the three sides. 5. Two sides are 70° and 80°, and the included angle 130°; find the remaining angles and side. 6. Two sides are 89° 16' 54" and 52° 39' 5", the angle opposite the former is 70° 39'; find the remaining parts. 7. Given the latitude of Paris 48° 50' 12", the latitude of New York 40° 17' 17", and the longitude of New York west of Paris 76° 20' 27", to find the distance between these points, along an arc of a great circle ; the earth being considered a sphere of a radius of 3956 miles. 8. How much would the last result be affected by an error of 2" in the given longitude ? in one of the given latitudes ?
334
TRIGONOMETRY.
CHAPTER XIV. LOGARITHMS. 906. Nearly all trigonometrical calculations are made by means of logarithms. To understand this chapter, the student must be acquainted with the algebraic theory of positive and negative exponents. He may refer to the algebra for an investigation of the principles and the methods of calculating tables. COMMON LOGARITHMS. 907. The Common Logarithm of a number is the exponent of that power of 10 which is equal to the num ber. Hence, The logarithm of 10 is 1, " " " 100 " 2, ". " " 1000 " 3, etc. Again,
the logarithm of 1 is 0, " " T\ or .1 " —1, u " T^ or .01 " -2, etc. CC
Numbers greater than unity have positive logarithms; numbers less than unity have negative logarithms. The powers of 10 have the positive integers for their log arithms, and the reciprocals of those powers have the
LOGARITHMS.
335
negative integers for their logarithms. No other num bers have integral logarithms. That part of a logarithm which is not integral is always expressed by decimals. CHARACTERISTIC. 908. The Characteristic of a logarithm is its in tegral part. The Mantissa of a logarithm is the decimal part. For convenience of calculation, it is an established rule that the mantissa of a logarithm is always positive, and only the characteristic of a negative logarithm is negative. To express this, the negative sign is written over the characteristic. Thus, log. .2 = 1.301030 = — 1 + .301030, log .08 = 2.903090 = — 2 + .903090. If any number is between 1 and 10, its logarithm is between 0 and 1 ; if a number is between 10 and 100, its logarithm is between 1 and 2, and so on ; the character istic of the logarithm is always one less than the number of integral places in the given number. If. the number is between 1 and .1, its logarithm is between 0 and — 1 ; hence, its characteristic is — 1. If the number is be tween .1 and .01, its logarithm is between — 1 and — 2; hence, its characteristic is — 2, and so on. The charac teristic of the logarithm of a fraction is numerically one more than the number of ciphers between the decimal point and the first significant figure of the given fraction written decimally. The student who has learned the theory of algebraic signs will perceive that the above rules are included in the following:
336
TRIGONOMETRY.
The characteristic of the logarithm denotes how many places the first significant figure of the number is to the left of the unit's place. The characteristics of logarithms are not given in the tables, but must be found as above. If this rule be taken conversely, it shows how to place the decimal point, when the number is found from its given logarithm.
TABLE OP LOGARITHMS. 909. Let c represent the characteristic and d the mantissa of any logarithm, and let N represent the number. By the definition,
10"+d
= N.
Multiplying by 10,
10c+1 + d= ION.
That is, if c -p' d is the logarithm of N, c -f- 1 + d is the logarithm of ION, the mantissa of each being d. Hence, multiplying a number by 10 does not change the mantissa of its logarithm, and it is the same when the number is multiplied or divided by any power of 10. In other words: if two numbers have the same significant figures, their logarithms have the same mantissas. For example, log.
5 = .698970,
log. 5000 = 3.698970, log. .005 = 3.69897C. The table in this work gives the mantissa of the log arithm of every number from 1000 to 11000. It follows "
\
LOGARITHMS.
337
that the mantissa of the logarithm of every number less than 11000 may be found in the table. The first three or four figures of each number are given in the left hand column (see Table); the next figure, at the head and at the foot of the several columns of mantissas. The mantissas in the column under 0 are given to six decimal places. The first and second deci mal figures of this column are understood to be repeated across the page, and for the spaces in the lines below. In the remaining columns, 1 to 9, only the last four of the six decimal figures of each mantissa are given. When the second decimal figure changes from 9 to 0, the remaining mantissas of the line are marked, to indi cate that, in these cases, the first two decimal figures are taken from the line below. The last column contains the difference between two successive mantissas, called the tabular difference. In all cases, the mantissa is only an approximation. The large tables of Adrien Vlacq give the logarithms to ten places of decimals of all numbers from 1 to 100000. The last figure is given within one-half a unit of its own order ; that is, if the first figure of the part not given is 5 or more, then the last figure given is increased by 1.
TO FIND THE LOGARITHM OP A GIVEN NUMBER.
910. If the significant figures of the number are the same as those of any number between 1000 and 11000, find the mantissa in the table and prefix the proper char acteristic. For example, to find the logarithm of 1245, find 124 in column N; in the same line and in column 5, find 5169 ; prefix .09 from column 0 ; then prefix the charac' TriS.—29.
338
TRIGONOMETRY.
teristic 3; and the logarithm of 1245 is 3.095169. ilarly, log. 124500 = 5.095169,
Sim
log. .0001245 = 4.095169. If the significant figures are those of a number less than 1000, annex ciphers to make a number between 1000 and 11000, and proceed as before. For example, the logarithm of 16 has the same mantissa as the log arithm of 1600, which is .204120. Therefore, the log arithm of 16 is. 1.204120. If the significant figures of the given number occupy more places than the numbers in the table, find the mantissa for the first four or five figures ; regard the remaining figures as a decimal fraction, and add to the mantissa already found the proportional part of the tab ular difference. For example, to find the logarithm of 3.1416. The mantissa of log. 3141 is . . . .497068, six-tenths of the tabular difference, 138, is 83, the characteristic being 0, 497151 is the logarithm sought. It is assumed that the mantissa of the logarithm of 3141.6 is the same as of 3141 increased by six-tenths of the difference between the mantissas of 3141 and 3142. To find the logarithm of 365.242. The mantissa of log. 3652 is = 562531, tab. diff. = 119 ; 119 X .42 = 50^ Therefore, log. 365.242 = 2.562581. All figures beyond the six places of decimals are re jected from the calculations, taking care that the last
LOGAHIiH.^S.
339
figure used shall be the nearest. Thus, six'tenths of 138 is nearer to 83 than to 82. When the tabular difference varies rapidly, as at the beginning of the table, there may be slight errors in its use, for the logarithms do not vary as the numbers. On this account, for all numbers between 10000 and 11000, it is better to use the last two pages of the Table instead of the first ten lines. If the given number has more than six significant figures, the seventh and subsequent figures rarely affect the first six places of the mantissa. Thus, the logarithm of 365.24224 is, to six places of decimals, the same as the logarithm of 365.242. TO FIND THE NUMBER, ITS LOGARITHM BEING KNOWN.
911. If the mantissa of the logarithm is the same as one in the table, take the corresponding number, and place the decimal point according to the rule of the characteristic. If the given mantissa is not in the table, find that mantissa in the table which is next less than the given one, and take the corresponding number. Annex to this, two figures of the quotient found by dividing by the tab ular difference, the excess of the given mantissa over the one used. Fix the decimal point by the rule of the characteristic. For example, to find the number whose logarithm is 4.016234. The next less mantissa is 016197, which has 10380 for its corresponding number (see page 364). The dif ference between it and the given mantissa is 37, and the tabular difference is 42.
340
TRIGONO.UE1RV.
Expressing the fraction \\ decimally, we have the fig ures 88 to be annexed to those already found, making 1038088, the significant figures of the required number. The characteristic 4 shows that the first significant figure should be in the fifth place. Therefore, 10380.88 is the number sought. As the logarithms are only approximations, so the number found can only be said to be true for six or seven places of figures. When a greater degree of ex actness is required, logarithms must be used of more than six decimal places. These may be calculated by means of Table II, and the formula given with it.
MULTIPLICATION AND DIVISION. 912. Let x and y represent the logarithms of M and N respectively. By the definition,
10* = M.
Similarly,
lO^N.
Multiplying the first by the second, 10*->" = MXN. Dividing the first by the second, 103:-2/= M-^N. That is, x-\-y is the logarithm of the product of M multiplied by N, and x — y is the logarithm of the quo tient of M divided by N. Hence, the following rules for multiplication and division by logarithms : To multiply, add the logarithm of the factors. sum is the logarithm of the product.
The
LOGARITHMS.
341
To divide, subtract the logarithm of the divisor from that of the dividend. The remainder is the logarithm of the quotient. For example, to find the product of 2, .000314, and 89.235. log. 2 = .301030, log. .000314 = 4.496930, log. 89.235 = 1.950535, The sum, 2.748495 is the logarithm of .0560396, which is the required product, true to six places of significant figures. Again, to divide 2 by .000314. log. 2 = log. .000314 = The remainder, of 6369.43, the quotient, true
.301030, 4.496930, 3.804100 is the logarithm to six places of figures.
Care must be exercised in the additions and subtrac tions, as the mantissas are all positive and the character istics sometimes negative. 913. It saves labor, instead of subtracting a log arithm, to add its arithmetical complement. The arith metical complement is the excess of 10 over the loga rithm. Let I represent any logarithm, then 10 — I is its complement. If 10 — I is added, the result is the same as when I is subtracted and 10 is added. There fore, Each time that an arithmetical complement is added, 10 must be subtracted from the result. When the log arithm is itself greater than 10, subtract it from 20 for the complement, and add 20 to the result.
r
342
TR1GONOMETRY.
If it were necessary to write out the logarithm in order to subtract it from 10, there would be little saving of labor, but the complement may be written at once, beginning at the left, and subtracting each figure of the given logarithm from 9, to the last significant figure which is to be subtracted from 10. This method is par ticularly useful when it is required to subtract several logarithms. n , , 3456 X 89123 I or example, to find the value of ?r„ —j^pi ' log. log. a. clog. a. clog.
3456 89123 9753 4321
=3.538574, = 4.949990, =6.010862, =6.364416,
log. 7.30873 = .863842. The sum is diminished by 20, for the complement twice used. Therefore, 7.30873 is the value of the given fraction.
INVOLUTION AND EVOLUTION. 914. Let y represent the logarithm of N. 102' = N. Raising both members to the x"i power,
Taking the a;th root of both members, 10* = yHS.
Then,
LOGARITHMS.
343
That is, xy is the logarithm of the xth power of N, and I is the logarithm of the xtb' root of N. Hence, these rules for involution and evolution by logarithms : To raise a number to a required power, multiply its logarithm by the exponent of the power. The product is the logarithm of the power. To extract any root of a number, divide its logarithm by the index of the required root. The quotient is the logarithm of the root. In making this division, if the characteristic of the given logarithm is negative, and is not exactly divisible by the divisor, then increase it by as many units as are needed to make it so divisible, prefixing the added num ber to the mantissa as an integer. The result is not affected by thus adding the same number to both the negative and positive parts of the logarithm. For example, to find the fourth root of J. log. .5 = 1.698970. This logarithm is equal to —4+3.698970, in which form it may be divided by 4. The quotient 1.924742 is the logarithm of .840896, which is the fourth root of \. 915. The positive or negative character of a factor is not considered in the use of logarithms. The proper sign can always be given to the result, according to the algebraic principles. In order that an arithmetical problem may be solved by logarithms, it should not contain any additions or subtractions. If, for example, it is required to find the sum of ^3 and j/2, each root may be found separately by the aid of logarithms, but the addition must be made afterward in the usual manner.
344
TRIGONOMETRY.
Mathematicians have given much attention to the con struction of such trigonometrical formulas as require only the operations of multiplication, division, involution, and evolution. For examples of this, see Articles 866 and seq. in Plane Triangles, and Articles 880 and seq. in Spherical Triangles. EXERCISES. 910.— 1. Calculate the value of these expressions: j/8932 X .045721i,
\/lbm -s- \ 10,
y\V X 14" -r- 1.256.
2. Find the area of a circle, the radius being 3 feet (500). 3. What is the diameter of a circle whose circumference is 314 feet 3 inches? 4. What is the area of a triangle whose sides are 417, 1493, and 1307 feet? (390.) 5. The diameter of the earth at the equator being 41850000 feet, what is the length in miles of one degree of longitude on the equator, there being 5280 feet in one mile? 6. The earth being a sphere with a radius of 20890000 ft., how many square miles are there in its surface? iiial exercises may be made upon the formulas of Art. 807.
TABLES OF
LOGARITHMS OF NUMBERS, From 1 to 11000,
LOGARITHMS OF 168 PRIME NUMBERS, To 15 places of Decimals,
NATURAL SINES AND TANGENTS, Fob every Ten minutes, and
LOGARITHMIC SINES AND TANGENTS, For every minute of the quadrant.
Num. 100, Log. 000.
1
2
3
101
000000 4321 8(i00 012837 7053
0434 4751 9026 3259 7451
0868 5181 9451 3680 7868
1301 5609 9876 4100 8284
1734 6038 .0300 4521 8700
2166 2598 6466 6894 .0724 .1147 4940 5360 9116 9532
3029 7321 .1570 5779 9947
3461 3891 7748 8174 .1993 .2415 6197 6616 .0361 .0775
432 428 424 420 410
103 108 107 108 109
021189 5303 9381 033424 7428
1603 5715 9789 3826 7825
2016 6125 .0195 4227 8223
2428 6533 .0800 4628 8620
2841 6942 .1004 5029 9017
3252 7350 .1408 5430 9414
3664 7757 .1812 5830 9811
4075 8164 .2216 6230 .0207
4486 8571 .2619 6629 .0602
4896 8978 .3021 7028 .0998
412 408 404 401 397
110 111 112 113 114
041393 5323 9218 053078 690.5
1787 5714 9006 3403 7286
2182 6103 9993 3846 7666
2576 6495 .0380 4230 8046
2969 6885 .0766 4613 8426
&362 7275 .1153 4996 8805
3755 7664 .1538 5378 9185
4148 8053 .1924 5760 9563
4540 4932 8442 88I30 .2309 .2691 6142 6.324 9942 .0320
393 390 386 382 379
115 116 117 118 119
080098 44.58 8186 071882 5547
1075 48S2 8557 2250 5912
1452 5206 8928 2617 6276
1829 5580 9298 2985 6640
2206 59.53 9668 3352 7004
2582 6326 .0038 3718 7368
2958 6699 .0407 4085 7731
3333 7071 .0776 4451 6094
3709 7443 .1145 4816 8457
4083 7815 .1514 5182 8819
376 373 369 367 364
120 121 122 123 124
079181 082785 (300 9903 093422
9543 3144 6716 .0258 3772
9901 3503 7071 .0311 4122
.0266 3861 7426 .0963 4471
.0826 4219 7781 .131.5 4820
.0987 4576 8136 .1667 5169
.1347 .1707 4934 5291 8490 8845 .2018 .2370 5518 itm
.2067 5647 9198 .2721 6215
.2426 6004 9332 .3071 6562
360 358 355 352 349
125 120 127 128 129
096910 100371 3804 7210 110390
7257 0715 4146 7549 0926
7601 1059 4487 7888 1263
7951 1403 4828 8227 1599
8298 1747 5169 8565 1934
8644 2091 5510 8903 2270
8990 2434 5851 9241 2605
9335 2777 0191 9.579 2940
9681 .0026 3119 3462 6531 6871 9916 .0233 3275 3WJ
346 344 341 338 335
130 181 182 133 131
113943 7271 120574 3852 7105
4277 7603 0903 4178 7429
4611 7934 1231 4501 7753
4944 8265 1560 4830 8076
5278 8595 1888 5158 8399
5611 8926 2216 5481 8722
5943 9253 2544 5?06 t045
6276 9586 2871 6131 9368
6608 9915 3198 6450 9690
6940 .024.5 3525 6781 .0012
333 330 328 325 323
185 136 137 138 139
130331 3539 6721 9879 143015
0355 3858 7037 .0194 3327
0977 1298 4177 4496 7354 7671 .0508 .0822 3639 3951
1619 4814 7987 .1136 4263
1939 5133 8303 .14.50 4574
2260 5451 8618 .1763 4885
2580 5769 8934 .2076 5196
2900 3219 6086 6403 9249 9564 .2389 .2702 5507 5818
321 318 316 313 311
140 141 142 143 144
146128 9219 152288 5330 8302
6438 9527 2594 5640 8664
6748 9835 2900 5943 8965
7058 .0142 3203
7676 .0756 3815 6852 9888
7985 .1063 4120 7154 .0168
8294 .1370 4424 74.57 .0469
8C03 .1676 4728 7759 .0709
8911 .1982 5032 8061 .1008
309 307 305 303
9286
7367 .0449 3510 6549 9567
143 140 147 148 149
161308 4353 7317 170262 3186
1667 4650 7613 0355 3478
1967 4947 7908 0848 3769
2266 5244 8203 1141 4060
2564 5541 8497 1434 4351
2863 5838 8792 1726 4641
3161 6134 9086 2019 4932
3460 6430 9380 2311 5222
3758 6726 9674 2603 5512
4055 7022 9968 2895 5802
299 297 294 293 291
1
2
3
4
5
6
7
8
9
D.
H. 100
101 102 10;:
N.
0
TABLE I.—LOGARITHMS
0
624ii
4
5
346
6
7
8
9
D.
301
;
OF NUMBERS. N.
1
2
150 151 152 153 154
176091 8977 181844 4691 7521
0
6381 9264 2129 4975 7803
6670 9552 2415 5259 8084
6959 9839 2700 5542
3
155 156 157 158 159
190332 3125 5900 8657 201397
0612 3403 6176 8932 1670
160 161 162 163 164
204120 6826 9515 212188 4844
165 166 167 168 169
4
5
Num. 199, Log. 300,
6
7
8
9
D.
aw
7248 .0126 2985 5825 8647
7530 .0413 3270 6108 8928
7825 .0699 3555 6391 9209
8113 8401 8689 .0986 .1272 .1558 3839 4123 4407 6674 6956 7239 9490 9771 .0051
288 287 285 283 281
0892 3681 6453 9208 1943
1171 3959 6729 9481 2216
1451 4237 7005 9755 2488
1730 4514 7281 .0029 2761
2010 4702 7556 .0303 3033
2289 5069 7832 .0577 3305
2567 5346 8107 .0850 3577
2846 5023 8382 .1124 3848
279 278 276 274 272
4391 7096 9783 2454 5109
4663 7365 .0051 2720 5373
4934 7634 .0319 2986 5638
5204 7904 .0586 3252 5902
5475 8173 .0853 3518 6166
5746 8441 .1121 3783 6430
6016 8710 .1388 4049 6694
6286 6556 8979 9247 .1654 .1921 4314 4579 6957 7221
270 269 267 266 264
217484 220108 2716 5309 7887
7747 0370 2976 5568 8144
8010 0331 3236 5828 8400
8273 0892 3496 6084 8657
8536 1153 3755 6342 8913
8798 1414 4015 6600 9170
9060 1675 4274 6858 9426
9323 1936 4533 7115 9682
9585 2196 4792 7372 9938
9846 2456 5051 7630 .0193
263 261 259 258 256
170 171 172 173 174
230449 2996 5528 8046 240.549
0701 3250 5781 8297 0799
0960 3504 60,33 8548 1048
1215 3757 6285 8799 1297
1470 4011 6537 9049 1546
1724 4264 6789 9299 1795
1979 4517 7041 9550 2044
2234 4770 7292 9800 2293
2488 5023 7544 .0050 2541
2742 5276 7795 .0300 2790
255 253 252 250 249
175 176 177 178 179
243038 5513 7973 250420 2853
3286 5759 8219 0664 3096
3534 6008 8464 0908 3338
3782 6252 8709 1151 3580
4030 6499 8954 1395 3822
4277 6745 9198 1638 4064
4525 6991 9443 1881 4306
4772 7237 9687 2125 4548
5019 5260 7482 7728 9932 .0176 2368 2610 4790 5031
248 246 245 243 242
180 181 182 183 184
255273 7679 260071 2451 4818
5514 7918 0310 2688 5054
5755 8158 0548 2925 5290
5996 8398 0787 3162 5525
6237 8637 1025 3399 5761
6477 8877 1263 3636 5996
6718 9116 1501 3873 6232
6958 9355 1739 4109 6487
7198 9594 1976 4346 6702
7439 9833 2214 4582 6937
241 239 238 237 235
185 186 187 188 189
267172 9513 271842 4158 6462
7408 9740 2074 4389 6692
7641 9980 2306 4620 6921
7875 .0213 2538 4850 7151
8110 .0446 2770 5081 7380
8344 .0679 3001 5311 7609
8578 .0912 3233 5542 7838
8812 .1144 3464 5772 8067
9046 .1377 3696 6002 8296
9279 .1009 3927 6232 8525
234 233 232 280 229
190 191 192 193 194
278754 281033 3301 5557 7802
8982 1261 3527 5782 8026
9211 1488 3753 6007 8249
9439 1715 3979 6232 8473
9667 1942 4205 6456 8696
9895 2169 4431 6681 8920
.0123 2396 4656 6905 9143
.0351 2822 4882 7130 9366
.0578 2849 5107 7354 9589
.0806 3075 5332 7578 9812
228 227 226 225 223
195 196 197 198 199
290035 2256 4466 6665 8853
0257 2478 4687 6884 9071
0480 2699 4907 7104 9289
0702 2920 5127 7323 9507
0925 3141 5347 7542 9725
1147 3363 5567 7761 9943
1369 1591 3584 3804 5787 6007 7979 8198 .0161 .0378
1813 2034 4025 4246 6226 6446 8416 8635 .0595 .0813
222 221 220 219 218
2
3
4
5
N.
0
1
347
6
7
8
9
D. '
K.
0
1
1
TABLtf I.— LOGARITHMS
Num. 200, Log. 301.
2
3
200 201 202 203 204
301030 3196 5*51 7496 9630
1247 1464 1681 3412 3628 3844 5566 5781 5996 7710 7924 8137 9843 .0056 .0268
205 206 207 200
311754 3867 5970 8083 320146
1966 4078 6180 8272 0354
2177 4289 6390 8481 0562
210 211 212 213 214
322219 4282 6336 8380 330414
2426 4488 6541 8583 0617
215 210 217 218 219
332438 4454 6460 8456 340444
220 221 222 223 224
4
5
«
7
8
9
D.
1898 4059 6211 8351 .0481
2114 4275 0425 8564 .0693
2331 4491 6639 8778 .0906
2547 4706 6854 8991 .1118
2764 4921 7068 9204 .1330
2389 4499 6599 8689 0769
2600 4710 0809 8898 0977
2812 4920 7018 9100 1184
3023 5130 7227 9314 1391
3234 5340 7436 9522 1598
3445 5551 7646 9730 1805
3656 5760 7854 9938 2012
211 210 209 208 207
2633 4694 6745 8787 0819
2839 4899 6950 8991 1022
3040 5105 71.55 9194 1225
3252 5310 7359 9398 1427
3458 5516 7563 9601 1630
3665 5721 7767 9805 1832
3871 5926 7972 .0008 2034
4077 6131 8176 .0211 2236
206 205 204 203 202
2640 4655 6660 8656 0642
2842 4856 6860 8855 0841
3044 5057 7060 9054 1039
3246 5257 7260 9253 1237
3447 5458 7459 9451 1435
3649 5658 7659 9650 1632
3850 5859 7858 9849 1830
4051 C059 8058 .0047 2028
4253 6260 8257 .0246 2225
202 201 200 199 198
342423 4392 6353 8305 350248
2620 4589 6549 8500 0442
2817 4785 6744 8694 0636
3014 4981 6939 8889 0329
3212 5178 71.35 9083 1023
3409 5374 7330 9278 1216
3606 5570 7525 9472 1410
3802 5766 7720 9666 1603
3999 5962 7915 9860 1796
4196 6157 8110 .0054 1989
197 196 195 194 194
225 226 227 228 229
352183 4108 6026 7935 9835
2375 4301 6217 8125 .0025
2568 4493 6408 8316 .0215
2761 4685 6599 8.506 .0401
29.54 4876 6790 8696 .0393
3147 5068 6981 8886 .0783
3339 5260 7172 9076 .0972
3532 5452 7E63 9266 .1161
3724 5043 7.354 94,56 .1350
3916 5834 7744 9646 .1539
193 192 191 190 189
230 231 232 233 231
361728 3612 5488 7356 9216
1917 3800 5675 7542 9401
2105 3988 5862 7729 9587
2294 4176 6049 7915 9772
2482 4363 6236 8101 9958
2671 4551 6423 8287 .0143
2859 4739 6610 8473 .0328
8048 4926 6796 8659 .0513
3236 5113 6983 8845 .0698
3424 5301 7169 9030 .0883
188 188 187 186 185
235 236 237 238 239
371068 2912 4748 6577 8398
1253 3096 4932 6759 8.580
1437 3280 5115 6942 8761
1622 3464 5298 7124 8943
1806 3647 5481 7.306 9124
1991 3831 5664 7488 9306
2175 4015 5846 7670 9487
2360 4198 6029 7852 9668
2544 4382 6212 8034 9849
2728 4.565 6394 8216 .0030
184 184 183 182 181
240 241 242 243 244
380211 2017 3815 5606 7390
0392 2197 3995 5785 7568
0573 2377 4174 5964 7746
0754 2357 1353 6142 7923
0931 2737 4533 6321 8101
1115 2917 4712 6499 8279
1296 3097 4891 6677 8456
1476 3277 5070 6856 8634
1656 3456 5249 7034 8811
1837 3636 5428 7212 8989
181 180 179 178 178
245 246 247 248 249
389166 390935 2697 4452 6199
9313 1112 2873 4627 6374
9520 1288 3048 4802 6548
9698 1464 3224 4977 6722
9875 .0051 .0228 1817 1641 1993 3400 3575 3751 5152 5326 5501 6896 7071 7245
.0405 2169 3926 5676 7419
.0582 2345 4101 5850 7592
.0759 2521 4277 6025 7766
177 176 176 175 174
20 ?
H.
0
1
2
3
4 348
5
6
7
8
2980 5136 7282 9417 .1542
217 216 215 213 212
9
D.
OF NUMBERS. N.
0
1
Num. 299, Log. 476.
2
3
4
8287 .0020 1745 3464 5176
8461 .0192 1917 3635 5346
8634 .0385 2089 3807 5517
8808 8981 .0538 .0711 2261 2433 3978 4149 5688 5858
9154 .0883 2005 4320 6029
9328 .1056 2777 4492 6199
9501 .1228 2949 4663 6370
173 173 172 171 171
5
6
7
8
9
D,
250 251 252 253 254
397940 9874 401401 3121 4834
255 256 257 258 259
400540 8240 9933 411620 3300
6710 6881 8410 8579 .0102 .0271 1788 1958 3467 3635
7051 8749 .0440 2124 3803
7221 8918 .0809 2293 3970
7391 9087 .0777 2461 4137
7561 9257 .0946 2629 4305
7731 9426 .1114 2796 4472
7901 9595 .1283 2964 4639
8070 9764 .1451 8132 4806
170 109 169 168 167
280 281 262 263 264
414973 6641 8301 9950 421604
5140 5307 6807 6973 8467 8033 .0121 .0286 1768 1933
5474 7139 8798 .0451 2097
5641 7306 8964 .0616 2261
5803 7472 9129 .0781 2426
5974 7638 9295 .0945 2590
6141 7804 9460 .1110 2754
6308 7970 9625 .1275 2918
6474 8135 9791 .1439 3082
167 166 165 165 164
265 266 267 268 269
423246 4882 6511 8135 9752
3410 3574 5045 5208 6674 6836 8297 8459 9914 .0075
3737 -5371 6999 8021 .0236
3901 5534 7161 8783 .0398
4065 5697 7321 8944 .0559
4228 5860 7486 9100 .0720
4392 6023 7648 9268 .0881
4555 6186 7811 9429 .1042
4718 6349 7973 9591 .1203
164 163 162 162 161
270 271 272 273 274
431364 2969 4569 6163 7751
1525 3130 4729 6322 7909
1685 3290 4888 6481 8067
1846 3450 5048 6040 8220
2007 3610 5207 6799 8384
2167 3770 5367 6957 8542
2328 3930 5520 7116 8701
2488 4090 5685 7275 8859
2649 4249 5844 7433 9017
2809 4409 6004 7592 9175
161 160 159 159 158
275 270 277 278 279
439333 440901 2480 4045 5604
9491 1036 2637 4201 5760
9643 1224 2793 4357 5915
9806 1381 2950 4513 0071
9904 1533 3106 4669 6226
.0122 1695 3263 4825 0382
.0279 1852 3419 4981 6537
.0437 2009 3576 5137 6692
.0594 2166 3732 5293 6848
.0752 2323 3889 5449 7003
1'58
280 231 282 283 231
447158 8703 450249 1786 8318
7313 8861 0403 1940 8471
7468 9015 0557 2093 3624
7623 91T0 0711 2247 8777
7778 9324 0865 2400 3930
7933 9478 1018 2553 4082
8088 9633 1172 2706 4235
8242 9787 1326 2859 4387
a397 9941 1479 3012 4540
8552 .0095 1633 3165 4692
155 154 154 153 153
285 286 287 288 289
454845 6336 7882 9392 460898
4997 6518 8033 9543 1048
5150 6670 8184 9691 1198
5302 6821 8336 9845 1348
5454 6973 8487 9995 1499
5606 7125 8638 .0146 1049
5758 7270 8789 .0296 1799
5910 7428 8940 .0447 1948
6062 7579 9091 .0597 2098
6214 7731 9242 .0748 2248
152 152 151 151 150
290 291 292 293 294
462308 8893 5383 6868 8347
2548 4042 5332 7010 8495
2697 4191 5680 7134 8043
2847 4340 5829 7312 8790
2997 4490 5977 7460 8938
3146 4039 6126 7608 9085
3296 4788 6274 7756 9233
3445 4936 6423 7904 9380
8594 5085 6571 8052 9527
3744 5234 6719 8200 9675
150 149 149 148 148
295 296 297 298
469822 471292 2756 4216 5671
9909 .0116 .0263 .0410 .0*557 .0704 1438 1585 1732 1878 2025 2171 2903 3049 3195 3341 3487 3633 4382 4508 4653 4799 4944 5090 5816 5962 6107 0252 6397 6542
.0351 2318 3779 5235 6687
.0998 2464 3925 5381 6832
.1145 2610 4071 5526 6976
147 146 146 146 145
7
8
299
N.
0
8114 9847 1573 3292 5005
1
2
3
4
5
349
6
9
157 157 156 155
D.
Num. 300, Log. 477,
TABLE I.—LOGARITHMS 2
3
4
5
6
7
8
9
D,
300 301 302 303 301
477121 8566 480007 1443 2874
7266 8711 0151 1586 3016
7411 8855 0294 1729 3159
7555 8999 0438 1872 3302
7700 9143 0582 2016 3445
7844 9287 0725 2159 3587
7989 9431 0869 2302 3730
81,33 9575 1012 2445 3872
8278 9719 1156 2588 4015
8422 9863 1299 2731 4157
145 144 144 143 143
305 306 307 308 309
484300 5721 7138 8.551 9958
4442 5863 7280 8692 .0099
4585 6005 7421 8833 .0239
4727 6147 7563 8974 .0380
4869 6289 7704 9114 .0520
5011 6430 7845 9255 .0661
5153 6572 7986 9396 .0801
5295 6714 8127 9537 .0941
5437 6855 8269 9677 .1081
5579 6997 8410 9818 .1222
142 142 141 141 140
310 311 312 313 314
491362 2760 4155 5544 6930
1502 2900 4294 5683 7068
1642 3040 4433 5822 7206
1782 3179 4572 5960 7344
1922 3319 4711 6099 7483
2062 3458 4850 6238 7621
2201 3597 4989 6376 7759
2341 3737 5128 6515 7897
2481 3876 5267 6653 8035
2621 4015 5406 6791 8173
140 139
315 316 317 318 319
498311 9687 501059 2427 3791
8448 9824 1196 2564 3927
8586 9962 1333 2700 4063
8724 .0099 1470 2837 4199
8862 .0236 1607 2973 4335
8999 .0374 1744 3109 4471
9137 .0'511 1880 3246 4607
9275 .0648 2017 3382 4743
9412 .0785 2154 3518 4878
9550 .0922 2291 3655 5014
138 137 137 136 136
320 321 322 323 324
505150 6505 7856 9203 510545
5286 6640 7991 9337 0679
5421 6776 8126 9471 0813
5557 6911 8260 9606 0947
.5693 7046 8395 9740 1081
5828 7181 8530 9874 1215
5964 7316 8664 .0009 1349
6099 7451 8799 .0143 1482
6234 7586 8934 .0277 1616
6370 7721 9068 .0411 1750
136
325 326 327 328 329
511883 3218 4548 5874 7190
2017 3351 4681 6006 7328
2151 3484 4813 6139 7460
2284 3617 4946 6271 7592
2418 3750 5079 6403 7724
2551 3883 5211 6535 7855
2684 4016 5344 6668 7987
2818 4149 5476 6800 8119
2951 4282 5609 6932 8251
3084 4415 5741 7064 8382
133 133 132 132
330 331 332 333 334
518514 9828 521138 2444 3746
8646 9959 1269 2575 3876
8777 .0090 1400 2705 4006
8909 .0221 1.530 2835 4136
9040 .0353 1661 2966 4266
9171 .0484 1792 3096 4396
9303 .0615 1922 3226 4526
9434 .0745 2053 3356 4656
9566 .0876 2183 3486 4785
9697 .1007 2314 3616 4915
131 131 131 130 130
335 336 337 888 339
525045 6339 7630 8917 530200
5174 6469 7759 9045 0328
5304 6598 7888 9174 0456
5434 6727 8016 9302 0584
5563 6856 8145 9430 0712
5693 C985 8274 9559 0840
5822 7114 8402 9687 0968
5951 7243 8531 9815 1096
0081 6210 7372 7501 8600 8788 9943 .0072 1223 1351
129 129 129 128
340 341 342 343 344
531479 2754 4026 5294 6558
1607 2882 4153 5421 6685
1734 3009 4280 5547 6811
1862 3136 4407 5674 6937
1990 3264 4534 5800 7063
2117 3:)91 4661 5927 7189
2245 3518 4787 6053 7315
2372 3645 4914 6180 7441
2500 3772 5041 6306 7567
2627 3899 5167 6432 7693
128 127 127 126 126
345 346 347 348
537819 9076 540329 1579 2825
7945 9202 0455 1704 2950
8071 9327 0,580 1829 3074
8197 9452 0705 1953 3199
8322 9578 0830 2078 3323
8448 9703 0955 2203 3447
8574 9829 1080 2327 3571
8699 9954 1205 2452 3696
8825 8951 .0079 .0204 1330 1454 2576 2701 3820 3944
126 125 125 125 124
2
3
4
5
6
7
K.
349
H.
0
0
1
1
350
8
9
139 1 139 138
ia5 135 134 134 13E
128 |
D.
OB NUMBERS.
Num. 399, Log. 601.
1
2
4
5
6
7
8
350 351 352 353 354
544068 5307 6543 7775 9003
4192 5431 6666 7898 9120
4316 5555 6789 8021 9249
4440 5678 6913 8144 9371
4564 5802 7036 8267 9494
4688 5925 7159 8389 9616
4812 6049 7282 8512 9739
4936 6172 7405 8635 9861
5060 5183 6296 6419 7529 7652 8758 8881 9984 .0106
124 124 123 123 123
355 a56 357 358 359
550228 1450 2668 3883 5094
0351 1572 2790 4001 5215
0473 1694 2911 4126 5336
0595 1816 3033 4247 5457
0717 1938 3155 4368 9578
0840 2060 3276 4489 5699
0962 2181 3398 4610 5820
1084 2303 3519 4731 5940
1206 2425 3640 4852 6061
1328 2547 3762 4973 6182
122 122 121 121 121
360 361 362 363 364
556303 7507 8709 9907 561101
6423 7627 8829 .0026 1221
6544 6664 7748 7868 8948 9068 .0146 .0265 1340 1459
6785 7988 9188 .0385 1578
6905 8108 9308 .0504 1698
7026 8228 9428 .0624 1817
7146 7267 8349 8469 9548 9667 .0743 .0863 1936 2055
7387 8589 9787 .0982 2174
120 120 120 119 119
365 366 367 368 369
562293 3481 4666 5848 7026
2412 3600 4784 5966 7144
2531 3718 4903 6084 7262
2650 3837 5021 6202 7379
2769 3955 5139 6320 7497
2887 4074 5257 6437 7614
3066 4192 5376 6555 7732
3125 4311 5494 6673 7849
3244 4429 5612 6791 7967
3362 4548 5730 6909 8084
119 119 118 118 118
370 371 372 373 374
568202 9374 570543 1709 2872
8319 9491 0800 1825 2988
8436 9608 0776 1942 3104
8554 9725 0893 2058 3220
8671 9842 1010 2174 3336
8788 9959 1126 2291 3452
6905 9023 .0076 .0193 1243 1359 2407 2523 3568 3684
9140 .0309 1476 2639 3800
9257 .0426 1592 2755 3915
117 117 117 116 116
375 376 377 378 379
574031 5188 6341 7492 8639
4147 5303 6457 7607 8754
4263 5419 6572 7722 8868
4379 5534 6687 7836 8983
4494 5650 6802 7951 9097
4610 5765 6917 8066 9212
4726 5880 7032 8181 9326
4841 5996 7147 8295 9441
4957 6111 7262 8410 9555
5072 6226 7377 8525 9669
116 115 115 115 114
380 381 382 383 384
579784 580925 2063 3199 4331
9898 1039 2177 3312 4444
.0012 1153 2291 3426 4557
.0126 1267 2404 3539 4670
.0241 1381 2518 3652 4783
.0355 1495 2631 3765 4896
.0469 1608 2745 3879 5009
.0583 1722 2898 3992 5122
.0697 1836 2972 4105 5235
.0811 1950 3085 4218 5348
114 114 114 113 113
385 386 387 388 389
585461 6587 7711 8832 9950
5574 6700 7823 8944 .0061
57386 5799 6812 6925 7935 8047 9056 9167 .0173 .0284
5912 7037 8160 9279 .0398
6024 7149 8272 9391 .0507
6137 7262 8384 9503 .0619
6250 6362 7374 7486 8496 8608 9615 9726 .0730 .0842
6475 7599 8720 9838 .0953
113 112 112 112
390 391 392 393 304
591065 2177 3286 4393 5496
1176 2288 3397 4503 5606
1287 2399 3508 4614 5717
1399 2510 3618 4724 5827
1510 2621 3729 4&34 5937
1621 2732 3840 4945 6047
1732 2843 3950 50.55 6157
1955 3064 4171 5276 6377
2066 3175 4282 5386 6487
111 HI 111 110 110
395 390 397 398 399
596597 7695 8791 9883 600973
6707 7805 8900 9992 1082
6817 7914 9009 .0101 1191
6927 8024 9119 .0210 1299
7037 8134 9228 .0319 1408
7146 8243 9337 .0428 1517
7256 7366 7476 8353 8462 8572 9446 9556 9665 .0537 .0646 .0755 1625 1734 1843
7586 8681 9774 .0864 1951
110 110 109 109 109
N.
H.
0
0
1
2
3
3
5
i
351
6
1843 2954 4061 5165 6267
7
8
9
9
D.
112
D.
,
1
Nnm, 400, Log. 602.
1
2
3
4
5
6
7
8
9
D.
401 402 403 404
602060 3144 4226 5303 6381
2101i 3253 4334 5413 6489
2277 3361 4442 5521 6596
2588 3469 4550 5028 6704
2494 a577 4658 5738 6811
2603 3686 4766 5844 6919
2711 3794 4874 5951 7026
2819 3902 4982 6059 7133
2928 4010 5089 6166 7241
3036 4118 5197 6274 7348
108 108 108 108 107
405 400 407 408 409
607455 8526 9594 610660 1723
7.502 86*3 9701 0767 1829
7669 8740 9808 0873 1936
7777 8847 9914 0979 2042
7884 8954 .0021 1086 2148
7991 9061 .0128 1192 2254
8098 8205 8312 9167 9274 9381 .0234 .0341 .0447 1298 1405 1511 2500 2466 2572
8419 9488 .0554 1617 2678
107 107 107 100 106
410 411 412 413 414
612784 3842 4897 5950 7000
2890 3947 5003 6055 7105
2996 4053 5108 6160 7210
3102 4159 5213 6265 7315
3207 4264 5319 6370 7420
3313 4370 5424 6476 7525
3419 4475 5529 6581 7629
3525 4581 5634 6680 7734
3630 4688 5740 6790 7839
3736 4792 5845 6895 7943
106 108 105 105 103
415 416 417 418 419
618048 9093 020138 1176 2214
8153 9198 0240 1280 2318
8257 9302 0344 1384 2421
8362 9408 0448 1488 2525
8466 9511 0552 1592 2628
8571 9615 0650 1695 2732
8676 9719 0760 1799 2835
8780 9824 0804 1903 2939
8884 8989 9928 .0032 0968 1072 2007 2110 3042 3146
104 104 104 104
420 421 422 423 424
025249 4282 5312 6340 7366
3353 4385 5415 6443 7468
3456 4488 5518 6546 7571
3559 4591 5621 6048 7673
3663 4095 5724 6751 7775
3766 4798 5827 6853 7878
3869 4901 5929 6956 7980
3973 5004 6032 70.58 8082
4076 5107 6185 7161 8185
4179 5210 6238 7263 8287
103 103 103 103 102
425 420 427 428 429
628389 9410 630428 1444 2457
8491 9.512 0530 1545 2559
8593 9013 0031 1647 2660
8695 9715 0753 1748 2761
8797 9817 0835 1849 2862
8900 9919 0936 1951 2963
9002 .0021 1038 2052 3064
9104 9206 .0123 .0224 1139 1241 2153 2255 3165 3266
9308 .0328 1342 2356 3387
102 102 102 101 101
430 431 432 433 434
633 468 4477 5484 0488 7490
3569 4578 5581 6588 7590
3670 4679 5685 6688 7090
3771 4779 5785 6789 7790
3872 4880 5886 6889 7890
3973 4981 5988 6989 7990
4074 5081 6087 708« 8090
4175 5182 0187 7189 8190
4376 5383 6388 7390 8389
101 101 100 100 100
4*5 438 437 438 439
638489 9486 640481 1474 2465
8589 9'386 0381 1573 2563
8689 9886 0680 1072 2602
8789 9785 0779 1771 2701
8888 9885 0879 1871 2800
8988 9984 0978 1970 2959
9088 .0084 1077 2089 3058
9188 .0183 1177 2108 3150
9287 9387 .02X3 .0382 1276 1375 2287 2366 3255 3354
loo
440 441 442 443 444
6 13 453 4439 5422 6404 7383
3551 4537 5521 6'302 7481
3650 4636 5619 6600 7579
3749 4734 5717 6698 7676
3847 4832 5815 6706 7774
3946 4931 5913 6894 7872
4044 5029 6011 6992 7969
4143 9127 0110 7089 8067
4242 5220 6208 7187 8165
4340 5324 6300 7285 8202
99 98 98 98 98
445 446 447 448 449
648360 9398 650308 1278 2246
8458 94:S2 0405 1375 2343
8555 9530 0502 1472 2440
8653 9627 0599 1569 2536
8750 9724 0696 1666 2633
8848 9821 0793 1762 2730
8945 9919 0890 1859 2826
9043 .0016 0987 1956 2923
9140 .0113 1084 2053 3019
9237 .0210 1181 2150 3116
97 97 97 97 97
1
2
3
4
5
6
N. 400
N.
0
TABLE I.—LOGARITHMS
0
352
7
4276 5283 6287 7290 8290
8
9
111'3
99 99 99 9!i
D.
' OF NUMBERS.
Num. 499, Log. 698.
2
3
4
5
6
7
8
9
451 452 453 454
653213 4177 5138 6098 7056
3309 4273 5235 6191 7152
3405 4369 5331 6290 7247
3502 4465 5427 6386 7343
3598 4562 5523 6482 7438
3695 4058 5019 6577 7534
3791 4754 5715 6673 7629
3888 4850 5810 6769 7725
8984 4946 5906 6864 7820
4080 5042 6002 6960 7916
96 96 96 96 90
455 456 457 458 459
658011 8107 8965 9060 9916 .0011 660865 0960 1813 1907
8202 9155 .0106 1055 2002
8298 9250 .0201 1150 2096
8393 9346 .0290 1245 2191
8488 9441 .0391 1339 2286
8584 9536 .0486 1434 2380
8679 9631 .0581 1529 2475
8774 9726 .0076 1623 2569
8870 9821 .0771 1718 2663
95 95 95 95 95
460 461 462 463 464
662758 3701 4642 5581 6518
2852 3795 4736 5675 6612
2947 3889 4830 5769 6705
3041 3983 4924 5802 6799
3135 4078 5018 5950 6892
3230 4172 5112 6050 6986
3324 4266 5206 6143 7079
3418 4360 5299 6237 7173
3512 4454 5393 6331 7266
3007 4548 5487 6424 7360
94 94 94 94 94
465 466 467 468 40>J
667453 8386 9317 670246 1173
7546 8479 9410 0339 1205
7640 8572 9503 0431 1358
7733 8005 9596 0524 1451
7826 8759 9689 0617 1543
7920 8852 9782 0710 1636
8013 8945 9875 0802 1728
8106 9038 9967 0895 1821
8199 9131 .0060 0988 1913
8293 9224 .0153 1080 2005
93 93 93 93 93
470 471 472 473 474
672098 3021 3942 4801 5778
2190 3113 4034 4903 5870
2283 3205 4126 5045 5962
2375 3297 4218 5137 6053
2467 3390 4310 5228 6145
2560 3482 4402 5320 6236
2052 &574 4491 5412 6328
2744 3666 4586 5503 6419
2836 3758 4677 5595 6511
2929 3850 4769 5687 0602
92 92 92 92 92
475 476 477 478 479
676691 7607 8518 9428 680358
6785 7098 8609 9519 0426
6878 7789 8700 0.17
6968 7881 8791 9700 0607
7059 7972 8882 9791 0698
7151 8003 8973 9882 0789
7242 8154 9064 9973 0879
7333 8245 9155 .0063 0970
7424 8336 9246 .0154 1060
7516 8427 9337 .0245 1151
91 91 91 91 91
480 481 4^2 481
681241 2145 3047 3917 4845
1332 2235 3137 4037 4935
1422 2320 3227 4127 5025
1513 2416 3317 4217 5114
1603 2306 3407 4307 5204
1693 2390 3497 4396 5294
1784 2686 3587 4486 5383
1874 2777 3677 4576 5473
1964 2867 3767 4660 5563
2055 2957 3857 4750 5052
90 90 90 90 90
485 486 487 488 489
685742 6036 7529 8420 9309
5831 0726 7618 8509 9398
5921 6815 7707 8598 9486
C010 6901 7796 8687 9575
I900 0994 7886 8776 9004
6189 7083 7975 8865 9753
6279 7172 8064 8953 9841
6368 7261 8153 9042 9930
6458 7351 8242 9131 .0019
6547 7440 9220 .0107
89 89 69 89 89
490 491 402
690196 1081 1905 2847 8727
0285 1170 20.53 2935 3815
0373 1258 2142 3023 3903
0462 1347 2230 8111 3991
05.50 143T, 2318 3199 4078
0039 1524 2406 3287 4166
0728 1612 2494 3375 4254
0816 1700 2583 3463 4342
0905 1789 2671 3551 4430
0993 1877 2759 3039 4517
694605 5482 6356 7229 8101
4693 5569 6444 7317 8188
4781 5657 6531 7404 8275
4868 5744 6618 7491 8362
4956 5832 6706 7578 8449
5044 5919 0793 7665 8535
5131 6007 6880 7752 8022
5219 6094 6908 7839 8709
5307 6182 7055 7920 8796
5394 6209 7142 6014 6883
1
2
3
4
6
6
7
Trii!
-ao.
IT. 450
4S':S
495
494 495 490 497 498 499
N.
0
0
1
» ao
;:53
8
san
9
D.
89 88
88 88 88 88 87 87 87 £7
D.
i
Nam. 500, Log. 698. N.
0
1
TABLE I.—LOGARITHMS 2
3
4
501 502 503 501
698970 9838 700704 1568 2431
9067 9144 9231 9924 .0011 .0098 07(Hi 0877 0963 1054 1741 1827 2517 2003 2089
9317 .0184 1050 1913 2775
505 500 507 508 509
703291 4151 5008 5864 6718
3377 4230 5094 5949 0803
3463 4322 .5179 6035 6888
3549 4408 5265 6120 6974
510 511 512 513 514
707570 8421 9270 710117 0903
7655 8500 9355 0202 1048
7740 8591 9440 0287 1132
515 516 517 518 519
711807 2050 3491 4330 5167
1892 2784 3575 4414 5251
520 521 522 523 524
716003 0838 7S71 8502 9331
525 520 527 528 529
500
5
6
7
8
9
9604 9751 .0531 .06i7 1395 1482 2258 234l 3119 3205
D. 87 87 86 86 86
9401 .0271 1136 1999 2861
9491 .0358 1222 2080 2947
9578 .0444 1309 2172 3033
3635 4494 5350 6200 7059
3721 4579 5436 6291 7144
3807 4665 5522 6370 7229
3893 4751 5007 0462 7315
3979 4837 5693 6547 7400
4065 4922 5778 6632 7485
86 80 80 85 85
7826 8676 9524 0371 1217
7911 8761 9609 0456 1301
7996 8846 9694 0540 1385
8081 8931 9779 0625 1470
8166 9015 9803 0710 1554
8251 9100 9948 0794 1039
8336 9185 .0033 0879 1723
85 85 85 85 84
1976 2818 3659 4497 5335
2060 2902 3742 4581 5418
2144 2980 3826 4665 5502
2229 3070 3910 4749 5580
2313 3154 3994 4833 5609
2397 3238 4078 4910 5753
2481 3323 4162 5000 5830
2566 3407 4246 5084 5920
84 84 84 84 84
6087 6921 7754 8585 9414
6170 7001 7837 8008 9497
6254 7088 7920 8751 9580
6337 7171 8003 8834 9063
6421 7254 8080 8917 9745
6504 7338 8109 9000 9828
6588 7421 8253 9083 9911
6671 7501 8330 9105 9994
6754 7587 8419 9248 .0077
83 83 83 83 83
720159 0980 1811 2031 3456
0242 1038 1893 2710 3538
0325 1151 1975 2798 3020
0407 1233 2058 2881 3702
0490 1316 2140 2903 3784
0573 1398 2222 3049 3800
0655 1481 2305 3127 3948
0738 1503 2387 3209 4030
0821 1640 2409 3291 4112
0903 1728 2552 3374 4194
83 82 82 82 82
530 531 532 533 534
724270 5095 5912 6727 7541
4358 5176 5993 6809 7623
4440 5238 6075 6890 7704
4522 5340 0156 6972 7785
4604 5422 6238 7053 7866
4685 5503 6320 7134 7948
4767 5585 0401 7210 8029
4849 5667 6483 7297 8110
4931 5748 0504 7379 8191
5013 5830 6646 7460 8273
82 82 82 81 8I
535 530 537 538 539
728354 8435 9105 9246 9974 .0055 730782 0803 1589 1609
8516 8597 9327 9408 .0130 .0217 0944 1024 1750 1830
8678 9489 .0298 1105 1911
8759 9570 .0378 1180 1991
8841 9651 .0459 1260 2072
8922 9732 .0540 1347 2152
9003 9813 .0021 1428 2233
9084 9893 .0702 1508 2313
81 81 81 81 81
540 541 542 543 544
732394 3197 3999 4800 5599
2474 3278 4079 4880 5679
2555 3358 4100 4900 5759
2635 3438 4240 5040 5838
2715 3518 4320 5120 5918
2796 3598 4400 5200 5998
2876 3679 4480 5279 0078
2956 3759 4500 5359 6157
3037 3839 4040 5439 6237
3117 3919 4720 5519 6317
80
545 540 547 548 549
730397 7193 7987 8781 9572
6476 7272 80S7 8860 9051
6550 7352 8140 8939 9731
6635 7431 822.5 9018 9810
6715 7511 8305 9097 9889
6795 7590 8384 9177 9908
6874 6954 7034 7113 7670 7749 7829 7908 8463 8543 8022 8701 9493 9250 9335 9414 .0047 .0120 .0205 .0284
80 79 79 79 79
2
3
4
N,
0
1
5
6
7
3
9
S0
|
80 80 80
D. J
354
Of NUMBERS.
Hum. 599, Log. 778.
D,
1
2
4
5
6
7
8
9
550 551 552 553 554
740363 1152 1939 2725 3510
0442 1230 2018 2804 3588
0521 1309 2096 2882 3667
0600 1388 2175 2961 3745
0678 1467 2254 3039 3823
0757 1540 2382 3118 3902
0836 1624 2411 3190 3980
0915 1703 2489 3275 4058
0994 1782 2568 8853 4136
1073 1860 2647 3431 4215
79 79 79 78 78
555 556 557 558 559
744293 5075 5855 6634 7412
4371 51.53 5933 6712 7489
4449 5231 6011 6790 7567
4528 5309 6089 6868 7645
4000 5387 6167 6945 7722
4684 5465 0245 7023 7800
4702 5543 6323 7101 7878
4840 5621 6401 7179 7955
4919 5699 6479 7256 8033
4997 5777 6556 7334 8110
78 78 78 78 78
560 561 562 568 564
748188 8903 9736 750508 1279
8266 9040 9814 0586 1356
8343 9118 9891 0663 1433
8421 9195 9968 0740 1510
8498 9272 .0045 0817 1587
8576 9350 .0123 0894 1664
8053 8731 8808 8885 9427 9504 9582 9659 .0209 .0277 .0554 .0431 0971 1048 1125 1202 1741 1818 1895 1972
77 77 77 77 77
565 566 567 508 569
752048 2816 3583 4348 5112
2125 2893 3660 4425 5189
2202 2970 3736 4501 5265
2279 3047 3813 4578 5341
2350 3123 3889 4654 5417
2433 3200 3966 4730 5494
2509 3277 4042 4807 5570
2586 3353 4119 4883 5046
2003 3430 4195 4960 5722
2740 3506 4272 5036 5799
77 77 77 76 70
570 571 572 573 574
755875 6636 7396 8155 8912
5951 0712 7472 8230 8988
6027 0788 7.548 8306 9063
6103 0804 7024 8382 9139
6180 6940 7700 8458 9214
6256 7010 7775 8533 9290
6332 7092 7851 8C09 9300
6408 7108 7927 8685 9441
6484 7244 8003 8761 9517
6560 7320 8079 8830 9592
70 76 76 76 70
575 576 577 578 579
759668 760422 1176 1928 2679
9743 0498 1251 2003 2754
9819 0573 1326 2078 2829
9894 0049 1402 2153 2904
9970 0724 1477 2228 2978
.0045 0799 1552 2303 3053
.0121 0875 1027 2378 3128
.0196 0950 1702 2453 3203
.0272 1025 1778 2529 3278
.0347 1101 1853 2604 3353
75 75 75 75 75
580 581 582 583 584
763428 4176 4923 5609 6413
3503 4651 4998 5743 0487
3578 4326 5072 5818 6502
8858 4400 5147 5892 6636
3727 4475 5221 5906 6710
3802 4530 5290 6041 6785
3877 4024 5370 0115 6859
3952 4099 5445 6190 6933
4027 4774 5520 0264 7007
4101 4848 5594 6338 7082
75 75 75 74 74
585 586 587 588 589
767156 7898 8638 9377 770115
7230 7972 8712 9451 0189
7304 8040 8780 9525 0203
7379 8120 8800 9599 0336
7453 8194 8931 9673 0410
7527 8268 9008 9746 0484
7601 8342 9082 9820 0557
7675 8416 9156 9894 0631
7749 8490 9230 9968 0705
7823 8564 9303 .0042 0778
74 74 74 74 74
590 591 592 593 594
770852 1587 2322 3055 3786
0926 1001 2395 3128 3800
0999 1734 2408 3201 3933
1073 1808 2542 3274 4006
1146 1881 2615 3348 4079
1220 1955 2688 3421 4152
1293 2028 2762 3494 4225
1367 2102 2835 3567 4298
1440 2175 2908 3640 4371
1514 2248 2981 3713 4444
74 73 73 73 73
595 596 597 598 599
774517 5240 5974 C701 7427
4590 5819 6017 6774 7499
4663 5392 6120 6840 7572
4736 5403 6193 6919 7644
4809 5538 0285 6992 7717
4882 5010 6338 7064 7789
4955 5083 6411 7137 7862
5028 5750 6483 7209 7934
5100 5829 0556 7282 8006
5173 5902 6029 7354 8079
73 73 73 73 72
1
2
3
4
5
6
7
8
9
N.
II.
0
0
3
355
D.
1
1 | Num. 630, Log. 778,
1
2
3
4
5
8224 8947 9009 0389 1109
8290 9019 9741 0401 1181
ssos
001 C02 603 604
778151 8874 £536 780317 1037
9091 9813 0533 1253
8441 9103 9885 0CO5 1324
8.513 8585 9236 9308 9957 .0029 0677 0749 1396 1468
605 60j 607 608 609
781755 2473 3189 3904 4617
1827 2544 3200 3975 4089
1899 2016 3332 4046 4700
1971 2688 3403 4118 4831
2042 2759 3475 4189 4902
2114 2831 3546 4261 4974
610 611 612 613 614
785330 6041 6751 7460 8168
5401 0112 6382 7531 8239
5472 6183 0893 7002 8310
5543 6254 6964 7673 8381
5015 0325 7035 7744 8451
5086 0390 7100 7815 8522
615 616 617 C18 619
788875 9581 790285 0988 1091
8946 9551 0350 1059 1761
9016 9722 0426 1129 1831
9087 9792 0490 1199 1901
620 021 622 623 024
792392 3092 3790 4488 5185
2462 3102 3800 4558 5254
2532 3231 3930 4027 5324
625 62o 027 628 629
795880 0574 7268 7900 8051
5949 0044 7337 8029 8720
630 031 032 033 031
799.341 080029 0717 1404 2089
635 030 637 638 639
N.
0
TABLE I.—LOGARITHMS 7
8
9
8658 6380 .0101 0821 1540
8730 9452 .0173 0893 1612
8802 0524 .0245 0965 1684
72 72 72 72 72
2186 2902 3618 4332 5045
22:8 2974 3689 4403 5116
2329 3040 3701 4475 5187
2401 3117 3832 4546 5259
72 72 71 71 71
5757 0467 7177 7885 8593
5828 6538 7248 7950 8003
5899 0009 7319 8027 8734
5970 0080 7390 8098 8804
71 71 71 71 71
9157 9803 0507 1209 1971
9228 9299 9933 .0004 0337 0707 1340 1410 2041 2111
9309 .0074 0778 1480 2181
9440 .0144 0848 1550 2252
9510 .0215 0918 1020 2322
71 70 70 70 70
2802 3301 4000 4097 5393
2672 3371 4070 4767 5403
2742 3441 4139 4830 5532
2812 3511 4209 4900 5602
2882 3581 4279 4970 5672
2952 3651 4349 5045 5741
3022 3721 4418 5115 5811
70 70 70 70 70
0019 6713 7400 8098 8789
0088 0782 7475 8167 8858
6158 0852 7545 8230 8927
0227 0921 7014 8305 8996
6297 0990 7083 8374 9065
0360 7660 7752 8443 9134
6430 7129 7821 8513 9203
0505 7198 7890 8J82 9272
69 69 69 09 09
9409 0098 0780 1472 2158
9478 0167 0854 1541 2220
9547 0236 0923 1009 2295
9010 0305 0992 1078 2303
9685 0373 1001 1747 2432
9754 0442 1129 1815 2500
9823 0511 1198 1884 2568
9892 0580 1260 1952 2037
9901 0048 1335 2021 2705
69 69 69 69 69
802774 3457 4139 4821 5501
2842 3525 4208 4889 5569
2910 3594 4276 4957 5637
2979 3662 4344 5025 5705
3047 3730 4412 5093 5773
3116 3798 4480 5101 5841
3184 3807 4548 5229 5908
3252 3935 4616 5297 5976
3321 4003 4685 5365 0044
3389 4071 4753 5433 0112
68 68 68 68 68
640 641 042 643 644
806180 7535 8211 8886
6248 6926 7603 8279 8953
6316 0994 7670 8346 9021
6384 7081 7738 8414 9088
6451 7129 7806 8481 9156
6519 7197 7873 8549 9223
6587 7264 7941 8616 9290
6655 7332 8008 8084 9358
6723 7400 8070 8751 9425
0790 7467 8143 8818 9492
68 68 68 67 67
645 646 647 648 649
809500 810233 0904 1575 2245
9627 0300 0971 1642 2312
9694 0367 1039 1709 2379
9762 0434 1100 1770 2445
9829 0301 1173 1843 2512
9896 0569 1240 1910 2579
9904 0836 1307 1977 2640
.0031 0703 1374 2044 2713
.0098 0770 1441 2111 2780
.0165 0837 1508 2178 2847
67 67 67 67 67
1
2
3
4
5
6
O0J
N.
oa58
0
loo
6
7
8
9
D,
D.
01 NUMBERS.
Rum. 699, Log. 845.
2
3
4
5
6
7
8
9
050 051 C52 053 054
812913 8581 4248 4913 5578
2980 3648 4314 4980 5044
3047 3714 4381 5046 5711
3114 3781 4447 5113 5777
3181 3848 4514 5179 5843
3247 8914 4581 5246 5910
3314 8981 4647 5312 £976
3381 4048 4714 5378 C042
3448 4114 4780 5445 6109
8514 4181 4847 5511 6175
67 (,7 67 66 66
055 050 C57 058 059
816241 6901 7565 8226 8885
6308 0970 7031 8292 8951
6374 7036 7698 8858 9017
C440 7102 7764 8424 9088
60i6 7160 7830 8490 9149
6573 6039 7235 7301 7896 .7902 8556 8622 9215 9281
0705 7367 8028 8088 9346
6771 7433 8094 8754 9412
6838 7499 8160 8820 9478
66 66 66 66 66
600 001 002 663 601
819.544 820201 0988 1514 2168
9610 0267 0924 1570 2233
9676 0333 0989 1645 2299
9741 0390 1055 1710 2364
9807 0464 1120 1775 2430
9873 0530 1186 1841 2495
9939 .C004 .0070 0595 0661 0727 1251 1317 1382 1006 1072 2037 2560 2620 2691
.0136 0792 144S 2103 2756
66 66 66 65 65
665 666 007 668 669
822822 3474 4126 4776 5426
2887 3539 4191 4841 5491
2952 3605 4256 4906 5556
3018 3670 4321 4971 5621
3083 3735 4380 5030 5680
3148 3800 4451 5101 5751
3213 3865 4510 5166 5815
3279 3930 4581 5231 5880
3344 3996 4646 5296 5945
3409 4061 4711 5361 6010
65 65 65 65 65
670 671 672 073 674
826075 6723 7369 8015 8660
6140 6787 7434 8080 8724
6204 6852 7499 8144 8789
6269 6917 7563 8209 8853
6334 6981 7628 8273 8918
6399 7046 7692 8338 8982
6464 7111 7757 8402 9046
6528 7175 7821 8467 9111
6593 7240 7886 8531 9175
6658 7805 7951 8595 9230
65 65 65 64 64
675 C7C 077 678 679
829G04 0917 830589 1230 1870
9368 .0011 0053 1291 1934
9132 .0075 0717 1358 1998
9497 .0139 0781 1422 20C2
9561 .0204 0845 1486 2126
9625 .0208 0C09 1:50 2189
9690 .0332 0973 1614 2253
9754 .0300 1037 1678 2317
9818 .0460 1102 1742 2381
9882 .0525
64 64 64
lt'06 2445
a
680 C81 682 08E 081
832509 3784 4421 5056
2573 3211 3848 4484 5120
2637 3275 3912 4548 5183
2700 3338 3075 4611 5247
2764 3402 4039 4675 5310
2828 3400 4103 4739 5373
2892 3530 4166 4802 5437
2956 3503 4230 4866 5500
3020 3657 4294 4920 5564
3083 3721 4357 4093 5627
64 64 64 64 63
685 Oil 687 08£ 689
835C01 0324 C957 7:88 8219
5754 0387 7020 7052 8282
5817 6451 7083 7715 8345
5881 6514 7146 7778 8408
5944 6577 7210 7841 8471
6007 6641 7273 7904 8534
6071 C704 7330 7967 8597
6184 6767 7309 8030 8CC0
6197 6b80 7402 8093 8723
6261 0894 7525 8156 8786
63 63 13 63 63
690 C81 092 693 691
838849 9478 840106 0733 1359
8912 9541 0100 0700 1422
8975 0C04 0232 08o0 1485
9038 9007 0204 0021 1547
9101 9729 0357 0984 1610
9164 9702 C420 1040 1S72
9227 9855 0482 1109 1735
9289 6918 0545 1172 1797
9352 9981 0608 1234 1860
9415 .0043 0671 1297 1922
03 63 63 63 63
695
841985 2000 3233 8855 4477
2047 2672 3295 3918 4539
2110 2734 3357 3080 4001
2172 2700 3420 4042 4664
2235 2859 3482 4104 4726
2297 2021 3544 4166 4788
2360 2983 360li 4220 4850
2422 3046 3660 4291 4012
2484 3108 3731 4353 4974
2547 3170 3793 4415 5036
62 62 62 62 62
1
2
3
4
5
6
7
0
go; 697 098 090
N.
U 17
0
1
c57
8
na
9
D.
04
E.
Hum, 700, Log. 845.
0
TABLE I.—LOGARITHMS
1
2
3
4
5
6
7
701 702 703 704
845098 5718 6337 6955 7573
5160 5780 0399 7017 7634
5222 5842 6461 7079 7090
5284 5904 6523 7141 7758
5346 5960 6585 7202 7819
5408 6028 6046 7264 7861
5470 6090 6708 7326 7943
5532 6151 6770 7388 8004
5594 6213 6832 7449 8060
5650 6275 0891 7511 8128
62 62 62 62 62
705 TiM 707 708 709
848189 8805 9419 850033 0646
8251 8866 9481 0095 0707
8312 8928 9542 0156 0769
8374 8989 9604 0217 0830
8435 9051 9005 0279 0891
8497 9112 9726 0340 0952
8559 9174 9788 0401 1014
8620 9235 9849 0462 1075
8082 9297 9911 0524 1130
8743 9358 9972 0585 1197
62 61 61 61 61
710 711 712 713 714
851258 1870 2480 3090 3698
1320 1931 2541 3150 3759
1381 1992 2602 3211 3820
1442 2053 2663 3272 3881
1503 2114 2724 3333 3941
1564 2175 2785 3394 4002
1625 2236 2846 3455 4063
1686 2297 2907 3516 4124
1747 2358 2968 3577 4185
1809 2419 3029 3637 4245
61 61 61 61 61
715 716 717 718 719
854306 4913 5519 6124 6729
4367 4974 5580 6185 6789
4428 5031 5640 8245 .6850
4488 5095 5701 0306 6910
4549 5156 5761 6366 6970
4610 5216 5822 6427 7031
4670 5277 5882 6487 7091
4731 5337 5913 6548 7152
4792 5398 6003 6608 7212
4852 5459 6064 6668 7272
61 61 01 60 60
720 721 722 723 724
857332 7935 8537 9138 9739
7393 7995 8597 9198 9799
7453 8050 8657 9258 9859
7513 8116 8718 9318 9918
7574 8176 8778 9379 9978
7634 8238 8838 9439 .0038
7694 8297 8898 9499 .0098
7755 8357 8958 9559 .0158
7815 8417 9018 9619 .0218
7875 8477 9078 9679 .0278
60 60 60 60 60
725 720 727 728 729
860338 0937 1534 2131 2728
0398 0990 1594 2191 2787
0458 1056 1654 2251 2847
0518 1116 1714 2310 2906
0578 1176 1773 2370 2986
0637 1236 1833 2430 3625
0697 1295 1893 2489 3085
0757 1355 1952 2549 3144
0817 1415 2012 2608 3204
0877 1475 2072 2068 3203
60 60 C0 60 60
730 731 732 733 734
863323 3917 4511 5104 5696
3382 3977 4570 5163 5755
3442 4030 4630 5222 5814
3501 4098 4089 5282 5874
3561 4155 4748 5341 5933
3620 4214 4808 5400 5992
3680 4274 4867 5459 0051
3739 4833 4920 5519 6110
3799 4392 4985 5578 6169
3858 4452 5045 5637 6228
59 59 59 59 59
735 .866287 738 6878 737 7467 738 8056 739 8644
6346 6937 7528 8115 8703
6405 0990 7585 8174 8762
0405 7055 7044 8233 8821
6524 7114 7703 8292 8879
6583 7173 7762 8350 8938
6642 7232 7821 8409 8997
6701 7291 7880 8408 9058
6760 7350 7939 8527 9114
6819 7409 7998 8586 9173
59 59 59 59 59
740 741 742 743 744
869232 9818 870404 0989 1573
9290 9877 0462 1047 1631
9349 9935 0521 1106 1690
9408 9466 9525 9994 .0053 .0111 0579 0638 0690 1104 1223 1281 1748 1806 1805
9584 9642 .0170 .0228 0755 0813 1339 1398 1923 1981
9701 .0287 0872 1456 2040
9760 .0345 0930 1515 2098
59 59 58 58 58
745 746 747 748 749
872156 2739 3321 3902 4482
2215 2797 3379 3980 4540
2273 2855 3437 4018 4598
2331 2913 8495 4070 4050
2389 2972 3553 4134 4714
2448 3030 3611 4192 4772
2506 3088 3669 4250 4830
2564 3140 3727 4308 4888
2622 3204 3785 4366 4945
2681 3262 3844 4424 5003
58 58 58 58 58
1
2
3
4
5
6
7
8
9
N. 700
N.
0
358
8
9
D.
D.
OF NUMBERS.
Hum. 799, Log. 903.
3
4
750 751 752 753 754
875061 5640 6218 6795 7371
5119 5698 6276 6853 7429
5177 5756 6333 6910 7487
5235 5813 6391 6968 7544
5293 5871 6449 7026 7602
5351 .5929 6507 7083 7659
755 756 757 758 759
877947 8522 9096 9669 880242
8004 8579 9153 9726 0299
8062 8637 9211 9784 0356
8119 8694 9268 9841 0413
8177 8752 9325 9898 0471
8234 8809 9383 9956 0528
760 761 762 763 764
880814 1385 1955 2525 3093
0871 1442 2012 2581 3150
0923 1499 2069 2638 3207
0985 1.556 2126 2695 3264
1042 1613 2183 2752 3321
1099 1670 2240 2809 3377
1156 1727 2297 2866 3434
1213 1784 2354 2923 3491
1271 1841 2411 2980 3548
1328 1898 2468 3037 3605
57 57 57 57 57
765 766 767 768 769
883661 4229 4795 5361 5926
3718 . 3775 4285 4342 4852 4909 5418 5474 5983 6039
3832 4399 4965 5531 6096
3888 4455 5022 5587 6152
3945 4512 5078 5644 6209
4002 4569 5135 5700 6265
4059 4625 5192 5757 6321
4115 4682 5248 5813 6378
4172 4739 5305 5870 6434
57 57 57 57 56
770 771 772 773 774
886491 7054 7617 8179 8741
6547 7111 7674 8236 8797
6604 7167 7730 8292 8853
6660 7223 7786 &S48 8909
6716 7280 7842 8404 8965
6773 7336 7898 8460 9021
6829 7392 7955 8516 9077
6885 7449 8011 8573 9134
6942 7505 8007 8629 9190
6998 7561 8123 8685 9246
56 56 56 56 56
775 776 777 778 779
889302 9862 890421 0980 1537
9358 9918 0477 1035 1593
9414 9974 0533 1091 1649
9470 .0030 0589 1147 1705
9526 .0086 0645 1203 1760
9582 .0141 0700 1259 1810
9638 .0197 0756 1314 1872
9694 .0253 0812 1370 1928
9750 .0309 0868 1420 1983
9800 .0365 0924 1482 2039
56 56 56 56 56
780 781 782 783 784
892095 2651 3207 3762 4316
2150 2707 3262 3817 4371
2206 2762 3318 3873 4427
2262 2818 3373 3928 4482
2317 2873 3429 3984 4538
2373 2929 3484 4039 4593
2429 2985 3540 4094 4648
2484 3040 3595 4150 4704
2540 3096 3651 4205 4759
2595 3151 3706 4261 4814
56 56 56 55 55
785 786 787 788 789
894870 5423 5975 6526 7077
4925 5478 6030 6581 7132
4980 5533 6085 6636 7187
5036 5588 6140 6692 7242
5091 5644 6195 6747 7297
5146 5699 6251 6802 7352
5201 5754 6306 6857 7407
5257 5809 6361 6912 7462
5312 5864 6416 6967 7517
5367 5920 6471 7022 7572
55 55 55 55 55
790 791 792 793 794
897627 8176 8725 9273 9821
7638 8231 8780 9328 9875
7737 8286 8835 9383 9930
7792 8341 8890 9437 9985
7847 7902 7957 8396 8451 8506 8944 8999 9054 9492 9547 9602 .0039 .0094 .0149
8012 8561 9109 9656 .0203
8067 8122 8615 8070 9164 9218 9711 9766 .0258 .0312
55 55 55 55 55
795 796 797 798 799
900367 0913 1458 2003 2547
0422 0968 1513 2057 2601
0476 1022 1567 2112 2655
0531 1077 1622 2166 2710
0586 1131 1676 2221 2764
55 55 54 51 54
1
2
3
4
H.
N.
0
0
1
2
35'J
5
0640 1186 1731 2275 2818
5
6
7
8
9
5409 5987 6564 7141 7717
5466 6045 6622 7199 7774
5524 6102 6680 7256 7832
5582 6160 6737 7314 7889
58 58 58 58 58
8407 8464 8981 9039 9555 9612 .0127 .0185 0699 0756
57 57 57 57 57
8292 8349 8866 8924 9440 9497 .0013 .0070 0585 0642
0695 1240 1785 2329 2873
0749 1295 1840 2384 2927
0804 1349 1894 2438 2981
0859 1401 1948 2492 3030
6
7
8
9
D.
D.
Num. 800, Log. 903.
N.
0
1
TABLE I.—LOGARITHMS 2
4
5
6
7
8
9
800 801 802 803 804
903090 3633 4174 4716 5256
3144 3687 4229 4770 5310
3199 3741 428E 4824 5364
3253 3795 4337 4878 5'418
3307 384!i 4391 49*2 5472
8361 3904 4445 4986 5520
3416 3958 4499 5040 5580
3470 4012 4553 5094 5634
3524 4066 4607 5148 5688
3578 4120 4661 5202 5742
54 54 54 54 54
803 80« 807 808 809
905796 6335 6874 7411 7949
5850 6389 6927 7465 8002
5904 0443 6981 7519 8056
5958 6497 7035 7573 8110
0012 6551 7089 7626 8163
6000 6604 7143 7680 8217
6119 6058 7196 7734 8270
6173 6712 7250 7787 8324
6227 6760 7304 7841 8378
0281 6820 7358 7895 8431
54 54 54 54 54
810 811 812 813 814
908485 9021 9356 910091 0624
8539 9074 9610 0144 0678
8592 9128 9003 0197 0731
8646 9181 9716 0251 0784
8699 9235 9770 0301 0838
8753 9289 9823 0358 0891
8807 9342 9877 0411 0944
8860 9396 99.30 0464 0998
8914 9449 9984 0518 1051
8967 9503 .0037 0571 1101
54 54 53 53 53
815 816 817 818 819
911158 1690 2222 2753 3281
1211 1743 2275 2806 3337
1264 1797 2328 2859 3390
1317 1850 2381 2913 3443
1371 1903 2445 2966 3496
1424 1956 2488 3019 3549
1477 2009 2541 3072 3602
1530 2063 2594 3125 3655
1584 2116 2647 3178 3708
1637 2169 2700 3231 3761
53 53 53 53 53
820 821 822 823 824
913814 4313 4872 5400 5927
3867 4396 4925 5453 5980
3920 4449 4977 5505 6033
3973 4502 5030 5538 6085
4026 4555 5083 5011 0138
4079 4608 5136 5664 6191
4132 4660 5189 5710 6243
4184 4713 5241 5769 6296
4237 4766 5294 5822 6349
4290 4819 5347 5875 6401
53 53 53 53 53
825 826 827 828 829
916454 6980 7506 8030 8555
6507 7033 7538 8083 8007
6559 7085 7611 81.33 8659
6612 7138 7663 8188 8712
6664 7190 7716 8240 8764
6717 7243 7768 8293 8816
6770 7295 7820 8345 8869
6822 7348 7873 8397 8921
6875 7400 7925 8450 8973
6927 7453 7978 8502 9026
53 53 52 52 52
830 831 832 833 834
919078 9001 920123 0645 1166
9130 9653 0176 0697 1218
9183 9706 0228 0749 1270
9235 9758 0280 0801 1322
9287 9810 0332 0853 1374
9340 9862 0384 0900 1426
9392 9914 0436 0958 1478
9444 9967 0489 1010 1530
1096 .0019 0541 1062 1582
9549 .0071 0593 1114 1634
52 52 52 52 52
835 836 837 838 839
921686 2206 2725 3244 3762
1T38 2258 2777 3290 3814
1790 2310 2829 »S48 3865
1842 2362 2881 3399 3917
1894 2414 203:3 3451 3969
1946 2466 2985 3503 4021
1998 2518 3037 8555 4072
2050 2570 8089 3007 4124
2102 2622 3140 3058 4176
2154 2074 3192 3710 4228
52 52 52 52 52
840 841 842 843 844
924279 4790 5312 5828 6342
4331 4848 5364 5879 6394
4383 4899 5415 5931 6445
4434 4951 5467 5982 6497
4486 5003 5518 6548
4538 5054 5570 6085 6000
4589 5106 5621 6137 6651
4641 5157 5673 6188 6702
4693 5209 5725 6240 0754
4744 5261 5776 0291 6805
52 52 52 51 51
843 846 847 848 849
916857 7370 7883 8396 8908
6908 7422 7935 8447 8959
6959 7473 7986 8498 9010
7011 7524 8037 8549 9001
7062 7576 8088 8601 9112
7114 7627 8140 8652 9163
7165 7678 8191 8703 9215
7216 7730 8242 8754 9266
7268 7781 8293 8805 9317
7319 7832 8345 8857 9368
51 51 51 51 51
1
2
3
4
5
6
7
8
9
N.
0
3
Imt
360
D.
D.
OP NUMBERS.
n..
0
1
2
3
4
5
Num. 899, Log. 954.
6
7
8
9
D.
850 851 852 853 854
929419 9930 930440 0949 1458
9470 9981 0491 1000 1509
0521 .0032 0542 1051 1560
9572 .0083 0592 1102 1010
9623 .0134 0643 1153 1661
9674 .0185 0094 1204 1712
855 856 857 858 859
931966 2474 2981 3487 3993
2017 2524 3031 3538 4044
2068 2575 3082 3589 4094
2118 2626 3133 3639 4145
2169 2677 3183 3690 4195
2220 2727 3234 3740 4246
2271 2778 3285 3791 4296
2322 2820 3335 3841 4347
2372 2879 3386 3892 4397
2423 2980 3437 3943 4448
51 51 51 51 51
800 861 862 863 864
934498 5003 5507 6011 . 6514
4549 5054 5558 6001 0504
4599 5104 5008 6111 6614
4050 5154 5658 6162 6665
4700 5205 5709 6212 6715
4751 5255 5759 6262 6765
4801 5300 5809 0313 6815
4852 5356 5860 0363 6865
4902 5406 5910 6413 6916
4953 5457 5960 6463 6966
to
805 866 867 868 860
937016 7518 8019 8520 9020
7086 7568 8089 8570 9070
7117 7618 8119 8620 9120
7167 7668 8169 8670 9170
7217 7718 8219 8720 9220
7267 7769 8269 8770 9270
7317 7819 8320 8820 0320
7367 7869 8870 9369
7418 7919 8420 8920 9419
7468 7960 8470 8970 9469
50 50 50 50 50
870 871 872 873 874
939519 940018 0516 1014 1511
9569 0068 0566 1064 1561
9019 0118 0016 1114 1011
9669 0108 0666 1163 1660
9719 0218 0710 1213 1710
9769 0267 0765 1263 1760
0819 0317 0815 1313 1809
9869 0307 0865 1362 1859
9918 0417 0915 1412 1909
9968 0467 0964 1462 1958
50 50 50 50 50
875 876 877 878 870
942008 2504 3000 3495 3989
2058 2554 3049 3544 4038
2107 2603 3090 3593 4088
2157 2653 3148 3643 4137
2207 2702 3198 3092 4180
2250 2752 3247 3742 4230
2366 2801 321(7 3791 4285
2355 2851 3340 3841 4335
2105 2001 8396 3890 4384
2455 2950 3445 3939 4433
50 \ 50 49 49 49
880 881 882 883 884
944483 4976 5469 5981 6452
4532 5025 5518 6010 0301
4581 5074 5507 6059 0551
4631 5124 5616 6108 6600
4680 5173 5665 6157 6649
4729 5222 5715 0207 6608
4779 5272 5704 6256 6747
4828 5321 5813 0305 6700
4877 5370 5862 6354 6845
4927 5419 5912 0403 6894
49 49 49 49 49
898 880 887 888 889
940943 7434 7924 8413 8002
6992 7483 7973 8402 8951
7041 7532 8022 8511 8999
7090 7581 8070 8560 9048
7140 7630 8119 8609 9097
7189 7670 8168 8657 9146
7238 7728 8217 8706 9195
7287 7777 8266 8755 9244
7386 7820 8315 8804 9202
7385 7875 8304 8853 9341
49 49 49 49 49
890 801 892 893 894
949390 9878 950365 0851 1338
9439 9926 0414 0900 1380
9488 9536 9975 .0024 0462 0511 0949 0997 1435 1483
9585 .0073 0560 1040 1532
9634 .0121 0008 1095 1580
9683 .0170 0657 1143 1629
9731 .0219 0700 1192 1677
9780 .0267 0754 1240 1726
9829 .0316 0803 1289 1775
49 49 49 40 49
895 896 897 898
951823 2308 2792 3276 3760
1872 2350 2841 3325 3808
1920 2405 2889 3373 3850
1969 2453 2938 3421 3905
2017 2502 2980 3470 3953
2066 2550 3034 3518 4001
2114 2599 3083 3566 4049
2163 2647 3131 3615 4098
2211 2696 3180 3663 4146
2260 2744 3228 3711 4194
48 48 48 48 48
2
3
4
5
6
7
8
9
899
N.
0
1
Trig.—31.
361
9725 9776 9827 9879 .0250 .0287 .0338 .0380 0745 0796 0847 0898 1254 1305 1356 1407 1763 1814 1865 1915
sno
51 51 51 51 51
50 50 50 50
D.
Num. 900, Log. 954.
N.
|
1
2
3
4
5
6
7
8
9
900 901 902 903 901
954243 4725 5207 5688 6168
4291 4773 5255 5736 6216
4339 4821 5303 5784 6265
4387 4869 5351 5832 6313
4435 4918 5399 5880 6361
4484 4966 5447 5928 6409
4532 5014 5495 5976 6457
4580 5062 5543 6024 6505
4628 5110 5592 6072 6553
4677 5158 5640 6120 0601
48 48 48 48 48
905 906 907 908 909
956649 7128 7607 8086 8564
6697 7176 7655 8131 8612
6745 7224 7703 8181 8659
679;} 7272 7751 8229 8707
6840 7320 7799 8277 8755
6888 7368 7847 8325 8803
6936 7416 7894 8373 8850
6984 7464 7942 8421 8898
7032 7512 7990 8468 8946
7080 7559 8038 8516 8994
48 48 48 48 48
910 911 912 913 914
959041 9089 9518 9566 9995 .0042 960471 0518 0946 0994
9137 9814 .0090 0566 1041
9185 9661 .0138 0013 1089
9232 9709 .0185 0661 1136
9375 9423 9852 9900 .0328 .0376 0804 0851 1279 1326
9471 9917 .0423 0899 1374
48 48 48 48 47
915 916 917 918 919
961421 1895 2369 2843 3316
1469 1943 2417 2890 3363
1516 1990 2464 2937 3410
1563 2018 2511 2985 3457
1611 2085 2559 3032 3504
1658 2132 2600 3079 3552
1706 2180 2653 3126 3599
1753 2227 2791 3174 3646
1801 2275 2748 3221 3693
1848 2322 2795 3268 3741
47 47 47 47 47
920 921 922 92s 924
963788 4260 4731 5202 5672
3835 4397 4778 5249 5719
3882 4354 4825 5296 5766
3929 4401 4872 5543 5813
3977 4448 4919 5390 5860
4024 4495 4966 5437 5907
4071 4542 £013 5484 5954
4118 4590 5001 5531 6001
4105 4637 5108 5578 6048
4212 4684 51.55 5025 6095
47 47 47 47 47
925 92« 927 928 929
966142 6611 7080 7548 8016
6189 6658 7127 7595 8062
6236 0705 7173 7042 8109
628:5 6752 7220 7688 8156
0320 0799 7267 7735 8203
6376 6845 7314 7782 8249
6423 6892 7301 7829 8296
0470 6939 7408 7875 8343
6517 6986 7454 7022 8390
6564 7033 7E01 7909 8438
47 47 47 47 47
930 931 032 933 934
968483 89.50 9416 9882 970317
8530 8990 946:} 9928 0393
8576 862} 9043 9090 9509 9556 9975 .0021 0440 0486
8670 9136 9602 .0068 0533
8716 9183 9649 .0114 0579
8763 9229 9695 .0161 0626
8810 9276 9742 .0207 0672
8856 8903 9323 9369 9789 9835 .0254 .0200 0719 0765
47 47 47 47 46
935 938 937 939
970812 1276 1740 2203 2660
0988 1322 1786 2249 2712
0904 1309 1832 2295 2758
0951 1415 1879 2342 2804
0997 1401 1925 2388 2851
1044 1508 1971 2434 2897
1090 1554 2018 2481 2943
1137 1601 2064 2527 2989
1183 1647 2110 2575 3035
1229 1693 2157 2619 3682
46 46 46 46 46
940 941 942 943 944
973128 3590 4051 4512 4972
3174 3636 4097 4558 5018
3220 3682 4143 4604 5064
3266 3728 4189 4650 5110
3313 3774 4235 4696 5156
3359 3820 4281 4742 5202
3405 3866 4327 4788 5248
3451 3913 4374 4834 5294
3497 3959 4420 4880 5340
3543 4005 4466 4926 5386
46 46 46 46 46
945 946 947 948 949
975432 5891 6350 6808 7266
5478 5937 6396 6854 7312
5524 5983 6442 6900 7358
5570 6029 6488 6946 7403
5616 6075 C533 6992 7449
5602 6121 6579 7037 7495
5707 6167 6625 7083 7541
5753 6212 6671 7129 7586
5799 6258 6717 7175 76152
5845 6304 6763 7220 7678
46 46 46 46 46
1
2
3
4
5
6
7
8
9
938
S.
0
TABLE I.—LOGARITHMS
0
9289 9328 9757 9804 .02i3 .0280 0709' 0756 1184 1231
D.
D.
OF NUMBERS.
Bum. 999, Log. 9£9,
1
2
3
4
5
6
7
8
9
950 977724 951 8181 952 8037 9093 953 954 . 9548
7709 8220 8683 9138 9594
7815 8272 8728 9184 9039
7861 8317 8774 92» 9U85
7906 8363 8819 0275 9730
7952 8409 886.5 9321 0776
7998 8454 8911 9366 9821
8043 8500 8956 9412 9807
8089 8546 9002 9457 9912
8135 8591 9047 9303 9958
46 46 46 46 46
955 956 957 938 959
930003 0458 0912 1308 1819
0049 0303 0957 1411 1804
0094 0549 1003 1450 1909
0140 0594 1048 1501 1954
0185 1093 1547 2000
0231 0698 1139 1592 2045
0270 0780 1184 1687 2090
0322 0770 1229 1683 2135
0307 0821 1275 1728 2181
0412 0807 1320 1773 2220
45 45 45 45 45
900 901 902 903 934
982271 2723 3175 3020 4077
2310 2709 3220 3071 4122
2362 2814 3205 3710 4107
2407 2859 8310 3762 4212
2452 2904 3356 3807 4257
2497 2949 3401 3852 4302
2543 2994 3446 3897 4347
2588 3040 3491 8942 4392
2633 3085 3530 3987 4437
2078 3130 3581 4032 4482
45 45 45 45 45
985 9S7 908 909
984527 4977 5420 5875 0324
4572 5022 5471 5920 6369
4617 5067 5510 5965 6413
4662 5112 5561 6010 0458
4707 5157 5606 6055 6503
4752 5202 5651 6100 6548
4797 5247 5690 6144 6593
4842 5292 5741 6189 6637
4887 5337 5786 6234 6082
4932 5382 5830 6279 6727
45 45 45 45 45
970 971 972 973 974
986772 7219 7000 8113 8559
0817 7204 7711 8157 8004
6861 7309 7756 8202 8648
6906 7353 7800 8247 8093
6951 7398 7845 8291 8737
6996 7443 7890 8336 8782
7040 7488 7934 8381 8820
7085 7532 7979 8425 8871
7130 7577 8024 8470 8916
7175 7622 8068 8514 8900
45 45 45 45 45
975 970 977 978 979
989003 9450 9895 990339 0783
9049 9494 9939 0383 0827
9094 9539 9983 0428 0871
9138 9,383 .0028 0472 0916
9183 9628 .0072 0516 0960
9227 9672 .0117 0501 1004
9272 9717 .0161 0605 1049
9316 9761 .0206 0650 1093
9361 9806 .0250 0094 1137
9405 9850 .0294 0738 1182
45 44 44 44 44
980 981 983 984
991220 1009 2111 2554 2995
1270 1713 2156 2598 3039
1315 1758 2200 2642 3083
1359 1802 2244 2686 3127
1403 1840 2288 2730 3172
1448 1890 2333 2774 3216
1492 1935 2377 2819 3260
1536 1979 2421 2863 3304
1580 2023 2405 2907 3348
1625 2067 2509 2951 3392
44 44 44 44 44
985 980 987 988 989
993430 3877 4317 4757 5190
3480 3921 4361 4801 5240
3524 3905 4405 4845 5284
3568 4009 4449 4889 5328
3013 4053 4493 4933 5372
3657 4097 4537 4977 5416
3701 4141 4581 5021 5400
3745 4185 4625 5065 5504
3789 4229 4669 5108 5547
3833 4273 4713 5152 5591
44 44 44 44 44
990 991 992 993 994
995635 0074 0512 0949 7386
5679 6117 6555 6993 7430
5723 6161 6599 7037 7474
5767 0205 6643 7080 7517
5811 6249 6687 7124 7561
5854 6293 6731 7168 7005
5898 6337 6774 7212 7648
5942 6380 6818 7255 7692
5986 6424 6862 7299 7736
6030 6408 6906 7343 7779
44 44 44 44 44
995 900 997 998
997823 8259 8095 9131 9505
7867 8303 8f39 9174 9609
7910 8347 8782 9218 9652
7954 8390 8820 9261 9696
7998 8434 8809 9305 9739
8041 8477 8913 9348 9783
8085 8521 8956 9392 9820
8129 8564 9000 9435 9870
8172 8608 9043 9479 9913
8216 8652 9087 9522 9957
44 44 44 43 43
2
3
4
5
6
7
8
9
0
9:;e
9.S2
999
N.
0
1
0M0
1 353
D.
D.
.—Uuni
H.
1000, Log. 000. 0
TABLE I.—LOGARITHMS
1
2
3
4
5
6
7
8
1000 1001 1002 1003 1001
000000 0434 0868 1301 1734
0043 0477 0911 1344 1777
0087 0521 0954 1388 1820
0130 0564 0998 1431 1863
0174 0608 1041 1474 1907
0217 0051 1084 1517 1950
0200 0694 1128 1561 1993
0304 0738 1171 1604 2030
0347 0781 1214 1647 2080
0391 0824 1258 1690 2123
43 43 43 43 43
1005 1006 1007 1008 1009
002166 2598 3029 3461 3891
2209 2641 3073 3504 3934
2252 2684 3116 3547 3977
2296 2727 3159 3590 4020
2339 2771 3202 3033 4063
2382 2814 3245 3676 4106
2425 2857 3288 3719 4149
2408 2900 3331 3762 4192
2512 2943 3374 3805 4235
2555 2980 3417 3848 4278
43 43 43 43 43
1010 1011 1012 1013 1014
004321 4751 5181 5609 6038
4364 4794 5223 5652 6081
4407 4837 5266 5695 6124
4450 4880 5309 5738 6166
4493 4923 5352 5781 6209
4536 4906 5395 5824 6252
4579 5009 5438 5867 6295
4622 5052 5481 5909 6338
4665 5095 5524 5952 6380
4708 5138 5567 5995 6423
43 43 43 43 43
1015 1016 1017 1018 1019
006466 6894 7321 7748 8174
6509 6936 7364 7790 8217
6552 6979 7408 7833 8259
6594 7022 7449 7876 8302
6637 7005 7492 7918 8345
6680 7107 7534 7961 8387
6723 7150 7577 8004 8480
6765 7193 7620 8040 8472
6808 7236 7662 8089 8515
6851 7278 7705 8132 8558
43 43 43 43 43
1020 1021 1022 1023 1024
008600 9026 9451 9876 010300
8643 9008 9493 9918 0342
8685 9111 9536 9961 0385
8728 9153 9578 .0003 0427
8770 9196 9621 .0045 0470
8813 9238 9603 .0088 0512
8856 9281 9706 .0130 0554
8898 &323 9748 .0173 0597
8941 9366 9791 .0215 0039
8983 9408 0833 .0258 0081
43 42 42 42 42
1025 1026 1027 1028 1029
010724 1147 1570 1993 2415
0766 1190 1613 2035 2458
0809 1232 1655 2078 2500
0851 1274 1697 2120 2542
0893 1317 1740 2102 2584
0936 1359 1782 2204 2626
0978 1401 1824 2247 2009
1020 1444 1866 2289 2711
1003 1480 1909 2331 2753
1105 1528 1951 2373 2795
42 42 42 42 42
1030 1031 1032 1033 1034
012837 3259 3680 4100 4521
2879 3301 3722 4142 4563
2922 3343 3764 4184 4605
2964 3385 3800 4226 4647
3006 3427 3848 4268 4689
3048 3469 3890 4310 4730
3090 3511 3932 4353 4772
3132 3553 3974 4395 4814
3174 3590 4010 4437 4850
3217 3638 4058 4479 4898
42 42 42 42 42
1035 1036 1037 1038 1039
014940 5380 5779 6197 6616
4982 5402 5821 6239 6657
5024 5444 5863 6281 6699
5066 5485 5904 6323 0741
5108 5527 5946 6365 6783
5150 5569 5988 6407 6824
5192 5011 6030 6448 6866
5234 5053 0072 0490 6908
5276 £695 0114 0532 0950
5318 5737 6156 6574 6992
42 42 42 42 42
1040 1041 1042 1043 1044
017033 7451 7868 8284 8700
7075 7492 7909 8326 8742
7117 7534 7951 8308 8784
7159 7576 7993 8409 8825
7200 7018 8034 8451 8867
7242 7659 8076 8492 8908
7284 7701 8118 8534 8950
7326 7743 8159 8576 8992
7367 7784 8201 8617 9033
7409 7820 8243 8659 9075
42 42 42 42 42
1045 1046 1047 1048 1049
019116 9532 9947 020361 0775
9158 9573 9988 0403 0817
9199 9615 .0030 0444 0858
9241 9656 .0071 0486 0900
9282 9698 .0113 0527 0941
9324 9739 .0154 0568 0982
9366 9781 .0195 0610 1024
9407 9822 .0237 0651 1065
9449 9490 9864 9905 .0278 .0320 0693 0734 1107 1148
42 42 41 41 41
2
3
N.
0
1
4
6
364
6
7
3
9
9
D.
D.
'i
01 NUMBERS. 0
1
Num. 1099, Log .041.
2
3
4
5
6
7
8
9
1050 1051 1032 1053 1054
021189 1603 2016 2428 2841
1231 1644 2057 2470 2882
1272 1685 2098 2511 2923
1313 1727 2140 2552 2964
1355 1768 2181 2593 3005
1396 1809 2222 2035 3047
1437 1851 2263 2076 3088
1479 1892 2305 2717 3129
1520 1933 2346 2758 3170
1501 1974 2387 2799 3211
41 41 41 41 41
1055 1056 1057 1058 1059
023252 3664 4075 4486 4896
3294 3705 4116 4527 4937
3335 3740 4157 4508 4978
3376 3787 4198 4609 5019
3417 3828 4239 4650 5000
3458 3870 4280 4091 5101
3499 3911 4321 4732 5142
3541 3952 4363 4773 5183
3582 3993 4404 4814 5224
3023 4034 4445 4855 5205
41 41 41 41 41
025306 5715 6125 0533 6942
5347 5756 6165 6574 6982
5388 5797 0206 6015 7023
5429 5838 0247 0050 7034
5470 5879 0288 6697 7105
5511 5920 0329 6737 7146
5552 5961 0370 6778 7180
5593 6002 6411 0819 7227
5634 6043 6452 6860 7268
5674 6084 6492 6901 7309
41 41 41 41 41
1065 1060 101>7 1068 1069
027350 7757 8104 8571 8978
7390 7798 8205 8612 9018
7431 7839 8246 8653 9059
7472 7879 8287 8093 0100
7513 7920 8327 8734 9140
7553 7961 8368 8775 9181
7594 8002 8409 8815 9221
7635 8042 8449 8856 9262
7076 8083 8490 8890 9303
7716 8124 8531 8937 9343
41 '41 41 41 41
1070 1071 1072 1073 1074
029384 9789 030195 0C09 1004
9424 9830 0235 0640 1045
9405 9871 0270 0081 1085
9500 9911 0310 0721 1126
9546 9952 0357 0702 1166
9587 9992 0397 0802 1200
9027 .0033 0438 0843 1247
9668 .C073 0478 0883 1287
9708 9749 .0114 .0154 0519 0559 0923 0964 1328 1368
41 41 40 40 40
1075 1070 1077 1078 1079
031408 1812 2210 2019 3021
1449 1853 2256 2659 3062
1489 1893 2296 2699 3102
1530 1933 2337 2740 3142
1570 1974 2377 2780 3182
1010 2014 2417 2820 3223
1651 2054 2458 2800 3263
1691 2095 2498 2901 3303
1732 2135 2538 2941 3343
1772 2175 2578 2981 3384
40 40 40 40 40
1080 1081 1082 1083 1084
03342i 3826 4227 4028 5029
3404 3866 4207 4609 5069
3504 3906 4308 4709 5109
3544 3040 4348 4749 5149
3585 39.% 4388 4789 5190
3625 4027 4428 4829 5230
3005 4067 4408 4869 5270
3705 4107 4508 4909 5310
3745 4147 4548 4949 5350
3786 4187 4588 4989 5390
40 40 40 40 40
1085 1086 1087 1088 1089
035430 5830 0230 6629 7028
5470 5870 0209 0009 7008
5510 5910 0309 0709 7108
5550 5950 6349 6749 7148
5590 5990 0389 0789 7187
5030 0030 0429 6828 7227
5670 6070 6469 6808 7267
5710 0110 6509 6908 7307
5750 6150 6549 6948 7347
5790 6190 6589 0988 7387
40 40 40 40 40
1090 1091 1092 1093 1094
037420 7825 8223 8020 9017
7400 7805 8262 86C0 9057
7506 7901 8302 8700 9097
7546 7944 8342 8739 9136
7580 7984 8382 8779 9176
7626 8024 8421 8819 9216
7665 8064 8461 8859 9255
7705 8103 8501 8898 9295
7745 8143 8541 8938 9335
7785 8183 8580 8978 9374
40 40 40 40 40
1005 1096 1097 1098 1090
039414 0811 040207 0602 0998
9454 9850 0240 0642 1037
9493 9890 0286 0681 1077
9.533 0929 0325 0721 1116
9573 9969 0365 0761 1156
9612 .0009 0405 0800 1195
9652 .0048 0444 0840 1235
9692 .0088 0484 0879 1274
9731 .0127 0523 0919 1314
9771 .0167 0563 0958 1353
40 40 40 40 39
1
2
3
4
N,
1060 1001 1002 1063 1064
IT.
-
0
6 3(55
6
7
8
9
D.
D.
TABLE II.—LOGARITHMS OF PRIME
H.
Logarithm.
N.
Logarithm.
N.
Logarithm.
2 3 5 7 11
30102 47712 69897 84509 04139
99956 12547 00643 80100 26851
63981 19662 36019 14257 58225
238 239 241 251 257
36735 37839 38201 39967 40993
59210 79009 70425 37214 31233
26019 48138 74868 81038 31295
547 557 563 569 571
73798 74585 75050 75511 75663
73263 51951 83948 22863 61082
33431 73729 51346 95071 45848
13 17 19 23 29
11394 23014 27875 38172 46239
33523 89213 38009 78360 79978
06837 78274 52829 17593 98956
263 269 271 277 281
41995 42975 43296 44247 44870
57484 22800 92908 97690 63199
89758 02408 74400 64449 05080
577 587 593 599 601
76117 76863 77305 77742 77887
58131 81012 46933 68223 44720
55731 47614 64263 89311 02740
31 37 41 43 47
49136 56820 61278 63316 67209
16938 17240 38507 84555 78579
34273 66995 19735 79587 35717
283 293 307 311 313
45178 46686 48713 49276 49554
64355 76203 83754 03890 43375
24290 54109 77186 26838 46448
C07 613 617 619 631
78318 78746 79028 79169 80002
86910 04745 51610 06190 93592
75258 18415 33242 20118 44134
53 59 61 67 71
72427 77085 78532 82607 85125
58696 20116 98350 48027 83487
00789 42144 10767 00826 19075
817 331 337 347 349
50105 51982 52762 51032 54282
92622 79937 99008 94747 54269
17751 75719 71339 90874 59180
641 643 647 653 659
80685 80821 81090 81491 81888
80295 09729 42806 31812 54145
18817 24222 68700 75074 94010
73 79 83 89 97
86332 89762 91907 94939 98677
28601 70912 80923 00066 17342
20456 90441 76074 44913 66245
353 359 367 373 379
54777 55509 56466 57170 57863
47053 44485 60642 88318 92099
87823 78319 52089 08688 68072
661 673 677 683 691
82020 82801 83058 83442 83917
14594 50642 86686 07036 80473
85640 23977 85144 81533 74198
101 103 107 109 113
00432 01283 02938 03742 05307
13737 72217 37776 64979 84434
82643 05172 85210 40024 83420
383 389 397 401 409
58319 58994 59879 60314 61172
87739 96013 05067 43726 33080
68623 25708 63115 20182 07342
701 709 719 727 733
84571 85064 85672 86153 86510
80179 62351 88903 44108 39746
66659 83067 82883 59038 41128
127 131 137 139 149
10380 11727 13672 14301 17318
37209 12956 05671 48002 62884
55957 55764 56407 54095 12274
419 421 431 433 439
62221 62428 63447 63048 64246
40229 20958 72701 78963 45202
66295 35668 60732 53365 42121
739 743 751 757 761
86864 87098 87563 87909 88138
44383 88137 99370 58795 46567
94826 60575 04168 00073 70573
151 157 163 167 173
17897 19589 21218 22271 23804
69472 96524 76044 64711 61031
93169 09284 03958 47583 28795
443 449 457 481 463
64640 65224 65991 66370 66558
87262 63410 62000 09253 09910
23070 03323 69850 89648 17953
769 773 787 797 809
88592 88817 89597 90145 90794
63398 94939 47323 83213 85216
01431 18325 59065 96112 12272
179 181 191 193 197
25285 25767 28103 28555 29446
30309 85748 33072 73090 62261
79893 09185 47728 07774 61593
467 479 487 491 499
66931 68033 68752 69108 69810
68805 55134 89612 14921 05456
66112 14563 14634 22968 23390
811 821 823 827 829
90902 91434 91539 91750 91855
08542 31571 98352 55095 45305
11156 19441 12270 52547 50274
199 211 223 227 229
29885 32428 34830 35602 35983
30764 24552 48630 58571 54823
09707 97693 48161 93123 39888
503 509 521 523 541
70156 70671 71683 71850 73319
79850 77823 77232 16888 72651
55927 36759 99524 67274 06569
839 853 857 859 863
92376 93094 93298 93399 93601
19608 90311 08219 31638 07957
28700 67523 23198 31212 15210
3titj
NUMBERS LESS THAN 1000. Logarithm. 877 881 683 8S7 907 911
94299 94497 94590 94792 95700 95951
95933 59084 07035 86198 72870 83769
66041 12048 77.509 31726 60095 72998
H. 919 929 9;i7 941 017 953
Logarithm,
Logarithm. 96331 96601 97173 97;i58 97634 97909
55113 57139 95908 96234 90790 29006
86111 93642 87778 27257 03273 38320
967 971 077 O83 001 097
98542 98721 98989 99255 99607
64740 92299 45637 35178 36544 51583
83002 08005 18773 32136 85275 11656
In the above table, only the mantissas are given ; the characteristics may be found by the rule (908). By means of these logarithms, the logarithm of any number may be found with equal accuracy. If the given number be the product of any of the prime numbers in the table, its logarithm may be found by addition (912). For example, log. 6 = log. 2 + log. 3= .77815 12503 83643; log. 1001= log. 7 + log. 11+ log. 13 = 3.00043 40774 79319. These results may err in the last figure ; the loga rithm of 6 to fifteen figures, has the last figure nearer to 4 than to 3. When the given number is not the product of numbers in the table, its logarithm may be calculated by the fol lowing formulas : M = .43429 44819 0325; 1 log. n = log. (n - 1) + 2 M (^^I + 3 z-n 067781 10-SI 2-H 55907I 01-81 12-H 085801 88-81 03-4I 09:3I?I 98-81 7l-4l II228I "3l 88 L"I-H 398081 80-SI H00!.-8f6 -81 U. 60-14 375418 l7-8I L0-11 97558I 27-SI io-ii 6sI est 09-8I :o-H nss7si 00-81 99-81 208118 Hi-8I 3.l 99 1 89S98I 19'8l lo7os1 80-81 69'81 16-81 630061 99-18 S9-8I 39H19 89*3I 4ossere 98-81 I5'!:I 84-18 KI80I -8I 81 "8I 18 89390I 9f8I 67-81 807HSI 84-8I 97-81 90995I I4-8I -8I 17 30F90I 88-81 17-81 8S397I 9i:-3I 69-81 H8019 83-18 99-81 W8819 -si 39 W81 189791
"euiso^
„\,\i auBjoo
nun
'J L s
o m n
so-n
*-I 00-H 59905I 78-14 7sessi 75-14 i7CIWI Kfll WH i0sso5'I 17«I16 19'H 1302291 18-H 007715 "" 81-14 1231816 07-14 095815 cf„!98W9I 97-H 800101 1 2l'4 s7-n 1K.W1 |380i16 80-14 64I116 177.,9t | 30-14 932016-0 499916'9 88-H io-H 858316 -n us 2897fll lii-n 3H316 001891 7s-H 88-H 1 00919 183016 13-14 'H 451'I11 715701 22-14 ss-n 078ftTI 290I71 oriI os-h 91571U 98971I 91*14 n-14 088091 79277I ::I-H n-14 98589I HS73I orn 24-H 029716 CCT74t 7o-H l 39-1I 7K-o7ro I '.o-i .1*9'" '"" 391 „ fi8'H 988711 01.7371 70073I 800S71
..1.1.H
srn
n fl 11 5I 11 71 81 01 f:S is rs 83 12
':' !|s 7s
ss 112 00 is 32 38 K
ss
!''.' 7s
ss 80 oi-
n si-
st ii 15 H4 U
l» 49 09
I9 52 89 54
ss 56 7S 89 09
,,..1d
912
ZC8KVi 201561 530519 0I711CI II7519 02308I 010019 078019 999020 19H20 342220 nostis'i 977"20
82-81 26-81 23-SI 21-8I -SI 81 -3l 91 -8I SI
irsi 08-8I 90-81 Hi-8I
io-si
c« re.,.
suooro
O'l
035020 458120 015320 I97320 zs7eoz 025013 001520 720920 301720 717S20 oiosos-o 200103 200213 801I21 5I3113 19313 1
O9 8'J
00-31 90-31 01-31 02-31 900120 "' .i'l0707iC sorcre ™;E!l58W20 819H3 222020 89W21 "f2I 299003 085721 0T21 oo7on-i 899GI5"9 W2I 12 2615 9ss7n 5-7-12 3H821 1oz-13 "2I 73 ssoen osgsis 71-3l 8818I3 0I70I2 89"! zmzz 794521 "31 99 22 1 373 888513 KT12 :oo:ii3 352033 20-3I 158912 eo'zi 80*223 009721 703623 57-3I 6 ' 328122 6388131. "3l 98 0II192 915523 89-3I ososss 8I8O21 03-31 007922 819203 8-1-21 I71722 736122 91-12 083522 15I223 ll"! iossss looozz ."I 21"! 909822 .033 ''2':o 39-3I oreizs 983082 31 028I33 020522 .5s-zi 990383-o 888922-( ::8T.! 265323 s7soss I8-3I II7322 98S823 23-12 804823 458123 26-31 487822 085123 re-zi ossssz 1S9523 33-12 2502',:3 nmsz 20-31 H118723 8400::3 srzi 2081S3 147813 9C12 :78883 112423 W12 229083-6 37I332'I 3I"2l 710812 998332 00"! 18I413 259123 xra 65sIre 015383 .5o'zi oiosre 730983 '31 80 isesre 507923
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983223 ossire 07'II 67ssre 850823 95-II 6l39re 079683 |„liI>I 8Ua1UC
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81 71 91 i_i
I1 81 P1 1I 01 1i 8 7 9 5 4 8
z i
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SINES AND TANGENTS.
io° 8ine
10 11 12 13 14 15 IB 17 I8 19 20 21 22 23 21 25 23 27 28 29 30 31 32 33 31 8o 38 37 38 39 '10
a 42 43 44 15 48 47 48 40 50 51 52 53 54 65 50 57 58 59 60
9.239070 11.93 240386 11.91 241101 11.89 211814 11.87 212523 11.85 243237 11.83 243317 11.81 244698 11.79 215333 11.77 2100i9 11.75 218775 11.73 9.217478 11.71 248181 11.69 248883 11.67 249583 11.65 250282 11.63 250080 11.61 251677 11.59 252373 11.58 253037 11.56 253701 11.54 9.254453 11.52 255144 11.50 255834 11.48 253523 11.48 257211 11.44 257893 11.42 258583 11.41 259238 11.39 259951 11.37 230333 11.35 9.231314 11.33 231994 11.31 232373 11.30 233351 11.28 264027 11.28 234703 11.24 235377 11.22 238051 11.20 236723 11.19 237393 11.17 9.288003 11.15 238734 11.13 239402 11.12! 270039 11.10 270735 11.03 271400 11.03 272014 11.05 272720 11.03 273338 11.01 274019 10.99 9.274708 10.98 275337 10.98 270025 10.94 276681 10.92 277337 10.91 277991 10.89 278045 10.87 279297 10.83 279948 10.84 280590 I Cosine
T9°
PPl"
Tung. HI" 1 9.240319 12.30 247057 12.28 247794 12-26 248530 12.24 249264 12.22 249998 12.20 250730 12.18 251461 12.17 252191 12.15 252920 12.13 253348 12.11 9.254374 12.09 255100 12.07 255824 12.05 250547 12.03 257269 12.01 257990 12.00 258710 11.98 259429 11.93 200146 11.04 260383 11.92 9.231578 11.90 262292 11.89 203005 11.87 283717 11.85 281428 11.83 263138 11.81 265847 11.79 288555 11. 237231 11.76 267967 11.74 9.238871 11.72 269375 11.70 270077 11.69 270779 11.67 271479 11.65 272178 11.64 272370 11.02 273573 11.60 274289 11.58 274964 11.57 9.275358 11. 270351 11.53 277043 11.51 277731 11.50 278121 11.48 279113 11.4: 279801 11.45 280488 11.43 281174 11.41 281858 11.40 9.282542 11.38 283225 11.36 283907 11.35 281588 11.33 285268 11.31 285947 11.30 286021 11.28 287301 11.26 287977 11.25 283652
M. 9.280599 281248 284897 282544 283190 283830 28441-0 285124 285766 286408 287048 9.287688
10.82 10.81 10.79 10.77 10.76 10.74 10.72 10.71 10.69 10.67 10.66 10.64 10.63 288964 10.61 289600 10.59 290230 10.58 290870 10.56 291504 10.54 292137 10.53 292768 10.51 293399 10.50 9.294029 10.48 294058 10.46 295286 10.45 295913 10.43 296539 10.42 297164 10.40 297788 10.39 298412 10.37 10.36 299655 10.34 9.300276 10.32 300895 10.31 301514 10.29 802132 10.28 302748 10.20 30,3364 10.25 803079 10.23 304593 10.22 305207 10.20 305819 10.19 9.300430 io.i307041 10.10 307050 10.14 308259 10.13 308867 10.11 309474 10.10 310080 10. 0! 310385 10.0; 311289 10.00 311893 10.04 9.312495 10.03 31309: 10.01 313098 10.00 31429; 9.98 314897 9.9; 31549: 310092 9.90 310089 9.94 317284 9.93 9.91 317879
PPPM Ootnng. l'Pl'
Tri-;.—32.
11"
PPl"
iV»
377
PPl 'Pang. 9.288052 11. 23 289320 11.22 289999 11.20 290071 1.18 291342 11.17 292013 11.15 292082 11.14 293350 11.12 294017 11.11 294084 11.09 295349 11.07 9.296013 11.06 296677 11.04 297339 11.03 298001 11.01 298662 11.00 299322 10.98 209980 10.96 300638 10.95 301295 10.93 301951 10.92 9.302607 10.90 303201 10.89 303914 10.87 304507 10.86 305218 10.84 305869 10.83 306519 10.81 307168 10.80 307816 10.78 308163 10.7 9.309109 10.75 309754 10. 310399 10 - 311042 10.71 311685 " 10 31232' 10.68 312968 10.07 313008 10.65 31424 10.64 314885 10.02 9.315523 10.01 316159 10.60 316795 10.58 3174 10. 318064 10. 31869' 10.54 319330 10.53 319961 10.51 320592 10.50 321222 10.48 9.321851 10.4 322479 10.45 323100 10.44 023733 10.43 324358 10.41 321083 10.40 325607 10.39 320231 10.37 326853 10.30 327475
PPl" Co tana
37 36 35 34 33 32 31 30 29 28 27 20 25 21 23 22 21 20 19 I8 17 16 15 14 13 12 11 10 9 8
PP1'
TS-
TAI1LE IV.— LOGAR1THM1C
12°
Mil ''
PlT
lun-:
PPr'i »l.
! Pl-P I
U.327175 828095 8190861" 319058 J
828715 32933! :Vi\ 50
320210 320810 321430 322010 323107 323194 323780
7 8
9 1 i 11 12 13 1t 15 10 17 1-i 111 20 21 22 23 24
1.821300 1121950
325534l"' 320117 321700 327281 327802 32s! 142 329021 329599 1.330179 330753 331320 831903 832478 333051
'ri 20 27 -N
29 30 31 32 33 .",1 35 33 37 38 30 40
'mm 331195 83170' 33583; 1.335900 3304' 337043 337010 338176 838742
£0 51
839307 3!9871 8104841 310990 '.341558! 342119 842679 318239 343797 344355 311912 315409 340024 310579 1.347184
52 53 54 55 50 57 58 59 60
347087 348240 318792 319343 319803 350143 350992 351540 352088
'a 42 43 44 45 40 47 '18 49
"57|7 77'
829353 1 J"
831187 JJ 33211' ' ' 333033 833640 2,i ai 9.331259 334871 335182 330093 330702 817311 £77919 83SJ27 339133 839739 9.310314 310948 811552 312155 342757 343358 343958 311.538 315157 315755 9.3401:3 310949 317545 848141 3487X5 310329 319922 85051! a51100 851697 9.3.52287 852876 353465 854053 351040 355227 355813 350398 350982 3575C0 9.a58149 358731 339313 359893 300174 361053 301632 862210 362787. 303364
i.K8
373114 9.373933
s»»£S
374452 371970 375487 370003 370519 377035 377.340 378003 378577 9.379089 879G01 38C113 3;-U"21 £1134 , 8.49 38104 8821.52 8. 17 382661 8.83108 8.45 383075 l Conine. 1 PPP
Cotang. i PP1"
378
13° 1'iiir.
9.31,8304 353940 304515 8K0D0 303004 36G237 300810 307382 307933 | 808324 8C9094 9.309063 370232 870799 371367 371933 872199 873064 373029 37119:; 37-1750 9.875310 375881 370442 877603 877503 878122 378681 379239 370797 30i8C1 9.380910 881100 882020 882575 883129 383082 884234 3S1780 885337 885888 9.886438 380987 317530 388084 388031 880178 389724 890270 300815 891300 9.8!11903 892147 392989 803531 394073 894014 895154 895094 3Gi233;„ 896771 16,
C0
so 38 C7
r.u 55 'A 53 51
so 49 48 47 40 15 41 43 42 41 40 89 88 30 85 31 33 82 81 80 2i 28 26 25 L1 21 21 20 19 18 17 16 15 11 13 12 ll' 10 9 8 7 6 5 4 3 2 1 0
Cotaug. i PPl"! ?.I.
7H"
14°
9 10 11 12 13 14 13 13 17 18 19 20 21 22 2.3 21 25 20 27 28 29 80 31 o2 33 31 85 30 37 88 39 40 '41 42 '43 44 45 -Hi 47 '18 49 .50 51 62 53 ol 55 56 57 58 .59
, 38337. ' 384182 381087 385192 385097 380201 3S0701 387207 387709 388210 388711 .389211 389711 390210 390780 391200 391703 302199 392095 893191 393035 .394179 394073 395100 3950-58 390150 390041 397132 397021 338111 398300 .399088 399575 400032 400549 401035 401520 402005 402189 402972 401155
. 40 mi 401120 401901 405382 405802 408341 40J820 407299 407777 408254 .408731 400207 400082 410157 4100-32 411100 411579 412052 412524 412990 Cusin
74 J
15°
SINES AND TANGENTS. Mil
PP1" 1.44 8.43 8.42 8.41 8.40 8.39 8.38 37 8,3i 1.35 i.31 8.32 8.31 8.30 8.28 8.27 1.20 8.25 8.24 8.23 1.22 8.21 8.20 8.20 8,18 .17 8.17 16 15 8.14 13 8.12 8.11 8.10 8.09 8.08 8.07 8.00 8.05 8.04 8.03 8.02 8.01 8.00 7.99 7.98 7.07 7.0 i 7.95 7.94 7.94 7.93 7.92 7.91 7.90 7.89 7.88 7.87 7.86
'Pn i
9.390771 397309 897846 398383 398919 3994.55 399990 400524 401058 401591 402124 9.402056 403187 403718 464249 404778 405308 405836 403304 403892 407419 9.407945 408471 409521 410045 410509 411092 411615 412137 412658 9.413179 413699 414219 414738 415257 415775 410293 410810 417320 417842 9.418358 418873 419387 419901 420415 420927 421440 421952 422403 422974 9.423484 423993 421503 425011 425519 420027 426534 427041 427547 428052
PP1" y\
M.
PPP 1.412990 7.85 413407 7.81 413938 7.83 414408 ,88 414878 Z 7.82 415347 7.81 415815 7.80 416283 7.79 410751 r8 417217 417684 7.76 1.418150 7.75 418615 74 419079 7.73 419544 7.73 420007 7.72 420470 7.71 420933 7.70 421395 7.09 421 .857 7.08 422318 7.07 1.422778 ,07 423238 '.66 423097 .65 424156 .64 421015 .63 425073 .02 425530 .61 425987 .60 426443 ,1-0 426899 .59 9.427354 '.58 427S09 .57 428263 .50 42871 429170 .55 '.54 429023 486075 .53 430527 7.53 .52 480978 7.51 431429 3.431879 .£0 -C0 432329 .49 432778 .48 433220 ,47 433075 .46 434122 ,4.5 434-;oii .45 4X010 .14 435402 .43 43-5008 .42 3.436353 .41 431*798 .10 437242 .40 437080 ,89 438129 38 438572 ' .37 439014 L 30 439450 '.35 439897 .35 440338
8.06 ,8.00 8.95 8.94 8.93 .02 .91 .90 ,8.89 8.8,8 8.87 8.80 8.85 84 83 8.82 8.81 8.80 8.79 t.78 1.77 8.70 8.75 8.74 8.71 73 8.72 8.71 8.70 8,00 8.08 8.07 8.00 8.0-5 8.04 8.04 8.63 8.02 8.61 8.00 8.-59 8..58 8.57 8.50 8.55 8,55 8.54 8.53 8.52 8.51 8.50 8.49 8.48 8f8 8.17 8.40 8.45 -8.44 8f3 8.43
1'P1" 9.428052 8.42 . 428558 ,8.41 429002 8.40 429560 8.89 430070 8.38 4» 573 8.88 431075 8.37 431577 8.80 432079 8.35 432580 8.34 433080 8.33 9.433580 8.32 434080 8.82 434579 8.31 435078 8.30 435570 8.20 430073 8.28 430570 8.28 437007 8.27 437503 8.20 438059 8.25 9.438554 8.24 430048 8.23 439543 8.23 440036 8.22 440529 8.21 441022 8.20 441514 8.19 442000 8.19 442497 8.18 442988 -8.17 9.443479 8.16 443968 8.10 444458 8.15 444947 8.14 445435 8.13 445923 8.12 440441 8.12 440£98 8.11 447384 8.10 447870 8.09 9.448350 8.09 448841 8.08 449320 8.07 449810 8.06 450294 8.06 450777 8.05 451260 01 451743! 8.03 452225 8.02 452700 8.02 9.453187 8.01 453008 8.00 454148 7.00 454028 7.09 455107 455580; ,.' 450004 J7.97 7.06 7.90 457019 7.95 457496
60 59 58 57 50 55 54 53 52 51 £0 40 '18 47 40 45 '11 48 42 41 40
37 30 85 34
83 32 31 80 20
28 27 20 25 24 23 22 21 20 19 18 17 10
15 11 13 12
11 10 9 8 7 6 5 4 3 2 1 0
lPPl" M.
Cotang. PP1"
379
74"
'
TABLE IV.—LOGARITHMIC
16° ii'i'i' 9.440338| 440778 :'£ 441218''™ 441058|!'Jf 442096™!
lPPl
PfV
9.457496. 457973 '' 458449 458925 459400
0.405035 6.88 466348 0.88 400701 0.87 467173 0.80 407585 6.89 407990 0.85 468407 0.84 468817 6.83 469227 6.83 469637 6.82 470046 0.81 9.470455 6.80 470863 80 471271 0.7!i 471079 0.78 472086 0.78 472492 0.77 472898 0.77 473304 0.70 473710 0.75 474115 0.74 9.474519 0.74 474923 0.73 47552' 0.72 475750 0.72 470133 6.71 470530 0.70 470038 0.70 477340 6.69 477741 8.68 478142 9.478542 6.67 0.07 478942 6.66 479342 0.05 479741 6.65 480140 0.01 480539 0.03 480937 0.03 481334 0.02 481731 0.02 482128 0.01 9.482525 0.00 482921 6.59 483310 0.50 483712 0.58 48410 6.57 484501 0.57 484895 485289 6.56 6.55 485082 6.55 480075 6.54 9.48040: 6.53 486860 6.53 487251 0.52 487643 6.51 488034 0.50 488424 0.50 488814 6.50 489204 6.49 489593 6.48 48C98:
*amVz s 9 10 11 12 13 11 15 16 17 18 19 20 21 22 2! 24 25 20 27 2S 29 30 31 02 33 34 35 36 '37 38 39 40 41 '42 43 44 15 -1O 47 48 4D 50 51 52 53 51 55 50 57 58 59 60
442!i73!^0 443410 lm~ 4i'isr '•-** 7.27 444284 .27 444720 .28 9.445155 .25 445590 7.24 446025 7.23 440459 23 440803 22 447323 7.21 447759 7.20 448191 7.20 448023 7.10 449054 7. IS 9.449485 7.17 449915 .17 450345 .10 450775 .15 451201 .11 4'51032 7.13 452000 7.13 452 488 7.12 452915 .11 453342 .10 9.453788 7.10 454101 7.03 454010 455044 7.03 7.07 455409 7.(17 455803 450E10 7.03 7.05 45S739 7.05 457102 7.01 457584 9.458006 7.03 .02 458 42: .01 458848 450208 .00 7.00 459388 400 108 6.99 8.98 40o; 400040 0.98 6.97 401301 401782 6.96 0.95 9.462189 402310 6.95 6'91 463032 0.03 403448 6.93 403864 6.92 404279 6.91 404094 465108 8.90 40552; 6.90 O.SL) 46593'' Cosini
Ti"
400349 400823 1 401297 401770 402242 l' 9.402715 463180 403658 404128 404599 405089 405539 400008 466477 460945 9.407413 407880 408347 408814 409280 409740 470211 470376 471141 471005 9.472039 472532 472995 473457 473919 474381 474842 475303 475703 470223 9.470083 477142 477001 478059 47851 47897, 479432 479889 480345 480801 9.481257 481712 48210 482021 483075 483529 483982 484435 484887 485539
IPP1"! Cotan'
Cosine.
380
err
17° I'm 9.4853391 485791 480242 486093 487143 487593 488048 488492 488941 469390 9.490286 490733 4911*0 401027 492073 • 492519 492965 493410 493854 494299 9.494743 495180 495030 ' 4960731' 496515 C 490957 17, 497399 497841 498282 498722 9.499103 499003 90C042 £00481 £00920 5013T19 £01797 £02235 £02672 £03100 9. £03540 £03982 £04418 £04854 505289 505724 500159 E06G93 £07027 C07460 9.507893 £08526 £08759 £09191 £09622 510054 510485 510910 511340 511770
C0 59 58 57 56 55 54 53 52 51 50 49 48 47 40 45 44 43 42 41 40
35 31 83 32 81 30 20 28 27 20 25 24 28 22 21 20 19 18 17 10 15 14 13 12 11 10 9 8 7 0 5 4 8 2 1
0
Cotnng.
T2°
S1NES AND TANGENTS.
IS PPl"!
M. 1.48!M82 490371 490759 491147 49153i 491922 49231)8
6f8 ii. 17 0.47 6.48 8.45 6.45 6.44 6.43 493081 6.43 493100 0.42 493351 6.42 9.494236 0.41 494021 3f0 495005 3.40 495383 3.33 495772 3.33 498151 3.33 493537 6.87 498919 3.37 497301 6.38 497082 6.38 9.493031 6.35 498114 3. .31 498325 3.33 499201 3.33 499581 3.32 499933 3.32 500342 6.31 500721 3.30 501033 3.33 501470 3.23 9.501851 3.23 502231 3.23 502307 3.23 502981 3.27 503330 3.23 503735 3.23 501110 6.25 501135 3,23 501300 6.21 505234 8.23 9.505808 3.22 505931 3.22 508351 3.22 503727 6.20 5070J9 6.20 507171 3.23 507843 6.19 508214 3.13 508385 3.13 50895B 3.17 9.50932 3.17 50939 3.16 510335 6.15 510431 3. 15 510303 15 511172 0.14 511510 3.13 511907 0.13 512: 3.12 512342 Cosine.
71"
PP1
Tn
PP1
1»° P1T'
9.512042 0.12 513009 3.11 513375 3.10 513741 3.10 514107 0.09 514472 3.08 514837 0.08 515202 6.07 515500 3.07 515930 3.03 516294 6.05 9.516057 3.05 517020 6.01 517382 3.64 517745 3.03 518107 3.02 518468 0.03 518829 0.02 519190 3.01 519551 3.00 519911 0.00 9.520271 3.00 520631 5.99 520990 5.98 521349 5.97 52170' 5. 08 522000 5.07 522424 5.90 522781 5.95 523138 5.95 52349. 5.95 9.523852 .3.01 524208 5.93 524564 5.03 524920 5.02 525275 5.02 525630 5.00 525984 5.91 520339 5.00 526693 5. SC 527046 5.00 9.527400 5.89 5277,53 5.87 528105 5284.58 5.88 5.S7 528810 5. 85 529101 5.80 529513 5.85 529804 5^85 530215 5.84 530505 5.83 9.530915 5.83 531205 5.82 531014 5.82 531963 5.82 532312 5.81 532661 5.80 533009 5.80 53335; 5.80 533704 5.79 534052
9.511770 7.10 512200 7.13 512335 7.15 513064 7.14 513103 14 513921 7.13 514349 7.1.3 514777 7.12 515204 7.12 515031 7.11 510057 7.111 9.510481 7.10 513910 7.09 517335 7.03 517761 7.03 518130 .08 518610 7.07 519031 7.0 i 519458 7.03 519382 7.03 520305 .05 9.520723 7.01 521151 7.03 521573 7.03 521995 03 522117 7.02 5228:38 7.02 523259 7.01 523380 7.01 521100 7.0ii 521520 6.99 9.521940 6.99 525359 6.98 525778 3.93 523197 6.97 520015 6. 07 5270:33 8.08 527451 6.93 527838 6.95 52823: 6.95 528702 3.111 9.529119 6.93 52953: 6.93 529951 6.93 530300 3.32 530781 3.01 531198 6.91 531011 6.90 532025 6.90 532439 0.89 532853 3. 83 9.533230 1!.,3,3 533879 3. 83 53409: 3.87 531,501 3.87 531910 3. 83 535323 0.86 535739 3. S3 530150 3. 83 630531 3. 81 5.33072 Cotang. PP1
iiiill11
381
33iii2
":"
9.536972 3.81 537382 3.83 537792 3. 83 5,38202 3.82 538011 6.82 539020 6.81 539429 3. 81 539837 6.80 540245 6.80 540053 3.79 541061 3.70 9.541468 0.78 541875 3.78 542281 3.77 542688 3.77 543094 3.73 543499 3.73 543905 3.75 544310 3.75 544715 3.74 545119 6.74 9.545524 0.73 545928 8.73 546331 3.72 546735 8.72 547138 8.71 547540 8.71 547943 8.70 54834= 8.70 548747 6.69 549149 8.1)0 9.519550 8. 1is 549951 6.68 550352 6.67 550752 8.87 551153 .60 551552 8.88 551952 6.65 552351 65 552750 6.65 553149 8.1,1 9.553548 8.81 553940 6.63 554344 6.63 554741 6.62 555189 8.82 5555,3i 8.81 55503;; 8.81 550329 8.30 556725 60 557121 6.59 9.5575! 8.50 55791.3 6.59 558308 6.58 558703 6.58 550097 8.57 559491 8.57 559885 6.56 560279 8.58 660673 0.55 561066 1'ntiiUR.
PPl
60 59 58 57 56 55 51 53 52 51 50 13 48 47 46 45 14 13 12 11 40 30 38 37 .30 35 31 33 32 fil 30 29 28 27 28 25 21 23 22 21 20 13 18 17 16 15 14 13 12 11 10
o
i,
2 1 0
TABLE IV.—LOGAR1THMIC
20°
Tang
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 -40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
.534052 534399 5.77 534745 5.77 535092 5.77 535438 5.70 535783 5.70 53612U 5.75 536474 5.74 536818 5.74 537163 5.73 537507 5.7:; .537851 5.72 538194 5.72 538538 5.71 538880 5.71 539223 5.70 539565 5.70 539907 5.70 540249 540590 5.09 5.68 540931 5.08 .541272 5.07 541613 5.07 541953 5.68 542293 5.05 542032 5.65 542971 5.65 543310 5.65 543649 5.64 543987 5.63 544325 5.63 .544663 5.02 545000 5.62 545338 5.01 545674 5. CI 546011 5.00 510347 5.00 546683 5.00 547019 5.59 547354 5.58 547689 5.58 .548024 5.57 548359 5.57 548693 5.50 549027 5.55 549390 ~uh~> 54U693 5.55 550026 5.55 550359 5.54 550692 5.53 551024 5.53 .551350 5.52 551687 5.52 552018 5.52 552349 5.51 552680 5.50 553010 5.50 553341 5.50 553670 5.50 554000 5.49 554329 Cosine.
C90
PP1'
9.561060 501459 561851 502244 562636 5S3028 563419 503811 564202 564593 564983 9.565373 565763 566153 566542 566932 567320 567709 568480 568873 9.569261 569648 570035 570422 570! 571195 571581 571967 572352 572738 9.573123 573507 573892 574276 574660 575044 575427 575810 576193 576576 9.576959 577341 577723 578104 578486 578867 579248 579029 580009 580389 9.580769 581149 681528 581907 582280 582665 583044 583422 583800 584177 Cotang.
PPl' 0.55 6.54 0.54 6.53 0.53 0.53 0.52 0.52 6.51 6.51 6.50 6.50 0.49 0.49 0.19 0.48 0.48 0.17 6.47 0.40 0.40 0.45 0.45 0.15 0.44 0.44 0.43 6.13 0.42 1.42 0.42 0.41 0.41 0.40 0.40 0.39 6.39 39 0.38 0.38 6.37 6.37 6.36 S.86 6. 36 6.35 0.35 6.34 0.34 6.34 6.33 6.33 0.32 0.32 0.32 0.31 6.31 0.30 6.30 0.29
PPP 9.554329 554658 554987 555315 £55643 555971 556299 556626 556953 557280 557606 9.557932 558258 958583 558909 559234 559'558 559883 56020; 560531 560855 9.561178 561501 561824 562146 562468 562790 563112 563433 563755 564075 9.564396 564716 505030 505350 565070 505995 566314 566032 566951 567269 9.507587 567904 508222 56853!. 568851 569172 509488 509804 570120 570433 9.570751 571060 571380 571095 572009 572323 572636 572950 573263 573575 Cosine.
5.48 5.48 5.47 5.47 5.47 5.40 5.45 5.45 5.45 5.44 5.43 5.43 5.42 5.42 5.42 5.41 5.41 5.40 5.40 5.40
5.38 5.37 5.37 5.37 5.30 5.85 5.35 5.85 5.34 5.34 5.33 5.33 5.33 5.32 5.32 5.31 5.31 5.30 9.30 5.29 5.29 5.28 5.28 5.27 5.27 5.27 5.20 5.25 5.25 5.25 5.24 5.24 5.23 5.23 5.23 5.22 5.21 5.20
21° Tans 9.584177 584555 584932 585309 585686 586002 586439 586815 587190 587560 587941 9.588310 588691 589066 589'440 589814 590188 590502 590935 591308 591681 9.592054 592426 592799 593171 593542 593914 594285 594656 59502; 595398 9.595768 596138 596508 590878 597247 597610 597985 598354 598722 599091 9.599459 . 599827 600194 6C05C2 000929 C01290 601603 602029 602395 602761 9.603127 C03493 C03858 C04225 C04588 604053 60531' 605682 606040 C06410
M. 6.29 6.29 6.28 6.28 6.27 6.27 0.27 0.20 0.20 0.25 0.25 C.25 0.24 0.24 0.23 6.23 0.23 0.22 0.22 0.22 0.21 0.21 0.20 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.17 0.17 0.10 6.16 0.10 0.15 0.15 0.15 6.14 0.14 6.18 0.13 0.13 0.12 0.12 0.11 6.11 6.11 0.10 0.10 0.10 6.09 6.09 6.09 6.08
00 59 58 57 56 55 54 53 52 51 50 -19 48 47 40 45 4'1 -13 42 41 40 39 38 37 36 35 31 33 32 31 30 29 28 27 20 25 24 23 22 21 20 19 18 17 16 15 14 18 12 11 10
u.os 0.07 fi.07 0.07 0.00
PPl" Cotang. PI'l"
fcS"
22 M. 0 1 2 3 1 9
SINES AND TANGENTS. lPPi 9.573575l 573888 :™, 574200.?™ 5i4-i12i- ,„.
a !"i is 7 8 9 10 11 12
vt u 15 l|i 17 I8 19 29 21 22 23 'J! 25 20 27 28 2O 30 31
33 39 40
'il 42 43 11 15 41,i 17 48 49 50 51 52 53 54 55 50 57 58 59 CC
575758 \" 570069?-" 576379 |?"" 570689 .1 >. 17 9.570999 ..10 :309 i.15 577018 .- .i. lo 57792' 5.15 578230 ': 5.15 578515 5. 14 578853 179102 .i.18 ... 579470 *}? 579777 «'» 9.580085 ?*, 680382?*, o. 1. 58009 ,.l1 58100 581312 5.11 1.10 581618 581924 i.10 582229 5.09 582535 i.09 i.08 582840 9.58314-: i.03 i.08 583119 583751 5.07 584058 5.07 584331 5.06 584005 5.00 584908 ,05 5.05 585272 585574 5.05 585877 5.01 9.580179 5.04 5.03 580482 5.03 586783 5.03 587085 5.02 587386 5.02 G87688 587989 5.01 583289 5.01 5.01 588590 5.00 588890 5.00 9.589190 1.99 589489 4.93 589789 1.98 590988 1.98 590337 1.98 590 18O 1.97 590D84 4.97 591282 1.97 591580 !.97 591878 i'..si
6V
Tuna
PP 1"
9.609440 0.00 00877: 0.08 00713: 0.0-5 607.500 0.05 607863 ii.lll 608225 6.04 608588 6.04 008950 6.03 009312 6.03 609074 6.03 610030 6.02 9.610397 0.02 610759 0.112 61U20 6.01 0114.80 6.01 611841 6.01 612201 i1.110 012-501 8.00 612921 6.00 013281 5.!0 613641 5.90 9. 01 4000 5.98 014359 98 614718 98 615077 5.: i7 6154:35 97 615793 5.07 610151 5.96 018509 5.96 618807 5.98 617224 .95 9.017582 5.95 617939 5.95 018295 5.94 618052 5.94 619008 5.94 619364 5-93 619720 5.93 020079 5.93 620432 5.92 020787 5.92 9.621142 5-92 621497 5.91 621852 5.91 622207 5.90 622561 5.90 622915 5.90 623209 5.89 623623 5.89 023970 5.89 024330 5.88 9.021083 5.88 025030 5.88 625388 5.87 625741 5.87 62ii093 5.87 626445 5.86 620797 5.80 027149 5.80 627501 5.85 627852
60 59 58 57 50 55 51 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 88 37 33 35 34 33 :ji
31 30 20 28 27 20 25 21 23 22 21 20 10 18 17 16 15 14 13 12 I1 10 9 8 7 6 5 4 3 2 1
0
1.591878 592176 .95 502473 1.05 592770 1.05 59:!067 1.94 59:1363 1.01 593659 1.98 593955 1.93 594251 1.93 594547 1.02 594842 1.595137 1.02 0..' 1.02 595432 1.01 595727 1.00 590021 1.90 590315 4.90 590609 4.90 590903 4.89 £97196 4.89 597490 4.88 597783 4.88 .598075 4.87 598368 4.87 598000 4.87 598952 4.80 999244 4.80 599530 4.85 599827 4.85 000118 4.85 600409 4.85 000700 4.84 600990 4.83 001280 4.83 601570 4.83 601800 4.83 002150 4.82 002439 4.82 602728 4.82 003017 4.81 003305 4.81 603594 4.80 .003882 4.80 004170 4.79 004457 4.79 604745 4.79 605032 4.78 605319 4.78 005000 4.78 605892 4.77 000179 4.77 000465 4.76 .6067 i7-il 4.70 60'17036 4.70 17322 00732 4.75 007007 4.75 007892 4.75 608177 1.71 008401 4.74 008745 4.73 609029 4.73 009313
23° Tun 9.027852 . 60 5.85 028203 59 5. 85 028554 58 5.8--i 628905 57 5.84 629255 56 5.84 629600 5.83 55 629956 54 5.83 i0i300 53 5.83 52 680656 5.83 631005 51 5.82 631855 82 £0 9.631704 40 5.82 032053 48 81 632402 '17 81 632750 46 5.81 45 633099 5.80 63344 ':i 5.80 633795 43 5.80 634143 42 5.70 41 634490 5.70 634838 40 5.79 9.635185 39 5.78 635532 38 5.78 635879 37 5.78 636226 36 5.77 036572 35 5.77 030919 31 5.77 037205 33 5.77 037011 32 5.70 037950 31 5.70 038302 80 5.70 9.038047 29 5.75 28 638992 5.7.i 27 639337 5.75 20 639082 '4 640027 26 5.74 21 640371 5.74 23 640716 5.73 22 641000 73 641404 21 73 20 041747 5.72 19 9.042091 5.72 18 042434 5.72 17 642777 72 16 043120 5.71 15 643463 1 14 643800 5.71 18 644148 5.70 12 044490 5.70 11 044832 0 10 045174 5.09 9 9.645516 5.69 8 045857 5.69 7 646199 5.09 0 640540 5.08 646881 5 5.08 4 647222 5.08 3 647502 5.67 2 047003 5.67 1 048243 5.07 0 618583;
PP1" Cntim
ll'l'l" i iitlln!
383
|PP1"
«ii
TABLE IV.—LOGARITHMIC
24° PPP
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 28 27 28 29 30 31 32 33 31 35 33 37 38 39 !0
a 42 43 44 45 46 17 -I8 49 50 51 52 53 54 55 56 57 58 59
.609313 1.73 009597 1.72 609880 4.72 610164 4.72 610447 1.71 610729 1.71 611012 4.70 611294 4.70 611576 4.70 611858 4.70 612140 1.69 ,612421 1.69 612702 4.68 612983 1.68 613264 4.(17 613545 1.67 61382.= 4.O7 614 10': 4.66 614385 1.66 614665 4.fii 614944 1.65 ,615223 1.65 615502 4.65 615781 4.64 616030 1.64 616338 4.64 616616 4.63 616891 4.63 617172 1.62 617450 1.62 617727 1.62 618004 1.01 618281 4.61 618558 1.61 018331 4. C0 619110 4.60 619383 1.60 619382 1.59 019933 4.59 620213 1.59 620488 1.58 .620763 4.58 021038 4.57 621313 1.57 621587 1.57 621861 4.56 622135 1.50 622409 1.50 6221)82 4-55 622950 1.55 623229 4.55 .623302 4.54 623774 1.54 624047 1.54 624319 1..33 624591 4.53 624833 1.53 625135 4.52 625406 4.52 625677 4.52 Cosiiv.
ti.V
PPI
Tang. 9.648583 648923 649263 649002 649942 6.50281 650620 650959 651297 651636 651974 9.652312 652050 652988 653326 653863 654000 651337 654074 655011 655348 9.655684 650020 05!>35fi 656092 657028 057364 657699 658034 658339 058704 9.059939 059373 659708 6ii0042 600376 600710 661013 631377 601710 602013 9.602370 662709 633042 603375 603707 664039 664371 664703 665035 665366 9.665698 666029 000300 666691 667021 667352 667682 668013 008343 Cotnnc.
8in,'. 9.625948 626219 626490 620760 627030 627300 627570 627840 628109 628378 628647 9.628916 629185 629453 629721
PPP
I o in
PPI"
4.51 4.51 4.50 4.50 4.50 4.50 4.50 4.49 4.49 4.48 4.48 4.17 4.47 4.17 4.4(1 4.40 630257 4.40 C30524 4.46 630792 4.45 631059 4.45 631326 4.45 9.631503 4.44 631859 4.44 632125 4.44 632392 4.43 632658 4.43 632923 4.43 633189 4.42 633454 4.42 633719 4.42 633984 1.41 9.634249 4.41 634514 4.40 634778 4.40 635042 4.40 635306 4.40 635570 4.39 635834 4.39 636097 4.38 636360 4.38 636623 1.38 9.636886 4.37 637148 4.37 637411 4.37 637673 4.37 637935 4.36 638197 4.36 638458 4.30 638720 4.35 638981 4.35 639242 4.35 9.639503 4.34 639764 4.34 640024 4.34 640284 4.33 640544 1.33 6408O4 4.33 641064 4.32 641324 4.32 641583 4.32 641842
I'l'l
384
25° Tang.
PPl" -M^
9.608673 5.50 669002 5.49 609332 B.49 5.49 669991 5.48 670320 5.48 670649 5.48 670977 5.48 671306 5.47 671635 .5.47 671903 5.47 9.672291 5.47 672619 5.46 672947 5.46 673274 5.46 673602 5.46 673029 5.45 674257 5.45 674584 5.45 674911 5.44 675237 5.44 9.675504 5.44 675890 5.44 676217 5.43 676543 5.43 670869 5.43 677194 5.43 077520 5.42 677840 5.42 678171 42 678496 5.42 9.078821 5.41 679146 3.41 679471 5.41 679795 5.41 680120 5.40 680444 5.40 680768 5.40 681092 5.40 681410 39 681740 9.39 9.682003 39 682387 5.39 082710 .5.38 683033 5.38 683350 5.38 683679 5.38 684001 684324 684040 5.37 684968 5.37 9.685290 5.30 685012 9.30 685934 5.36 C80255 5.30 686577 5.35 686898 9.*5 687219 5.35 687540 5.35 687861 31 688182
60 59 58 56 55 54 .53 52 51 50 49 48 47 40 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 2!i 28 27 20 25 24 23 22 21 £0 1!i 18 17 16 1e 14 13 12 II 10 9 8 7 0 5 4 3 2 1 0
PPl"
Ol"
26 M.
S1NUS AND TANGENTS. 8in
PP1'
9.641842 4.31 042101 4.31 042360 4.30 642618 1.30 642877 1.30 64:3135 4.30 043393 1.30 043650 1.29 643908 1.29 044165 1.29 044423 4.28 9.614680 1.28 044936 4.28 645193 1.27 645450 1.27 645708 4.27 045932 1.26 016218 4.26 69I474 4.23 616729 1.25 646984 4.29 9.647240 4.21 047494 4.21 647749 1.21 048001 4.23 618258 1.23 048512 1.23 .048708 1.23 049020 1.23 619274 1.22 649527 1.22 9.049781 4.22 050034 4.22 650287 1.21 050330 1.21 650792 1.21 631914 1.20 0:1237 1.20 651519 1.20 651800 1.10 6.52052 1.10 9.05230! 1.18 652555 1.18 652803 1. 18 053057 4. 1S 653308 1.17 053558 1.17 053898 1.17 6.51059 1.17 094309 1.13 654558 1. Hi 9.6.51808 4.10 055058 4.18 035307 4.15 655-550 4. 15 65580.5 4. 15 050051 1.14 050302 1.14 650551 1.14 650799 4.13 057017 Co8i1K
63°
P1T
Tun".
r| JH._
PPi'
PP1"
0.ossis:
'.6.57017 65720.5 637.542 657700 658037 658284 6.58531 658778 059025 059271 059517 1.659763 C60009 000255 000.501 C60740 060991 061230 001481 661726 661970 1.002214 6621,59 66270: 662946 063190 603433 06307' 063920 06416S 604406 1.664648 661891 665133 66.5375 665017 66.58,59 666100 606342 000583 666824 1.66706.5 067305 667546 007780 6ti8027 068267 668506 668716 668986 609225 1.609404 669703 669942 070181 070119 670658 670896 671134 671372 671009
5.34 688.502 5.31 08882.3 5.31 689143 5.33 689463 5.33 689783 5.83 690103 5.33 696423 5.33 690742 5.82 691002 5.32 691381 5.32 9.691700 .3.31 092019 .3.31 692338 31 602650 31 69297. 5.31 693293 30 693612 .5.30 693930 5.30 094248 5.30 694566 5.20 9.094883 5.29 695201 5.29 09.5518 5.29 695836 3.20 696153 5.28 69B170 5.28 693787 5.28 697103 5.28 697420 5.27 697736 5.27 9.698053 5.27 698389 5.27 098583 5.28 000001 5.26 009316 5.23 699632 5.26 699947 .3.21f 700203 9.25 700578 700893 5.25 5.25 9.701208 5.24 701523 '5.24 701837 5.24 702152 '3.21 702166 5.21 702781 5.23 7030*5 5.23 703409 5.23 703722 5.23 701030 5.22 9.701350 5.22 701663 '3.22 704976 5.22 709290 5.22 703603 5.21 705916 5.21 703228 5.21 706341 5.21 708854 .5.21 707106 Ootn
Trig.—33.
PPP
1.13 4.13 1.12 1.12 4.12 1.12 i.ll 1.11 1.11 1.10 1.10 1.10 1.09 1. 111i 4.09 1.00 1.08 4.08 1.08 1.07 1.117 1.117 1.07 4.08 1.06 1.08 1.0.5 1.05 1.0.5 1. 1 1.5 1.01 4.01 1.111 4.03 4.03 1.03 1.12 1.02
'1.112 4.02 1.01 1.01 4.01 4.01 4.011 4.li0 4.00 3.1:0 3.09 3.99 3.99 3. O8 8.98 3. 1 i,S 3.07 3.07 3.07 3.07 3.96 3.06 PPP
385
PP1 07166 07478 07790 08102 08414 08720 09037 09349 00600 09971 10282 10303 10904 11215 11525 11836 12146 12458 12766 13076 13386 13096 14005 14314 14621 14933 1.5242 15551 15860 10108 16477 107&5 17093 17401 17700 ltti17 18325 18633 18010 19248 19555 19862 20169 20170 20783 '21089 '21396 '21702 22009 2231.5 22621 22027 '23232 23.538 '23844 '24140 244.54 24760 25065 25370 '25674
5.20 20 5.20 5.20 5.19 5. 10 5.19 5.10 5.19 5.18 5.18 5.18 5.18 5.18 9.17 5.17 5.17 5.17 5.10 5.16 5. 1D 5.16 5.16 5.15 5.15 5. 15 5. 15 9.11 9.14 5.14 9.14 5. 1 1 5.13 5.13 5.13 5.13 5.13 5.12 5.12 9.12 9.12 5.12 5.11 5.11 5. 1 1 5. 1 1 5. 1 i 5.10 5.10 5.10 5.10 5.10 9.00 5.00 5.09 5.09 5.09 5.08 5.08 9.08
00 59 5s 67 50 5.5 .31 53 52 :,i .Ml
'111 ';s 1T 40
45 11 43 42 '11 40 39 U8 37 ;M 85 :;i 33 32 31 30 20 28 -7 20 2"i 21 23 22 21 29 19 is 17 16 15 !1 13 12 11 10 9 8 7 ii 5 1 3 2 1 0
Cotttng. PP1" M
62"
a«°
TABLE IV.—LOGARITHMIC
» i
2
:; 1 6 6 7 8 9 1D 11 12 18 14 15 Iti 17 18 111 20 21 22 i;: ;
24 25 28 27 28 29 30 31 32 33 31 35 38 87 38 39 40 -11 42 43 44 45 48 47 48 49 50 51 52 53 54 55 56 57 58 39 60
1.1(71609 671847 672081 672121 072558 672795 873032 673268 678505 673741 673977 1.674213 674448 671681 671919 675155 675390 675621 675859 676094 676328 1.676502 676790 677030 677264 677498 677731 677961 678197 678430 1.678895 679128 679.592 679824 680050 680288 680519 680750 1.681213 681443 681674 681905 682135 682365 682.595 682825 683055 683284 1.683514 683743 684201 684430 684658 684887 685115 685343 685571
1.085571 685799 686027 686254 686182 686709
9.725074'. 725979 '-' 726284 I?' 726.588/' 726892! 727197 !
727801 1, 687163 687389 687616 687843 1.688069 688295 688521 688747 668972 689198 689423 689648 689873 690098 1.690323 690548 690772 690990 091220 691444 691008 691892 692115 692339 1.C92562 692785 693008 693231 693453 693676
727805? 7281091': 728412!': 7287101' 9.729020 729323 729026 729929 730233 7305*5) ?' 780838!?' 731141 ? 731444 |? 781746* 9.732048? 732351 ?' 732653"?' 7829551?' 733257!?' 733558 '?' 733860 i? 784162? 731163 ? 734764 ?' 9.735060 735367 735668 , 735969 ?' 730269 ?' 736570 ?' 736870 ?' 737171 ' ' 737471 737771 . 9.738071 ?' 738371 ? 738671 *' 738971 \ 739570 739870 740169 740408 740767 9.741060 741365 741664
, * *' V ' *' ' *'
741962 j
742261 J' 742559 , 742858 j
743156 J 743454 ~ 743752 ' CotRim. ppi" M
ei°
:l.Ml 8.79 3.79 8.79 3.79 8.78 8.78 8.78 3.78 3.77 8.77 8.77 .'1.77 3.70 3.76 3.70 8.78 3.75 8.78 3.75 3.75 8.74 3.74 3.74 8.74 3.7:: 8.78 3.73 3.73 3.72 3.72 8.72 8.71 3.71 3.71 3.71 3.70 3.70 3.70 3.70 3.09 3.09
694120 694.342 694564 1.694786 695007 695229 695450 8.68 695671 3.88 3.08 696113 3.08 696334 3.07 696554 3.07 696775 3.07 1.696995 3.07 697215 3.06 697435 3.00 697654 3.00 697874 3.00 3.05 698313 8.65 698532 3.05 698751 3.05 698970
739271 J"
386
29°
P 1T
M
PPP 9.743752 744050 744348 744645 744943 745240 745538 745835 746132 746429 740720 9.747023 747319 747010 747913 748209 748505 748*01 749097 749393 749089 9.749985 750281 750570 750872 751167 751402 751757 752052 752347 752642 9.752937 753231 753520 753820 754115 754409 754703 754997 755291 755585 9.755878 756172 756465 756759 757052 757345 757038 757931 758224 758517 9.758810 759102 759395 759687 759979 760272 760564 760856 761148 761439
1.96 1.96 1.96 4.98 1.96 4.96 1.95 4.95 4.95 1.95 1.95 1.94 4.94 4.94 4.94 4.94 1.93 1.93 4.93 4.98 4.93 4.93 1.92 4.92 4.92 4.92 I. 92 1.92 4.91 4.91 4.91 4.91 4.91 4.91 4.90 4.110 4.90 4.90 4.00 4.90 4.89 4.89 4.89 4.89 4.89 4.89 1.88 4.88 4.88 4.88 4.88 4.88 4.87 4.87 4.87 4.87 4.87 1.87 4.80 4.80
C0 59 58 57 56 55 54
53 52 51 50 19 48 47 40 45 44 43 12 41 40 39 88 37 38 35 34 33 32 31 30 29 28 27 20 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
Cosine. | PP1" Cotanir. PP1" M.
6O0
3O0
SINES AND TANGENTS. 1'1T
o 1 2 3 -1 5 6 7 8 g 10
n 12 18 11 16 16 17 I8 111 20 21 22 23 21 2."i 20 27 28 2!i 30 31 .32 33 ;:i 35 36 ,'!7 38 39 40 41 -42 43 11 45 43 '17 '18 49 G0 51 62 63 54 5o 56 57 58 59 60
3.65 8.64 8.64 699626 3.64 699844 8.63 700002 700280 3.68 3.63 700198 3.68 700716 3.63 700033 3.62 701151 3.62 .701308 3.62 70158c 3.62 701802 3.62 702010 8.62 702230 3.61 70245: 3.61 702069 3.60 702885 3.60 703101 3.60 70331 3.60 .703533 3.60 703749 3.59 703904 3.59 704179 3. 59 704395 3.58 704610 3.58 704825 3.58 705040 3.58 705254 8.58 705409 8.67 .705683 8.57 705898 8.57 708112 3.67 706328 3.56 708539 8.58 706753 8.58 706907 3.55 707180 8.55 707393 8.55 707006 3.55 .707819 8.65 708035 8.55 70821" 3.54 7084.58 3.54 708070 3.53 708882 3.63 709094 3.63 709300 3.53 709518 3.58 709730 8.52 ,709941 3.52 710153 3.52 710304 3.52 710575 3.52 710786 1.52 710997 3.51 711208 3.51 711419 3.50 711029 8.50 711839 Cosine. 1P1T
&9°
1.761439 761731 762023 762314 762606 762897 763188 763479 763770 704001 764352 1.761643 764933 765221 765514 705805 706385 7666l 766965 707255 9.76754c 707834 768124 768414 708703 708992 709281 709571 769860 770148 9.770437 770726 77101E 771303 771592 771880 772168 772457 772745 773033 9.773321 773608 773898 774184 774471 774759 775010 77.5333 775621 775908 9.770195 776482 776708 777055 777342 777628 777915 778201 778488 778774 Cotiing
31° P1T
TiiIlC.
,711839 712050 712260 712469 712679
3.50 3.50 3.50 3.50 3.50 3.49 713098 3.49 713308 3.49 713517 3.48 713726 3.48 713935 3.48 1.714144 3.48 714352 3.47 714861 3.47 714709 3.47 714978 3.47 71.5186 ''3.47 715:394 3.46 715002 3.46 715809 3.46 710017 1.716224 3.46 3.45 716432 8.45 716639 3.45 716846 3.45 717053 3.45 717259 3.44 717460 3.44 717073 3.44 717879 3.43 718085 3.43 '.718291 3:43 718197 3.43 718703 3.43 718909 3.43 719114 3.42 719320 1.42 719525 3.42 719730 3.42 719935 3.41 720140 3.41 .720345 3.41 720549 3.41 720754 3.40 720958 3.40 721162 3.40 721366 3.40 721570 3.40 721774 3.40 721978 3.39 722181 3.39 .72238-5 3.39 722588 3..38 722791 O.ii8 722994 3.88 723197 3.38 723100 3.38 723603 3.37 723805 3.37 724007 3.37 724210
so
M. | M.
~387~
Cosine.
Taiig. 1.778774 779060 779346 779632 779918 780203 780u5 781060 781346 781631 ,781916 782201 782486 782771 788056 783341 783620 783910 784195 784479 9.7■84764 f85048 '8-5332 '85616 786184 785900
4.77 4.77 '7 4.76 4.76 4.76 4.76 4.76 4.76 4.75 4.75 4.75 1.7.5 4.75 4.75 4.75 4-75 4.74 4.74 4.74 4.74 4.74 4.73 4.73 4. ■*' j~
786408 4.73 786752 !3 787030 4.73 787319 4.72 9.787003 i2 787880 788170 4.72 78845:i 4.72 7887.% 4.72 789019 .72 789302 4.72 789585 4.71 789868 4.71 790151 4.71 9.790434 1.71 790710 4.71 790999 4.71 791281 4.70 791563 4.70 791846 4.70 792128 4.70 792410 4.70 792692 4.70 792974 4.70 9.793256 4.70 793538 4.69 793819 4.60 794101 4.69 794383 4.011 794664 4.09 794946 4.69 795227 4.09 795508 4.68 79.3789
58 57 58 55 .54 53 52 .31 50 49 48 47 46 45 44 '43 '12 41 40
37 30 35 34 83 ::j
31 30 29 28 27 20 25 21 28 22 21 20 19 18 17 16 15 14 18 12 11 10 9 8 7 0 5 4 3 2 1
0
PP1"| Cotniig. PP]'
58°
TABLE IV.—LOGAR1THM1C
34° 8ine 9.724210 :;.:r7 721112 72 11ill 72isi«ii*-;" 725017** 725819|** 725120 1,*™ 725022',:;.' 725823!^ 720024 **' 72622-,'*-:*' 9.72912.,,* 72B39B ;*:,i 726827;*'" 3.31 727027 i.34 727228 'i:'» 727428 72762.8 :: 3.:» 727828,' :;.:t; 728027 9.728427 72W26i 1 3.32 3.32 728825 3.32 729021 , , 729;22; „. 7290"i1, 7298201::,: 730018|*^ 7302i7|** 9.730115:*" 73tW13!:!-~ 730811j** 731009:*™ 73120)'r~ 7314W329 731802!329 731799;*^ 731990 *~ 73219:: " 3.28 7,'i23»0 3.28 732587 3.28 732781 3.28 732980 3.27 7;;3I77 3.27 73*17:; .: 733509 I'i'^l 73370-5 * '
i:^i;3.2o ^3.26 734i«.., „. -.,-,..- ;o.2o 7351oo|., 9735330 1*£ 7355251 „",735719 i 733914 13,-?? 786M»r M.
Cosine. I PI'!'
Tmiir.
PPP
9.79.57S9' 1.08 790070, 1.08 790351 1.68 798632 1.08 7900!3 1.08 797194 1. 0.8 797474 1.1,8 797755 1.08 7980»; 1.07 798310 1.07 798590 1.67 9.798877 1.67 799157 1.07 799437 1.07 799717 1.07 799997 1.07 800277 i.oo 800557 1.06 800830 1.06 801111i 1.66 801390 1.06 9.801075 1.06 801955 1.00 802231 1.05 802.513 1.05 802792 1.65 803072 4.65 803351 4.65 803030 1.65 803909 1.65 804187 1.65 9.801100 1.65 804745 1.61 805023 1. 01 80.5302 1. 01 805580 1.04 805859 1.01 800137 4.64 806415 1.03 800003 4.63 800971 1.63 9.807210 1.63 807527 1.03 807.805 1.03 808083 1.03 £08301 4.03 808038 4.03 808910 4.02 809193 4.62 800171 1.02 8007 18 4.02 9.810025 1.02 810302 1.02 810580 4.02 810S57 1.02 81113! 1.00 811410 1.01 811087 4.61 811904 1.01 812241 1.01 812517! Cotang.
M. 00 59 58 57 50 55 54 53 52 51 50 49 48 47 40 45 44 43 42 41 40 39 38 37 30 35 34
83 32 31 30 29 28 27 20 25 24 23 22 21 20 19 18 17 10 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 H.
M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10 17 18 19 20 21 22 23 24 25 20 27 28 2O 30 31 32 33 31 35 36 :i7 38 09 40 41 42 43 44 45 40 47 48 49 50 51 52 53 54 oo 56 57 58 59 00 M.
388
Mil"
9.730!09 730303 736498 730092 736880 737080 737274 737407 73700! 737855 738048 9.73824! 738431 738027 738820 739013 739200
PlT'l :;.LV1
3.24 3.24 0.23 3.23 3.23 3.23 3.23 3.22 3.22 3.22 3.22 3.22 3.22 ',.21 2 '£
739590*20 739783 ,. nn 739975 "" 1.20 9.710107 3.20 740359 3.20 740550 3.19 710712 3.19 710931 3.19 711125 3.19 711310 3.19 741508, „ , 741699*" 7418893-8. 1.742080 *W 3.18 742271 3.18 742102 3.17 742052 3.17 712842 3.17 743033 3.17 74322] 3.17 743413 3.16 713002 3.10 743792 1.7-13982 3.10 3.10 744171 3.16 744361 3. 15 744550 3-15 744730 3.15 714028 3.15 715117 3.15 745306 13.14 745404 3.14 745083 ., 9.715871*}* 7400001* 3.14 3.13 746430 |3.13 710024 3.13 746812 3.13
746248 :'
747187;*}| / l/o62
33 « '1'nng. ;i'P|-' 9.812517 812794 1.61 l.ol 813070 1.61 813347 l.oo 813623 1.60 813899 1.60 814170 4.60 8144.52 4.00 814728 4.10 815601 1.1.0 81.52£0 1.1,0 9.815555 815831 4.C0 l.oo 810107 1.50 810382 4.59 816658 4.59 810933 4.59 817200 1.59 817481 1.59 817759 4.59 818035 1.50 9.818310 4.58 818585 1..38 818800 4.58 810135 1.58 810110 1.58 810084 4.58 819959 4.58 £20234 4.58 820508 1.- 8 82078;i 4.57 9. 82105" 4.57 £21332 1.57 821006 1.57 821880 4.57 822154 4.57 822420 4.57 822703 4.57 822977 4.50 82323) 4.50 823321 4.56 9.823708 4.50 824072 4. 50 621345 4.56 821019 4.50 824803 4..50 825100 4.56 825430 4.56 825713 4.55 825080 4.55 8202.50 '1.55 9.820532 4.55 820805 4.55 827078 4.55 827351 4.55 827024 4.55 827807 4.65 828170 4.54 828442 4.54 82871.5! 4.54 828087;
i.D 50 .5.8 .37 .50 .35 .31 53 .52 51 .50 -1: l 18 '17
'i,i 45 4! '13 '12 44 40
::-!i 38 37
:;,;
32 31 30 20 28 .7 20 25 24 22
'_i ::i 19 I8 17 10 15 14 13 12 11 10 9 8 7 0
Cosine. | PP1" Culung, lPP1" M.
56°
34° 8ine
0 1 2 3 1 5
M.
05°
PP1"
9.7475i12 12 717749 3.12 747938 3. 12 748123 .3. 12 748310 3.11 748197 3.11 748383 3.11 748870 3.11 719338 3.11 749243 3.10 749429 3.10 9.749313 3.10 749301 3. 10 749387 3.09 750172 3.09 750353 3.09 750543 3.09 750723 3.09 750914 3.03 751099 .3.03 751284 3.08 9.751439 3.03 751651 03 751839 3.03 752923 3.07 752203 3.07 752392 3.07 752570 '3.07 752709 .3.07 752944 '3.07 753123 .3.03 9.753312 3.06 753495 3.06 753879 3.03 753332 3.05 754043 3.05 754229 3.05 754412 3.05 751595 3.05 754778 '3.01 754930 3.01 9.755143 3.04 755328 '3.01 755508 3.01 755990 3.04 755872 3.03 750054 3.03 756236 3.03 750418 3.03 756300 3.03 758782 3.02 9.75698: 3.02 757144 3.02 757326 3.02 757507 3.02 757688 3.02 757869 '3.01 758050 3.01 758230 3.01 758111 3.01 75.3591
SINES AND TANGENTS. ppr PP1"
Tung
9.8289S 1..51 829280 1.51 829532 1.51 829805 4.51 83007 1.51 830319 4.53 830621 1.53 83089,3 1.53 83110D 4.53 831437 4.53 831709 1.53 9.8319S1 4.5.3 832253 4.53 83251 1.53 832798 4.53 833338 4.52 833339 1.52 833811 4.52 833832 4.52 834151 4.52 83412; 1.52 9.834693 4.52 834937 4.62 835238 1.52 835509 4.52 835780 4.52 838051 4.51 838322 4.51 838593 4.51 833804 1.51 837131 1.51 9.837405 1.51 837675 4.51 837946 4.51 8:38216 1.51 838487 4.50 838757 4.50 839027 1.50 839297 4.50 839508 1.50 839838 1.50 9.810103 1.50 840378 1.50 840348 1.50 840917 1.50 841187 1.19 811457 4.49 841727 1.49 841998 1.19 812286 4.49 842535 1.19 9.842805 1.19 843074 4.49 843343 4.49 843012 1.19 843882 4.18 844151 4.48 844420 4.48 844089 4.48 844958 1.-18 815227
|PP1"| Cotaug.
PP1"
389
35° 1:111ir .845227 4.18 845196 4.48 845764 846033 846302 846570 846839 847108 817376 847644 847913 1.848181 848449 848717
9.75&591 7.58772 758952 759132 759312 759492 759672 759852 760031 760211 760390 9.760569 760748 760927 761106 761285 761464 761642 761821 761999 762177 9.762356 762534 762712 762889 763067 763245 763422 763600 763777 763954 9.764131 764308 764485 764662 761838 765015 765191 765367 765514 765720 9.765896 766072 766247 766423 766598 766771 766949 767121 767300 767175 9.707619 767824 767999 768173 768348 768522 768697 708871 769045 769219
3.01
Cosine.
PP1" | Cotana
.3.00 8.00 3.00 3.00 3.00 3.00 2.99 2.9!i 2.99 2.99 2.98 2.98 2.98 2.98 2. !18 2.9.8 2.97 2.97 2.97 2.97 2.97 2.97 2.96 2.96 2.96 2.96 2.90 2.95 2.95 2.95 2.95 2.95 2.95 2.94 2.91 2.04 2.94 2.94 2.93 2.93 2.93 2.93 2.93 2.93 2.92 2.92 2.92 2.92 2.92 2.91 2.91 2.91 2.01 2.91 2.90 2.90 2.90 2.90 2.90
849254 849522 819790 850057 850325 850593 9.850861 851129 851.396 851661 851931 852199 852466 852733 853001 853268 9.853535 854069 854336 854603 854870 855137 855404 855671 855938 9.856204 856471 856737 857004 857270 857537 857803 858336 858602 9.858868 859134 859400 8-59932 860198 860464 860730 860995 861261
35 34 33 32 81 80
29 2,3 27 20 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 ' 7 0 5 '1 3 2
1 0
54°
TABLE IV.— LOGARITHMIC
36°
Tang. | PF1
o
9.7692111
2.90 2. Nil 709500 2.89 709740 2. Ml 709913 2. Nil 770087 2. V1 770200 2.KN 770433 2.88 770000 2.88 7707711 2.88 770952 2.88 9.771125 2.88 771298 2.87 771470 2.87 771043 2.87 771815 2.87 771987 2.87 772159 2.87 772331 2.87 772503 2.81! 772075 2.86 9.772847 2.86 773018 2.86 773190 2. sii 773301 2.85 773533 2.85 773704 2.85 773875 2.85 774010 2.85 774217 2.85 774388 2.84 9. //4558 2.81 774729 2.84 774899 2.81 775070 2. 81 775240 2.,S1 775410 2.83 775580 2.83 as 775750 2.88 775920 so 2.83 776090 40 2.83 9.776259 41 2.88 42 776429 2.82 776598 43 2.82 U 776768 2.82 776937 45' 2.82 40 777106 2.82 777275 47 2.82 777444 48 2.81 777013 49 2. 81 50 777781 2.81 9.777950 51 2. 81 778119 52 2.81 5sr 778287 2.,S1i 51 778455 2. 80 778624 55 2. 80 778792 56 2. 80 57 778900 2.80 779128 58 2.80 779295 59 2.80 779463 60 l 2 3 1 5 6 7 8 6 10 11 12 13 11 15 16 17 18 19 20 21 22 23 24 25 20 27 28 29 3U 31 32 33 31 35 30 .17
C.'sil
53°
P1T
9.861201 801527 801792 802058 802323 802589 802854 80.3119 863385 863050 803915 9.804180 804445 804710 864975 805240 865505 865770 866035 806300 866564 9.860829 867094 867358 807623 807887 868152 868410
9.869473 869737 870001 870205 870529 870793 871057 871321 871585 871849 9.872112 872376 872640 872903 873167 873430 873694 873957 874220 874484 9.874747 875010 875273 875537 875800 876003 876326 876589 876852 877114
9.779463 779631 779798 779900 780133 780300 780107 780031 780801 780908 781 131 9.7»1301 781408 781031 781800 781900 782132 782298 782164 782630 782790 9.782901 783127 783292 783158 783623 783788 783953 781118 781282 781447 9.784612 784: 781941 785105 785269 785433 785597 785761 785925
1.43 1.43 1.43 4f2 4.42 1.12 1.42 1.42 1.42 1.42 1.42 1.12 1.12 1.12 1.12 1. 11 1.11 1.11 4.11 1.11 1. 11 1. 11 4.11 1.11 1. 1I 1.11
1.40 1. 1d 1.40 1.40 4.10 1.40 1.40 1.40 1. 1i1 1.10 1.1i1 4.40 1.10 1.10 1.89 1.89 1.89 4.89 1.89 1.89 1.89 1.89 1.89 1.89 1.39 1.39 1.89 1.88 4.88 1.88 4.38 1.38 1.38 1.38
9.786252 786410 786579 786742 786906 787069 787232 787395 78755' 787720 1.787883 788015 788208 788370 788532 788094 788856 789018 789180 789342
Untung, PPl" M.
M. | Cosine.
390
2.79 2.79 2.79 2.79 2.79 2.7S 2.78 2.78 2.78 2.78 2.78 2.78 2.77 2.77 2.77 2.77 2.77 2.77 2.70 2.78 2.70 2.70 2.70 2. 75 2.75 2.75 2.75 2.75 2.75 2.75 2.71 2.71 2.71 2.74 2.71 2.73 2.78 2.73 2.73 2.73 2.78 2.78 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.71 2.70 2.70 2.70 2.70 2.70 2.70
3?° Tung. [PP1 9.877114 4.38 877377 4.38 877640 4.38 877903 4.38 878105 4.38 878428 4.38 878691 4.38 878953 1.88 879216 4.37 879478 1.37 879741 1.87 9.880003 1.37 880205 1.87 880528 1.87 880790 1.37 881052 | 4. 87 881314 4.37 881577 1.87 881839 1.87 882101 4.87 9.882025 4.36 882887 4.36 883148 4.36 883110! 1.38 883072 1.36 883931
SSSSf884719*"^ 884980,* 9.885242^ 88,., ik, 88002C7™ |l.36 880519 886811 l4.35 14.35 4.35 887333 i.35 887594 4.35 9.887855 4.35 888116 1.35 888378 4.35 888039 1.35 888900 4.35 889101 4.85 889421 4.35 889682' 1.35 889943 4.35 890201 ! 4.35 9.8901C5 |4.35 890725 4.34 4.31 891217 4.34 891507 4.34 891768 4.34 892028 4.31 892289 4.34 892519 4.34 892810
C0 59 58 57 .50 55 51 53 52 51 50 49 18 47 10 45 11 43 42 1I 40 89 38 87 86 35 84 88 32 81 80
29 28 27 26 25 21 23 22 21 20 19 I8 17 10 15 I1 18 12 11 10 9 8 7 0 5 4
8 2 1 0
PP1"| Cotung. PP1
&»°
3S°
S1NES AND TANGENTS. 8ine.
FP1"
9.789342 2.69 789501 2.69 789«i5 2.69 789827 2.69 2.69 790149 2.69 790310 2.68 790171 2.68 790832 2.68 790793 2.68 7909.54 2.88 9.791115 2.68 791275 2.68 791430 2.67 791590 2.67 791757 2.07 791917 2.67 792077 2.67 792237 2.67 792397 2.66 792557 2.66 9.79271C 2.66 792876 2.06 793035 2. 06 793195 2.05 793354 2.05 793514 2.65 793673 2.05 793832 2.65 793991 2.65 794150 2.61 9.794308 2.64 79446' 2.64 794620 2.64 794784 2.64 794942 2.64 795101 2.61 795259 2.63 79541' 2.63 795575 2.03 795733
2. as
79C049 796206 796364 798521 796679 798836 796993 7971,50 797307 9.797464 797621 797777 797934 798091 798247 798403 798560 798716 798872 Cosine
51°
2.63 2.63 2.63 2.62 2.62 2.62 2.62 2.62 2.02 2.02 2.61 2.61 2.61 2.61 2.61 2.61 2.60 2.00 2.00
Tang.
"P1" .M
M.
9.892810 893070 893331 893591 893851 894111 894372 894632 894892 895152 895412 9.895672
8inn. 9.798872 799028 799184 799339 799495 799651 799806 799962 800117 800272 800427 9.800582 800737 800892 80104' 801201 801356 801511 801665 801819 801973 9.802128 802282 802436
896192 898452 896712 896971 897231 897491 897751 898010 9.898270
899019 899308 899568 899827 900087 900346 900605 9.900864 901124 901383 901642 901901 902160 902420 902879 902938 903197 9.903456 903714 903973 904232 904491 904750 905008 905267 905526 905785 9.906043 906302 906500 906819 907077 907336 907594 907853 908111 908309
802743 802897 808050 803204 80335' 803511 9.803664 803817 803970 804123 804276 804428 £04581 804734 804886 605039 9.805191 805343 805495 10564' 605799 805951 806103 £06254 806106 80655' 9.800709 806860 £07011 807163 807314 807465 807615 807766 807917 808067
PP1" C'otaiiB. PP1" II
391
Cosine.
39° PP1
Tantr.
2.60 2.60 2.60 2.00 2.59 2.59 2.59 2.59 2.59 2.58 2.58 2.58 2.58 2.58 2.58 2.58 2.58 2.57 2.57 2.57 2.57 2.57 2.57 2.56 2.56 2.56 2.56 2. 56 2.56 2.55 2.55 2.55 2.55 2.55 2.55 2.54 2.54 2.54 2.54 2.54 2.54 2.54 2.53 2.53 2.53 2.53 2.5,'! 2.53 2.53 2.52 2.52 2.52 2.52 2.52 2.52 2.52 2.51 2.51 2.51 2.51
9.908369 908628 908886 909144 909402 909060 909918 9101 910435 910693 910951 9.911209 9114S 911725 911982 912240 912498 912756 913014 913271 913529 9.913787 914044 914302 914560 91481' 915075 915332 915590 915847 916104 9.916362 916619 916877 917134 917391 917648 917906 918163 918420 918677 9.918934 919191 919448 919705 919902 920219 920476 920733 920990 921247 9.921503 921700 922017 922274 922530 922787 923044 923300 923557 923814
PP1" Cotan«
60 59 58 57 56 55 54 53 52 51 50 49 -is 47 40 45 44 43 42 41 40 89
as 37 36 35 31 83 82 31 30 29 28 27 26 25 21 23 22 21 20 19 18 17 16 15 I1 I8 12 11 10 9
s 7 6 5 4 3 2 1 0 PP1" M.
8ini'.
P1T
9.808007 2.51 808218 2.51 808308 808519 2.51 808069 2.50 2.50 808819 808909 2.50 800119 2.50 809209 2.50 2.50 809419 809569 ! .50 2. Mi 9.809718 l'i 809868
810017 2*35 810167," 810310 1,4* 810^248 810014 1*48 810763 ,r! 810012! "i8 8U061~* 9.811210-,-" 811:1581** 8L1.507 V ,_ 811653.^1 811801 1 7„ 811952' *" 812100,,' 812248' *47 812398!,™ 814514 2« 9.812692!*™ 812810i246 812988,™ ..,,.„- 2.46 8131.W 813283 *™ 813*» ," 813578 *™ 813725 **! 813872 *" 814019 ,-4? 9.81416« ~~ 814813,!? 814480 ,t814607 *™ 2.44 814753 2.44 814900 2.44 815046 2.44 815193 2.44 815339 2.44 815485 2. 43 1.815632 2.43 815778 2.43 81592! 1. 18 816000 , ,., 816215*™ 810301!,,;, 818807 , 'S 816652|*" 8167981 ;-4; 8169431*^ Cosine
4'.*"
41 c
TABLE IV.— LOGAR1THM1C
4O0
|PPl
Tung.
PP1"
9.923814 921070 924327 924583 924810 925090 925352 925009 925861 926122 926878 9.920634 920890 92714' 927403 927050 927015 928171 928427 928684 928940 9.929190 929452 929708 929961 930220 9301' 930731 93098; 931243 93149» 9.9317.55 932010 932266 932522 932778 933033 933289 93354.: 933800 934056 9.934311 934567 934822 935078 935833 935589 93584 ; 936100 936355 936611 9.936860 937121 937377 937632 937887 938142 938388 988653 938908 939103 Cotang. |PPl
M.
51.
PP1" 1.816943 2.42 817088 2.42 817233 2. 12 817379 2.42 817524 2.41 817668 2.41 817813 2.41 817958 2.11 818103 2.4] 818217 2.11 818.392 2.11 1.818536 2.11 818681 2. 10 81882", 2.10 818969 2.40 819113 2.40 819257 2.40 819401 2. 10 819545 2.40 819689 2.:',!i 819832 i.81!i»76 2.89 820120 2.39 2. 89 820263 2,89 82040« 2.39 820550 2.38 820093 2.88 8201 820979 2.88 2.88 821122 8212051 r™ 2.38 1.82140'; 2.38 821550 2.:« 8211l93 2.37 821835 821977 2-£ 822120 A'61 2.37 822262 2.37 822104 2.37 822546 822688^ 8228301*?7 822972 82.3114 823255 82.3397 823539 823080 823821 828963 824104 1.824245 824386 824527 824i;68 824808 821949 825090 825230 825371 825511
** tf„ *™ *™ ** f-f. i-f, **J 2.35 2.85 2.35 2.35 2.35 2.34 2.31 2.34 2.84 2.31 2.34
Tung.
PP1"
9.939163 4.25 939418 1.25 939073 4.25 939928 4.25 940183 1.25 940439 4.25 940694 1.25 940949 4.25 941201 4.25 94145!; 1.25 941713 9.941968 L 25 4.25 942223 4.25 942478 4. 25 942733 4.25 942988 1.25 943243 943498 4.25 4.25 943752 944007 1.25 1.25 944202 1.25 9.944517 4.25 944771 4.25 945026 4.25 945281 1.24 945535 4.24 945790 4.24 946045 4.24 946299 4.21 940554 4.24 946808 4.24 9.947063 4.24 947318 4.24 947572 4.24 947827 4.24 948081 4.24 948335 4.24 948590 4.24 948844 1.24 949099 4.24 949353 1.24 9.949608 4.24 949862 4.24 950116 4.24 950371 1.24 95062 1.21 950879 4.24 951133 4.24 951388 4.24 951642 4.24 951890 1.952150 * 1.24 952405 4.24 ' 952659 4.24 952913 1.24 953167 4.23 953421 4.23 953675 4.23 953929 4.23 954183 4.23 «54437
! PP1" Cotiuig.
1.0 59 58 57 58 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38
35 34 83 32 31 80 29 28 27 2« 23 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 2 1 0
PP1"
48°
8ine. 9.825511 825051 825791 825931 828071 823211 828351 823191 826631 826770 826910 9.827049 827189 827328 827467 827606 827745 827884 828023 828162 828301 9.828439 828578 828716 823855 823993 829131
PP1'
2.31 2.33 2.33 2.33 2.33 2.33 2.83 2.33 2.33 2.33 2.33 2.32 2.33 3.33 3.33 3.32 2.32 2.32 2.31 3.31 2.31 2.31 2.31 2.31 3.30 2.30 2.30 3.30 829107 2.3l) 829545 2.30 829383 2.30 9.829821 2.30 829959 2.29 830097 3.29 830234 3.21J 830372 2.39 830509 2.29 830646 2.29 830784 2.29 839921 2.23 831058 2.23 9.831195 3.23 831332 3.33 831469 2.23 831608 2.23 831742 2.28 831879 2.23 832011 2.28 832152 2.27 832283 2.27 832125 2.27 9.832561 2.27 83269: 2.37 832333 3.27 832969 2.27 833105 2.28 833241 2.23 833377 2.26 833512 2.26 833648 2.28 833783
Tung.
M.
9.954437 4 954691 4. 954946 4, 955200 4. 955454 4. 955708 1. 955961 4, 956215 4 956169 4. 950723 4, 956977 1 9.957231 I. 957485 4, 957739 4. 957993 1. 958247 4. 958500 4. 958754 I. 959008 4, 959232 1. 959516 !. 9.959769 4 930023 i 930277 1. 960530 4. 960784 4. 981038 1 931292 4 931545 4 981' 4 982052 4 9.982306 4 962560 4. 982813 4 98306 4. 903320 4. 963574 4. 963828 4 904081 1 984335 904588 4 1 9.904842 1 965095 4 985319 4 965002 4. 965855 1 930109 4. 960332 4 960616 4 966869 4. 967123 4 9.987376 i 967629 4. 987883 1. 968138 4. 908389 4, 908643 4. 4 909149 1.22 969403 1.22 969650
8i |11
9.833783 833919 834054 834189 834325 834460 834595 834730 834863 834999 835134 9.835269 gqcmaq 835403 835538 835672 835807 835941 836209 836343 836477 9.836611 836745 836878 837012 837146 837279 837412 837546 837679 837812 9.837945 838078 838211 838344 838477 838610 838742 838875 839007 839140 9.839272 839404 839.536 839068 839800 839932 846064 840196 810328 810159 9.840591 840722 840854 840985 811116 811217 841378 811509 811640 841771
P Pi'
47 "
43°
SINES AND TANGENTS.
42°
Cosine.
393
PP1'
PP4"
9.969656 2.20 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 j?*|j ^'^4 2.24
2.21 2.24 2.21 2.24 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2.22 2.22 3.22 2.22 2-22 2.22 2.22 2.22 2.22 2.21 2.21 2.21 2.21 2.21 2.21 2.21 2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.19 2.19 2.19 2.19 2.19 2.19 2.19 2.18 2.13 2.13 2.18 2.13
970162 970416 970669 970922 971175 971429 971G82 971935 972188 9.972441 972695 972948 973201 973154 973707 973960 974213 974466 974720 9.974973 975226 975479 975732 975985 976238 976491 970744 976997 977250 9.977503 977756 978009 978262 978515 978768 979021 979274 979527 979780 9.980033 980286 980538 980791 981044 98129 981550 981803 982056 982309 9.982562 982814 983067 983573 983826 984079 984332 931584 984837
4.22 4.22 1.22 4.22 1.22 1.22 4.22 1.22 4.22 4.22. 4.22 4.22 4.22 1.22 1.22 1.22 4.22 4.22 1.22 '1.22 4.22 4.22 4.22 4.22 1.22 4.22 1.22 1.22 4.22 4.22 1.22 4.22 4.22 1.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.21 4.21 4.21 4.21 4.21 '1.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 1.21
PP1" Cotang. | PIT'
46°
*4iL M, U 1 2 :i '1
'. i. 7 8
t 111 1l 12 l:i 11 I9 111 I8
111 20 21 22 23 21 20 2ii 27 28 2!i
8ine
1 PIT
9.841771 2.18 841902 ;;2.is 842033 "2.18 s-12103 2.18 842294 2.17 812421 2.17 8120V, 8120.V 2.17 '.'|2.17 84281.' !2.17 8421111 ; 2. 17 843071 2.17 9.M3201 843331 l 2. 17 8431611 ''2.16 84372.-1 "' 2.18 84385." 2.16 843118 1, 2.16 814114i; 84424317?? 2. lo 814372 2.15 9.844502 2.15 844031 2. 15 844760 2.15 844889 2.15 845018 2.1. 84514'
845270 1 7 J! 845405,^
m
845533;£J* 845682|,"
31 1f.'
9.815790--J* 845919 ,"
;;:; 34 39 3ij 87 38
;;:i 'id
u 42 43
-11
™: : U'° 2 14 810301 ™" 846432,--" 846560 1 ;'" 846688"}? 846816- ,, 840944 |"" 9.84707117" 847199™,,
Hsu
!.-i 10 47 '18 19 '",0 51 52 53 51 6j 56 r,7 .18 69 1Ki
9.984837 985090 985343 985506 985848 986101 986354 986607 986860 9H7112
1 1 i 1
t 4 1 1
\
987369 !,' 9.987018,
987871 ;
988123! »ss37ii 988629 988882 989131 9893K7! 989640! 989893 9.990145 990398 990891 990903 991156 991409! 991662 991914 9! 12167
60 59 58 :,7 50 5.i 54 5.3 .'.2 51 50 111 48 17 40 45 11 13 42 '11 1l1 39 38 irr
992420 9.992072 99292.5 993178 993431 993083 993936! 994189 994441 994094 994947 9.995199 995452 995705 99595! 998210 900463 996715 998968 997221 997473 9.997720 997979 098231 908484 998737
847709- f 84783617 " 847964^ 848091:7" 848218,^" 9.848345 /-" 'J. 12 818472 2.12 848599 8187211 2.11 2.11 848852 2.11 818970 2.11 849108 999242 2.11 849232 90940.: 2.11 819359 99974; 2.10 819489 10.000000 -p:'
45°
TABLE IV.—LOGAR1THM1C PPP M. 8in... PPP
Tumr.
9.849485 840041 84W738
2.10 2. in
849990 7 J" 85011o£" 8-102427" 850308,7"
85049.-1!7 J K06I9209 800749" 890996!^ 8.51121 -"" 2.09 851246 2.09 851372 2.09 851497 2.11N 851022 2.08 851747 2.08 851872 2.08 8.51997 2.08 9.852122 2.08 85224' 2. OS 852371 2.08 852496 2.07 852620 2.07 852745 2.07 8528C9 2.07 892904 2.07 893118 2.117 863242 2.07 9.853366 2.07 858490 2.07 893614 2.07 853738 2.00 893862 2.00 893986 854109 2.06 2.00 854233 2.00 854350 894480 2.06 2.00 9.894603 2.00 854727 2.09 854890 2.09 854973 2.09 S99096 2.00 895219 2.09 895342 2.09 899165 2.09 8- -j ;\8 859711 9.895833 899956 856078 896201
45° Thnb.
I P1T'
10.000000 1.21 000253 1.21 000909 1.21 000758 1.21 0010111 1.21 001263 1.21 001510 1.21 001760 4.21 002021 1 |4.21 002274 4.21 002.527 1.21 10.002779 1.21 003032 1.21 003285 1.21 003537 1.21 003790 1.21 004043 1.21 004295 1.21 004548 1.21 064801 1.21 009053 1.21 il0.C05306 1.21 000599 1.21 009811 1.21 006064 1.21 006317 1.21 006569 1.21 000822 1.21 007075 007328 !,„, 4.21 007580 4.21 10.007833 1.21 008086 4.21 008338 1.21 008591 4.21 008844 1.21 009007 4.21 009349 1.21 009002 1.21 009899 4.21 010107 1.21 10.010300 1.21 010613 1.21 010800 1.21 0IUI8: 1.21 011371 1.21 011024., ,„ n.,c— 14.21 01187,
012129 *';{
012382:^J 012635k,\ 10.012888 y'f. 2.01 018140 4 £ 2.01 018888!?,J 2.01 013040 : ' f 2.04 013899 !*-f{ 2.06 2.04
2.01 2.01 2.0-1 2.03
014404 | ' t\ 014657 *'£ 0H0ioij,:
2.03 0151C3i "^
PP1" Cotnng. |PP1"| M.
l"otnnc.
394
'—440-
Tans
m°* 2.03 857178 2.03 857300 2.03 857422 2.03 857543 2.03 857065 2.02 857780 2.02 857908 2.02 858029 2.02 858151 10 2.02 9.858272 2.02 858393 12 2.02 858514 18 2.02 858035 14 02 858756 15 2.02 8588; 16 2.01 858998 17 2.01 850110 IS 2.01 850239 19 2.01 859360 20 2.01 21 9.859480 2.01 859601 22 2.01 859721 23 2.01 859842 24 2.00 850002 25 2.00 800082 20 2.00 860202 27 2.00 8(10322 28 2.00 800442 20 2.00 800562 80 2.00 31 9.800682 2.00 800802 32 1.99 860922 33 1.99 861041 34 1.99 861161 35 1.99 801280 30 1.00 801400 37 1.99 861519 38 1.00 801638 3!) 1.00 861758 40 1.98 41 9.801877 1.98 801990 42 1.98 862445 13 1.08 862234 11 1.08 862353 45 1.08 862471 46 1.98 -17 802590 M fi8 18 802709 1.98 862827 10 1.08 862946 50 1.07 51 9.803064 1.97 863183 52 1.07 863301 53 1.07 863419 54 1.97 863538 55 1.97 863656 56 1.07 57 863774 1.07 803892 58 1.07 864010 59 1.00 804127 00
n
Cosine
43"
47°
SINES AND TANGENTS.
46°
M.
PPl"
10.015163 4.21 015416 4.21 015668 1.21 015921 4.21 016174 4.21 016427 4.21 016680 4.21 016933 4.21 017186 4.21 017438 4.21 017691 4.21 10.017944 4.21 018197 4.21 018450 4.21 018703 4.21 018950 4.22 019209 4.22 019462 4.22 019714 1.22 019967 1.22 020220 1.22 10.020473 1.22 020726 1.22 020979 1.22 021232 4.22 021485 1.22 021738 1.22 021991 1.22 022244 1.22 022497 1.22 022750 1.22 10.023003 1.22 023250 K22 023509 1.22 023762 1.22 02401 1.22 024268 1.22 024521 1.22 024774 1.22 025027 1.22 025280 1.22 10.025534 1.22 025787 1.22 020040 1.22 020293 1.22 026546 1.22 026799 1.22 027052 1.22 027305 1.22 027559 1.22 027812 4.22 10.028065 1.22 028318 4.22 028571 1.22 028825 1.22 029078 4.22 029331 4.22 029584 4.22 029838 4.22 030091 1.22 030344
8ine.
PP1"
9.864127 1.96 864245 1.96 864363 1.96 864481 1.90 864598 1.96 864716 1.96 864833 1.90 864950 1.95 865008 1.95 865185 1.05 865302 1.95 9.805410 1.95 865536 1.95 865053 1.95 865770 1.95 865887 1.95 866004 1.94 866120 1.94 800237 1.94 800353 1.91 866470 1.91 9.866586 1.94 866703 1.94 866819 1.94 866935 1.93 867051 1.93 867167 1.98 867283 1.93 807309 1.93 867515 1.93 867631 1.93 9.867747 1.93 867802 1.03 867078 1.O2 868093 1.92 868209 92 808324 1.92 868440 1.92 868555 1.92 868070 1.92 868785 1.92 9.868900 1.92 800015 1.92 809130 1.91 869245 1.91 1.91 869474 1.91 [.91 860701 1.91 869818 1.91 869933 1.91 9.870047 1.90 870101 1.90 870276 1.90 87U390 1.90 870504 1.00 870618 1.90 870732 1.90 870840 1.90 870960 1.00 871073
PP1" CotanK. PPl" M
395
Cosine.
Tftne. PP1" 10.030344 4.22 030597 1.22 030851 4.22 031104 4.22 031357 1.22 031611 1.22 031864 1.22 032117 4.22 032571 1.22 032624 4.22 032877 1.22 10.033131 1.22 03*384 1.22 033088 1.22 033801 4.22 034145 1.22 034398 1.22 034651 1.22 034005 1.22 085158 1.23 035412 4.23 10.035665 |4.23 035010 4.23 030172 1.23 036420 1.23 030080 4.23 030933 1.28 037187 1.23 037440 4.23 037694 4.28 037048 1.23 10.038201 4.23 038453 4.23 038708 1.23 03896'2 4.23 030210 1.2: 039470 1.23 039723 1.23 039977 1.23 040231 4.23 040484 1.23 10.040738 1.23 040992 1.23 041246 4.23 041500 4.23 041753 1.23 042007 1.23 042261 4.23 042515 1.23 042769 1.23 043023 1.23 10.043277 4.23 043531 1.23 043785 4.23 044039 1.23 044292 4.23 044540 1.23 044800 1.23 045054 1.23 045309 1.23 045503
60 59 58 57 56 55 54 53 52 51 50 49 48 17 46 45 .11 43 42 -II 40 30 38 37 86 85 34 33 32 81 30 20 28 27 20 25 24 23 22 21 20 19 18 17 16 15 14 I8 12 [I 10 9 8 7 0 5 4 8 2 1 0
PPl" Cotnng. PP1" M
42°
4S° 0 1 2 3 4 5
M.
41°
49°
TAULE IV.—LOGAR1THMIC
M.
I'l'l"! 9.871073 1.90 871187 1.89 871301 1.89 871414 1.89 871528 1.89 871041 1.89 871755 1.89 871888 1.89 871981 1.89 872095 1.88 872208 1.88 9.872521 1.88 87243 1 1.88 872547 1.88 872059 1.88 872772 1.88 872885 1.88 872098 1.88 873110 1.88 873223 1.87 878335 1.87 9.873448 1.87 873500 1.87 873072 I.87 873784 1.87 87&890 1.87 874000 1.87 874121 1.87 874232 1.87 874344 1.80 87445O 1.86 9.874508 1.88 874080 1.86 874791 1.88 874903 1.86 875014 [1.86 875120 1.85 875237 1.85 875348 1.85 875459 1.85 875,571 1.85 9. 875082 ! 1.85 875703 1.85 875004 1.85 870014 1.85 876125 1.85 876230 1.85 876347 1.84 876457 1.84 876508 1.81 876678 1.81 9.876789 1.84 876899 1.84 877010 1.84 877120 1.84 877230 1.83 877340 1.83 877450 1.83 877500 1.83 877070 1.83 877780 Cosine.
lang. |PPl"| 10.045503 4.25 045817 4.23 046071 4.25 040325 4.23 046579 4.25 040833 4.24 04708. 4.24 047341 1.24 047505 4.24 047850 4.24 048104 4.24 10.048358 4.24 048612 4.24 04880; 4.24 049121 4.24 049375 4.24 049029 4.24 049884 4.24 050138 4.24 050392 4.24 05004' 4.24 10.050901 4.24 051156 4.24 051410 4.24 051665 1.24 051919 1.24 052173 4.24 052428 1.24 052682 4.24 052937 1.24 053192 4.24 10.053440 4.21 053701 1.24 053955 1.24 054210 1.24 054465 1.24 054719 4.25 054974 4.25 055229 1.25 055483 4.25 055738 4.25 10.055093 1.25 056248 1.25 056.502 4.25 050757 4.25 057012 4.25 057207 4.25 057522 4.25 057777 4.25 058032 4.25 058287 4.25 10.058541 4.25 058796 4.25 059051 4.25 059300 4.25 059.501 4.25 059817 1.25 060072 4.25 060327 4.25 000582 4.25 000837
I'lT
M. I
9.877780 877890 877999 878109 878219 878328 878438 878547 878656 878700 878875 9.878984 879093 879202 879311 879420 879529 879037 879746 879855 879903 9.880072 880180 880289 880397 880505 880613 880722 880830 880938 881040 9.881153 881201 881369 881477 881584
PP1" Cotang. I PP1" M
00
1.83 1.83 1.83 1.83 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.80 1.79 1.79 1.79 1.79 881799 1.79 881907 1.79 882014 1.79 882121 1.79 9.882229 1.79 882330 1.78 838448 1.78 882550 1.78 882057 1.78 882704 1.78 882871 1.78 882977 1.78 883084 1.78 883191 1.78 9.883297 1.77 883404 11.77 883510:. K883017 ,1' 888723, „ 883829," 883936 !„ 8840421, " 884148 \'LL 884254
M.
Cosine.
396
Tans 10.000887 001095 001347 001602 001858 062113 062308 062023 062879 003134 003389 10.003045 008900 064156 064411 06460' 064922 005178 06543:5 005089 005944 10.066200 066455 066711 06696' 067222 007478 067734 067990 00824 008501 10.068757 000013 069209 069525 009780 070030 070292 070548 070804 071000 10.071316 071573 071829 072085 072341 072597 072853 073110 073360 073022 10.073878 074135 074391 074648 074904 075160 075417 075673 075930 070186
PP1" Cotang. |PP1" M.
40Q
M. 0 1 2 3 4 5
9 10
u 12 18 14 15 16 17 18 19 20 21 22 23 21 25 26 27 28 29 80 31 32 33 31 35 36 87 38 89 '10 11 42 43
'n 45 46 17 '18 49 50 51 92 58 51 55 56 57 53 59
8il
PP1"|
9.881251 884360 831468 884572 884677 881783 881889 881994 885100 885205 885311 9.885416 88552: 88562; 885732 88588: 885942 i 88801 886152 8882. 888362
1.77 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.70 1.75 1.7.5 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.71 1.71 888571 1.71 886676 1.71 836780 1.74 838885 1.71 886089 1.71 887003 1.71 887193 1.71 887302 1.71 837403 1.73 9.837510 1.73 887614 1.73 887718 1.73 837822 1.73 88792 i 1.73 838030 1.73 838131 1.73 883237 1.73 83,8311 1.73 883U1 9.888513 1.73 1.72 883651 1.72 883755 1.72 888858 1.72 883981 1.72 839081 1.72 889168 1.72 889271 1.72 889371 1.72 889477 1.71 9.889579 1.71 889882 1.71 889785 1.71 889888 1.71 1.71 890093 1.71 1.71 1.71 890100 1.71 890503 Cosine. |PP1'
39°
r>i°
S1NES AND TANGENTS.
5O0
Tang,
9.890503 1.70 89060: ;^i i.7o 89070: 1.70 890809 1.70 890911 1.70 891013 1.70 891115 1.70 891217 1.70 891319 1.70 891421 1.70 891523 1.69 9.891624 1.69 891726 1.69 891827 1.69 891929 1.69 892030 1.69 892132 1.69 892233 1.69 892334 1.69 892435 1.68 892536 1.68 9.802638 1.68 892739 1.68 892839 1.68 892940 1.68 8a3011 1.68 893142 1.68 893243 1.68 893343 1.68 893444 1.07 893544 1.67 9.893645 1.67 893745 1.67 893846 1.67 893946 1.67 894046 1.67 894146 1.67 894246 1.C7 894346 1.67 894446 1.07 894546 1.66 9.894646 1.66 894746 1.66 894846 1.66 . 894945 1.66 895045 1.66 895145 1.66 895244 1.66 895343 895443 1.66 895542 1.65 1.65 9.895041 1.1l5 895741 1.65 895810 1.65 895939 1.65 8960:38 1.65 896187 1.65 1.6.5 89633: 1.65 896433 1.05 896532
10.076186 4.27 076443 4.28 076700 4.28 076956 4.28 077213 4.28 077470 4.28 077726 4.28 077983 4.28 078210 4.28 078197 4.28 078753 4.23 10.079010 4.28 079287 4.23 079521 1.28 079781 1.28 030038 4.23 030233 4.23 030352 4.23 080309 4.23 031086 4.23 081323 4.23 10. 031580 1.28 031837 1.2i 082094 1.29 082352 1.2i 082609 1.29 032838 1.29 033123 1.211 083381 1.29 033838 1.29 033893 1.2i 10.031153 1.2i 031110 1.29 081068 1.2i 03192: 4.29 035183 1.29 085140 1.29 085093 4.29 085956 4.29 088213 4.29 036471 1.29 10.038729 1.29 036986 1.30 037211 1.80 087502 1.30 037760 1.80 088018 4.30 088275 1.80 088533 4.30 088791 1.30 039649 1.30 10.039307 1.80 089565 1.30 089323 1.80 090082 1.80 090340 1.30 090598 4.80 090856 1.30 091114 1.30 091372 4.30 091631 Cotiing.
PPl'
1PPP
PP1
PP1"
1'PP
397
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SINES AND TANGENTS.
54° PIT 9.9079'58 1.58 908049 1.53 908141 1.53 9082)3 1.53 908324 1.53 908416 1.52 908507 ,52 908599 1.52 908690 1.52 908781 1.52 908873 1.52 9.9089W 1.52 909055 1.52 909140 1.52 9092)7 1.52 909328 1.52 909419 1.51 909>10 1.51 909601 1.51 909691 1.51 909782 1.51 9.909873 .51 909963 1.51 910031 1.51 910144 910235 1.51 1.51 910325 910415 1.50 910306 1.50 910596 1.50 910686 50 9.910776 1.50 910366 1.50 9109.36 1.50 911046 1.50 1.50 911130 1.50 911226 1.50 91131i 1.50 911490 911495 1.49 911584 1.49 9.911674 1.49 911763 1.49 1.49 91185:i 1.49 911942 912031 1.49 1.49 912121 912210 1.49 1.49 912299 1.48 912388 1.48 912477 M8 9.912566 1.48 912655 1.48 912744 1.48 912833 1.48 912922 1.48 913010 1.48 913099 1.48 913187 1.48 913276 1.47 913365 Cosin'*.
35°
TaiiK.
PP1" M.
10.138739 1.13 139005 1.13 139270 4.43 139536 4.43 139802 4.43 140068 4.43 140331 1.43 140600 1.43 140866 4.43 141132 4.43 141398 4.44 10.141604 4.44 141931 4.44 142197 4.44 142463 4.44 142730 4.44 142993 1.44 143233 4.44 143529 4.44 143796 1.41 144062 4.45 10.144329 4.45 144593 1.45 14486) 4.45 145130 4.45 145397 1.45 145604 1.45 143931 4.45 146198 4.13 146485 1.45 146732 1.45 10.146999 4.48 1472ii7 1.48 141534 4.48 147801 4.48 148089 1.48 148330 1.46 148601 4.48 148871 4.48 149139 1.48 149407 4.46 10.149J75 4.48 149943 4.47 150210 1.47 150478 1.47 150746 4.47 151014 4.17 151283 1.17 151551 4.47 151819 4.47 152087 4.47 10.152356 1.17 152624 4.47 152892 4.47 153161 4.48 153480 4.48 153098 4.48 153967 '4.'48 154236 4.48 154504 4.48 154773
8ini'.
55° PIT'
9.913365 1.17 913453 1.47 913541 47 913630 47 913718 1.47 913806 1.47 913804 1.17 913982 1.47 914070 1.47 914158 47 914246 1.47 9.914334 1.40 914422 1.1(1 914510 1.46 914598 1.46 914685 1.46 914773 1.48 914860 1.46 914948 1.46 915035 1.48 915123 1.45 9.915210 1.45 915297 1.45 915385 1.45 915472 1.45 915559 1.45 915646 1.45 915733 1.'45 915820 1.45 915907 1.45 915994 1.45 9.916081 1.45 9161K 1.15 916254 1.W 916341 1.44 916427 1.44 916514 1.44 916600 1.44 916687 1.44 916773 1.44 916859 1.44 9.916946 1.44 917032 1.44 917118 1.43 917201 1.43 917290 1.43 917376 1.13 917462 1.43 917548 1.48 917634 1.43 917719 1.43 9.917805 1.43 917891 1.48 9179; 1.42 918062 1.42 918147 1.42 918233 1.42 918318 1.42 918404 1.42 918489 1.42 918574
pit
COsl i
S\)IJ
P1T
1an 10.154773 4.48 155042 4.48 155311 4.48 155580 4.48 155849 4.48 156118 4.49 156388 4.49 156657 1.49 156926 4.49 157195 4.49 157465 4.49 10.157734 1.49 158004 4.49 158273 4.49 158543 1.48 158813 4.50 159083 4.50 159352 4.50 159622 4.50 159892 4.50 160162 4.50 10.160132 1.50 160703 1.50 160973 4.50 161243 1.50 161513 1.51 161784 1.51 162054 4.51 162325 4.51 16259: 1.51 162806 1.51 10.163136 1.51 16340: 1.51 163678 1.51 163949 4.52 164220 1.52 164491 1.52 164762 1.52 16503;) 1.52 165301 4.52 165575 1.52 10.165810 1.52 166118 1.52 166389 4.52 166661 1.52 166932 1.53 167204 1.53 167475 4.53 16774' 4.53 168019 4.53 168291 1.53 10.168563 4.53 168835 1.53 16910' 1.53 169379 1'53 169651 4.54 169923 1.54 170195 1.54 170468 -1.54 170740 4.54 171013
80 59 58 57 58 55 54 53 52 51 50 49 48 47 46 45 '44 43 42 41 40 39 38 37 36 35 31 33 32 31 30 29 28 27 26 25 24 23 22 21 20 10 is 17 16 15 II 13 12 11 10 (i
8 7 6 5 4 3 2 1 0
PIT
34'
Tin
0 1 2 3 4
5 6 7
8 9 1D
11 12 13 H 15 Hi 17 13 19 20 21 22 23 21 25 26 27 28 29 90 31 32 33 31 35 86 37 38 39 40 11 42 '13 '11 15 40 47 '18 49 50 51 52 53 51 55 56 57 58 59 60
1.9185741. 918859:, 918745 !! 918830;! 918915. 919001 J 919085 919109 919251
919389| 919421 1.919508 919593 919077 919702! 919840 919931 920015 | j 920099 920181 920208 1.920352 920430 920520 920C01 920888 920772 920850 9209:® 921023 921107 1.921190 921274 921357 921441 921524 921007 921691 921774 921857 921940 1.022023 922108 922189 922272 922355 922438 922520 922003 922080 922708 1.922851 922933 923016 923098 923181 923263 923345 923427 923509 923591 Cosine.
33°
5,o
TABLE 1V.—LOGARITHMIC
56°
10.171013 171285 171558 171830 172103 172376 172649 172922 173195 173468 173741 10.174014 174287 171501 174834 175107 175381 175055 175928 170202 170476 10.176749 177023 177297 177571 177810 178120 178394 178008 178943 179217 10.179492 179766 180011 180316 180590 180885 181140 181415 181090 181965 10.182241 182516 182791 183087 183342 183018 183893 184109 184445 181720 10.184990 185272 185548 185824 180101 180377 180653 180930 187206 187483
T1'l
lM':'
9.923591;. ~ 4.61 1.51 4.51 4.55 1.55 1.55 1.55 4.55 4.55 1.55 4.55 4.65 1.56 1.56 1.56 4.68 1.56 1.56 1.56 1.56 1.58 4.50 1.57 1.57 1.67 1.57 4.57 1.57 1.57 1.57 4.58 4.58 4.158 4.58 4.58 4.5-8 4.68 4.68 4. 58 4.59 4.59 4.59 1.59 1.59 4.59 4.59 4.59 1.60 1.60 4.60 4.60 1.00 4.60 4.00 4.60 4.604.60 4.61 4.01 4.61
10.187483 ,
923073 | J' ^ 923755:, "„_ 923837 ' * 1.86 923919 1.38 924001 1.36 924083 1.86 924164 1.36 924246 1.30 924328 1.88 924409 1, 9.924491 ij'™ 924572 1,a0 1.86 924654 1.36 924735 1.85 924K11. 1.35 924897 1.35 024979 1.35 92E01 0 1.35 925141 1.85 925222 1.85 9.925803 1.85 925384 1.85 925405 1.85 925545 1.35 925620, j ,,1 925711' 1.84 925788 1.84 92E808 1.84 925949 1.31 926029 1.31 9.026110 1.34 920190 1.84 920270 1.34 926351 1.31 026431 1.34 926511 1.83 920591 1.83 926071 1.83 926751 1.83 920831 1.83 9.020911 1.33 926991 1.33 927071 1.83 927151 1.33 927231 1.33 927810 1.33 927390 1.83 927470 1.32 927549 1.82 927629 1.32 9.927708 1.32 927787 1.32 927867 1.32 927946 1.32 928025 1-32 928104 1.32 928183 1.32 928263 1.32 928342 1.32 928420
PP1" CotniiB. PP1" M.
M.
400
Cosine.
188030 188313 188590 188866:. 189143 , 189420: . 169698:* 189975 '* 190252s l lon^oo|'3 19080' 191084 191362 191639 191917 192195 192473 192751 193029 10.193307 193585 193863 194141 194420 194098 104977 195257 195534 195813 10.196091 196370 196649 196928 197208 19748 197766 198045 198325 198604 10.198884 199104 199443 199723 200003 200283 200563 200843 201123 201464 10.201684 201964 202245 202526 202806
4, 4 4 4 4 -1 4 4 4 4 4 4 4. 1 4 4 4 4 4, 4. 4, 4 4 1 4 4 4, 4 4 4 4. 4 4 4 4 4. 4. 4, 4, 4. 4, 4. 4. 4. 4, 203368 4. 203649 4 203930 204211
PP1" Cotunz. PP1" M
32"
S1NKS AND TANGENTS.
58° 8in,
9 10 11 12 13 14 15 16 17 13 lit 20 21 22 23 21 25 26 27 28 29 30 31 32 33 31 35 36 37 38 39 40 41 '12 43 11 45 40 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Tang.
9.928420 92849!1 928578 928857 928730 923815 928893 928972 929050 929129 929207 9.929288 929304 929442 929521 929599 929077 929755 92D833 9299U 929989 9. 930067 930145 930223 930300 930378 930156 930533 930811 930088 930700 9.930843 930921 930998 931075 931152 931229 931300 931383 931460 931537 9.931614|* 931691 1 . 931768;} 931845! ! 931921 ! } 9319981 { 932075 ! { 932151 1 } 932228 { 932301!! 9.982380.! 932157 932533 932009 932085 932762 032838 932914 932990 933086
10.204211 201492 204773 205054 205336 205617 205899 200181 206162 203744 207026 10.207303 207590 207872 208154 203437 208719 209001 209281 209566 209849 10.210132 210115 210698 210981 211204 211547 211830 212114 212397 212081 10.212904 213218 213532 213316 214100 214381 214668 214952 215236 215521 10.215805 216090 210374 2166-59 216944 217220 217514 217799 218081 218339 10.218654 218940 219225 219511 219797 220082 220368 220054 220940 221220
PPl" 9.933000 933141 933217 933203 933369 933445 933520 933590 933671 933717 933822 9.933898 933973 934048 934123 934199 934274 934349 934424 934499 934574 9.934049 934723 934798 934873 934948 935022 935097 935171 935246 935320 9.935395 935469 935543 935018 93.5692 935766 935840 935914 935988 930002 9.936136 936210 936284 930357 930431 930505 930578 936652 936725 930799 9.930872
1.68 4.69 1.69 4.69 4.69 4.69 4.00 4.69 1.70 4.70 4.70 4.71) 1.70 1.70 4.70 1.70 1.71 4.71 1.71 4.71 4.71 4.71 4.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 4.73 1.73 1.73 1.73 1.73 1.73 1.7.1 1.78 1.71 1.71 1.71 1.74 1.74 1.75 1.75 1.75 4.75 1.75 1.73 1.7.3 1.75 1.76 1.76 1.76 1.76 1.70 1.76 4.77 1.77 1.77
937019 937092 937165 937238 937312 937385 937458 937531
10.221220 221512 221799 22208: 222872 222658 222045 223232 223518 223805 224092 10.224379 224067 224954 225241 225529 225811. 220101 220392 220070 220967 10.227255 227.543 227832 228120 228408 228697 228985 229274 229563 229852 10.230140 230429 230719 231008 231297 231580 231876 232166 232455 232745 10.233035 233325 2330 15
Trig.—34.
'101
4.78 4.78 1.78 4.78 1.78 1.78 1.79 1.79 1.79 1.79 1.70 1.79 4.79 4.80 1.80 1,80 4.80 4.80 4.80 4.80 4.81 4.81 4.81 4.81
4.81 1.81 4.82 1.82 4.82 4.82 4.82 4.82 4.88 1.88 1.83 4.88 4.83 1.83 4.83 233905 4. 81 21119.5 4.84 234480 4.84 234770 1.81 235007 4.84 235397 4.85 235048 1.85 10.235939 1.85 230230 4.85 236521 4.85 236812 4.85 237103 4.85 237394 4.86 237086 4.80 237977 4.86 S38269 4. S0 2-18561 Cotung.
31"
4.77 4.77 4.77 1.77 4.78
1'l'1"
60 59
58 57 56 55 54 53 52 51 50 4!i 48 47 46 15 44 43 42 41 40 89 38 37 30 35 31 33 32 81 30 29 28 27 26 2.5 21 23 22 21 20 19 I8 17 16 15 14 13 12 11 10
9 8 7 0 5 4 3 2 1 0
TA11LK IV.— LOGAR1THMIC
60°
Tung. .21 i 2 3 4 5 ii 7 8 9 10 11 12 L3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 80 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 .'H1 51
939123 1!
'a 53 54 55 56 57 58 59 60 Cosine.
a»u
1T1"
.11.
|PP1" 9.9418111 1.17 941889' 1.16 941!ii!l 1.10 942029 1 1.16 942099! 1. 10 942109l 1.111 942239 9423081 Lie
10.2i8.VU 238852 239144 23943|1 2i!r728 240021 210313 240605 240898 2411911 24148:: 10.241776 242099 212362 242655 242948 243241 243535 243828 244122 24441 10.21470!1 245003 245297 215591 24588.5 240180 216474 246769 2470113 247358 10.247653 217948 218243 218538 248833 219128 219421 219719 250015 250311 10.250007 250903 251199 251495 251791 252087 252384 252081 2.52977 258274 10.253571! 253808 254165 251402 254700 255057 25,5355 255052 , 255950 k 2502481
Lie 942J78 li6 942448 Lie 942517 9.942587 Lie 1.18 942650 Lie 942720 1.16 942795 1.15 942804 1.15 942934 1.15 943003 1.15 943072 1.15 943141 1.15 943210 1.15 9.943279 1.15 943348 1.15 943417 1.15 943480 1.15 943555 1.15 943624 ! 1.15 943i1931 1. 14 0437011 1.14 943830 1. 11 943899 1.14 9.948967! 1.14 94403li' 1.11 944104 1.14 944172 1.14 944211 1.14 944309 1.14 944377 1.1! 044440 1.14 944514 1.14 944582 1-13 9.944030 1.13 944718 1.13 944780 1 . 13 944854 1.13 944922 1.18 944990 1.13 945058 1.13 945125 1.13 945193 1.13 94.5201 1.13 9.945328 1.13 945390 1.13 945104 1.12 945531 1.12 945598 1.12 945000 1.12 945733 1.12 945800l 1.12 9458081 1.12 945935
61° lung. 10.250248 236546 97 !17 250844 257142 97 07 257441 97 257739 97 258938 98 258336 '.18 258635 98 258S34 98 259238 98 10.259532 99 259831 11ii 200130 99 200430 99 200729 99 201029 99 261329|' 00 201029!?' 00 261929 * 1:l1 262229 i 2' iH1 10.262529* 00 202829 1 'f' i11 2631301'.'' 01 263430!°' 01 203731!'.'' 111 264031 (11 204332 112 204033 i12 264934 112 205236 02 10.205537 02 205838 02 200140 . 03 200442'° ,03 200743,? ,03 2070451? 03 26734' 08 207649 ,04 207952 ,111 268254 .04 10.208550 .01 208859 04 209162 05 209405 05 209707 i15 270071 i15 270374 05 270077 06 270980 1.0 271284 i»i 10.271588 IHi 271891 06 272195 i17 272499 07 272803 07 273108 i17 273412 117 273710 . 08 271021 ?' i18 2743261°
5 4 3 2 1 0 CoBini'. iPPl"! L'otnug. |PP1"| 1M.
PP1" Cotang
401'
as»
62° ' PPl"
ii l 2 3 1
9.945935 1.12 946002 1.12 940069 1.12 948138 1.12 946203 1.12 946270 1.11 940337 1.11 946401 l.U 946471 1.11 946538 1.11 946664 1. 11 9.946671 1.11 946738 1.11 946801 l.U 946871 l.U 946937 l.U 947001 l.U 947070 l.U 947136 1.10 947203 1.10 947269 1.10 9.947335 1.10 947401 1.10 947467 1.10 9475*3 1.10 947600 1.10 94766"i 1.10 947731 1.10 947797 1.10 947803 1.10 947929 1.10 9.94799: 1.09 9480B0 948126 1.09 1.0!) 948192 l.0ii 948257 1.0!i 94832i 948388 1.09 1.0:i 948454 1.0!i 94*519 1.0O 948581 1.0O 9.948650 1.09 94871 1.09 948780 1.08 94884c 1.08 948910 1.08 948975 l.0S 949040 1.08 94910"i l.iii 949170 1.08 94923"i 1.03 9.949300 1.08 949364 1.08 949429 1.08 949494 1.08 949558 1.08 949623 1.08 949688 1.07 949752 1.07 949816 1.07 949881
7 8 9 111 11 12 13 14 15 Hi 17 18 IS 20 21 22 23 24 25 28 27 28 29 30 31 32 33 "1 3i 36 38 39 40 41 42 43 11 45 Hi 17 '18 49 50 51 52 53
.-.! 55 50 57 58 59 i10
Cosine. 1l'PP
«?u
68°
SINES AND TANGENTS.
11.
iPPl"! Tan 10.274326 5. 08 274630 5.08 274935 5.08 275240 5.0O 275546 5.09 275851 5.09 276156: 5.09 276462: 5.09 276768 5.10 277073 5. 10 277379 10 10.277685 " 10 277991 i'i .10 278298 5.11 278604 5.11 278911 5.11 279217 5.11 279524 5.11 279831 5. 12 280138 5.12 280115 5.12 10.280752 5. 12 2810 i0 5.12 281367 13 281675 13 28198:! 13 282291 5.13 282599 5.13 282907 5.14 2*3215 5. 1 1 2*3523 5. 1 1 10.28:3832 5. 1 1 284140 5.11 281149 5.15 284758 5.15 285007 5. 15 285376 5.15 285686 5.18 285995 .10 286301 .16 286614 .10 10.286924 .16 287234 .17 287544 5. 17 2878.54 5. 17 288164 5.17 288175 5. 18 2887K 5.18 289096 5. 18 28940: 5.18 289718 3. 18 10.290029 5.19 290340 5.1O 290651 5.19 290963 3. 1!i 291274 5.19 291586 5.20 291898 5.20 292210 5.20 292522 5.20 202834
M. tio 59 58 57 58 55 51 53 52 51 50 4!i '18 47 4li 45 44 43 '42 41 '10 39
as 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 I8 17 16 15 14 13 12 11 10
Cotuiij
1PPl"
'.949881 _ 10.292*14 949945 '£ 293146 950010 ■£ 293459 930074-07 293772 294084 gsoias '£ 950202: "^ 294397 950206 1-u/ 5 22 294710! 1.07 5.22 295024! 29 950330 1. 07 295337 ! 050394 1.07 i.22 950458 2956150 5 22 1.07 295964 950522 1.06 10.296278 .950580 5.23 1.08 9..0650 296561 5.23 1.06 950714 296905 1.06 5-23 9507781 297219 1.06 297534 5.24 950841 1.116 - iif 5.24 297848 95090"i' 1.06 5.24 298163 24 950968 1.06 298477 951032 1.06 298792 951096 1.06 951159 299107 1.06 10.299422 .951222 5.25 1.08 299737 951286 :5.26 1.06 300053 951349 1.05 5.26 300368 951412 '5.26 1.05 800684 951476 5.26 1.05 300999 951539 5.26 1.05 301315 951602 1.05 5.27 301631 951665 5.27 1.05 301947 951728 5.27 1.05 302204 951791 5.27 1.05 .051854 10.302580 !5.28 1.05 951917 30289; 5.28 1.05 303213 951980 1.05 '5.28 95204,'! 303530 5.28 1.05 952106 303847 5.29 1.05 952168 304164 1.01 |5.29 952231 304482 1.01 5.29 304799 952294 1.01 5.29 952356 30511' 1.01 5.29 952419 305434 5.30 1.04 10.305752 '.952481 i!o.80 1.04 952544 ;:ooo7o 5.80 1.04 952600 306388 1.04 .5.80 952609 30670; 5.31 1.04 952731 307025 5.31 1.04 95279;; 307344 5.31 1.04 952855 307662 5.31 1.04 307981 952918 1.04 5.31 308300 952980 5.32 1.03 308019 035042 5.32 1.03 .953104 10.308938 1.03 5.32 309258 953166 1.03 5.38 953228 309577 1.03 5.a'B 309897 953290 5.33 1.03 310217 953352 1.03 6.33 310537 953413 5.33 1.03 3108.57 953475 5.31 1.03 953537 311177 5.84 1.03 311498 953599 !5.34 1.03 953060 311818
.11.
?1! -3 £ £5 | °° £ ^ 52 rr 51 50 *° « % f4o 44 ;l 4i 42
* s
lii8ilM
403
PPl"
iPPl
37 36 35 34 83 82 31
l-i.t.mg. iPPl"| M.
"~ 26°
8ine. 11 1
2
'; 4 5 9 7 8 9 10 11 12 13 11 15 16 17 I8 19 20 21 22 23 24 25 26 27 2S 29 80 31 32 Si :si 85 86 87
;;s 39 40 '11 '42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00
1'l'l"!
Tiuig.
Tun::.
PPl'
PPl
10.331327 5.50 331057 5.50 331987 5.50 332318 5.51 332648 5.51 382979 5.51 333309 5.51 333040 5.52 333971 5.52 334302 5.52 334634 5.53 10.334965 E CO 335297 !'.- "
10.311818 5.31 1.03 312139 1.02 5.35 312100 5.35 1.02 312781 5.35 02 313102 5.35 1.02 313423 5.36 1.02 313745 5.36 .02 314000 02 314388 02 314710 5.37 02 315032 5.37 .02 10.315354 5.37 02 315070 02 5.87 315999 112 5.38 316321 01 5.88 316644 5.38 01 316967 111 5.88 317290 5.39 .01 317613 5.89 01 317937 (11 5.89 318200 5.3!i 111 10.318581 .01 5.40 318908 5.40 01 319232 5.40 ,01 319550 955126|} 5.40 01 3198£0 955186 { 5.41 i1I 320205 955217 |) 5.41 01 955307 320529 5.41 01 955308 320854 5.41 ,01 955428 321179 5. 12 00 055488 321501 5.42 00 10.321829 .055548 5.42 00 322154 9.55009 5.42 .00 322480 955069 5.43 322800 955729 00 5.43 00 323131 955789 5.43 00 323457 955849 5.43 00 323783 955909 5.44 00 324110 955909 5.44 00 324436 956029 5.44 050080 ,00 324763 5.44 00 .950148 10.325089 5.45 00 950208 325116 5.45 99 950208 325743 5.45 99 9.50327 320071 5.46 99 950387 326398 5.46 956447 99 320726 5.40 950500 90 327053 5.46 327381 956500 99 5.47 99 950025 327709 5.47 950084 99 328037 5.17 .99 .950744 10.328305 5.47 99 956803 328094 5.48 99 950802 329023 5.48 99 956921 329351 5.48 98 950981 329680 5.48 98 957040 330009 5.49 957099 330339 5.49 957158 330608 5.49 957217 330908 5.50 33132 957270 .953600 953722 953783 953845 9589011 95390S 954029 954090 954152 954213 954274 .954*1.5 951396 954457 9.54518 954579 0511110 951701 954702 954823 954883 .9,54944 955005 955005
Cosinn
2.V
6r,°
TABLE IV.—LOGAR1THM1C
«l°
335629:^g 335961 °5.51 336293 5.54 330025 5.54 336958 5.54 337291 5.55 337624 5.55 337957 5.55 10.338290 5.56 338023 5.56 ,56 339290 5.57 339624 339958 ?-57 5.57 340202 5.57 340627 5.58 340961 5.58 341296 5.58 10.341031 5.58 341900 59 342301 5.59 342036 5.59 342972 5.59 343308 5.00 348644 .00 ,00 344310 5.01 344652 5.61 10.344989 5.61 84532C 5.61 345663 5.02 340000 5.62 340337 5.62 340074 5.63 347012 5.63 347350 347088 9-M 5.63 348020 5.64 10.348304 5.64 348703 5.64 349041 5.65 349380 '349719 £'* 5.65 350058
59 58 57 .50 55 54 .53 52 51 90 40 48 47 46 4.5 44 43 42 41 40 39 38 37 30
as 34 33 32 31 30 29 28 27 20 25 24 23 22 21 20 19 I8 17 10 15 14 13 12 11 10 9 8
350737 5-66
351077 "J 351417 °-66 PPl"
Cotana. 1PPi"
404
24°
SINES AND TANGENTS.
66° 8in.. M. 0 9.960730 1
2 3 4
Cosh
Sim
PP1"
10. 351417
960780 960843 960809 960955 961011 961087 961123 961179 981235 961290 9.981316 961402 981458 961513 961569 961621 981680 961735 981791 961816 9.981902 981957 982012 962067 962123 962178 982233 962288 962343 962393 9.982453 982508 982502 962817 932672 982727 982781
982945 9.962909 983054 963103 903163 983217 933271 963325 983379 983431 983 188 9.983512 983593 963650 963701 983757 983811 983885 963919 963972 9J4021i
S3"
Tang.
9.984026 964080 964133 904187 964240 964294 964347 964400 964454 96450: 964560 9.964613 964666 964720 904773 964826 964879 964931 964984 96503' 965090 9.965143 965195 965248 965301 965353 965400 965458 965511 965563 965015 9.965608 905720 965772 965824 965870 905929 905981 966033 960085 966136 9.966188 966240 080292 966344 960305 90044' 906499 966550 900602 906053 9.906705 066756 966808 960859 960910 900901 907013 967004 967115 9(>711i6
351757 a52097 ar)2438 352778 353119 353400 353801 354143 354484 354826 355168 355510 355852 356194 356537 358880 357223 357586 357909 358253 10. 358596 358940 359284 359629 359973 360318 360663 301008 361353 10-302014 362735 363081 363428 363774 364121 364468 364815 365162 10-335510 305857 366205 366553 366901 367250 367598 367947 368296 368645 10-368995 309344
89
370044 370394 370745 371095 371446 371797 372148
PP1" Cnimig.
Cosin
405
67° PPl"!
Tun*. 1.372148 372499 372851 373203 373555 373907 374259 374612 374964 375317 375670 U76024 370377 376731 377085 377439 377793 378148 378503 378858 379213 1.379508 379924 380280 380636 381348 381705 382061 382418 382776 1.383133 383491 383849 884207 384505 384923 885282 385641 380000 386459 1.386719 387079 387439 387799 388159 388520 388880 389241 389003 1.390320 390088 391050 391412 391775 392137 392500 392863 393227 393590
PP1" Cotana
PP1" 5.85 5.86 5.80 5.86 5.87 5.87 5.87 5.88 5.88 5.88 5.89 5.80 5.89 5.90 5.90 5.90 5.91 5.91 5.92 5.92 5.92 5.93 5.93 5.93 5.94 5.94 5.94 5.95 5.95 5.95 5.96 5.96 5.96 5.97 5.97 5.97 5.98
5.99 5.99 6.00 6.00 6.00 0.01 0.01 6.01 6.02 6.02 6.02 6.03 6.03 6.03 0.01 6.0! 6.01 6.05 6.05 6.06 0.06
PP1"
60 59 58 57 .56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39
38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 I8 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
«s°
TABLE IV.— LOGAR1THM1C 8im',
o
i 2 3 1
.". i; 7 s 9 in 11 12 13 14 15 Hi 17 1S 19 -0 21 22 23 21 25 26 27 28 2! 1 30 81 32 33 34 35
:;:; 37 38 39 10 11 12 '43 11 45 46 47 48 1:1 50 51 52 53 54 55 56 57 58 50 G0
9.907100 907217 907208 987319 9073711 007421 987471 907522 907573 907021 907074 9.907725 907775 987828 967870 987927 007077 008027 008078 008128 008178 9.908228 988278 908320 968379 068429 968470 96*528 068578 968028 908678 9.968728 908777 968827 968877 908920 008970 960025 909075 960124| 069173i 9.009223; 0li9272; 909321 ! 999370 900420 060100 060518 069507 909610 009005 9.060714 960762 060811 000800 009900 969957 970006 970055 070103 970152 Cosine.
21"
!PPl" 1
69°
Tang. |PPl 10.393590 li.1Ki 393954 0.07 394318 0.07 394083 0.07 395017 0.08 395412 6.08 395777 0.09 3961421 ,
,970152 070200 070240 070207 970315 970394 970142 970490 970588 070580 0700.35 .070083 070731 070770 970827 970874 970022 070970! 971018 971000 971113 .071101 071208 071250 971303 971351 971398 971446 971493 971540 971588 .971035 971682 971729 971770 971823 971870 971917 971904 972011 972058 .972105 972151 972108 972245 972291 072338 972385 972431 972478 072524 .072570 972017 0720C3 072700 072755 072802 972848 972801 972040 972989
396B73| '"" 3072i: i 0.10 6.10 10.397005 6.10 307971 6.11 398337 6.11 398701 6.11 399071 6.12 6.12 6.13 400173 6.13 '100511 6.18 400909 6.14 10.401278 6.14 401040 6.15 402015 6.15 402384 6.15 402753 6.16 403122 6.16 403492 6.16 403862 404232 6.17 G.17 401002 10.404973i1!i 6.18 405344 6.18 405715 6.18 406080 8.19 406458 6.19 406829 6.20 407201 0.20 407574 6.21 407940 6.21 408319 6.22 10.408092 0.22 409005 0.22 409438 0.23 409812 6.23 410180 0.23 410500 6.24 410034 6.24 4113001 6.25 411084! 6.25 412050 6.25 10.412434 6.26 412810 0.26 413185 0.27 413561 0.27 413038 6.27 414314 0.28 414091 0.28 415068, 0.29 41.5445! 0.20 415823
10.415823 .81 6.29 410200 .81 416578 6.80 .81 410950 6.80 .81 6.31 417335 .80 6.31 417714 .Ml 6.82 .80 6.32 418472 .80 6.32 418851 .80 6.33 419231 .80 6.33 419611 .80 6.34 10.419991 .80 8.84 420371 .80 6.34 420752 .80 6.35 421133 6.35 .80 421514 .80 6.86 421896 .80 6.36 422277 .80 6.88 422059 6.87 .80 423041 .7O 6.37 423424 .70 6.38 10.423807 .70 424190|!i-^ .70 4240,0, „ .70 424958 "/IS .711 4253401 "'* .70 42o/24| ,ft .70 426108!^™ .70 426493 "'" .70 42C877!"-" .7!) 427262°-?; .70 10.427648!"-*;; .70 428033!"':f .70 428419!^: 4288051"'*; 429191!"-" 429578«« 429965"-!; 430852j°-*_
«n»?-* 4811271*-* 10.431514 °-*° 431002!"- ' 432291"-*' 432080|"-^ 433008!"-^° 433458!"-™ 433847|»-^ 434237;"-?° 434027 \6-fn 435011 6 51 10.435407°-", 4&5798|"-^ 430189|"-„ ' 43058ll°-?f 4309721°-^ 437364|°-^
60 59 58 57 56 55 51 53 52 51 90 49 48 17 46 15 44 13 42 41 40 30 08 37 30 35 34 83 32 31 30 20 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4
:; 438149;"-^ 438541°-:? 438984
2 1 0
PP1"| CotKiig. iPPi"
Cutuiig. i P1'P
4Ub
ao»
SINES AND TANGENTS.
70° M. 0 1 2 :i '1 5 6 7 8 9 10 1I 12 13 14 15 Iii 17 I8 19 20 21 22 2.'1 21 25 29 27 28 2! 1 80 31 32 33 34 35 36 37
as 39 40 41 42 43 II 45 46 17 18 '19 50 51 52 53 54 55 56 57 58 59 CC
8ine, 9.972986 973032 973078 973124 973109 973215 973261 973307 973352 973398 973114 9.973189 978535 973580 97362.5 973671 973716 973761 973807 973852 973897 9.973942 973987 971032 974077 974122 974167 974212 974257 974302 97431 9.974391 974430 974481 974525 974570 974014 974059 974703 974748 974792 9.974836 971880 974925 974909 975013 975057 975101 975145 975189 975233 9.975277 975321 97.5365 975108 975452 975496 975539 975583 97:5027 975070 Cosim
19J
Tang. PPl" 10.438934 6.55 439327 6.56 439721 6.56 440115 6.57 440509 6.57 440903 6.58 441297 6.58 441692 6.59 442087 6.59 442183 6.59 442879 6.60 10.443275 6.60 443671 6.61 444067 6.61 444461 6.62 444861 6.62 445259 6.63 445658 8.63 446054 6.64 446152 6.64 446851 6.65 10.447250 0.05 447649 6.65 448018 0.66 448118 0.66 448817 6.67 449218 6.67 449348 6.68 450049 6.68 450450 6.69 450851 6.69 10.451253 6.70 451655 6.70 452057 6.71 452460 6.71 452862 6.72 453265 6.72 453669 6.73 454072 6.73 454476 6.74 454881 6.74 10.455285 8.75 455690 6.75 450095 6.76 456501 6.76 456908 6.77 457312 0.77 457719 6.78 4,58125 6.78 458532 6.79 458939 6.79 10.459347 6.80 459755 6.80 460163 6.81 460571 6.81 460980 0.N2 461389 6.82 461798 6.83 462208 6.83 462618 6.84 403028 PPHCotang.
M. 9.975670 975714 975757 975800 975844 975887 975930 975974 976017 976000 970103 9.970140 976189 976232 970275 970318 976361 976404 970440 970489 976532 9.976574 970617 970000 976702 976745 976787 976830 976872 970914 97695' 9.976999 977041 977083 977125 97716: 977209 977251 977293 977335 9773: 9.977419 977461 977503 977544 977586 977028 977609 977711 977752 977794 9.977835 977s; 977918 977959 978001 978012 978083 978121 978105 978200
Pl
Cosine
4117
71° Tiing.
PP1"
10.463028 6.84 403439 6.85 463850 6.85 464201 6.86 461072 6.86 465084 6.87 465496 6.87 405908 6.88 400321 6.88 406734 6.89 407147 6.89 10.407561 6.90 467975 0.90 468389 6.91 468804 6.91 409219 6.92 409034 6.93 470049 0.93 470405 6.93 470881 6.94 471298 6.95 10.471715 6.95 472132 6.96 472549 6.96 472907 6.97 473385 6.97 473803 6.! 474222 6.98 471041 0.99 475000 6.99 475480 .00 10.475900 .01 470320 7.01 470741 7.02 477162 7.02 477583 7.03 478005 7.03 478427 7.03 478849 7.04 479272 7.05 479695 7.05 10.480118 7.06 480542 7.06 480960 7.117 481390 7.08 481814 7.08 482239 7.09 482065 7.09 483090 7.10 483510 7.111 483043 7.11 10.484309 7.12 484790 7.12 485223 7.13 485051 7.13 480079 7.14 486507 7.14 480930 7.15 487365 7.16 487794 7.16 488224 Cntang.
iid
59 58 57 56 55 51 53 52 51 50 19 48 '17 46 45 '11 43 42 11 40 39 88 37 36 35 34 33 32 31 30 2I1 28 27 20 25 24 28 22 21 20 19 18 17 111 15 14 13 12 11 10 9
PPl" M. I
—15°
TABLE IV.—LOGARITHMIC
T2° 0 1 2 3 '1 5 6 7 H
e 10
n 12 13 14 15 16 17 18 111 20 21 22 23 21 2"i 26 27 2.8 29 30 31 32 33 31 85 36 37 38 39 40 11 42 43 '11 45 46 47 48 49 50 51 52 53 54 55 50 57 58 59 00
978737 978777 978817 978858 978898 978939 978979 979019 1.979059 979100 979140 979180 979220 979200 979300 979340 979380 979420 1.079459 979499 979539 979579 979018 979658 979697 979737 979776 979810 1.979855 979895 979934 979973 980012 980052 980091 980130 980109 980208 1.980247 980325 980364 980403 980442 980480 980519 980558 980590
Tang. H'lT'i 10.4.88224 488654 489081 489515 489940 490378 490809 491241 491074 492107 492540 10.492973 493407 493841 494276 494711 495146 495582 496018 4964.54 496891 10.497328 497765 498203 498641 499080 499519 499958 500397 500837 501278 10.501718 502159 502001 503043 503485 50:3927 504370 504814 505257 505701 10.500146 506590 507035 507481 507927 508373 508820 509267 509714 510162 10.510610 511059 511508 511957 512407 512857 513307 513758 514209 514601
M, 60 5: i 58 57 56 55 51 53 52 51 50 49
9.980596 980635 980673 980712 980750 9807K9 980827 980866 980904 980942 980981 9.981019 981057 981095 981133 981171 981209 981247 981285 981323 981361 9.981399 981436 981474 981512 981549 981587 981625 981662 981700 981737 9.981774 981812 981849 981886 981924 981961 981998 982035 982072 982109 9.98214C 982183 982220 082257 982294 982331 982!67 982404 982441 982477 9.982514 982551 982587 982624 982660 982696 982733 982769 98281*5 982842 Cosine.
Cutuiiir.
M"i
8iiii-
403
T8° Tung. | P1T 10.514001|. „ 515113i'-?° 515565 ''" Blt018i'-» 616471 ''" 516925il-*
517879XL'0' 517833; !"" 518288:i-'ir 518743 ',oa 7.60 519199 7.60 10.519655 60 52011 ljl' 61 520568 7.62 521025 7.63 521483 7.63 521941 7.111 7.65 522858 7.65 523317 7.611 523777 7.67 10.524237 7.67 524697 7. iis 525158 7.611 525619 7.70 526081 7.70 526543 7.70 527005 7.71 527468 7.72 527931 7. 73 528395 7.73 10.528859 7.71 529324 7.75 529789 7.75 530254 7.76 530720 7.77 531186 7.78 631653 7.7N 532120 7.79 532587 7. S0 533055 7. S0 10.533523 7. 81 533992 7.82 534461 534931 535401 7M 535872u'4,_ 586842;-?? 536814 7.86 537285 7.87 537758 7.88 10.538230 7.8S 538703 7.89 539177 7.00 539651 7.90 540125 7.111 540000 7.02 541075 7.03 541551 7.93 542027: 7.01 542504! Coin
1'1M
16"
iN. S AND TANGENTS.
T4°
'inng.
M. 0 1 2 3 4 5 C 7 8 0 10 11 12 13 14 15 16 17 13 19 20 21 22 23 21 23 28 27 28 29 30 31 32 S3 34 35 30 37
9.082842 982878 982914 982950 983022 983058 983094 983130 983160 983202 9.983238 983273 983309 983315 983381 983116 983152 933137 983523 983358 9.983594 983029 983061 983700 983735 983770 988805 983875 983911 9.983946 983981 981015 981050 981085 981120 981155 981190 981221 981259 9.981294 984328 981303 984397 981432 981106 984500 981535 981509 981603 9.984638 984672 984700 981740 981774 981808 984842 981876 981910 984944 Cosine
15"
i'Pl"
10.542501 542981 643158 543933 541114 541893 54537 545852 540332 540813 547294 10.547775 548257 548740 549223 54970: 550190 550374 551159 551644 552130 10.552616 553102 553380 551077 551503 555053 555512 550032 550521 557012 10.557503 557994 558180 558978 559471 559904 560157 500952 501446 501941 10.562437 562933 563430 503927 564421 561922 505421 505920 560420 566920 10.507420 567921 508423 50892. 56942 569930 570434 670938 571442 571948 PPl" i'otana.
Trig.—35.
M. 0 1 2 3 1 5 0 7 8 9 10 11 12 13 11 15 10 17 18 19 20 21 22 23 21 25 26 27 28 211 80 31 32 33 34 35
8ine 9.984944 984978 985011 985045 985079 985113 98CH0 88Clf0 985213 98524" 985280 0.985314 985317 985381 985414 98514' 985480 985514 98554" 985580 985013 9.985646 985679 9i'5712 085745 985' 985811 985843 985876 985909 985942 9.985974 C860C
9a.u» 980072 986101 986137 986169 980202 980234 980266
980427 986459 986491 986523 980555 980587 986651 980714 986740 986778 986809 980841 986873 Conine.
409
PPl" M
10.571948 572453 572959 573466 573973 574481 574989 575497 576007 570516 577020 10.577537 578018 678560 576073 679585 580099 580613 581127 581642 582158 10.582674 583190 583707 584225 584743 585262 585781 586301 586821 587342 10.587863 688385 588908 &D431 589955 690479 591004 691529 692055 592581 10.593108 593630 594104 694692 595222 595751 696282 590813 597344 597876 10.598109 598942 699476 600010 600545 601081 601617 602154 602691 603229
986331 986363
PPi" M
T.V lunar
PP1
Cotang. PP1" M. |
.JC M. | 0 1 li
:: i
;; 7
s 0 11 11
'.: i ; 1 1 l . 1 '! 17 13 V.\ L'J LI L2 23 21
TABLE IV.—LOGAR1THMIC .8in 9.933901 08C030 08C937
'1nn g
ll-Pl" fll.
10.603223 6037071 06430C osiaos 601810 D87030 C05380 087C01 6059i 087092 C004C9 087121 C07011 987155 C0755.' 087180 cosoo; C87217 608040 9.987218 10.609185 087279 C097:» .'"2 057310 C18270 .52 087811 010822 .52 C87372 C11308 D871C3 enoio C87131 C12104 087105 01,3013 987100 oiajos 98752C 014112 9.98755; 10.014003 9S7588 615214 687618 615701, 987049 610318 010871 987070 087710 01742. 087740 617080 087771 018531 087801 619090 087832 619646 9.937802 10.620203 987892 020701 987922, 021319! 021878 987053 987983 622437 983013 622997 988043 623558 988073 021119 988103 621081 988133 025211 10.025807 9.968163 988193 020371 988223 620930 988252 6: 501 |J 98828: 628087I"' 98831! 028633 J 988312 C2«201 ! J 988371 629768 ?' 988101 630337 *' 988130 630906 „ 9.988180 10.631476 „ 988489 C3aM7!?' 988519 6320181 ^ 988548 683100 „ 988578 633763 g 634336 «'
98803 '
988721
9.C88721 988753 988782 98V 11
c::ss.:o O8i8C0 088808 Cr8C27 CSS956 988055 CS9C11 9.0fc'6042 086071 939100
9.'!'r>" 98LI57 9£9180 98921 ! 9891' 13 089271 989301/ 9.980328 089350 989385 089413 9894U 989409 989497; 0805251 089553 989582 9.989010 089037 989005 989693:i 989721 989719 989777 989801 989832 989800 9.98.9887 089915 089942 98C970 9S9097 B0C0C5 9C60.",2 CO6079 990107 090131 9.990101 990188 990215 990243 990270 990297 990321 900351 990378 990101
631910! o 635185 6800C0, 030030| iVrtnng. iPP1"
2SU
.' i : i
Cofiinc.
410
i'1'i
.19 .19 .1S .18 .18 .13 .18 .18 .18 .18 .18 .48 .18 .48 .48 .18 .18 .17 .17 .47 .47 .47 .47 .17 .47 .17 .17 .17 .47 .17 .47 .47 .47 .46 .16 .46 .48 .48 .48 .46 .46 .46 .46 .48 .46 .46 .40 .46 .46 .45 .45 .15 .45 .15 .45 .45 .45 .45 .45 .15
T7° Tung.
|Pl'l"i hi.
10.C3SCCGl 037213 9.61 i 9.02 C877CC 9.03, C38308 o.cc C3£947 9.60' 63952C 9.07 610107 9.C8[ 610C87 9.09 6412C9 9.71 611851 9.71 642431 9.73 10.643018 9.71 643602 611187 9.7 611773 9.7,, 61.5300 9.7C 615947 9.80 610535 9.81 017121 9.82 017713 9.83 618303 9.85 10.018894 9.80 049480 9.87 650078 9.88 650071 o.oo: 651265 9.91 C51859 9.92 652155 9.03 053051 9.94 653647 9-96| 3* 654215 97 1 19 10.654843 9.1'8| 055442 9.99 656012 10. 00 1 650042 10.02 657243 10.03| 6578151 10.01: 058118 ! 10.00 0500521 10.07 6590501 10.08 660201I 10.10 10.000807| 10.11 6014731 10.12 602081 ; 10.13 0020891 10. 15 6632081 10.10 603907 i 10.17 064518! 10.10 665129 10.20 005741 10.21 600351 10.23 10.000907 10.21 607582 10.25 008197 10.20 C08813 10.28 609430 10.29 670017 10.30 070000 10.32 071285 10.33 671005 10.35 672525
rPl" Cntang. PP1" M
ii.u
SINES AND TANGENTS.
7S°
sine
'Pnni
0 1 2 3 1 5 C 7 8 9 10 11 12 13 14 15 10 17 I8 19 20 21 22 23 21 2"i 20 27 28 20 8C 8I 32 33 31 35 30 37 33 89 40 41 '12 43 44 45 40 47 48 49 90 51 52 53 £1 59 50 67 68 69 C0
M.
11^
9.990401 090431 990458 990489 990511 990538 990069 930091 09031s 090819 090071 9.990097 990721 990750 990777 990803 990829 990855 990.882 990908 990031 9.990000 990980 991012 091038 901004 991090 091115 991141 9911C; 991193 9.991218 991244 091270 091295 991321 991340 991372 991307 991422 991448 9.091473 991498 991521 091549 991574 991599 991024 991049 991674 991099 9.991724 991749 991774 991799 991823 991848 991873 991897 991922 0i1947 Co--
10..072929 .45 073147 .45 073709 .45 074393 .45 675017 .11 675642 .11 676267 .1! 670894 .11 677521 .44 678149 .11 678778 .11 10..679408 .11 .44 680670 .11 081303 .11 081930 .11 082570 .11 683205 .11 683841 .11 684477 .44 685115 .43 ).689753 ,43 686392 .43 687032 .43 687673 .43 688315 .43 688958 .43 689801 .43 600246 .48 090891 .43 091537 .48 ).692184 .43 692832 .43 693181 .13 694131 .43 664782 .13 695433 .42 690080 .42 690739 .42 007393 .42 C98049 .42 ). 696705 .42 099302 .42 700020 .42 70007; .42 701338 .42 701099 .42 702001 .12 703323 42 703987 .VI 764619 .42 0.700310 .11 705983 .11 700090 .11 707318 .11 707987 .11 708058 .11 709329 .41 710001 .41 710074 .11 711348
9.991947 991971 991990 992020 992044 992009 992093 992118 992142 992106 992190 9.992214 992239 992263 992287 992311 992335 992359 902382 992406 992430 9.992454 992478 992501 90292.-i 992549 992572 992590 992019 992843 992006 9.992090 992713 992730 992799 992783 992800 992829 992852 992879 992898 9.992921 992944 99296 992990 993013 993030 993099 993081 993101 9931: 9.993140 993172 993195 99321 993240 993202 993284 903307 993329 993351 Cosinfl.
i'ntnni*
411
T90 Timg. PP1' 711348 11.25 712023 11.26 712699 11.28 713376 11.30 711053 11.31 714732 11.33 715112 11.35 710093 11.30 710775 11.38 717458 11.40 718142 11.41 10. .718820 11.43 719512 11.45 720199 11.47 720887 11.48 721576 11.50 722266 11.51 72295^ 11.53 723049 11.55 724342 11.57 725030 11.58 .725731 11.60 726427 11.62 727124 11.64 727822 11.05 728521 11.61 729221 11.69 729923 11.70 730025 11.72 731329 11.74 732033 11.70 .732739 11.78 733445 11.79 734153 11.81 7348C2 11.83 735572 11.85 73028:: 11.8 730995 11.89 737708 11.90 738422 11.92 739137 11.94 10.1 '39854 11.90 740571 11.98 741290 12.00 742010 12.01 742731 13.05 743493 12.09 744170 12.01 744960 12.09 745C20 12.11 74089S 12.15 10.747080 12.10 747809 12.17 748539 12.18 749270 12.20 750002 12.22 750730 12.24 751470 12.20 752200 12.28 752943 12.30 799S81 PIT' Cntnnc
60
on 58 67 56 55 54 53 52 51 59 49 48 47 46 45 1! 43 42 41 40 39 38 37 36 35 84 33 32 31 30 20 28 27 2|i 20 21 23 22
21 2D 19 18 17 10 15 14 10 12 11 10 9 8 7 6 6 1 3 2 1 0
PP1"| M.
io°
TABLE IV.—LOGARITHM1C
SO"
|P1'l" ii
9.,993351
1
993374 993390 993118 993110 903102 993184 993500 993528 993550 993572 .993594 093310
2 3 -1 5 6 7 8 9 10 11 12 13 14
15 1D 17 18 19 20 21 22 23 24 2"i 26 27 28 29
80 31 32 :ii 84 35 30 87 88 89 40 41 42 43 44 45 40 '17 48 49 50 51 53 51
£3 59 C0
»«:;38 99301,0 993081 993703 Itn:172". 993740 903708 993789 ,993811 093832 093854 993875 993897 993918 003939 993900 993982 901003 .994021 094045 994000 094087 991108 994129 094150 994171 D94191 994212 .994233 994254 994274 094295 004310 994330 994357 994377 994398 994418 .994438 994459 994479 994499 994519 094.540 994500 994580 994C00 094020 i?ii8ll11".
10.753081 754421 755101 755903 750640 757390 758135 7.58882 759029 700378 701128 10.701880 702032 703380 764141 704897 705055 706414 707171 7177935 70.8098 10.709401 770227 770993 771701 772529 773300 774071 774844 775018 770,393 10.777170 777948 778728 779508 780290 781074 781858 782044 783432 781220 10.785011 785802 786.595 787380 788185 788082 780780 790580 791381 792183 10.792987 793793 794060 795408 796218 797029 797841 798055 7110171 S60287 Com
8i uft.
9.994020 094040 994000 994080 994700 094720 994739 994759 094770 994798 994818 0.994838 004857 094877 994890 004910 994935 994955 994974 094993 995013 9.095032 005051 09.5070 995C89 C95108 995127 995140 S95105 005184 995203 9.C0G222 995241 99-3200 D95278 995207 995310 995334 995353 995372 095390 9.905409 995427 995440 995464 995482 99.5501 995519 995537 995555 995573 9.995391 90.3010 C05028 995640 095064 995081 995699 993717 995735 99.57,53
12.32 12.31 12.36 12.38 12.40 12.12 12.44 12.40 12.48 12.50 12.52 12.54 12.50 12.58 12.00 12.02 12.05 12.07 12.09 12.71 12.73 12.75 12.77 12.79 12.81 12.81 12.80 12.88 12.90 12.92 12.94 12.97 12.99 13.01 13.03 13.00 13.08 13.10 13.12 13.15 13.17 13.19 13.21 13.24 13.28 13.28 13.3! 13.33 13.35 13.38 13.40 13.42 13.45 13.47 13.49 13.52 si.54 13.57 13.59 si. 0l
prr
CoRim1
412
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10.800283 801 100 801926 60274' 803570 804394 805220 £06047 €06870 607700 8CR538 10.809371 810200 811042 811880 812720 813501 814403 815248 816003 816941 10.817789 818640 819492 820345 821201 £22058 822916 £23770 824038 S25501 10.826366 827233 628101 £28971 £29843 830710 831591 632408 £33340 634220 10.835108 835002 830877 837764 838653 830543 840435 841329 642225 84312i 10.8440T2 844023 845820 810731 847637 848540 819450 850308 851282 £.52197
13.64 13.06 13.69 13.71 13.74 13.76 13.79 13.81 13.84 I3.86 13.89 13.91 13.93 13.06 13.99 14.02 14.04 14.07 14.00 14.12 14.15 14.17 14.20 14.23 14.25 14.28 14.31 14.33 14.30 14.39 14.42 14.44 14.47 14.50 14.53 14.55 14.58 14.61 14.64 14.67 14.70 14.73 14.70 14.79 14.81 14.84 14.87 14.90 14.03 14.90 14.99 15.02 15.05 15.08 15.11 15.14 15.17 15.20 15.23 15.20
PP1" Cotang. PP1"
SINES AND TANGENTS.
S3"
o 1 2 3 4 5
e 7 8 9 10 11 Z2 13 14 15 10 17 18 19 20 21 22 23 21 25 28 27 28 23 3D 31 32 33 51 83 3j 37 33 3D 40 41 42 43 44 45 49 47 48 '19 50 51 52 13 51
Sine. 9.995753 995771 995788 995808 995823 995811 995859 995876 995894 995911 995928 9.995948 995903 995930
Tang.
P1'i'
10, 8i2197
993015 993032 993D19 938088 998033 993100 9.938117 998131 998151 998183 993185 993232 993219 933235 998252 996239 9.933233 933302 998318 933335 993331 998J38 996381 998403 993117 993433 9.993149 993103 993132 99319S 990514 996530 993513 998582 996378 998591 9.998310 993825 933811 993357 99J
996804 9.996919 996934 996964 996979 997CC9 997024 997039 997053 9.897068 997083 097112 997127 997141 997156 997170 997185 997199 9.997214 997228 997242 997257 997271 997285 997299 997313 99732; 997341 9.997355 997369 997383 997397 997411 997425 997439 997452 997466 997480 9.997493 99750' 997520 997534 997547 997561 997574 997588 997601 997614
910836
PPl"! Coting, ppi"
"35*
S30 '1':.
9.996751 996766 996782 996797 998812 996828 996843 996858 990871
15.29 15.32 15.35 15.39 15.42 15.45 15.48 15.51 15.55 15.58 15.01 13.64 15.67 15.71 15.74 15.77 15.81 15.81 15.87 15.91 15.94 13.97 16.01 10.01 13.07 16.11 16.15 10.13 10.22 13.25 13.29 16.32 13.33 10.33 16.43 16.46 10.50 10.54 10.58 10.61 16.65 16.69 16.72 16.76 16.80 16.84 18.87 16.91 16.95 16.99 17.03 17.0; 17.11 17.15 17.19 17.22 17.27 907734 17.30 903772 17.34 909313 17.38
853115 854031 854956 855879 856804 857731 858660 859591 860524 881458 10, ,862395 883333 861274 865216 830161 887107 833958 839033 839959 870913 10. 871370 872323 873789 874751 875716 878333 877652 878!23 879593 830371 831513 832523 833599 881193 835479 836167 837457 833119 839111 830111 ,891440 892111 833144 894150 89,5158 893168 897431 893196 899513 900532 13,.991554 902578 9030O 901633 905664
993704 996720 993735 993751
V"
M.
910856 911902 912950 914000 915053 916109 917167 918227 919290 920356 921424 ,922495 923503 92464*4 925722 926803 92788' 928973 930062 931154 932248 10,,933345 934444 935547 936652 937760 938870 939984 941100 942219 943341 '944465 945593 946723 947856 948992 950131 951273 952118 953566 954716 10,.955870 957027 95818 999349 960515 961684 962856 964031 965209 966391 10, 967575 968763 969954 971148 97234." 973545 974749 975956 977166
M. 1 Cosino. | PP1" Cotan
4ii>
PPi
17.43 17.47 17.51 17.55 17.59 17.63 17.67 17.72 17.76 17.80 17.84 17.89 17.93 17.97 18.02 18.06 18.10 18.15 18.19 18.24 18.28 18.33 18.37 18.42 18.46 18.51 18.55 18.00 18.65 18.70 18.74 18.79 18.84 18.89 18.93 18.98 19.03 19.08 19.13 19.18 19.23 19.28 19.33 19.38 19.43 19.48 19.53 19.58 19.64 19.69 19.74 19. 19.85 19.90 19.95 20.00 20.00 20.11 20.17 20.23 PPI"
6°
TABLE IV.— LOGAR1THMIC
S4° M. 11 1 2 3 4
in 11 12 i:1 11 15 in 17 18 111 211 21 22 2: 24
'-'."" 26 27 28 20 ;:n 31
'''a 83 ;;i 35 36 37 38 39 111 1I -42 43 44 45 10 17 48 49 50 51 52 53 54 55 56 57 58 59 ill
9.997014 007628 09764 1 997654
v.nux 997680 997603 99770ii 997719 997732 997745 9.907758 997771 997784 997797 907800 907S22 9978.1.5 997847 997860 997872 9.997885 997807 907910 907022 997935 997947 097050 097072 907981 9079911 9.998008 098020 998032 098011 998050 908008 998080 998002 008104 008110 9.998128 998139 998151 098 UK 998174 908186 998197 908200 998220 998212 9.998243 098255 908200 998277 998280 998300 998311 998322 998333 998344 Cosine
'lang. 1" M. 10.978380 20.28 979507 980817 20.33 0821111 20.10 20.45 983208 20.51 984498 20.56 985732 9809i;0 20.62 20.68 988210 20.71 989454 20.80 990702 20.85 10.991953 20.01 9!ffi08 20.07 994460 21.03 905728 21.09 990993 21.15 998202 21.21 909535 21.27 11.000812 21.:34 002002 21.10 003376 21.40 11.001063 21.52 005955 21.58 007250 21.65 008549 21.71 000851 21.78 01 1 158 21.81 012108 21.01 013783 21.07 015101 22.01 010423 22.10 11.017719 22.17 011Xi7!i 22.23 020114 22.30 021752 22.37 023094 22.44 02 14 10 ! 22.51 02.5701 22.57 027145 22.65 028501 22.71 020867 22.70 11.031234 22.86 032606 22.03 033081 23.00 035301 23.07 030745 23.14 038131 23.22 039527 23.20 010025 23.37 012320 23.44 0437:33 23.51 11.045144 23.00 046559 23.66 017079 23.71 040103 23.82 050832 23.011 052286 23.07 053705 24.05 055148 24.13 050506 24.21 0580 18
31.
1"P1'" rolling. 1 PP1
i1
1 2
:: 1
! PP1' .998344 998355 998300 998377 908388 998399 908410 998421 998431 998142 998453 '.998464 098474 998185 998495 908506 998516 908527 998537 908548 908558 '.998568 998578 908580 908599 998009 908i110 !108020 008630 908640 998859 '.00866l1 908670 908680 908000 998708 998718 998728 008738 908747 908757 .008760 908770 008785 908795 908801 998813 008823 998832 898841 098851 .008800 908869 908878 908887 998890 9080O5 098914 098023 908032 008011
85° Tang. 11.058048 059500 000968 062435 083907 00.5384 066866 068353 069815 071342 072844 11.074351 075804 077381 078004 080432 081966 083505 | 085040 086590 088151 11.089715 001281 092853 004430 096013 007602 000107 100797 102401 104010 11.105634 107258 108888 110524 112107 113815 115470 117131 118708 120471 11.122151 123838 125531 127230 128036 180649 132368 131001 135827 13750" 11.130314 141008 142820 14459" 146372 148154 149043 151710 153515 155350
PP1"; Cotung.
A 14
60 59 58 57 56 .55 51 53 52 51 50 40 is 47 40 45 44 43 -12 11 10 39 38 37 36 35 34 33 32 31 8O 20 28 27 26 25 24 23 22 21 20 19 is 17 16 15 14 13 12 11 10
11 8
M.
86°
SINES AND TANGENTS. 8un
9.99S941 998950 99895.8 998970 998984 998993 999002 999010 999019 999027 9.9990311 999014 999053 999001 999039 999077 999080 999094 999102 999110 9.999118 999120 999134 999142 999150 099158 999100 999174 999181 999189 9.999197 99920i 999212 9119220 999227 999235 999242 999250 999257 999205 9.999272 999279 999287 999294 999301 999308 99931. 999322 999329 999330 9.999343 999350 999357 999364 999371 ' 999378 999384 999391 999398 999404 Co->in i".
'Pm
11.155350 .15 30.32 157175 .15 80. 46 159002 .15 311.57 160837 .14 30.70 102079 .14 80.&3 104529 .14 30.90 160387 .14 31.10 108252 .14 31.23 170120 .14 31.30 172008 .14 31.50 173897 .14 31.03 11.175795 .14 31.77 177702 .11 31.91 179010 .14 32.05 181.539 .14 32.19 1&5471 32.33 .14 185111 .14 32.48 187359 .14 32.02 189317 .13 32.77 191283 .13 32.92 193258 .13 33.07 11.195242 .13 33.22 197235 .13 33.37 199237 33.52 .13 201248 33.08 .13 203209 .13 33.83 205299 .13 33.99 207338 .13 31.15 209387 .13 34.31 211440 31.47 . 13 213514 .13 31.04 11.215592 .13 31.80 217080 34.97 .13 219778 ,15.14 .13 221880 .35.31 .13 224005 .13 35.48 220134 .13 35.65 228273 .13 35.83 230122 .12 30.00 232583 .12 30.18 234754 .12 36.36 11.236985 .12 30.55 239128 .12 30.73 241332 .12 30.92 243.547 .12 37.10 245773 .12 37.29 248011 .12 37.49 250200 .12 37.08 252521 .12 37.87 254793 .IS 38. 07 257078 .12 38.27 11.259374 38.48 .12 201083 .12 38.08 264004 .11 38.89 200337 .11 39.09 268688 .11 39.30 271041 .11 39.52 273412 39.71 .11 275790 39.95 .11 278194 .11 10.17 280601 Cotiui".
St"
|PPl"
9.999404 999411 999418 999424 999431 999437 999443 999450 999450 999403 999469 9.999475 9994S1 999487 999493 999500 999500 999512 999518 999524 999529 9.999535 999541 999547 999553 999558 999504 999570 999575 999581 999586 9.9H9592 99959 999C03 999608 999014 999819 999024 1199029 999635 999640 9.999045 9996.50 999055 999060 99906: 999070 999675 99!.080 9096.S5 999089 9.9991l94 999699 999704 999708 999713 999717 999722 999720 999731 999735
.11 .11 .11 .11 .11 .11 .11 .11 .11 .111
.10
11.280604 283028 285406 28791 290:382 2,1281,0 295354 297861 360383 302919 305471 11.308037 310019 313216 315828 318456 321100 323701 826437 329130 331840 11.334567 337311 310072 342851 345648 348463 351291 354147 357018 359907 11.362810 365744 368092 371060 374048 8776K
.10 .111 .111 .10 .10 .10 .10 .10 .10 .10 .10 .10 .10 .111 .09 .09 .09 .09 .09 .09 .09 .09 .09 .00 .09 .09 .09 888738 .09 380811 .09 389906 .09 11.393022 .us 396161 .08 399323 .08 402508 .08 405717 .us 408949 .08 41220;"i .08 415486 .08 418792 .08 422123! .08 11.425480 .08 428863' '*' .iis 432273 o0 .08 435709 .08 439172 .08 442004 .us 440183 .117 449732 .117 453309 .07 456916
60 59 58
51 53 52 51 50 49
48 47 46 45 11 43 42 11 40 39 38 37 86 35 84 33 82 31 30 211 28 26 25 24 23 L"J
21 20 19 is 17 16 15 14 13 12 11 10 9 8 7
« 5 1 3 2 1 )i
PPI"
.13
2°
ss° 0 1 2 8 1 5 6 7 8 D 10 11 12 13 11 15 16 17 18 19 20 21 22 28 21 25 28 27 28 29 30 31 32 33 84 35 36 37 38 39 40 '1I '12 43 44 45 40 '17 48 49 50 51 52 53 51 55 56 57 58 69 60
TABLE IV.— S1NES AND TANGENTS. 1.999735 999710 999744 999748 999753 999757 999761 99976.5 999709 999771 909778 1.999782 99978C 999790 999794 999797 099801 999895 999809 999813 99981C 1.999820 999824 999827 999831 999834 999838 990841 999811 999818 999851 1.999854 999858 999801 999864 999807 999870 999873 999870 999879 999882 1.999885 999888 999891 999894 999897 999899 999902 99900' 99990; 999910 1.999918 999915 999918 999920 999922 999925 999927 999929 999932 999931
PPl
PPPIV 1.450911: 60.02 400553 61.13 101221 0l.tt." 407920 62.18 471051 02.72 47.5114 03.2i 479210 63.82 483039 04.39 480902 01.91: 490800 05.55 4917a'i 00. 15 .498702 00.70 502707 07. 38 5011750 08.01 510.830 08.05 514050 09.31 519108 09.98 523307 78.60 527510 71.35 531828 72.00 530151 72.79 .540519 73.52 54 1039 | 71.28 5493871 75.05 553890 75.83 558110 76.63 563038 77.45 587085 78.29 572382 79.14 577131 80.02 581932 80.91 .580787 81.82 591093 82.70 590002 83.71 601685 81.70 600700 85.70 011908 80.72 017111 87.77 022378 88.85 027708 89.95 633105 91.08 .038570 92.24 614105 93.43 049711 94.05 055390 95.90 601144 97.19 600975 98.51 072880 99.87 078878 101.3 081954 102.' 091110 104.2 .697306 105. 703708 107.2 710144 108.9 710677 110.5 723309 112.2 730044 114.0 736885 115.8 748885 117.7 750898 119.7 758079
9.991!931 999931! 999938 999910 1i99942 999944 099946 099948 999950 999952 999954 9.999950 999958 999959 999961 999903 099064 099960 990908 999909 999971 9.099972 !.99973 899975 199976 999977 999979 999980 999981 999982 999983 9.996085 999980 99998-
416
T;i
999989 999989 999990 D99991 999992 999993 9.999993 999994 999995 999995 99999ii 999996 89999; 99999; 999998 999998 9.999999 999999 999999 990999 10.000000 000000 000000 000000 000009 000000
11.75.-079 705379 772t05 780359 788047 795874 £03844 811964 8^0237 828072 837273 11.840648 855004 864149 873490 881:037 892797 902783 913003 923469 931194 11.945191 950473 908055 979956 992191 12.004781 017747 031111 04-1S00 059142 12.073800 089100 101901 121292 138326 156056 174540 193845 214049 235239 12.257510 280997 305821 332151 300180 390143 422328 457091 494880 530273 12.582030 683183 6911 7.58122 837301 934214 13.059153 235244 536274 Infinite.
Cosine.
Cntiuii!.
S9° PPl" 121.7 123.8 125.9 128.1 130.4 132.8 135.3 137.9 140.0 143.3 140.2 149.3 152.4 155.7 159. 1 102.7 160.4 170.3 174.4 178.7 183.3 1S8.0 193.0 198.3 203.9 209.8 210.1 222.7 229.8 237.3 215.4 254.0 203.2 273.2 283.9 295.5 308.0 321.7 336.7 353.2 371.2 391.3 413.7 438.8 407.1 499.4 536.4 579.4 129.8 689.9 702.0 852.5 900.5 1116 1320 1015 2082 2935 5017
o»
TABLE Y. PRECISE CALCULATION OF FUNCTIONS. The proportional parts, as given in Table IV, are sufficient for ordinary use. When precision is desired the following rules should be observed: I. In finding the logarithmic function of an angle expressed in degrees, minutes, and seconds, derive it from that function which is nearest to it, whether greater or less; for, the proportional parts, being only approximations, should be multiplied by as small a number as possible. II. In finding the angle from its given function, use that loga rithm which differs least from the one given, subtracting or adding as the case may be. III. To find the logarithmic sine of an angle of less than 2° 36' : reduce it to seconds; add the logarithm of the number of seconds to the logarithmic sine of one second, which is 4.685575; from this sum subtract the difference in the following table correspond ing to the number of seconds ; the remainder is the required loga rithmic sine within one millionth. IV. Conversely, to find the angle when the given logarithmic sine is less than 8.656702: first, find the angle approximately by Table IV; reduce this to seconds; add to the given sine the differ ence in the following table corresponding to the number of sec onds; from this sum subtract 4.685575; the remainder is the loga rithm of the required number of seconds within one. V. To find the logarithmic tangent of an angle less than 2° 36' : reduce it to seconds; add to the logarithm of the number of sec onds the logarithmic tangent of one second, which is 4.685575; to this sum add the difference in the table (p. 419 and 420) corres ponding to the number of seconds; the sum is the required loga rithmic tangent within one millionth. VI. To find the angle when the given logarithmic tangent is less than 8.657149, which is the tangent of 2° 36': first find the angle approximately by Table IV; reduce it to seconds; subtract from the given tangent the difference in the table corresponding to (he number of seconds: from this remainder subtract 4.685575; the remainder is the logarithm of the required number of seconds within one. VII. To find the logarithmic cotangent of an angle less than 2° 36' : reduce it to seconds ; subtract the logarithm of the number of seconds from the logarithmic cotangent of one second, which is 15.314425; from this remainder subtract the difference in the table corresponding to the number of seconds; the remainder is the required logarithmic cotangent within one millionth. VIII. To find the angle when the given logarithmic cotangent is greater than 11.342851, the cotangent of 2° 36': first find the angle approximately by Table IV; reduce it to seconds; add to the given cotangent the difference in the table corresponding to the number of seconds; subtract this sum from 15.314425; the remainder is the logarithm of the required number of seconds within one.
_—
TABLE V.—AIDS TO F0B THE SIXE8 0F SMALL ANGLE8. a : i '
;
i iir.
i.
0" 9. 15' 50" 20' 20" 23' 60" 27'
0 640 950 1220 1130 1 :20
Anel; s. 1° 29' 30' 31' 32' 33' 34'
50" 50" 40" D0" 30" 20"
8econds. Dill'. 5590 5450 5500 5550 5610 5500
50 51 52 53 51
Diff.
Ancles. 2=
30" 8' 10" 8' 45" 9' 20" 10'
7650
7C96
10' 40"
7725 7768 7806 7M0
11' 11' 12' 13' 13'
15" 50" 80" 5" 40"
7956 7985 8020
14' 15' 15' 10' 16'
20" S3" 10" 45"
8060 8100 8135 8170 8205
17' 17' 18' 19' 19'
20" 66" 30" 5" 40"
8210 8275 8310 8345 8380
20' 20' 21' 22' 22'
15" 50" 25" 35"
8415 si.'.o s is.-, 8520 8555
23' 23' 21' 21' 25'
10" 45" 20" 55" 50"
8590 8025 801,0 8C95 8730
20' 26' 27' 27' 28'
35" 5" 40" 10"
8760 8705 8825 £800 8890
28' 29' 29' 80' 50'
45" 15" 50" 20" 65"
8925 8955 8990 Bl',20 E055
31' 52' 32' 33' 83'
25" 80" 5" 35"
0085 9120 9150 9185 1215
31' 31' 85' 35' 30'
5" 40" 10" 40" 15"
iJii
2210 2370
35' 30' 30' 87' 38'
"0" 40" 30"
5710 6700 5810 f 8i 0 6010
41' iff' at 20" 45' io" 47' 4*40"
2190 21,00 2710 2820 2920
39' 40' 41' 41' 42'
30" 20" 10" 50" 10"
5970 0020 0070 0110 0100
50' 20" 52' 53' 30" 55' 50' 30"
3020 3120
30" 10"
3390
43' 44' 45' 45' 40'
0210 0250 0500 C3T0 0390
58' 50 20" 1° 00' 40" 2' 3' 20"
3 180 3500 3010 8720 3800
47' 48' 48' 49' 50'
20"
4' 5' 7' 8' 9'
3880 3950 1020 41100 1160
51' 51' 50" 52' 50" 63' 10" 51'
ooco
10' 30" 11' 40" 12' 50" 14' 15'
4230 4300 4370
51' 55' 50' 56' 57'
10" 20" 10" 50 ' 50"
0880 0920 0070 7010 7050
16' 17' I8' 19' 20'
10" 10" 10" 20" 20"
4570 4030 4i100 4700 1820
58' 58' 59' 2° 00'
10" 50" 50" 10" 50"
7000 7150 7170 7210 72",0
21' 22' 23' 21' 23'
20" 20" 20 ' 20 ' 10"
4880 40l0 5000 5000 5110
1' 40" 2' 20"
7500 7510
3' 55" 4' 10"
7115 7150
2C' 27' 28' 29' 20'
10" 10" 10"
5170 5250 5200 53 10 5300
4' 5' 0' 6' 7'
7150 7550 7570 7010 7050
29' 50" 82, CC"
ay
17!10 ! CM 21 ii
37' 20" 89' 80''
40" 50" 10" 20"
50"
3210
:::;oo
mo 4500
10"
50" 30"
50" 30" 20"
C110 01M1 C:.'.0 0570 0520
50 57 58 69
to 01 02 63 64 65
66 67 68 C9 70 71 72 73 74
7875 7!. 1n
75
50" 50" 10" 50" 30"
i,710 0750 0700 0810
70 78 79 80 81 82 83
8i 85 86 87 88
89 90 !11 02
93 94 05 05 97 08 99
100 101 102 103 104 105
9215 9280 9310 93l0
100 107 108 109 110 111 112 113 114 115 11C 117 118 119 120 121 122 123 124 125 126 127 128 129 150 151 132 133 SI1 135 133 137 138 139 110 Ml 142 113 111 153 140 147 118 119
PREC1SE CALCULATIONS. F0B TANGENT8 AND C0TANGENT8 0F SMALL ANGLK8. Angles.
8econds.
Diff.
Angles.
8econds.
Diff.
Angles
8econds. | Diff.
40"
3810 8850 3890 3930 3900 4000
5410 5430 5400 5490 5520 5540
7' 20" 7' 50" 8' 80"
4040 4070 4110
5570 5590
»'
4140
V 40"
4180
5650 5670
1° 3' 4' 4' 5' 6' 0'
30" 10" 50" 30"
0" 7' 10" 11' 10" 14' 10" 17' 10'
0 430 670 850 1020 1140
21' 23' 21' 50" 20' 30" 27' 50"
1260 13£0 1400 1590 1670
29' 80' 32' 33' 31'
10" 20"
17C0 1810 1920 19S0 2000
10' 10' 11' 12' 12'
30"
35' 36' 37' 38' 80'
30" 40" 50" 50" 50"
2130 2200 2270 2330 2390
13' 13' 11' 14' 15'
10" 40" 10" 50" 20"
40' 41' 42' 43' 44'
50" 50" 50" 50" 40"
2450 2610 2570 2630 2680
15' 50" 16' 20" 17' 17' 30" 18'
4550
4080
6050
45' 40' 47' 48' 49'
40" 80" 20" 10"
2710 2790
'1710 '1711i 4770
2940
18' 30" 19' 19' 80" 20' 20' 80"
0070 6100 0120 6150 6170
49' 50' 51' 52' 53'
50" 40" 30" 20"
2990 3010 8090 31 H1 3180
21' 21' 30" 22' 22' 30" 23'
481 1i 4890 4920 4950 4980
6190 0220 0240
53' 54' 55' 56' 56'
50" 40" 20"
3230
23' 30" 24' 24' 30" 25' 25' 30"
5010 5040 5670 5100 5130
6320 0340 6360 6380
50"
82si 3320 3300 3410
57' 30" 58' 10" 58' 50" 59' 30" i 0'20"
3450 3490 3530 3570 8020
20' 20' 20' 27' 2?
30" 50" 20" 50"
5160 5190 5210 5240 5270
0430 6450 6480 0500 0520
3700 8730 8770 3810
28' 28' 29' 29' 80'
20" 40" 10" 40" 10"
5800 5820 5350 5380 5410
6510 6500 0580 0010 0030
1' 1' 2' 2' 3'
20" 40"
40" 10" 50" 30"
2810 2890
20" 50" 30"
4220
4250 12! n l 4320 4350 4390 4!_ii 4450
4490 4520
4580 41 Lli 'li 50
4S1 11 4830
419
51 :21 1
5700 57211 5750 5770 5800 5830 5850 5880 5900 5!180 .5050 51170 6000 i1i21i
i279
6290
0l00
100 101 1112 1113 1111 105 106 1117 1D8 109 1111 111 112 113 111 115 118 117 118 119 120 121 122 128 124 125, 120 127 128 121i I80 181 132 133 134 135 130 137 138 139 140 111 142 118 144 145 146 147 148 149
TABLE V.— A1DS TO PREC1SE CALCULATIONS. F0R TANGENT8 AND C0TANGENT8 0F 8MALL ANGLE8. Anglet.
Diir.
a 7' 40"
i«.-|i orro l.t.10 6720
i,;w 17Mi i"7Ni lMM
Ml!.
Angicl. 701.0 70>0 7i.05 7710
150 151 l'.'J 1.x! 1 51 155
8' 8' 8' 8' 9-
15" 30" 50" 10"
i--; 157 158 159
9& 10' 10' 10'
30" W" io" 20" 40"
7770 7790 7.-10 7820 7840
10' 11' 11' 11' 12'
55" 15" 35" 55" 15"
7855 7K75 7,-95 7015 7!M5
12' 12' 13' 13' 13'
85" 55" 15" 35" .50"
7955 7075 7995 8015
14' 14' 14' 15' 15'
10" 30" 45" 5" 20"
8050 8070 8085 8105 8120
15' 15' 10' 10' 16'
40" 55" 15" 30" G0"
8140 M55 8175 8190 8210
17' 17' 17' 18' 18'
5" 25" 40"
8225 8215 82i 0 8280 8205
18' 18' I«, 19' 19'
35" 55" 15" 30" 45"
an 15
2C 20' 20' 20' 21'
5" 20" 40" 55" 15"
8405 8420 8II0 8455 8175
21' 21' 22' 22' 22'
30" 45" 5" 20" 35"
8190 8.-05 8525 8540 8555
1, -.Ji
:: :o 7750
100
c-to 0MW 1,910
0930 i.970 r.070 i,9:i0 7olli 70M 7i1. .0
101 102 103 101 105 160 107 108 100
sum
170
7080 7100 71 JO 7149 7180 7180 7200 7220 7-.' 10 7200 7280 7300 7320 73 10 73i ;o
171 172 17:: 174 175 170 177 178 179 180 181 182 183 184 185
7880 7100
7-420 7110
7480
'10"
7480
7600 7520 7540 7500 6' 6' 7' T r
20" 40" 20" 40"
186 187 188 189 100
7580 7i,ii0 7020 7010 7000
191 102 193 101 105 196 107 198 199
15"
8335 8355 8370 8385
420
20 201 202 203 201 205 206 207 2l iS 21 'O 210 211 212
213 214 215 210 217 218 210 220 221 222 228 221 225 220 227 228 220 2.l0 231 282 233 234 235 2.30 287 238 230 210 211 212 213 211 245 246 217 2 18 219
Adfili'8. 2° 22' 35" 22' 55" 23-10" 23' 80" 2* 45" 24' 24' 24' 24' 2.7 25-
Biff. 8555 8575 8590 8010 8625 8640
20" 37' 55" 10" 25"
6660 81i75 8695 8710 8725
25M5" 26i
8715 87i0 87-0 87! 15 8.-10
26' 20" 26' 35" 26' 50" 27' 27' 27' 28' 28'
10" 25" 45"
8-30 8815 88ii5 88,-0
15"
28' 35" 28/50" 29' 10" 29'25" 29' 40"
8015 8930 8950 8965 8080
80" 3C 30' 30' 81'
15" K0" £0" 5"
9000 !.| 15 !l 30 9050 1005
81' 31' 81' 32' 32"
20" 35" 55" 10" 25"
9080 9096 !1115 91.30 9145
32. 32' 33' 33' 33'
40" 55" 15" 30" 45"
9160
34' 34' 15" 34' 30" 34' 45"
9240 9255 9270 9285 9300
a& 35' 35' 35' 36' 36'
20" 35" 50" 5" 20"
9175 9105 9210 9225
9320 8835 9850 9365 9380
250 251 252 2.53 251 255 250 257 258 2.9 200 2C1 202 263 204 205 2"0 2,7 2t8 2C« 270 271 272 273 274
270 277 278 27!1 280 2.-1 282 283 2,-4 285 280 287 288 280 200 291 292 293 294 285 2% 297 298
c\ -
f*
I To avoid fine, this book should be returned on ' or before the date last stamped below
in 5/3-/ T774
623170
-
