GIFT OF Dr. Horace Ivie ^T^-r—ZJ^-^ ECLECTIC EDUCATIONAL SERIES. TREATISE GEOMETRY AND TRiadKOIETBt COLLEGES, SCHOOLS, AND PRIVATE STUDENTS. WRITTEN FOR THE MATHEMATICAL COl'RSE OF JOSEPH RAY, M.D., BY ELI T. TAPPAN, M.A., P R F E S S K OF MATHEMATICS, OHIO UNIVERSITY. NEW-YORK ..-. CINCINNATI .:• CHICAGO AMERICAN BOOK COMPANY FROM THK PRESS OF VAN ANTWERP, BRAGG, A CO. GIFT OF . T^ Ray'S Series, (i^^ EMBRACING A Thorough and Progressive Course in Arithmetic, Algebra, and the Higher Mathematics. Friiiiary Aritliiuetic. lutiilectiial AHtlwuftic. Itiidhnents of Ai'ithiuetie; Hig:li('r Aritliiiiotie. Test Examples in Arithmetic New Elementary Algebra. y.i'i'l'rii'. AritJiMetJc. I New Higher Alg:el)ra. Plane and ^oliftS' 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 Propc- SITIOX. 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 CoNCLUSiox. 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. (9) 10 ELKMENTS OF GEOMETRY. 3. A proposition is either fh'oretical, that is, it de> dares, that a certain property belongs to a certain thing; or it d,s* jt?raf'^?va'(,:i?b(ai'Ji,is, it declares that something can be , do.rie; ' • P'rop'ositi'^ns t^rc citiicr demonstrable^ that is, they may b3 established by the aid of reason ; or they are mdemon- sfrable, 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 Avhich 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 — redudio 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. GEiNEKAL AXIOMS. U 5, Remark. — Tlie 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 lias 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- tify, are equal to each other. T. If the same 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 tvhole is equal to the sum of all the parts. 1 0. The ivhole is greater than a part. EXERCISE. 11, What is the hypothesis of the first axiom ? Ans. If sev- eral quantities are each equal to the same quantity. 12 ELEMENTS OF GEOMETRY. What is the subject of the first axiom ? Ayis. Several quan- tities. What is the condition of the first axiom ? Ayis. 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 Ave 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 Avith 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 5, may be written , . A ratio is usually written a : h, 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^ ]/3, or ti. 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 : h '. : c '. d, and is read, « is to 5 as c is to c?, but it is sometimes written, a '. h = c : d^ or It may be, h^d' all of which express the same thing: that a contains h 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 : h : : 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 : h : 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 meaiis. 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. 17. 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 tivo may be the extremes and the other two the means of a proportion. Let aXd=^bXe represent the equal products. If we divide by b and d, we have (1st.) (2.1.) a b~ '-%' "'^ a : b : : c : d. If we divide by c and d, we have a_ e b = d^ ^^' a : c : : b :d. If we arrange the equal products thus: ^-Xt'^ = aXd, RATIO AND PROPORTION. 15 and then divide by a and c, we have h: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 he 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. TherCxWe (18), a and d may be taken as either the extremes or the means of a new proportion. 20. When Ave 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 tivo quantities have the same multiplier, the multiples will have the same ratio m the given quantities. Let a and b represent any two quantities, and m any multiplier. Then the identical equation, 7nXaXb = mXbXa, gives the proportion, mXa : mXb :: a: b (18). 23. Proposition. — In a series of cqiial 7'atios, the sum of the antecedents is to the suyn of the consequents as any antecedent is to its consequent. 16 ELEMENTS OF GEOMETRY. Let a : b :: c : 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-i-c-\-e+g), Therefore (18), a^c-\-e-^g : b^d-\-f^h :: a : b. This is called proportion by Composition. 24, Proposition. — The difference bekveen 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 eqi^tion, b_d, b~d The remaining equation, a — b c — d T'^'^dT' 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 potvers are in proportion, also their same roots. Thus, if we have a : b then, a"^ : 6^ also, i/a : |/5 : c : d, : c": d^; : i/c : \/d. These principles are corollaries of the second gen- eral axiom (7), since a proportion is an equation. THE SUBJECT STATED. }J 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 Geoni. — 2 18 ELEMENTS OF GEOMETRT. 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 pirts 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. SO. 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 followinjr 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, Ave 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 he made fo have any extent tvhatever. 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. Vy^o may 20 ELEMENTS OF GEOMETRY. conceive of surfaces without solids, and of lines witL surfiices. 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 he 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 difter 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. 39. 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- tremities of the line. Figures are usually designated by naming some of their points, as the line AB, and the figure CDEF, or simply the figure DF. 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 h. LINES. 44. A Straight Line is one which has the same di- rection throughout its whole extent. A B c D \ \ - \ \ F a E 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. 47. Problem. — A straight line may he made to pass through any two points. 48. Problem — There may he a straight line from any pointy in any direction, and of any exte7it. 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. — hi one direction from a point, there can he only one straight line. 50. Corollary — From one point to another, there can be only one straight line 51. Theorem — If a straight line have two of its points commo7i ivith anolher straight line, the tivo lines must coin- cide throughout their inutual 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 tw^o 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 straight line is the shortest which can join tivo points. Therefore, the distance from one point to another is reckoned along a straight line. 55. There have now been given two postulates and two axioms. The science of geometry rests upon these four simple truths. The possibility of every figure defined, and the truth of every problem, depend upon the postulates. Upon the postulates, with the axioms, is built the demonstration of every principle. SURFACES. 56. Surfaces, like lines, are classified according to their uniformity or change of direction. A Plane is a surface which never varies in direction. A Curved Surface is one in which there is a change of direction at every point. THE PLANE. 25 THE PLANE. 5T. The plane surface and the straight line have the same essential character, sameness of direction. The plane is straight in every direction that it has. A straight line and a plane, unless the extent be specified, are always understood to be of indefinite extent. 58. Theorem — A straight line which has tivo points in a plane, lies wholly in it, so far as they both extend. For if the line and surface could separate, one or the other would change direction, which by their definitions is impossible. 59. Theorem Two planes having three points com- mon, and not in the same straight line, coincide so far as they both extend. Let A, B, and C be three points which are not in one F,,-x straight line, and let these points ^ ,^.-.::r.'r.*....\ \c be common to two planes, which ^\ may be designated by the letters \ m and p. Let a straiorht line \^ pass through the points A and B, a second through B and C, and a third through A and C. Each of these lines (58) lies wholly in each of the planes m and p. Now it is to be proved that any point D, in the plane m, must also be in the plane p. Let a line extend from D to some point of the line AC, as E. The points D and E being in the plane m, the whole line DE must be in that plane ; and, therefore, if produced across the inclosed surface ABC, it will meet one of the other lines AB, BC, which also lie in that plane, say, at the point F. But the points F and E Geom. — 3 26 ELEMENTS OF GEOMETRY. are both in the plane p. Therefore, the whole line FD, including the point D, is in the plane p. In the same manner, it may be shown that any point which is in one plane, is also in the other, and therefore the two planes coincide. 60. Corollary. — Three points not in a straight line, or a straight line and a point out of it, fix the position of a plane. 61. Corollary — That part of a plane on one side of any straight line in it, may revolve about the line till it meets the other part, when the two will coincide (53). EXERCISES. 62. When a mechanic wishes to know whether a line is straight, he may apply another line to it, and observe if they coincide. In order to try if a surface is plane, he applies a straight rule to it in many directions, observing whether the two touch throughout. The mason, in order to obtain a plain surface to his marble, applies another surface to it, and the two are ground together until all unevenness is smoothed away, and the two touch throughout. "What geometrical principle is used in each of these operations ? In a diagram two letters suffice to mark a straight line. Why? But it may require three letters to designate a curve. Why ? DIVISION OF SUBJECT. 03. By combinations of lines upon a plane. Plane Figures are formed, which may or may not inclose an area. By combinations of lines and surfaces, figures are DIVISION OF SUBJECT. 27 formed in space, which may or may not inclose a vol- ume. In an elementary work, only a few of the infinite va- riety of geometrical figures that exist, are mentioned, and only the leading principles concerning those few. Elementary Geometry is divided into Plane Geome- try, which treats of plane figures, and Geometry in Space, which treats of figures whose points are not all in one plane. In Plane Geometry, we will first consider lines with- out reference to area, and afterward inclosed figures. In Geometry in Space, we will first consider lines and surfaces which do not inclose a space ; and after- v^^ard, the properties of certain solids. 28 ELEMENTS OF GEOMETRY. PLANE GEOMETRY. CHAPTER III. STRAIGHT LINES. f»4. Problem — Straight lines may he added together^ and one straight line may he suhtracted from another. For a straight line may be produced to any extent. Therefore, the length of a straight line may be increased by the length of another line, or two lines may be added together, or we may find the sum of several lines (35j. Again, any straight line may be applied to another, and the two will coincide to their mutual extent. One line may be subtracted from another, by applying the less to the greater and noting the difference. 65. Problem — A straight line may he multiplied hy any number. For several equal lines may be added together. GO. Problem — A straight line may he divided by another. By repeating the process of subtraction. 67. Problem. — A straight line may he decreased in any ratio., or it may he divided into several equal parts. This is a corollary of the postulate of extent (35). PROBLEMS IN DRAWING. 29 PROBLEMS IN DRAWING. 68. Exercises in linear drawing afford the best applications of the principles of geometry. Certain lines or combinations of lines being given, it is required to construct other lines which shall have certain geometrical relations to the former. Except the paper and pencil, or blackboard and crayon, the only instruments used are the ruler and compasses; and all the required lines must be drawn by the aid of these only. The reason for this rule will be shown in the following chapter. The ruler must have one edge straiglit. The compasses have two legs with pointed ends, which meet when the instrument is shut. For blackboard work, a stretched cord may be substituted for the compasses. 69, With the rxiler^ a straight line may be drawn on any plane surface, by placing the ruler on the surface and drawing the pen- cil along tiie straight edge. A straight line may be drawn through any two points, after placing the straight edge in contact with the points. A terminated straight line may be produced after applying the straight edge to a part of it, in order to fix the direction. f O, With the compasses^ the length of a given line may be taken by opening the legs till the fine points are one on each end of the line. Then this length may be measured on the greater line as often as it will contain the less. A line may thus be produced any required length. 71. The student must distinguish between the problems of geometry and problems in drawing. The former state what can be done with pure geometrical magnitudes, and their truth de- pends upon showing that they are not incompatible with the nature of the given figure; for a geometrical figure can have any conceivable form or extent. The problems in drawing corresponding to those above given, except the last, " to divide a given straight line into proportional or equal parts," are solved by the methods just described. 72. The complete discussion of a problem in drawing includes, besides the demonstration and solution, the showing whether the problem lias only one solution or several, and the conditions of each. 30 ELEMENTS OF GEOMETRY. STKAIGHT LINES SIMILAR. TS. Theorem. — Any two straight lines are similar fig- ures. For each has one invariable direction. Hence, two straight lines have the same form, and can differ from each other only in their extent (37). 74. Any straight line may be diminished in any ratio (67), and may therefore be divided in any ratio. The points in two lines which divide them in the same ratio are homologous points, by the definition (37). Thus, if the lines AB and ED are divided at ^ 2 ? the points C and F, so jj F j) that AC : CB :: EF : FD, then C and F are homologous, or similarly situated points in these lines ; AC and EF are homologous parts, and CB and FD are homologous parts. 75. Corollary — Two homologous parts of two straight lines have the same ratio as the two whole lines. For, AC+CB : EF+FD : : AC : EF (23). That is, AB : ED : : AC : EF. Also, AB : ED : : CB : FD. 76. Problem in Drawing. — To find the ratio of tivo given straight lines. Take, for example, the lines b and c. If these two lines have a common multiple, that is, a line which contains each of them an exact number of times, let x be the number of times that b is contained in the least common multiple of the two lines, and y the number of times it contains c. Tlien x times b is equal to ?/ Limes c. BROKEN LINES. 3 J Therefore, from a point A, draw an indefinite straight line AE. Apply each of the given lines to it a number of times in sue- ression. The ends of the two lines will coincide after x applica- tions of b, and y applications of c. If the ends coincide for the first time at E, then AE is the least common multiple of the two lines. The values of x and y may be found by counting, and these express the ratio of the two lines. For since y times c is equal to X times b, it follows that b : c : : y : x, which in this case is as 3 to 5. It may happen that the two lines have no common multiple. In that case the ends will never exactly coincide after any number of applications to the indefinite line; and the ratio can not be ex- actly expressed by the common numerals. By this method, however, the ratio may be found within any desired degree of approximation. Iflf, But this means is liable to all the sources of error that arise from frequent measurements. In practice, it is usual to measure each line as nearly as may be with a comparatively small standard. The numbers thus found express the ratio nearly. Whenever two lines have any geometrical dependence upon each other, the ratio may be found by calculation with an accu- racy which no measurement by the hand can reach. BROKEN LINES. T8. A curve or a broken line is said to be Concave on the side toward the straight line which joins two of its points, and Convex to the other side. TO. Theorem. — A broken line which is convea- toward another line that unites its extreme points, is shorter than that line. The line ABCD is shorter than the line AEGD, to- ward which it is convex. 32 ELEMENTS OF GEOMETRY. Produce AB and BC till they meet the outer line in F and H. Since CD is shorter than CHD, it follows (8) that the line ABCD is shorter than ABHD. For a simi- lar reason, ABHD isi shorter than AFGD, and AFGD is shorter than AEGD. There- fore, ABCD is shorter than AEGD. The demonstration would be the same if* the outer line were curved, or if it were partly convex to the inner line. EXERCISE. 80. Vary the above demonstration by producing tlie lines DC and CB to the left, instead of AB and BC to the right, as in the text; also, By substituting a curve for the outer line; also, By letting the inner line consist of two or of four straight lines. 81, A fine thread being tightly stretched, and thus forced to assume that position which is the shortest path between its ends, is a good representation of a straight line. Hence, a stretched cord is used for marking straight lines. The word straight is derived from ^'- stretch^'' of which it is an obsolete participle. ANGLES. 82, An Angle is the difference in direction of two lines which have a common point. 83. Theorem. — The two lines which form an angle lie in one planer and determine its position. For the plane may pass through the common point and another point in each line, making three in all. These three points determine the position of the plane (60). . •>^;iiai^> ANGLES. 33 DEFINITIONS. H4. Let the line AB be fixed, and the line AC revolve m a plane about the point A; thus taking every direction from A in the plane of its revolution. The angle or difference in direc- tion of the two lines will in- -^ ^ crease from zero, when AC coincides with AB, till AC takes the direction exactly opposite that of AB. .. \ \ \ j / / / .. If the motion be contin- "C VM\ I //vvv-'' x'' ued, AC will, after a com- plete revolution, again co- C incide with AB. The lines which form an angle are called the Sides, and the common point is called the Vertex. The definition shows that the angle depends upon the directions only, and not upon the length of the sides. 85. Three letters may be used to mark an angle, the one at the vertex being in the middle, as the angle BAC. When there can be no doubt what angle is intended, one letter may answer, as the angle C. ^ It is frequently convenient to mark angles with letters placed between the sides, as the angles a and b. Two angles are Adjacent when they have the same vertex and one common side between them. Thus, in the last figure, the angles a and b are adjacent; and, in the previous figure, the angles BAC and CAD. 86. A straight line may be .regarded as generated S4 ELEMENTS OF GEOMETRY. by a point from either end of it, and therefore every straight line has two directions, which are the opposite of each other. We speak of the direction from A to B as the direction AB, and of the direction from B to A as the direction BA. One line meeting another at some other point than t:ie extremity, makes two angles B with it. Thus the angle BDF is the difference in the directions DB and DF ; and the angle BDC C D F is the difference in the directions DB and DC. When two lines pass through or cut each other, four angles are formed, each direction of one line making a difference with each direction of the other. The opposite angles formed by two lines cutting each other are called Vertical angles. A line which cuts another, or which cuts a figure, is called a Secant. PROBLEMS ON" ANGLES. 87. Angles may be compared by placing one upon the other, when, if they coincide, they are equal. Problem. — One angle may he added to another. Let the angles ADB and BDC be ad- jacent and in the same plane. The angle ADC is plainly equal to the sum of the other two (9). D Problem. — An angle mag he suhtraded from a greater one. For the angle ADB is the difference between ADO and BDC. ANGLES. 35 It IS equally evident that an angle may be a multiple or a part of another angle ; in a word, that angles are quantities which may be compared, added, subtracted, multiplied, or divided. But angles are not magnitudes, for they have no ex- tent, either linear, superficial, or solid. ANGLES FORMED AT ONE POINT. 88. Theorem. — The sum of all the successive angles formed in a j)lane upon one side of a straight line, is an invariable quaniity; that is, all such sums are equal to each other. If AB and CD be two straight lines, then the sum of all the successive angles at E is equal to the sum of all those at F. For the line AE may be placed on CF, the point E on the point F. Then EB will f^ill on FD, for when two straight lines coin- cide in part, they must coincide tliroughout their mutual extent (51). Therefore, the sum of all ~~ the angles upon AB exactly coincides with the sum of all the angles upon CD, and the two sums are equal. 89. When one line meets another, making the adjacent angles equal, the angles are called Rkhit Angles. One line is Perpendicular to the other when the angle which they make is a right angle. Two lines are Oblique to each other when they make an angle which is greater or less than a rip^ht angle. 36 ELEMENTS OF GEOMETRY. OO. Corollary — All right angles are equal. For each is half of the sum of the angles upon one side of a straight line. By the above theorem, these sums are always equal, and (7) the halves of equal quantities are equal. 91. Corollary. — The sum of all the successive angles formed in a plane and upon one side of a straight line, is equal to two right angles. 02. Corollary — The sum of all the successive angles formed in a \^^ plane about a point, is equal to four right angles. 93. Corollary — When two lines / cut each other, if one of the angles / thus formed is a right angle, the other three must be right angles. 94. In estimating or measuring angles in geometry, the right angle is taken as the standard. An angle less than a right angle is called Acute. An angle greater than one right angle and less than the sum of two, is called Obtuse. Angles greater than the sum of two right angles are rarely used in ele- mentary geometry. When the sum of two angles is equal to a right angle, each is the Complement of the other. When the sum of two angles is equal to two right angles, each is the Supplement of the other. 95. Corollary. — Angles which are the complement of the same or of equal angles are equal (7). 96. Corollary. — Angles which are the supplements of the same or of equal angles are equal. 97. Corollary. — The supplement of an obtuse angle is acute. ANGLES. 37 08. Corollary. — The greater an angle, the less is its supplcme.it. 99. Corollary — Vertical angles are equal. Thus, a and i are each supplements of e. 100. Theorem — When the sum of several angles in a plane having their vertices at one point is equal to two right angles, the extreme sides form one straight line. If the sum of AGB, BGC, etc., be equal to two right an- gles, then will AGF be one straight line. For the sum of all these angles being equal (91) to the sum of the angles upon one side of a straight line, it follows that the two sums may coincide (40), or that AGF may coincide with a straight line. Therefore, AGF is a straight line. EXERCISES. 101. Wliich is the greater angle, a or ^, and why? ^ — " What is the greatest number of" points ^^""^-'"^^ in which two straight lines may cut each other? In which three may cut each other? Pour? 10!2. The student should ask and answer the question "why" at each step of every demonstration ; also, for every corollary. Thus : Why are vertical angles equal ? Why are supplements of the same angles equal ? And in the last theorem: Why is AGF a straight line? Why may AGF coincide with a straight line? Why may the two sums named coincide ? Why are the two sums of angles equal ? 3S ELExMENTS OF GEOMETKV^. PERPENDICULAR AND OBLIQUE LINES. 103. Theorem. — There can he only one lijie through a given point perpendicular to a given straight line. For, since all right angles are equal (90), all lines ly- ing in one plane and perpendicular to a given line, must have the same direction. Now, through a given point in one direction there can be only one straight line (49). Therefore, since the perpendiculars have the same direction, there can be through a given point only one perpendicular to a given straight line. When the point is in the given line, this theorem must be limited to one plane. 104. Theorem — If a perpendicular and oblique lines fall from the same point upon a given straight line, the perpendicular is shorter than any oblique line. If AD is perpendicular and AC oblique to BE, then AD is shorter than AC. Let the figure revolve upon BE as — upon an axis (61), the point A falling upon F, and the lines AD and AC upon FD and FC. Now, the angle CDF is equal to the angle CD A, and both are right angles. Therefore, the sum of those two angles being equal to two right angles (100), ADF is a straight line, and is shorter than ACF (54). There- fore, AD, the half of ADF, is shorter than AC, the half of ACF. 105. Corollary. — The distance from a point to a straight line is the perpendicular let fall from the point to the line. PERPENDICULAR AND OBLIQUE LINES. 39 106. Theorem. — If a perpendicular and several oblique lines fall from the same point upon a given straight line, and if two oblique lines meet the given line at equal dis- tances from the foot of the perpendicular, the two are equal. Let AD be the perpendicular ^ and AC and AE the oblique lines, making CD equal to DE. Then AC and AE are equal. Let that portion of the figure ^ C D E F on the left of AD turn upon AD. Since the angles ADB and ADF are equal, DB will take the direction DF ; and since DC and DE are equal, the point C will fall on E. Therefore, AC and AE will coincide (51), and are equal. 107. Corollary — When the oblique lines are equal, the angles which they make with the perpendicular are equal. For CAD may coincide with DAE. 108. Theorem — Jf a line be perpendicular to another at its center, then every point of the perpendicular is equally distant from the two ends of the other line. For straight lines extending from any point of the perpendicular to the two ends of the other line must be equal (106). Let the student make a diagram of this. Then state what lines are given by the hypothesis, and what are constructed for demonstration. lOO. Corollary — Since two points fix the position of a line, if a line have two points each equidistant from the ends of another line, the two lines are perpendicular to each other, and the second line is bisected. The two points may be on the same side, or on opposite sides of the second line. 40 ELEMENTS OF GKOMETRY. no. Theorem — If a perpendicular and several oblique lines fall from the same point on a given straigld line, of two oblique lines, that which meets the given line at a greater distance from the perpendicular is the longer. If AD be perpendicular to BG, and DF is greater than DC, then AF is greater than AC. On the line DF take a part DE equal to DC, and join AE. Then let the figure revolve upon BG, the point A falling i / upon H, and the lines j / / AD, AE, and AF upon j/X' HD, HE, and HF. H Now, AEH is shorter than AFH (79) ; therefore, AE, the half of AEH, is shorter than AF, the half of AFH. But AC is equal to AE (106). Hence, AF is longer than AC, or AE, or any line from A meeting the given line at a less distance from D than DF. 111. Corollary — A point may be at the same distance from two points of a straight line, one on each side of the perpendicular; but it can not be at the same dis- tance from more than two points. 112. Theorem — If a line be perpendicular to another at its center, every point out of the perpendicular is nearer to that end of the line which is on the same side of the perpendicular. If BF is perpendicular to AC at its center B, then D, a point not in BF, is nearer to C than to A. Join DA and DC, and let the A BE perpendicular DE fall from D upon the line AC. PERPENDICULAR AND OBLIQUE LINES. 41 This perpendicular must fall on the same side of BF as the point D, for if it crossed the line BF, there would be from the point of intersection two perpendiculars on AC, which is impossible (103). Now, since AB is equal to BC, AE must be greater than EC. Hence, AD is greater than CD (110). The point D is supposed to be in the plane of ACF. If it were not, the perpendicular from it might fall on the point B. BISECTED ANGLE. 113. Theorem — Every point of the line which bisects an angle is equidistant from the sides of the angle. Let BCD be the given angle, / and AC the bisecting line. Then ^ / the distance of the two sides from \ //j any point A of that line is meas- \^''' / j ured by perpendiculars to the \Z_J sides, as AF and AE. C E D Since the angles BCA and DCA are equal, that part of the figure upon the one side of AC may revolve upon AC, and the line BC will take the direction of CD, and coincide with it. Then the perpendiculars AF and AE must coincide (103), and the point F fall upon E. Therefore, AF and AE are equal, and the point A is equally distant (105) from the sides of the given angle. APPLICATION. 114. Perpendicular lines are constantly used in architecture, carpentry, stone-cutting, nmcliinery, etc. I'he mason's square consists of two flat rulers made of iron, and connected togetlier in such a manner that both edges of one Geoni. — 4 42 ELEMENTS OF GEOMETUV are at right angles to those of the otlier. The carpenter's square is much like it, but one of the legs is wood. This instrument is used for drawing perpendicular lines, and for testing the correctness of right angles. The square itself should be tested in the following manner: On any plane surface draw an angle, as BAG, with the square. Extend BA in the same straight line to D. Then turn the square so that the edges by which the angle BAG was described, may be applied to the angle DAG. If the coincidence is exact, the square is correct as to these edges. Let the student show that this method of testing the square is according to geometrical principles. The square here described is not the geometrical figure of that name, which will be defined hereafter. B A MINIMUM LINE 115. Theorem. — Of any two lines wTiicJi may extend from hvo given points outside of a straight line to any point iyi it, those which are together least make equal an- gles with that line. Let CD be the line and A and B the points, and AEB the shortest line that can be made from A to B through any point of CD. Then it is to be proved that AEC and BED are etjual angles. Make AH perpendicular to CD, and produce it to F, making HF equal to AH. Now every point of the line CD is equally distant from A and F (108). Therefore, every line joining B to PARALLELS. 43 F through some point of CD, is equal to a line joining B to A through the same point. Thus, BGF is equal to BGA, since GF and GA are equal. So, BEF is equal to BEA. But BEA is, by hypothesis, the shortest line from B to A through any point of CD. Therefore, BEF is the shortest line from B to F, and is a straight line (54). Since BEF is one straight line, the angles FEH and BED are vertical and equal (99). But the angles FEH and AEH are equal (107). Therefore, AEH and BED are equal (6). 116. When several magnitudes are of the same kind but vary in extent, the least is called a minimum^ and the greatest a maximum, APPLICATION. When a ray of light is reflected from a polished surface, the incident and reflected parts of the ray make equal angles with the surface. We learn from this geometrical principle that light, when reflected, still adheres to that law of its nature which re- quires it to take the shortest path. PARALLELS. IIT. Parallel lines are straight lines which have the same directions. 118. Corollary — Two lines which are each parallel to a third are parallel to each other. 119. Corollary — From the above definition, and the Axiom of Direction (49), it follows that there can be only one line through a given point parallel to a given line. 120. Corollary — From the same premises, it follows that two parallel lines can never meet, or have a com- mon point. 44 ELEiVlEiM'« OF GEOMETRY. \2\» Theorem — Two parallel lines both lie in one plane and determine its position. The position of a plane is determined (60) by either line and one point of the other line. Now the plane has the direction of the first line and can not vary from it (56), and the second line has also the same direction (117) and can not vary from it (44). Therefore, the second line must also lie wholly in the plane. NAMES OF ANGLES. 122. When two straight lines are cut by a secant, the eight angles thus formed are named as follows: The four angles between the two lines are Interior; as,/, ^, h, and k. The other four are Ex- terior; as, b,c, I, and m. Two angles on the same side of the secant, and on the same side of the two lines cut by it, are called Cor- responding angles. The angles h and b are corre- sponding. Two angles on opposite sides of the secant, and on opposite sides of the two lines cut by it, are called Alternate angles. The angles / and k are alternate ; also, b and m. The student should name the corresponding and the alternate angles of each of the eight angles in the above diagram. Let him also name them in the diagram of the following theorem. 12]$. Corollary — The corresponding and the altern- ate angles of any given angle are vertical to each other, and therefore equal (99). PARALLELS. 45 PARALLELS CUT BY A SECANT. 134. Theorem. — When two parallel lines are cut hy a secant^ each of the eight angles is equal to its corresponding angle. If the straight lines AB and CD have the sam6 di- rections, then the angles FHB and FGD are equal. For, since the directions GD and HB are the same, the direction GF diifers equally from them. Therefore, the angles are equal (82). In the same manner, it may be shown that any two corresponding angles are equal. 125. Corollary — When two parallel lines are cut by a secant, each of the eight angles is equal to its altern- ate (128). 126. Corollary — Two interior angles on the same side of the secant are supplements of each other. For, since GHB is the supplement of FHB (91), it is also the supplement of its equal HGD. Two exterior angles on the same side of the secant are supplementary, for a similar reason. 127. Corollary — When a secant is perpendicular to one of two parallels, it is also perpendicular to the other, and all the angles are right. Let the student illustrate by a diagram, in this and in all cases when a diagram is not given. 128. Corollary — When the secant is oblique to the parallels, four of the angles formed are obtuse and are equal to each other ; the other four are acute, and equal ; und any acute angle is the supplement of any obtuse. 4« ELEMENTS OF GEOMETRY. ISO. Theorem. — When two straight lines, being in the same plane, are cut hy a third, making the corresponding angles equal, the two lines so cut are parallel. If AB and CD lie in the same plane, and if the angles AHF and CGF are equal, then AB and CD are parallel. For, suppose a straight line to pass through the point H, paralle] to DC. Such a line makes a corresponding angle equal to CGF, and therefore equal to AHF. This sup- posed parallel line lies in the same plane as CD and H (121); that is, by hypothesis, in the same plane as AB. But if it lies in the same plane with AB and makes the same angle with the same line EF, at the same point H, then it must coincide with AB. For, when two angles are equal and placed one upon the other, they coincide throughout. Therefore, AB is par- allel to CD. 130. Corollary — If the alternate angles are equal, the lines are parallel (123). 131. Corollary — The same conclusion must follow when the interior angles on the same side of the secant are supplementary. DISTANCE BETWEEN PARALLELS. 13!S. Theorem Two parallel lines are everywhere equally distant. The distance between two parallel lines is measured by a line perpendicular to them, since it is the short- est from one to the other. Let AB and CD be two parallels. Then any per- PARALLELS. 47 A E G B C F M H D peiidiculars to them, as EF and GH, are equal. From M, the center of FH, erect the perpendicular ML. Let that part of the figure to the left of ML revolve upon ML. All the angles of the figure being right angles, MC will fall upon MD. Since MF is equal to MH, the point F will fall on H, and the angles at F and H being equal, FE will take the direction HG, and the point E will be on the line HG. But since the angles at L are equal, the point E will also fall on LB, and being on both LB and HG, it must be on G. Therefore, FE and HG coincide and are equal. 133. Corollary — The parts of parallel lines included between perpendiculars to them, must be equal. For the perpendiculars are parallel (129). SECANT AND PARALLELS. 134. Theorem — If several equally distant parallel lines he cut by a secant, the secant will he divided into equal parts. If the parallels BC, DF, GH, and KL are at equal distances, then the parts ^ EI, 10, and OU of the secant AY are equal. For that part of the figure included between BC and DF may be placed upon and will coincide with that part between DF and GH ; for the parallels are everywhere equally distant (132). B \e c D \l F G \o H K v L 48 ELEMENTS OF GEOMETRY. Let them be so placed that the point E may fall upon I. Then, since the angles BEI and DIO (124) are equal, the line EI will take the direction 10. And since DF and GH coincide, the point I will fall on 0. Therefore, EI and 10 coincide and are equal. In like manner, show that any two of the intercepted parts of the line AY are equal. 135. Corollary. — Conversely, if several parallel lines intercept equal segments of a secant, then the several distances between the parallels are equal. 136. Corollary. — When the distances between the parallels are unequal, the segments of the secant are unequal. And conversely, when the segments of the secant are unequal, the distances are unequal. LINES NOT PAEALLEL MEET. 137. Theorem. — If two straight lines are in the same plane and are not parallel^ they will meet if sufficiently produced. Let AB and CD be two lines. Let the line EF, parallel to CD, pass through any point of _^^^ AB, as H. From H E- .:::::::::^[:^^ F let the perpendicular — ^ HG foil upon CD. J Since AB and EF ^ \^ ^ have different direc- tions, they cut each other at the point H. Take any point, as I, in that part of AB which lies between EF and CD, and extend a line IK parallel to CD through the plhnt I. Now divide HG into parts equal to HK until one of the points of division falls beyond G. Then along IIB, take parts equal to HI, as often as PARALLELS. 49 HK was taken along HG. Lastly, from each point of division of HB, extend a line perpendicular to HG. These perpendiculars are parallel to each other and to CD (129). These parallels by construction intercept equal parts of HB. Therefore (135), they are equally distant from each other. Hence, HG is divided by them into equal segments (134); that is, each one passes through one of the previously ascertained points of the line HG. But the last of these points was beyond the line CD, and as the parallel can not cross CD (120), the corre- cponding point of HB is beyond CD. Therefore, HB and CD must cross each other. ANGLES WITH PARALLEL SIDES. 138. Theorem. — When the sides of one angle are par- allel to the sides of another^ and have respectively the same directions from their vertices^ the two angles are equal. If the directions BA and DC are the same, and the directions DE and BF are the same, then the angles ABF and CDE are equal. For each of these angles ig equal to the angle CGF (124). 139. Let the student dem- onstrate that when two of the parallel sides have opposite di- rections, and the other two have the same direction, then the angles are supplementary. Let him also demonstrate that if both sides of one angle have directions respectively opposite to those of the other, then again the angles are equal. (Teoni. — 5 50 ELEMENTS OF GEOMETRY. ANGLES WITH PERPENDICULAR SIDES. 140. Theorem — Two angles which have their sides re- spectively perpendicular are equal or supplementary. If AB is perpendicular to DG, and BC is perpendicu- lar to EF, then the angle ABC is equal to one, and supplement- ary to the other of the i angles formed by DG and EF (86). Through B extend BI parallel to GD, and BH parallel to EF. Now, ABI and CBH are right angles (127), and therefore equal (90). Sub- tracting the angle HBA from each, the remainders HBI and ABC are equal (7). But HBI is equal to FGD (138), and is the supplement of EGD (139). Therefore, the angle ABC is equal or supplementary to any angle formed by the lines DG and EF. APPLICATIONS 141. The instrument called the T square consists of two straight and flat rulers fixed at ri";ht angles to each other, as in the figure. It is used to draw parallel lines. Draw a straight line in a direction per- pendicular to that in which it is required to draw parallel lines. Lay the cross-piece of (he T ruler along this line. The other piece of the ruler gives the direction of one of the parallels. The ruler being moved along the paper, keep' ing the cross-piece coincident with the line first described, any number of parallel lines may be drawn. PARALLELS. 51 What is the principle of geometry involved in the use of this instrument? 142* The uniform distance of parallel lines is the principle upon which numerous instruments and processes in the arts are founded. If two systems, each consisting of several parallel lines, cross each other at right angles, all the parts of one system included between any two lines of the other system will be equal. Tlie ordinary framing of a window consists of two systems of lines of this kind; the shelves and upright standards of book-cases and the paneling of doors also afford similar examples. 143. The joiner's gauge is a tool with which a line may be drawn on a board parallel to its edge. It consists of a square piece of wood, with a sharp steel point near the end of one wide, and a movable band, which may be fastened by a screw or key at any required distance from the point. The gauge is held perpen- dicular to the edge of the board, against which the band is pressed while the tool is moved along the board, the steel point tracing the parallel line. 144. It is frequently important in machinery that a body shall have what is called a parallel motion ; that is, such that all its parts shall move in parallel lines, preserving the same relative position to each other. The piston of a steam-engine, and the rod which it drives, re- ceive such a motion ; and any deviation from it would be attended with consequences injurious to the machinery. The whole mass of the piston and its rod must be so moved, that every point of it shall describe a line exactly parallel to the direction -^f the cylinder. 52 ELEMENTS OF GEOMETRY. CHAPTER IV. THE CIRCUMFERENCE. 145. Let the line AB revolve in a plane about the ^nd A, which is fixed. Then the point B will describe a line which returns upon itself, called a cir- cumference of a circle. Hence, the following definitions : A Circle is a portion of a . ^^ plane bounded by a line called a Circumference, every point of which is equally distant from a point Avithin called the Center. 146. Theorem. — A circumference is curved throughout. For a straight line can not have more than two points equally distant from a given point (111). 14T. Corollary. — A straight line can not cut a cir- cumference in more than two points. 148. The circumference is the only curve considered in elementary geometry. Let us examine the proper- ties of this line, and of the straight lines which may be combined with it. HOW DETERMINED. 149. Theorem. — Three points 7iot in the same straight line fix a circumference both as to position and extent. The three given points, as A, B, and C, determine ARCS AND RADII. 53 the position of a plane. Let the given points be joined by straight lines AB and A D "R BC. At D and E, the mid- — -y \e die points of these lines, let • / \C perpendiculars be erected i in the plane of the three q: points. ; /jj By the hypothesis, AB | / and BC make an angle at ! / B. Therefore, GD is not Jl perpendicular to BC, for / i if it were, AB and BC would be parallel (129). Hence, DG and EH are not parallel (117), since one is per- pendicular and the other is not perpendicular to BC. Therefore, DG and EH will meet (137) if produced. Let L be their point of intersection. Since every point of DG is equidistant from A and B (108), and since every point of EH is equidistant from B and C, their common point L is equidistant from A, B, and C. Therefore, with this point as a center, a circumference may be described through A, B, and C. There can be no other circumference through these three points, for there is no other point besides L equally distant from all three (112). Therefore, these three points fix the position and the extent of the circumference which passes through them. ARCS AND RADII. 150. An Arc is a portion of a circumference. A Radius is a straight line from the center to the circumference. A Diameter is a straight line passing through the center, and limited at both ends by the circumference. A Chord is ? straight line joining the ends of an arc. 54 ELEMENTS OF GEOMETRY. 151. Corollary. — All radii of the same circumference are equal. 15!S. Corollary — In the same circumference, a diame- ter is double the radius, and all diameters are equal. 153. Corollary. — Every point of the plane at greater distance from the center than the length of the radius, is outside of the circumference. Every point at a less distance from the center, is within the circumference. Every point whose distance from the center is equal to the radius, is on the circumference. 154. Theorem. — CircumferenccB ivTiich have equal radii are equal. Let the center of one be placed on that of the other. Then the circumferences will coincide. For if it were otherwise, then some points would be unequally distant from the common center, which is impossible when the radii are equal. Therefore, the circumferences are equal. 155. Corollary — A circumference may revolve upon, or slide along its equal. 156. Corollary. — Two arcs of the same or of equal circles may coincide so far as both extend. 157. Theorem. — Every diameter bisects the circumfer ence and the circle. For that part upon one side of the diameter may be turned upon that line as its axis. When the two parts thus meet, they will coincide; for if they did not, some points of the circumference would be unequally distant from the center. 158. A line which divides any figure in this manner, is said to divide it syr)imetricalhj ; and a figure which can be so divided is symmetrical. ARCS AND RADII. 55 159. Theorem — A diameter is greater than any other fihord of the same circumference. To be demonstrated by the student. 160. Problem Arcs of equal radii may he added to- gether, or one may he suhtracted from another. For an arc may be produced till it becomes an entire circumference, or it may be diminished at will (35 and 145). Therefore, the length of an arc may be increased or decreased by the length of another arc of the same ra- dius; and the result, that is, the sum or diiference, will be an arc of the same radius. 161. Corollary — Arcs of equal radii may be multi- plied or divided in the same manner as straight lines. 163. The sum of several arcs may be greater than a circumference. 163. Two arcs not having the same radius may be joined together, and the result may be called their sum ; but it is not one arc, for it is not a part of one circum- ference. APPLICATIONS. 164. The circumference is the only line which can move along itself, around a ct.iter, vvitliout suffering any change. For any line that can do this must, therefore, have all its points equally distant from the center of revolution ; that is, it must be a cir- cumference. It is in virtue of this property that the axles of wheels, shafts, and other solid bodies which are required to revolve witlun a hol- low mold or casing of their own form, must be circular. If they were of any other form, they would be incapable ot revolving with- out carrying the mold or casing around with them. 165. Wheels which are intended to maintain a carriage always at the same hight above the road on which they rol), must be cir- cular, with the axle in the center. aO ELEMENTS OF GEUMETKV. 166. The art of turning consists in the production of the cir- cular form by mechanical means. The substance to be turned is placed in a machine called a lathe, which gives it a rotary mo- tion. The edge of a cutting tool is placed at a distance from the axis of revolution equal to the radius of the intended circle. As the substance revolves, the tool removes every part that is further from the axis than the radius, and thus gives a circular form to what remains. PROBLEMS IN DRAWING. IGT. The compasses enable us to draw a circumference, or an arc of a given radius and given center. Open the instrument till the points are on the two ends of the given radius. Then fix one point on the given center, and the other point may be made to revolve around in contact with the surface, thus tracing out the circumference. The revolving leg may have a pen or pencil at the point. In the operation, care should be taken not to vary the opening of the compasses. 168. It is evident that with the ruler and compasses (69), 1. A straight line can be drawn through two given points. 2. A given straight line can be produced any length. 3. A circumference can be described from any center, with any radius. 169. The foregoing are the three postulates of Euclid. Since the straight line and the circumference are the only lines treated of in elementary geometry, these Euclidian postulates are a sui- ficient basis for all problems. Hence, the rule that no instruments shall be used except the ruler and the compasses (68). ITO. In the Elements of Euclid, which, for many ages, was the only text-book on elementary geometry, the problems in drawing occupy the place of problems in geometry. At present, the mathe- maticians of Germany, France, and America put them aside as not forming a necessary part of the theory of the science. English writers, however, generally adhere to Euclid 171. Problem — To bisect a given straight line. With A and B as centers, and with a radius greater than the half of AB, describe arcs which intersect in the two points D PROBLEMS IN DRAWING. 57 and E. The straight line joining these two points will bisect AB at C. Let the demonstration be given T) by the student (109 and 151). /\ 172. Problem.— To erect a perpendicular on a given straight line at a given point. Take two points in the line, one on each side of the given point, at equal distances from it. yC Describe arcs as in the last prob- lem, and their intersection gives one point of the perpendicular. Demonstration to be given by the student. ITS. Problem — To let fall a perpendicular from a given point on a given straight line. With the given point as a cen- ^ ter, and a radius long enough, j describe an arc cutting the given j line BC in the points D and E. j The line may be produced, if j necessary, to be cut by the arc in two places. With D and E as centers, and with a radius greater than the half of DE, describe arcs cutting each other in F. The straight line joining A and F is perpendicular to DE. Let the student show why. 174. Problem — To draw a line through a given point parallel to a given line. Let a perpendicular fall from the point on the line. Then, at the given point, erect a perpendicular to this last. It will be par- allel to the given line. Let the student explain why (129). 175. Problem. — To describe a circumference through three given points. The solution of this problem is evident, from Article 149. Xf 58 ELEMENTS OF GEOMETRY. 176, Problem — To find the center of a given arc or circumference. Take any three points of tlie arc, and proceed as in the last problem. XKH , Tlie student is advised to make a drawing of every prob- lem. First draw the parts given, then the construction requisite I'or solution. Afterward demonstrate its correctness. Endeavor to make the drawing as exact as possible. Let the lines be fine and even, as they better represent the abstract lines of geometry. A geometrical principle is more easily understood by the student, when he makes a neat diagram, than when his drawino; is careless. TANGENT. 178. Theorem — A straight line which is perpendicular to a radius at its extremity, touches the circumference in only one point. Let AD be perpendicular to the radius BC at its extremity B. Then it is to be proved that AD touches the circumference at B, and at no other point. If the center C be joined by straight lines with any points of AD, the perpendicular BC will be shorter than any such oblique line (104). Therefore (153), every point of the line AD, except B, is outside of the circumference. 17^. A Tangent is a line touching a circumference in only one point. The circumference is also said to be tangent to the straight line. The common point is called the point of contact. SECANT. 59 APPLICATION. 180. Tangent lines are frequently used in the arts. A com- mon example is when a strap is carried round a part of the cir- cumference of a wheel, and extending to a distance, sufficient tension is given to it to produce such a degree of friction between it and the wheel, that one can not move without the other. 181. Problem in Drawing. — To draiv a tangent at a given point of an arc. Draw a radius to the given point, and erect a perpendicular to the radius at that point. It will be necessary to produce the radius beyond the arc, as the student has not yet learned to erect a perpendicular at the e.vtremity of a line without producing it. SECANT. 182. Theorem. — A straight line which is oblique to a radius at its extremity, cuts the circumference in two points. Let AD be oblique to the radius CB at its extrem- ity B. Then it will cut the cir- cumference at B, and at some other point. From the center C, let CE fall perpendicularly on AD. On ED, take EF equal to EB. Then the distance from C to any point of the line AD be- tween B and F is less than the length of the radius CB (110), and to any point of the line be- yond B and F, it is greater than the length of CB. Therefore (153), that portion of the line AD between B and F is within, and the parts be- yond B and F are without the circumference. Hence, the oblique line cuts the circumference in two points. 60 ELEMENTS OF GEOxMETRY. 183. Corollary. — A tangent to the circumference is perpendicular to the radius which extends to the point of contact. For, if it were not perpendicular, it would be a secant. 184. Corollary. — At one point of a circumference, there can be only one tangent (103). CHORDS. 185. Theorem The radii being equal, if two arcs are equal their chords are also equal. If the arcs AGE and BCD are equal, and their radii are equal, then AE and BD are equal. For, since the radii are equal, the circumferences are equal (154) ; and the arcs may be placed one upon the other, and will coincide, so that A will be upon B, and E upon D. Then the two chords, being straight lines, must coincide (51), and are equal. 180. Every chord subtends two arcs, which together form the whole circumference. Thus thp chord AE sub- tends the arcs AOE and AIE. The arc of a chord always means the smaller of the two, unless otherwise expressed. 187. Theorem. — The radius tvhich is perpendicular to a chord bisects the chord and its arc. CHORDS. 61 Let CD be perpendicular at E to the chord AB, then will AE be equal to EB, and the arc AD to the arc DB. Produce DC to the circum- ference at F, and let that part of the figure on one side of DF be turned upon DF as upon an axis. Then the semi-circum- ference DAF will coincide with DBF (157). Since the angles at E are right, the line EA will take the direction of EB, and the point A will fall on the point B. Therefore, EA and EB will coincide, and are equal; and the same is true of DA and DB, and of FA and FB. 188. Corollary — Since two conditions determine the position of a straight line (52), if it has any two of the four conditions mentioned in the theorem, it must have the other two. These four conditions are, 1. The line passes through the center of the circle, that is, it is a radius. 2. It passes through the center of the chord. 3. It passes through the center of the arc. 4. It is perpendicular to the chord. 189. Theorem. — The radii being equals when two arcs are each less than a semi-circumference, the greater arc has the greater chord. If the arc AMB is greater than CND, and the radii of the circles are equal, then AB is greater than CD. Take AME equal to CND. Join AE, OE, and OB. Then AE is equal to CD (185). Since the arc AMB is less than a semi-circumference, the chord AB will pass between the arc and the center 0, Hence, it cuts the radius OE at some point I. 62 ELEMENTS OF GEOMETRY. Now, the broken line OIB is greater than OB (54), or its equal OE. Subtracting 01 from each (8), the C N remainder IB is greater than the remainder IE. Add- ing AI to each of these, we have AB greater than AIE. But AIE is greater than AE. Therefore, AB, the chord of the greater arc, is greater than AE, or its equal CD, the chord of the less. lOO. Corollary — When the arcs are both greater than a semi-circumference, the greater arc has the less chord. DISTANCE FROM THE CENTER. lOl. Theorem. — When the radii are equal, equal chords are equally distant from the center. Let the chords AB and CD be equal, and in the equal circles ABG and CDF; then the distances of these chords from the centers E and H will also be equal. CHORDS. 63 Let fall the perpendiculars EK and HL from the centers upon the chords. Now, since the chords AB and CD are equal, the arcs AB and CD are also equal (185) ; and we may apply the circle ABG to its equal CDF, so that they will coincide, and the arc AB coincide with its equal CD. Therefore, the chords will coincide. Since K and L are the mid- dle points of these coinciding chords (187), K will fall upon L. Therefore, the lines EK and HL coincide and are equal. But these equal perpendiculars measure the distance of the chords from the centers (105). If the equal chords, as MO and AB, are in the same circle, each may be compared with the equal chord CD of the equal circle CDF. Thus it may be proved that the distances NE and EK are each equal to HL, and therefore equal to each other. 192. Theorem. — When the radii are equal, the less of two unequal chords is the farther from the center. Let AB be the greater of two chords, and FG the less, in the same or an equal circle. Then FG is farther from the center than AB. Take the arc AC equal to FG. Join AC, and from the center D let fall the perpendiculars DE and DN upon AB and AC. Since the arc AC is less than AB, the chord AB will be between AC and the center D, and will cut the perpendicular DN. Then DN, the whole, is greater than DH, the part cut off; and DH is greater than DE (104). So much the (}4 ELEMENTS OF GEOMETRY. more is DN greater than DE. Therefore, AC and its equal FG are farther from the center than AB. 19S. Corollary — Conversely of these two theorems, when the radii are equal, chords which are equally dis- tant from the center are equal ; and of two chords which are unequally distant from the center, the one nearer to the center is longer than the other. 194. Problem in Drawing — To Used a given arc. Draw the chord of the arc, and erect a perpendicular at its center. State the theorem and the problems in drawing here used. 195. "The most simple case of the division of an arc, after its bisection, is its trisection, or its d-ivision into three equal parts. Tills problem accordingly exercised, at an early epoch in the prog- ress of geometrical science, the ingenuity of mathematicians, and has become memorable in the history of geometrical discovery, for having baffled the skill of the most illustrious geometers. "Its object was to determine means of dividing any given arc into three equal parts, without any other instruments than the rule and compasses permitted by the postulates prefixed to Euclid's Elements. Simple as the problem appears to be, it never has been solved, and probably never will be, under the above conditions." — Lardners Treatise. ANGLES AT THE CENTER. l^G. Angles which have their vertex at the center of a circle are called, for this reason, angles at the center. The arc between the sides of an angle is called the in- tercepted arc of the ajigle. 197. Theorem. — The radii being equal, any two angles at the center have the same ratio as their intercepted arcs. This theorem presents the three following cases: 1st. If the arcs are equal, the angles are equal. ANGLES AT THE CENTER. 65 For the arcs may be placed one upon the other, and will coincide. Then BC will coincide with AO, and DC with EG. Thus the angles may coincide, and are equal. The converse is proved in the same manner. 2d. If the arcs have the ratio of two whole numbers, the angles have the same ratio. Suppose, for example, the arc BD : arc AE : : 13 : 5. Then, if the arc BD be divided into thirteen equal parts, and the arc AE into five equal parts, these small arcs will all be equal. Let radii join to their respective cen- ters all the points of division. The small angles at the center thus formed are all equal, because their intercepted arcs are equal. But BCD is the sum of thirteen, and AOE of five of these equal angles. Therefore, angle BCD : angle AOE : : 13 : 5 ; that is, the angles have the same ratio as the arcs. Geom. — 6 66 ELEMENTS OF GEOMETRY. 3d. It remains to be proved, that, if the ratio of the arcs can not be expressed by two whole numbers, the angles have still the same ratio as the arcs ; or, that the radius being the same, the arc BD : arc AE : : angle BCD : angle AOE. If this proportion is not true, then the first, second, A / ^\^ / \ ^\ and third terms being unchanged, the fourth term is either too large or too small. We will prove that it is neither. If it were too large, then some smaller angle, as AOI, would verify the proportion, and arc BD : arc AE : : angle BCD : angle AOI. Let the arc BD be divided into equal parts, so small that each of them shall be less than EI. Let one of these parts be applied to the arc AE, beginning at A, and marking the points of division. One of those points must necessarily fall between I and E, say at the point U. Join OU. Now, by this construction, the arcs BD and AU have the ratio of two whole numbers. Therefore, arc BD : arc AU : : angle BCD : angle AOU. These last two proportions may be written thus (19) ; arc BD : angle BCD : : arc AE : angle AOI ; arc BD : angle BCD : : arc AU : angle AOU. METHOD OF LIMITS. 61 Therefore (21), arc AE : angle AOI : : arc AU : angle AOU; or (19), arc AE : arc AU : : angle AOI : angle AOU. But this last proportion is impossible, for the first antecedent is greater than its consequent, while the second antecedent is less than its consequent. There- fore, the supposition which led to this conclusion is false, and the fourth term of the proportion, first stated, is not too large. It may be shown, in the same way, that it is not too small. Therefore, the angle AOE is the true fourth term of the proportion, and it is proved that the arc BD is to the arc AE as the angle BCD is to the angle AOE. DEMONSTRATION BY LIMITS. 19S. The third case of the above proposition may be demonstrated in a different manner, which requires some explanation. We have this definition of a limit: Let a magnitude vary according to a certain law which causes it to ap- proximate some determinate magnitude. Suppose the first magnitude can, by this law, approach the second indefinitely, but can never quite reach it. Then the second, or invariable magnitude, is said to be the limit of the first, or variable one. 109. Any curve may be treated as a limit. The straight parts of a broken line, having all its vertices in the curve, may be diminished at will, and the broken line made to approximate the curve indefinitely. Hence, a curve is the limit of those broken lines which have all tlieir vertices in the curve. 68 ELEMENTS OF GEOMETRY. SOO. The arc BC, which is cut off bj the secant AD, may be diminished by successive bisections, keeping the remain- ders next to B. Thus AD, re- volving on the point B, may approach indefinitely the tan- gent EF. Hence, the tangent at any point of a curve is the limit of the secants which may cut the curve at that point. 301. The principle upon which all reasoning by the method of limits is governed, is that, whatever is true up to the limit is true at the limit. We admit this as an axiom of reasoning, because we can not conceive it to be otherwise. Whatever is true of every broken line having its vertices in a curve, is true of that curve also. What- ever is true of every secant passing through a point of a curve, is true of the tangent at that point. We do not say that the arc is a broken line, nor that the tangent is a secant, nor that an arc can be without extent; but that the curve and the tangent are limits toward which variable magnitudes may tend, and that whatever is true all the way to within the least pos- sible distance of a certain point, is true at that point. 202. Having proved (first and second parts, 197) that, when two arcs have the ratio of two whole numbers, the angles at the center have the same ratio, we may then suppose that the ra- tio of BD to BF can not be ex- pressed by whole numbers. Now, if we divide BF into two equal parts, the point of division will be at a certain METHOD OF INFINITES. 69 distance from D. We may conceive the arc BF to be divided into any number of equal parts, and by in- creasing this number, the point 0, the point of division nearest to D, may be made to approach within any con- ceivable distance of D. By the second part of the theorem (197), it is proved that arc BO : arc BF : : angle BCO : angle BCF. Now, although the arc BD is itself incommensurable with BF, yet it is the limit of the arcs BO, and the ansle BCD is the limit of the angles BCO. Therefore, since whatever is true up to the limit is true at the limit, arc BD : arc BF : : angle BCD : angle BCF. That is, the intercepted arcs have the same r?tio as their angles at the center. METHOD OF INFINITES. 203. Modern geometers have made much use of a kind of reasoning which may be called the method of infinites. It consists in supposing that any line cf def- inite extent and form is composed of an infinite num- ber of infinitely small straight lines. A surface is supposed to consist of an infinite number of infinitely narrow surfaces, and a solid of an infinite number of infinitely thin solids. These thin solids, nar- row surfaces, and small lines, are called infinitesimals. 204. The reasoning of the method of infinites is substantially the same in its logical rigor as of the method of limits. The method of infinites is a p'uch abbreviated form of the method of limits. . The student must be careful how he adopts it. For when the infinite is brought into an argument b}' the unskillful, the conclusion is very apt to be absurd It 70 ELEMENTS OF GEOMETRY. is sufficient to say, that where the method of limits can be used, the method of infinites may also be used with- out error. The method of infinites has also been called the 7nefhod of indivisibles. Some examples of its use will be given in the course of the work. ARCS AND ANGLES. We return to the subject of angles at the center. The theorem last given (197) has the following !S05. Corollary. — If two diameters are perpendicular to each other, they divide the whole circumference into four equal parts. 206. A Quadrant is the fourth part of a circumference. 20T. Since the angle at the cen- ter varies as the intercepted arc, mathematicians have adopted the same method of measuring both an- gles and arcs. As a right angle is the unit of angles, so a quadrant of a certain radius may be taken as the standard for the measurement of arcs that have the same radius. For the same reason, w^e usually say that the inter- cepted arc measures the angle at the center. Thus, the right angle is said to be measured by the quadrant; half a right angle, by one-eighth of a' circumference; and so on. ' ' APPLICATIONS. 20S. In tlie applications of geometry to practical purposes, the quadrant and tlie riglit angle are divided into ninety equal parts, each of which is called a degree. Each degree is marked ARCS AND ANGLES. 71 thus °, and is divided into sixty minutes, marked thus ^; a-nd each minute is divided into sixty seconds, marked tjius '\ Hence, it appears that there are in an entire circumference, or in the sum of all the successive angles about a point, 300°, or 21600^, or 1296 000^^ Some astronomers, mostly the Frencli, divide the right angle and tlie quadrant into one hundred parts, each of these into one hundred; and so on. 209, Instruments for measuring angles are founded upon the principle that arcs are proportional to angles. Sucl) instruments usually consist either of a part or an entire circle of metal, on the surface of which is accurately engraven its divisions into de- grees, etc. Many kinds of instruments used by surveyors, navi- gators, and astronomers, are constructed upon this principle. 210. An instrument called a protractor is used, in drawing, for measuring angles, and for laying down, on paper, angles of any required size. It consists of a semicircle of brass or mica, the circumference of which is divided into degrees and parts of a degree. PROBLEMS IN DRAWING. 211. Problem — To draw an angle equal to a given angle. Let it be required to draw a line making, with the given line BC, an angle at B equal to the given angle A. With A as a center, and any as- sumed radius AD, draw the arc DE cutting the sides of the angle A. With B as a center, and the same radius as before, draw an arc FG. Join DE. With F as a center, and a radius equal to DE, draw an arc cut- ting FG at the point G. Join BG. Then GBF is the required angle. For, joining FG, the arcs DE and FG have equal radii and equal chords, and therefore are equal (185). Hence, they sub- tend equal angles (197). 212. Corollary. — An arc equal to a given arc may be drawn in the same way. 72 ELEMENTS OF GEOMETRY. !S13. Problem — To draw an angle equal to the sum of two given angles. Let A and B be the given an- gles. First, make the angle DCE equal to A, and then at C, on the line CE, draw the angle ECF equal to B. The angle FCD is equal to ths sum of A and B (9). 214. Corollary. — In a similar manner, draw an angle equal to the sum of several given angles; also, an angle equal to the dif- ference of two given angles ; or, an angle equal to the supplement, or to the complement of a given angle. 215. Corollary. — By the same methods, an arc may be drawn equal to the difference of two arcs having equal radii, or equal to the sum of several arcs. 216. Problem. — To erect a perpendicular to a given line at its extreme point, without producing the given line. A right angle may be made separately, and then, at the end of the given line, an angle be made equal to the given angle. This is the method universally employed by mechanics and draughtsmen to construe? right angles and perpendiculars by the use of the square. SIT. Problem — To draw a line through a given point parallel to a given line. This has been done by means of perpendiculars (174). Tt may be done with an oblique secant, by making the alternate or the corresponding angles equal, ARCS INTERCEPTED BY PARALLELS. 218. An arc which is included between two parallel lines, or between the sides of an angle, is called inter- cepted. 210. Theorem — Two parallel lines intercept equal arcs of a circumference. INTERCEPTED ARCS. 75 The two lines may be both secants, or both tangents, or one a secant and one a tangent. 1st. When both are secants. The arcs AC and BD inter- cepted by the parallels AB and CD are equal. For, let fall from the center a perpendicular upon CD, and produce it to the circumference at E. Then OE is also perpendicular to AB (127), Therefore, the arcs EA and EB are equal (187); and the arcs EC and ED are equal. Subtracting the first from the second, there remains the arc AC equal to the arc BD. 2d. When one is a tangent. Extend the radius OE to the point of contact. This radius is perpendicular to the tangent AB (183). Hence, it is perpen- dicular to the secant CD (127), and therefore it bisects the arc CED at the point E (187). That is, the intercepted arcs EC and ED are equal. 3d. When both are tangents. Extend the radii OE and 01 to the points of contact. These radii being perpendicular a i? b (183) to the parallels, must (103 and 127) form one straight line. Therefore, EI is a diameter, and divides (157) the circumference into equal parts. But these equal parts are the arcs intercepted by the parallel tangents. C I D Therefore, in every case, the arcs intercepted by two parallels are equal. Oeom. — 7 74 ELEMENTS OF GEOMETRY. ARCS INTERCEPTED BY ANGLES. 320. An Inscribed Angle is one whose sides are chords or secants, and whose vertex is on the circum- ference. An angle is said to be inscribed in an arc, when its vertex is on the arc and its sides extend to or through the ends of the arc. In such a case the arc is said to contain the angle. Thus, the angle AEI is inscribed in the arc AEI, and the arc AEI con- tains the angle AEI. An angle is said to stand upon the arc intercepted between its sides. Thus, the angle AEI stands upon the arc AOL 221. Corollary. — The arc in which an angle is in- scribed, and the arc intercepted between its sides, com- pose the whole circumference. 222m Theorem. — An inscribed angle is measured by half of the intercepted arc. This demonstration also presents three cases. The center of the circle may be on one of the sides of the angle, or it may be inside, or it may be outside of the angle. 1st. One side of the angle, as AB, may be a diameter. Make the diameter DE, paral- lel to BC, the other side of the angle. Then the angle B is equal to its alternate angle BOD (125), which is measured by the arc BD (207). This arc is equal to OE (219), and also to EA (197). Therefore, the arc INTERCEPTED ARCS. 75 BD is equal to the half of AC, and the inscribed angle B is measured by half of its intercepted arc. 2d. The center of the circle may be within the angle. From the vertex B extend a diameter to the opposite side of the circumference at D. As just proved, the angle ABD is measured by half of the arc AD, and the angle DBC by half of the arc DC. Therefore, the sum of the two angles, or ABC, is measured by half of the sum of the two arcs, or half of the arc ADC. Bd. The center of the circle may be outside of the angle. Extend a diameter from the vertex as before. The angle ABC is equal to ABD diminished by DBC, and is, therefore, meas- ured by half of the arc DA di- minished by half of DC; that is, by the half of AC. 23S. Corollary. — When an inscribed angle and an angle at the center have the same intercepted arc, the inscribed angle is half of the angle at the center. 224. Corollary. — All angles in- scribed in the same arc are equal, for they have the same measure. 225. Corollary. — Every angle inscribed in a semi- circumference is a right angle. If the arc is less than a semi-circumference, the angle is obtuse. If the arc is greater, the angle is acute. 76 ELEMENTS OF GEOMETRY !SS6. Theorem — The angle formed hij a tangent and a chord is measured by half the intercepted arc. The angle CEI, formed by the tangent AC and the chord EI, is measured by half the intercepted arc IDE. Through I, make the chord 10 parallel to the tangent AC. The angle CEI is equal to its alternate EIO (125), which is measured by half the arc OME (222), which is equal to the arc IDE (219). Therefore, the angle CEI is measured by half the arc IDE. The sum of the angles AEI and CEI is two right angles, and is therefore measured by half the whole cir- cumference (207). Hence, the angle AEI is equal to two right angles diminished by the angle CEI, and is measured by half the whole circumference diminished by half the arc IDE ; that is, by half the arc lOME. Thus it is proved that each of the angles formed at E, is measured by half the arc intercepted between its sides. 227. This theorem may be demonstrated very ele- gantly by the method of limits (200). 228. Theorem. — Every angle whose vertex is within the circumference, is measured hy half the sum of the arcs intercepted between its sides and its sides pro- 0/ 7-.---\U duced. Thus, the angle OAE is meas- ured by half the sum of the arcs OE and lU. To be demonstrated by the student, using the previous theorems (219 and 222). INTERCEPTED ARCS. 77 S29. Theorem — Every angle whose vertex is outside of a circumference, and whose sides are either tangent or secant, is measured hy half the difference of the inter- cepted arcs. Thus, the angle ACF is measured by half the dif- ference of the arcs AF and AB ; the angle FCG, by half the difference of the arcs FG and BI; and the angle ACE, by half the difference of the arcs AFGE and ABIE. This, also, may be demon- strated by the student, by the aid of the previous theo- rems on intercepted arcs. PROBLEMS IN DRAWING. S30. Problem — Through a given point out of a cir- cumference, to draw a tangent to the circumference. Let A be the given point, and C the center of the given circle. Join AC. Bisect AC at the point B (171). With B as a center and BC as a radius, describe a circumference. It will pass through C and A (153), and will cut the cir- cumference in two points, D and E. Draw straight lines from A through D and E. AD and AE are both tangent to the given circumference. Join CD and CE. The angle CDA is inscribed in a semi- circumference, and is therefore a right angle (225). Since AD is perpendicular to the radius CD, it is tangent to the circumference (ITS). AE is tangent for the same reasons. 78 ELEMENTS OF GEOMETRY. ;S31. Problem. — Upon a given chord to describe an arc which shall contain a given angle. Let AB be the chord, and C the aM<;le. Make the angle DAB equal to C. At A erect a perpendicular to AD, and erect a perpendicu- lar to AB at its center (172). Produce these till they meet at the point F (137). With F as a center, and FA as a ra- dius, describe a circum- ference. Any angle in- scribed in the arc BGHA will be equal to the given angle C. For AD, being perpendicular to the radius FA, is a tangent (178). Therefore, the angle BAD is measured by half of the arc AIB (226). But any angle contained in the arc AHGB is also measured by half of the same arc (222), and is therefore equal to BAD, which was made equal to C. POSITIONS OF TWO CIRCUMFERENCES. 232. Theorem. — Two circumferences can not cut each other in more than two poiyits. For three points determine the position and extent of a circumference (149). Therefore, if two circumfer- ences have three points common, they must coincide throughout. S33. Let us investigate the various positions which two circumferences may have with reference to each other. Let A and B be the centers of two circles, and let these points be joined by a straight line, w^hich there- fore measures the distance between the centers. First, suppose the sum of the radii to be less than AB. TWO CIRCUMFERENCES. 79 Then AC and BD, the radii, can not reach each other. At C and D, where the curves cut tlie line AB, let perpendic- ulars to that line be erected. These perpendiculars are paral- lel to each other (129). They are also tangent respectively to the two circumferences (178). It follows, therefore, that CD, the distance between these parallels, is also the least distance between the two curves. S^. Next, let the sum of the radii AC and BC be equal to AB, the distance between the centers. Then both curves will pass through the point C (153). At this point let a perpendicular be erected as before. This per- pendicular CG is tangent to both the curves (178); that is, it is cut by neither of them. Therefore, the curves have only one common point C. 235. Next, let AB be less than the sum, but greater than the diiFerence, of the radii AC and BD. Then the point C will fall within the circum- ference DF. For if it fell on or outside of it, on the side toward A, then AB would be equal to or greater than the sum of the radii ; and if the point C fell on or out- side of the curve in the direction toward B, then AB would be equal to or less than the difference between the radii. Each of these is contrary to the hypothesis. For the same reasons, the point D will fall within the 80 ELEMENTS OF GEOMETRY circumference CG. Therefore, these circumferences cut each other, and have two points common (232). S36. Next, let the difference between the two radii AC and BC be equal to the distance AB. A perpendic- ular to this line at the point C will be a tangent to both curves, and they have a com- mon point at C. They have no other common point, for the two curves are both symmetrical about the line AC (158), and, therefore, if they had a common point on one side of that line, they would have a corresponding com- mon point on the other side; but this can not be, for they would then have three points common (232). 2S7* Lastly, suppose the distance AB less than the difference of the radii AC and BD, by the line CD. That is, AB + BD + DC = AC. A— ^ Join A, the center of the larger circle, with F, any point of the smaller cir- cumference, and join BF. Then AB and BD are to- gether equal to AB and BF, which are together greater than AF. Therefore, AD is greater than AF. Hence, the point D is farther from A than any other point of the circumference DF. It follows that CD is the least distance between the two curves. The above course of reasoning develops the follow- ing principles : )S38. Theorem. — Two circumferences may have, with reference to each other, five positions: TWO CIRCUMFERENCES. 81 1st. Each may he entirely exterior to the other, when the distance between their centers is greater than the su:n of their radii. 2d. They may touch each other exteriorly, having one point common, when the distance between the centers is equal to the sum of the radii. Sd. They may cut each other, having two points com- mon, when the distance between the centers is less than the sum and greater than the difference of the radii. 4th. One may be within the other and tangent, having one point common, when the distance between the centers is equal to the difference of the radii. bth. One may be entirely within the other, tvhen the distance between the centers is less than the difference of the radii. S30. Corollary — Two circumferences can not have more than one chord common to both. :^0. Corollary — The common chord of two circum- ferences is perpendicular to the straight line which joins their centers and is bisected by it. For the ends of the chords are equidistant from each of the centers, the ends of the other line (109). S4]. Corollary. — When two circumferences are tan- gent to each other, the two centers and the point of contact are in one straight line. ;^12. Corollary. — When two circumferences have no common point, the least distance between the curves is measured along the line which joins the centers. tMS. Corollary. — When the distance between the cen- ters is zero, that is, when they coincide, a straight line through this point may have any direction in the plane; and the two curves are equidistant at all points. Such circles are called Concentric. 82 ELEMENTS O;- GEOMETRY. 214. A Locus is a line or a surfivce all the points of which have some common property, which does not belong to any other points. This is also frequently called a geometrical locus. Thus, The circumference of a circle is the locus of all those points in the plane, which are at a given distance from a given point. A straight line perpendicular to another at its center is the locus of all those points in the plane, which are at the same distance from both ends of the second line. The geometrical locus of the centers of those circles which have a given radius, and are tangent to a given straight line, is a line parallel to the former, and at a distance from it equal to the radius. 24:5« The student will find an excellent review of the preceding pages, in demonstrating the theorems, and solving the problems in drawing which follow. In his efforts to discover the solutions of the more difficult problems in drawing, the student will be much assisted by the following Suggestions. — 1. Suppose the problem solved, and the figure completed. 2. Find the geometrical relations of the different parts of the figure thus formed, drawing auxiliary lines, if necessary. 3. From the principles thus developed, make a rule for the solution of a problem. This is the analytic method of solving problems. EXERCISES. 1. Take two straight lines at random, and find their ratio. Make examples in this way for all the problems in drawing. 2. Bisect a quadrant, also its half, its fourth; and so on. EXERCISES. 83 3. From a given point, to draw the shortest line possible to a given straight line. 4. With a given length of radius, to draw a circumference through two given points. 5. From two given points, to draw two equal straight lines which shall end in the same point of a given line. 6. From a point out of a straight line, to draw a second lii e making a required angle with the first. 7. If from a point without a circle two straight lines extend to the concave part of the circumference, making equal angles with the line joining the same point and the center of the circle, then the parts of the first two lines which are within the circumfer- ence are equal. 8. To draw a line through a point such that the perpendicu- lars upon this line, from two other points, may be equal. 9. From two points on the same side of a straight line, to draw two other straight lines which shall meet in the first, and make equal angles with it. 10. In each of the five cases of the last theorem (238), how many straight lines can be tangent to both circumferences? The number is different for each case. 11. On any two circumferences, the two points which are at the greatest distance apart are in the prolongation of the line which joins the centers. 12. To draw a circumference with a given radius, through a given point, and tangent to a given straight line. 13. With a given radius, to draw a circumference tangent to two given circumferences. 14. What is the locus of the centers of those circles which have a given radius, and are tangent to a given circle? 15. Of all straight lines which can be drawn from two given points to meet on the convex circumference of a circle, the sum of those two is the least which make equal angles with the tan- gent to the circle at the point of concourse. 16. If two circumferences be such that the radius of one is the diameter of the other, any straight line extending from their point of contact to the outer circumference is bisected by the inner one. 84 ELEMENTS OF GEOMETRY. 17. If two circumferences cut each other, and from either point of intersection a diameter be made in each, the extremities of these diameters and the other point of intersection are in the same straight line. 18. If any straight line joining two parallel lines be bisected, any other line through the point of bisection and joining the two parallels, is also bisected at that point. 19. If two circumferences are concentric, a line which is a chord of the one and a tangent of the other, is bisected at the point of contact. 20. If a circle have any number of equal chords, what is the locus of their points of bisection? 21. If any point, not the center, be taken in a diameter of a circle, of all the chords which can pass through that point, that one is the least which is at right angles to the diameter. 22. If from any point there extend two lines tangent to a circumference, the angle contained by the tangents is double the angle contained by the line joining the points of contact and the radius extending to one of them. 23. If from the ends of a diameter perpendiculars be let fall on any line cutting the circumference, the parts intercepted be- tween those perpendiculars and the curve are equal. 24. To draw a circumference with a given radius, so (hat the eiaes of a given angle shall be tangents to it. 25. To draw a circumference through two given poi.ivs., with the center in a given line. 26. Through a given point, to draw a straig\iv line, making equal angles with the two sides of a given angle. PROPERTIES OF TRIANGLES. 85 CHAPTER V. TRIANGLES. 246. Next in regular order is the consideration of those plane figures which inclose an area ; and, first, of those whose boundaries are straight lines. A Polygon is a portion of a plane bounded by straight lines. The straight lines are the sides of the polygon. The Perimeter of a polygon is its boundary, or the sum of all the sides. Sometimes this word is used to designate the boundary of any plane figure. S47. A Triangle is a polygon of three sides. Less than three straight lines can not inclose a sur- face, for two straight lines can have only one common point (51). Therefore, the triangle is the simplest polygon. From a consideration of its properties, those of all other polygons may be derived. 248. Problem. — Any three points not in the same straight line may he made the vertices of the three angles of a triangle. For these points determine the plane (60), and straight lines may join them two and two (47), thus forming the required figure. INSCBIBED AND CIRCUMSCRIBED. 249. Corollary — Any three points of a circumference may be made the vertices of a triangle. A circumfer- m ELEMENTS OF GEOxMETRY. ence may pass through the vertices of any triangle, for it may pass through any three points not in the same straight line (149). S50. Theorem. — Within every triangle there is a point equally distant from the three sides. In the triangle ABC, let lines bisecting the angles A and B be produced until they meet. The point D, where the two bisecting lines meet, is equally distant from the two sides AB and BC, since it is a point of the line which bisects the angle B (113). For a similar reason, the point D is equally distant from the two sides AB and AC. Therefore, it is equally distant from the three sides of the triangle. 251. Corollary. — The three lines which bisect the sev- eral angles of a triangle meet at one point. For the point D must be in the line which bisects the angle C (113). 252. Corollary — With D as a center, and a radius equal to the distance of D from either side, a circum- ference may be described, to which every side of the triangle will be a tangent. 25S. When a circumference passes through the ver- tices of all the angles of a polygon, the circle is said to be circumscribed about the polygon, and the polygon to be inscribed in the circle. When every side of a polygon is tangent to a circumference, the circle is inscribed and the polygon circumscribed. 254. The angles at the ends of J? one side of a triangle are said to be adjacent to that side. Thus, the PROPERTIES OF TRIANGLES. 8^ an^-lc3 A and B are adjacent to the side AB. The angle formed by the other two sides is opposite. Thus, the angle A and the side BC are opposite to each other. SUM OF THE ANGLES. 255. Theorem — The sum of the angles of a triangle is equal to two right angles. Let the line DE pass through the vertex of one an- gle, B, parallel to the op- ^ BE posite side, AC. ■••p^ Then the angle A is equal ^/^ \ to its alternate angle DBA A'^— ^C (125). For the same rea- son, the angle C is equal to the angle EBC. Hence, the three angles of the triangle are equal to the three consecutive angles at the point B, which are equal to two right angles (91). Therefore, the sum of the three angles of the triangle is equal to two right angles. !^6. Corollary. — Each angle of a triangle is the sup- plement of the sum of the other two. 257. Corollary. — At least two of the angles of a tri- angle are acute. 258. Corollary. — If two angles of a triangle are equal, they are both acute. If the three are equal, they are all acute, and each is two-thirds of a right angle. 259. An Acute Angled triangle is one which has all its angles acute, as a. A Right Axgled triangle has one of the ancrles riorht, as h. 88 ELEMENTS OF GEOMETRY. An Obtuse Angled triangle has one of the angles obtuse, as c. 260. Corollary — In a right angled triangle, the two acute angles are complementary (94). 261. Corollary — If one side of a triangle be pro- duced, the exterior angle thus B formed, as BCD, is equal to the sum of the two interior angles not adjacent to it, as A A- and B (256). So much the more, the exterior angle is greater than either one of the interior angles not adja- cent to it. 302. Corollary. — If two angles of a triangle are re- spectively equal to two angles of another, then the third angles are also equal. 26^. Either side of a triangle may be taken as the hase. Then the vertex of the angle opposite* the base is the vertex of the triangle. The Altitude of the triangle is the distance from the vertex to the base, which is measured by a perpen- dicular let fall on the base produced, if necessary. 264. Corollary — The altitude of a triangle is equal to the distance between the base and a line through the vertex parallel to the base. 265. .When one of the angles at the base is obtuse, the perpendicular falls outside of the triangle. When one of the angles at the base is right, the alti- tude coincides with the perpendicular side. When both the angles at the base are acute, the alti- tude falls within the triangle. Let the student give the reason for each case, and illustrate it with a diagram. PROPERTIES OF TRIANGLES. &9 LIMITS OF SIDES. S66. Theorem. — Each side of a triangle is smaller than the sum of the other two, and greater than their dif- ference. The first part of this theorem is an immediate conse- quence of the Axiom of Distance (54) ; that is, AC < AB + BC. ^ Subtract AB from both members of this inequality, and AC — AB < BC. That is, BC is greater than the diiference of the other sides. Prove the same for each of the other sides. 267. An Equilateral triangle is one which has three sides equal. An Isosceles triangle is one which has only two sides equal. A Scalene triangle is one which has no two sides equal. EQUAL SIDES. 268. Theorem. — When two sides of a triangle are equal, the angles opposite to them are equal. If the triangle BCD is isosceles, the angles B and D, which are opposite the equal sides, are equal. Let the angle C be divided into two equal parts, and let the divid- ing line extend to the opposite side of the triangle at F. Then, that portion of the figure upon one side of this line may be turned upon it as Geom.— 8 90 ELEMENTS OF GEOMETRY. upon an axis. Since the angle C was bisected, the line BC will fall upon DC; and, since these two lines are equal, the point B will fall upon D. But F, being a point of the axis, remains fixed; hence, BF and DF will coincide. Therefore, the angles B and D coincide, and are equal. 2GO. Corollary — The three angles of an equilateral triangle are equal. 270. In an isosceles triangle, the angle included by the equal sides is usually called the vertex of the trian- gle, and the side opposite to it the base. ISTl. Corollary.— If a line pass through the vertex of an isosceles triangle, and also through the middle of the base, it will bisect the angle at the vertex, and be perpendicular to the base. The straight line which has any two of these four con- ditions must have the other two (52). UNEQUAL SIDES. 2T2. Theorem. — When hvo sides of a triangle are une- qual^ the angle ojjposife to the greater side is greater than the angle opposite to the less side. If in the triangle BCD the side BC is greater than DC, then the angle D is greater than the angle B. Let the line CF bisect the an- gle C, and be produced to the side BD. Then let the triangle CDF turn upon CF. CD will take the g^ direction CB ; but, since CD is less than CB, the point D will fall between C and B, at G. Join GF. Now, the angle FGC is equal to the angle D, because PROPERTIES OF TRIANGLES 91 they coincide; and it is greater than the angle B, be- cause it is exterior to the triangle BGF (261). There- fore, the angle D is greater than B. 273. Corollary — When one side of a triangle is not the largest, the angle which is opposite to that side is acute (257). 274. Corollary. — In a scalene triangle, no two angles are equal. EQUAL ANGLES. 275. Theorem — If two angles of a triangle are equal, the sides opposite them are equal. For if these sides were unequal, the angles opposite to them would be unequal (272), which is contrary to the hypothesis. 276. Corollary — If a triangle is equiangular, that is, has all its angles equal, then it is equilateral. UNEQUAL ANGLES. 277. Theorem — If two angles of a triangle are une- qual, the side opposite to the greater angle is greater than the side opposite to the less. If, in the triangle ABC, the angle C is greater than the angle A, then AB is g greater than BC. For, if AB were not greater than BC, it would be either equal to it or less. If AB were equal to BC, the oppo- site angles A and C would be equal (268) ; and if AB Avere less than BC, then the angle C would be less than A (272) ; but both of these co.nclusions are contrary to the hypothesis. Therefore, AB being neither less than nor equal to BC, must be greater. 92 ELEMENTS OF GEOxVIETRY. 2T8. Corollary. — In an obtuse angled triangle, the longest side is opposite the obtuse angle; and in a right angled triangle, the longest side is opposite the right angle. 279. The Hypotenuse of a right angled triangle is the side opposite the right angle. The other two sides are called the legs. The student will notice that some of the above prop- ositions are but different statements of the principles of perpendicular and oblique lines. EXERCISES. 280. — 1. How many degrees are there in an angle of an equi- lateral triangle? 2. If one of the angles at the base of an isosceles triangle be double the angle at the vertex, how many degrees in each ? 3. If the angle at the vertex of an isosceles triangle be double one of the angles at the base, what is the angle at the vertex? 4. To circumscribe a circle about a given triangle (149). 5. To inscribe a circle in a given triangle (252). 6. If two sides of a triangle be produced, the lines which bi- sect the two exterior angles and the third interior angle all meet in one point. 7. Draw a line BE parallel to the base BC of a triangle ABC, 80 that DE shall be equal to the sum of BD and CE. 8. Can a triangular field have one side 436 yards, the second 547 yards, and the third 984 yards long? 9. The angle at the base of an isosceles triaigle being one- fourth of the angle at the vertex, if a perpendicular be erected to the base at its extreme point, and this perpendicular meet the opposite side of the triangle produced, then the part produced, the remaining side, and the perpendicular form an equilateral triangle. .10. If with the vertex of an isosceles triangle as a center, a circumference be drawn cutting the base or the base produced, then the parts intercepted between the curve and the extreniities of the base, are equal. EQUALITY OF TRIANGLES. 93 EQUALITY OF TRIANGLES. S81. The three sides and three angles of a trian- gle may be called its six elements. It may be shown that three of these are always necessary, and they are generally enough, to determine the triangle. THREE SIDES EQUAL. 282, Theorem. — Two triangles are equal when the three sides of the one are respectively equal to the three sides of the other. Let the side BD be equal to AI, the side BC equal to AE, and CD to EI ; then the two triangles are equal. Apply the line AI to its equal BD, so that the point A will fall upon B. Then I will fall upon D, since the lines are equal. Next, turn one of the triangles, if nec- essary, so that both shall fall on the same side of this common line. Now, the point A being on B, the points E and C are at the same distance from B, and therefore they are both in the circumference, which has B for its center, and BC or AE for its ra- dius (153). For a similar reason, the points E and C are both in the circumference, w^hich has D for its cen- ter and DC or IE for its radius. These two circumfer- ences have only one point common on one side of the line BD, which joins their centers (232). Hence. E and C are both at this point. Therefore (51), AE coincides 94 ELE^iENTS OF GEOMETRY. with BC, and EI with CD; that is, the two triangles coincide throughout, and are equal. 283. Every plane figure may be supposed to have two faces, which may be termed the upward and the downward faces. In order to place the triangle m upon Z, we may conceive it to slide along the plane without turning over; but, in order to place n upon Z, it must be turned over, so that its upward face will be upon the upward face of I. There are, then, two methods of superposition; the first, called direct, when the downward face of one figure is applied to the upward face of the other; and the second, called inverse, when the upward faces of the two are applied to each other. Hitherto, we have used only the inverse method. Generally, in the chapter on the circumference, either method might be used indif- ferently. TWO SIDES AND INCLUDED ANGLE. 284. Theorem — Two triangles are equal when they have two sides and the included angle of the one, respect- ively equal to two sides and the included angle of the other. If the angle A is equal to D, and the side AB to the side DF, and AC to DE, then the two triangles are equal. Apply the side AC to its equal DE, turning one tri- EQUALITY OF TRIANGLES. 95 angle, if necessary, so that both shall fall upon the same side of that common line. Then, since the angles A and D are equal, AB must take the direction DF, and these lines being equal, B will fall upon F. Therefore, BC and FE, having two points common, coincide; and the two triangles coincide throughout, and are equal. ONE SIDE AND TWO ANGLES. 2H5m Theorem. — Two triangles are equal when they have one side and two adjacent angles of the one, respect- ively equal to a side and the two adjacent angles of the other. If the triangles ABC and DEF have the side AC equal to DE, and the angle A equal to D, and C equal to E, then the triangles are equal. Apply the side AC to its equal DE, so that the ver- tices of the equal angles shall come together, A upon D, and C upon E, and turning one triangle, if neces- sary, so that both shall fall upon one side of the com- mon line. Then, since the angles A and D are equal, AB will take the direction DF, and the point B will fall some- where in the line DF. Since the angles C and E are equal, CB will take the direction EF, and B will also be in the line EF. Therefore, B ftvlls upon F, the only point common to the two lines DF and EF. Hence, the 96 ELEMENTS OF GEOMETRY. sides of the one triangle coincide with those of the other, and the two triangles are equal. S88. Theorem — Two triangles are equal when they have one side and any two angles of the one, respectively equal to the corresponding parts of the other. For the third angle of the first triangle must be equal to the third angle of the other (262). Then this be- comes a case of the preceding theorem. TWO SIDES AND AN OPPOSITE ANGLE. 287. Theorem — Two triangles are equal when one of them has two sides, and the angle opposite to the side which is equal to or greater than the other, respectively equal to the corresponding parts of the other triangle. Let the sides AE and EI, EI being equal to or C greater than AE, and the angle A, be respectively equal to the sides BC, CD, and the angle B. Then the tri- angles are equal. For the side AE may be placed on its equal BC. Then, since the angles A and B are equal, AI will take the direction BD, and the points I and D will both be in the common line BD. Since EI and CD are equal, the points I and D are both in the circumference whose center is at C, and whose radius is equal to CD. Now, this circumference cuts a straight line extending from B toward D in only one point; for B is either within or on the circumference, since BC is equal to or less than CD. Therefore, I and D are both at that point. EQUALITY OF TRIANGLES. 97 Hence, AI and BD are equal, and the triangles are equal (282). 288. Corollary — Two triangles are equal when they have an obtuse or a right angle in the one, together w^ith the side opposite to it, and one other side, respect- ively equal to those parts in the other triangle (278). The two following are corollaries of the last five theo- rems, and of the definition (40). 289. Corollary — In equal triangles each part of one is equal to the corresponding part of the other. 290. Corollary. — In equal triangles the equal parts are similarly arranged, so that equal angles are opposite to equal sides. EXCEPTIONS TO THE GENERAL RULE. 201. A general rule as to the equality of triangles has been given (281). There are two excep- tions. 1. When the three angles are given. For two very unequal triangles may have the angles of one equal to those of the other. 2. When two unequal sides and the angle opposite to the less are given. For with the sides AB and B BC and the angle A given, there are two triangles, ABC and ABD. 292. The student may show that two parts alone are never enough to determine a triangle. 98 ELEMENTS OF GEOMETRY. UNEQUAL TRIANGLES. 29^, Theorem — Whe7i two triangles have two sides of the one respectively equal to two sides of the other, and the included angles unequal, the third side in that triangle which has the greater angle, is greater than in the other. Let BCD and AEI be two triangles, having BC equal to AE, and BD equal to AI, and the angle A less than B. Then, it is to be proved that CD is greater than EI. Apply the triangle AEI to BCD, so that AE will coincide with its equal BC. Since the angle A is less than B, the side AI will fall within the angle CBD. Let BG be its position, and EI will fall upon CG. Then let a line BF bisect the angle GBD. Join EG. The triangles GBF and BDF have the side BF com- mon, the side GB equal to the side DB, since each is equal to AI, and the included angles GBF and DBF equal by construction. Therefore, the triangles are equal (284), and FG is equal to FD (289). Hence, CD, the sum of CF and FD, is equal to the sum of CF and FG (7), which is greater than CG (54). Therefore, CD is greater than CG, or its equal EL If the point I should fall within the triangle BCD or on the line CD, the demonstration would not be changed. 1394. Theorem. — Conversely, if two triangles have tivo sides of the one equal to two sides of the other, and the third sides unequal, then the angles opposite the third sides are unequal, and that is greater which is opposite th ' 302. Similar magnitudes have been defined : to« fee' those which have the same form while they differ in extent (37). 30S. Let the student bear in mind that the form of a figure depends upon the relative directions of its points, and that angles are differences in direction. Therefore, the definition may be stated thus : Two figures are similar when every angle that can be formed by lines joining points of one, has its corre- sponding equal and similarly situated angle in the other. ANGLES EQUAL. 304. Theorem. — Tivo triangles are similar, when the three angles of the one are respectively equal to the three angles of the other. This may appear to be only a case of the definition of similar figures ; but it may be shown that every angle that can be made by any lines whatever in the one, may have its corresponding equal and similarly situated angle in the other. 102 ELEMENTS OF GEOMETRY. Let the angles A, B, and C be respectively equal to the angles D, E, and F. Suppose GH and IR to be any ;tw.o.. lines in the triangle ABC. ' Join 10 and GB- From F, the point homologous to C, extend JL, making the angle LFE equal to ICB. Nusv, the triangles LFE and ICB have the angles B and E equal, by hypothesis, and the angles at C and F equal, by construction. Therefore, the third angles, ELF and BIC, are equal (262). By subtraction, the angles AIC and DLF are equal, and the angles ACI and DEL. From L extend LM, making the angle FLM equal to CIR. Then the two triangles FLM and CIR have the angles at C and F equal, as just proved, and the angles at I and L equal, by construction. Therefore, the third angles, LMF and IRC, are equal. Join RG. Construct MN homologous to RG, and NO homologous to Gil. Then show, by reasoning in the same manner, that the angles at N are equal to the corresponding angles at G ; and so on, throughout the two figures. The demonstration is similar, whatever lines be first made in one of the triangles. Therefore, the relative directions of all their points are the same in both triangles ; that is, they have the same form. Therefore, they are similar figures. SIMILAR TRIANGLES. 103 305. Corollary. — Two similar triangles may be di- vided into the same number of triangles respectively similar, and similarly arranged. 306. Corollary — Two triangles are similar, when two angles of the one are respectively equal to two angles of the other. For the third angles must be equal also (262). 307. Corollary — If two sides of a triangle be cut by a line parallel to the third side, the triangle cut off is similar to the original triangle (124). 308. Theorem. — Two triangles are similar, when the sides of one are parallel to those of the other; or, when the sides of one are perpendicular to those of the other. We know (138 and 139) that the angles formed by lines which are parallel are either equal or supplement- ary; and that the same is true of angles whose sides are perpendicular (140). We will show that the angles can not be supplementary in two triangles. If even two angles of one triangle could be respect- ively supplementary to two angles of another, the sum of these four angles would be four right angles ; and then the sum of all the angles of the two triangles would be more than four right angles, which is impos- sible (255). Hence, when two triangles have their sides respectively parallel or perpendicular, at least two of the angles of one triangle must be equal to two of the other. Therefore, the triangles are similar (306). 104 ELEMENTS OF GEOMETRY. SIDES PROPORTIONAL. S09. Theorem — One side of a triangle is to the ho- mologous side of a similar triangle as any side of the first is to the homologous side of the second. If AE and BC are homologous sides of similar tri- C E ^.. \^ Hzl „ \K AZ \i ^^ XD angles, also EI and CD, then, AE : BC : : EI : CD. Take CF equal to EA, and CG equal to EI^ and join FG. Then the triangles AEI and FCG are equal (284), and the angles CFG and CGF are respectively equal to the angles A and I, and therefore equal to the angles B and D. Hence, FG is parallel to BD (129). Let a line extend through C parallel to FG and BD. Suppose BC divided at the point F into parts which have the ratio of two whole numbers, for example, four and three. Then let the line CF be divided into four, and BF into three equal parts. Let lines parallel to BD extend from the several points of division till they meet CD. Since BC is divided into equal parts, the distances between these parallels are all equal (135). Therefore, CD is also divided into seven equal parts (134), of which CG has four. That is, CF : CB : : CG : CD : : 4 : 7. But if the lines BC and CF have not the ratio of two whole numbers, then let BC be divided into any SIMILAR TRIANGLES. • 105 number of equal parts, and a line parallel to BD pass through H, the point of division nearest to F. Such a line must divide CD and CB proportionally, as just proved; that is, CH : CB : : CK : CD. By increasing the number of the equal parts into which BC is divided, the points H and K may be made to approach within any conceivable distance of F and G. Therefore, CF and CG are the limits of those lines, CH and CK, which are commensurable with BC and CD ; and we may substitute CF and CG in the last propor- tion for CH and CK. Hence, whatever be the ratio of CF to CB, it is the same as that of CG to CD. By substituting for CF and CG the equal lines AE and EI, we have, AE : BC : : EI : CD. By similar reasoning it may be shown that AI : BD : : EI : CD. 310. Corollary. — The ratio is the same between any two homologous lines of two similar triangles. 311. This ratio of any side of a triangle to the ho- mologous side of a similar triangle, is called the linear ratio of the two figures. 312. Corollary The perimeters of similar triangles have the linear ratio of the two figures. For, AE : BC : : EI : CD : : lA : DB. Therefore (23), AE+EI+IA : BC+CD-f DB : : AE : BC. 313. Corollary. — If two sides of a triangle are cut by one or more lines parallel to the third side, the two sides 106 ELEMENTS OF GEOMETRY. are cut proportionally. For the triangles so formed are similar (807). S14. Corollary — When several parallel lines are cut by two se- cants, the secants are divided pro- portionally. For the secants being produced till they meet, form several simi- lar triangles. 31«>. Theorem. — Tf two sides of a triangle he cut pro- portionally hy a straight line, the secant line is parallel to the third side. Let BCD be the triangle, and FG the secant. A line parallel to CD may pass through F, and such a line must divide BD in the same ratio as BC (313). But, by hypothesis, BD is so divided at the point G. There- fore, a line through F parallel to CD, must pass through G, and coin- cide with FG. Hence, FG is parallel to CD. 316. Theorem — Two triangles are similar when the ratios between each side of the one and a corresponding side of the other are the same. Suppose AE : BC : : EI : CD : : AI : BD. Take CF equal to EA and CG equal to EI, and join FG. Then, CF : CB : : CG : CD. Therefore, FG is par- a- allel to BD (315), the triangles CFG and CBD are simi- lar (307), and CF : CB : : FG : BD. SIMILAR TIUANGLES. 107 But, by hypothesis, EA : CB : : AI : BD. Hence, since CF is equal to EA, EG is equal to AI, and the triangles AEI and FCG are equal. Therefore, the triangles AEI and BCD have their angles equal, and are similar. 31T. Theorem. — Two triangles are similar when two sides of the one have respectively to two sides of the other the same ratio, and the included angles are equal. Suppose AE :BC :: AI :BD; and let the angle A be equal to B. Take BE equal to AE, and BG equal to AI, and join EG. Then the triangles AEI and BEG are equal (284), and the angle BEG is equal to E, and BGE is equal to I. Since the sides of the triangle BCD are cut proportionally by EG, the angle BEG is equal to C, and BGE is equal to D (315). Therefore, the triangles AEI and BCD are mu- tually equiangular and similar. 318. If two similar triangles have two homologous lines equal, since all other homologous lines have the same ratio, they must also be equal, and consequently the two figures are equal. Thus, the equality of figures may be considered as a case of similarity. PROBLEMS IN DRAWING. 319. Problem. — To find a fourth proportional to three given straight lines. Let a be the given extreme, and b and c the given means. Take DG equal to a, the given extreme. Produce it, making 108 ELEMENTS OF GEOMETRY. DH equal to c, one of the means. From G draw GF equal to b. Then, from D draw a line ^ through F, and from H a line ^ parallel to GF. Produce these ^ two lines till they meet at the point K. FIK is the required fourth proportional. For the triangles DGF and DHK are similar (307). Hence, DG : GF : : DH : HK. That is, a : h :. C : HK. It is most convenient to make GF and HK perpendicular to DH. 3^0« Problem. — To divide a given line into parts having a certain ratio. Let LD be the line to be divided into parts proportional to the lines a, 6, and c. From L draw the line LE equal to the sum of a, ^, and c, making LF equal to a, FG equal to b, and GE equal to c. Join DE, and draw GT and FH parallel to DE. LH, HI, and ID are the parts required. The demonstration is similar to the last. 321. Problem — To divide a given line into any num- ber of equal parts. This may be done by the last problem; but when the given line is small, the following method is preferable. To divide the line AB into ten equal parts; draw AC indefinitely, and take on it ten equal parts. Join BC, and from the several points of division of AC, draw lines parallel to AB, and produce them to BC. The parallel nearest to AB is nine-tenths of AB, the next is eight-tenths, and so on. This also dejKnids upon similarity of triangles. SIMILAR TRIANGLES. 109 3S!S. Problem. — To draw a triangle on a given base, similar to a given triangle. Let this problem be solved by the student. IIIGHT ANGLED TRIANGLES. 3!33. Every triangle may be divided into two right angled triangles, by a perpendicular let fall from one of its vertices upon the opposite side. Thus the investi- gation of the properties of right angled triangles leads to many of the properties of triangles in general. 3S4. Theorem — If in a rigid angled triangle^ a per- pendicular he let fall from the vertex of the right angle upon the hypotenuse^ then, 1. Each of the triangles thus formed is similar to the original triangle; 2. Either leg of the original triangle is a mean propor- tional between the hypotenuse and the adjacent segment of the hypotenuse; and, 3. The perpendicidar is a mean proportional between the two segments of the hypotenuse. The triangles AEO and AEI have the angle A com- mon, and the angles AEI and ^ AOE are equal, being right angles. Therefore, these two triangles are similar (306) That the triangles EOI and EIA are similar, is proved by the same reasoning. Since the triangles are similar, the homologous sides are proportional, and we have AI : AE :: AE : AO; That is, the leg AE is a mean proportional between no ELEMENTS OF GEOMETllV. the whole hypotenuse and the segment AO which h adjacent to that leg. In like manner, prove that EI is a mean proportional between AI and 01. Lastly, the triangles AEO and EIO are similar (304), and therefore, AO : OE : : OE : 01. That is, the perpendicular is a mean proportional between the two segments of the hypotenuse. 325. Corollary. — A perpendicular let fall from any point of a circumference upon a diameter, is a mean proportional between the two segments which it makes of the diameter. (225) 32G. In the several proportions just demonstrated, in place of the lines we may substitute those numbers which constitute the ratios (14). Indeed, it is only upon this supposition that the proportions have a meaning. It is the same whether these numbers be integers or radicals, since we know that the terms of the ratio are in fact numbers. 327. Theorem — The second power of the length of the hypotenuse is equal to the sum of the second powers of the lengths of the two legs of a right angled triayigle. Let h be the hypotenuse, a the perpendicular let fall upon it, h and c the legs, and d and e the corresponding seg- ments of the hypotenuse made by the perpendicular. That is, these letters represent the num- ber of times, whether integral or not, which some unit of length is contained in each of these lines. By the second conclusion of the last theorem, we have SIMILAR TRIANGLES. 11 h : b : : b : d, and h : c : : c : c. Hence, (16), hd^=b^, and he = c'^. By adding these two, h {d-\- e) = b^ + c^. But d-\- e=^h. Therefore, It^ = P -h 328. Theorem. — If, in any triangle, a perpendieidar be let fall from one of the vertices upon the opposite side as a base, then the whole base is to the sum of the other two sides, as the difference of those sides is to the difference of the segments of the base. Let a be the perpendicular, b the base, c and d the sides, and e and i the segments of the base. Then, two right an- gled triangles are formed, in one of which we have a^^i' = d^; and in the other, a'^-{-e^ = c'^. Subtracting, i'^ — e'^ = d'^ — c^. Factoring, (** + ^) {i — e)==(d-\-c) {d — c). Whence (18), i + e : d-\-c :: d — c : i — e. 329. Theorem. — If a line bisects an angle of a trian- gle, it divides the opposite side in the ratio of the adjacent sides. If BF bisects the angle CBD, then CF : FD : : CB : BD. This need be demonstrated only in the case where the sides adjacent to the bi- sected angle are not equal. From C and from D, let perpendiculars DG and CH fall upon BF, and BF pro- "''\/ duced. Then, the triangles BDG and BCH are similar, for /!•? ELEMENTS OF GEOMETRY. thej have equal angles at B, by hypothesis, and at G and H, by construction. Hence, CB : BD : : CH : DG. But the triangles DGF and CHF are also mutually equiangular and similar. Hence, CF : FD : : CH : DG. Therefore (21), CF • FD : : CB : BD. 330. Problem in Drawing. — To find a mean propor- tional to two given straight lines. Make a straight line equal to the sum of the two. Upon this as a diameter, describe a semi-circumference. Upon this diame- ter, erect a perpendicular at the point of meeting of the two given lines. Produce this to the circumference. The line last drawn is the required line. Let the student construct the figure and demonstrate. CHORDS, SECANTS, AND TANGENTS 331. Theorem. — If two chords of a circle cut each other ^ the parts of one may he the extremes, and the parts of the other the means, of a proportion. Join AD and CB. Then the two triangles AED and CEB have the angle A equal to „ the angle C, since they are in- ''^^\ ^X scribed in the same arc (224). For the same reason, the angles i D and B are equal. Therefore, the triangles are similar (306); and we have (309), ]5\^ H.^ AE : EC : : DE : EB. ^ 332. Theorem. — If from the same point, without a cir- ;le, two lines cutting the circumference extend to the far- 'her side, then the whole of one secant and its exterior SIMILAR TRIANGLES. 113 part may he the extremes^ and the whole of the other secant and its exterior part may he the means, of a proportion. Joining BC and AD, the triangles AED and CEB are similar; for they have the angle E common, and the angles at B and D equal. Therefore, AE : EC : : DE : EB. S33. Corollary. — If from the same point there be a tangent and a secant, the tangent is a mean propor- tional between the secant and its exterior part. For the tangent is the limit of all the secants which pass through the point of meeting. 334. Problem in Drawing. — To divide a given straight line into two parts so that one of them is a mean propor- tional hettveen the whole line and the other part. This is called dividing a line in extreme and mean ratio. -jt^ Let AC be the given line. At C erect a perpendicular, CT, equal to half of AC. Join A I. Take ID equal to CI, \e and then AB equal to AD. The line AC is divided at the point B in extreme and mean ratio. That is, AC : AB : : AB : BC. With I as a center and IC as a radius, describe an arc DCE, and produce AI till it meets this arc at E. Then, AC is a tangent to this arc (178), and there- fore (333), AE : AC : : AC : AD. Or (24), AE — AC : AC : : AC — AD : AD. But AC is twice IC, by construction, and DE is twice IC, be- cause DE is a diameter and IC is a radius. Therefore, the first Gcom.— 10 /14 ELEMENTS OF GEOMETRY. term of the last proportion, AE — AC, is equal to AE — DE, which is AD; but AD is, by construction, equal to AB. Also, the third term, AC — AD, is equal to AC — AB, which is BC. And the fourth term is equal to AB. Substituting these equals, the pro- portion becomes AB : AC : : BC : AB. By inversion (19), AC : AB : : AB : BC. ANALYSIS AND SYNTHESIS. 335. Geometrical Analysis is a process employed both for the discovery of the solution of problems and for the investigation of the truth of theorems. Analy- sis is the reverse of synthesis. Synthesis commences with certain principles, and proceeds by undeniable and successive inferences. The whole theory of geometry is an example of this method. , In the analysis- oi a problem, what was required to be done is supposed to have been effected, and the con- sequences are traced by a series of geometrical con- structions and reasonings, till at length they terminate in the data of the problem, or in some admitted truth. See' suggestions, Article 245. In the synthesis of a problem, however, the last con- sequence of the analysis is the first step of the process, and the solution is effected by proceeding in a contrary order through the several steps of the analysis, until the process terminates in the thing required to be done. If, in the analysis, we arrive at a consequence which conflicts with any established principle, or which is incon- sistent with the data of the problem, then the solution is impossible. If, in certain relations of the given mag- nitudes, the construction is possible, while in other rela- tions it is impossible, the discovery of these relations is a necessary part of the discussion of the problem. ANALYSIS AND SYinTHESIS. 115 In the analysis of a theorem, the question to be de- termined is, whether the proposition is true, as stated ; .and, if so, how this truth is to be demonstrated. To do this, the truth is assumed, and the successive conse- quences of this assumption are deduced till they term- inate in the hypothesis of the theorem, or in some established truth. The theorem will be proved synthetically by retracing, in order, the steps of the investigation pursued in the analysis, till they terminate in the conclusion which had been before assumed. Tlijs constitutes the demon- stration. If, in the analysis, the assumption of the truth of the proposition leads to some consequence which conflicts with an established principle, the false conclusion thus arrived at indicates the falsehood of the proposition which was assumed to be true. In a word, analysis is used in geometry in order to discover truths, and synthesis to demonstrate the truths discovered. Most of the problems and theorems which have been given for Exercises, are of so simple a character as scarcely to admit of the principle of geometrical analy- sis being applied to their solution. S36. A problem is said to be determinate when it admits of one definite solution ; but when the same con- struction may be made on the other side of any given line, it is not considered a different solution. A prob- lem is indeterminate when it admits of more than one definite solution. Thus, Article 300 presents a case where the problem may be determinate, indeterminate, or insolvable, according to the size of the given angle and extent of the given lines. The solution of an indeterminate problem frequently 116 ELEMENTS OF GEOMETRY. amounts to finding a geometrical locus; as, to find a point equidistant from two given points ; or, to find a point at a given distance from a given line. EXERCISES. 337. Nearly all the following exercises depend upon principles found in this chapter, but a few of them de- pend on those of previous chapters. 1. If there be an isosceles and an equilateral triangle on the same base, and if the vertex of the inner triangle is equally distant from the vertex of the outer one and from the ends of the base, then, according as the isosceles triangle is the inner or the outer one, its base angle will be ^ of, or 2^ times the vertical angle. 2. The semi-perimeter of a triangle is greater than any one of -the sides, and less than the sum of any two. 3. Through a given point, draw a line such that the parts of it, between the given point and perpendiculars let fall on it from two other given points, shall be equal. What would be the result, if the first point were in the straight line joining the other two? 4. Of all triangles on the same base, and having their ver- tices in the same line parallel to the base, the isosceles has the greatest vertical angle. 5. If, from a point without a circle, two tangents be made to the circle, and if a third tangent be made at any point of the cir- <3umference between the first two, then, at whatever point the last tangent be made, the perimeter of the triangle formed by these tangents is a constant quantity. 6. Through a given point between two given lines, to draw a Jine such that the part intercepted by the given lines shall be bi- sected at the given point. 7. From'a point without two given lines, to draw a line such that the part intercepted between the given lines shall be equal to the part between the given point and the nearest line. 8. The middle point of a hypotenuse is equally distant from the three vertices of a right angled triangle. EXERCISES. 117 9. Given one angle, a side adjacent to it, and the difference of the other two sides, to construct the triangle. 10. Given one angle, a side opposite to it, and the difference of the other two sides, to construct the triangle. 11. Given one angle, a side opposite to it, and the sum of the other two sides, to construct the triangle. 12. Trisect a right angle. 13. If a circle be inscribed in a right angled triangle, the dif^ ference between the hypotenuse and the sum of the two legs is equal to the diameter of the circle. 14. If from a point within an equilateral triangle, a perpen- dicular line fall on each side, the sum of these perpendiculars is a constant quantity. How should this theorem be stated, if the point were outside of the triangle? 15. Find the locus of the points such that the sum of the dis- tances of each from the two sides of a given angle, is equal to a given line. 16. Find the locus of the points such that the difference of the distances of each from two sides of a given angle, is equal to a given line. 17. Demonstrate the last corollary (333) by me?ins of similar triangles. 18. To draw a tangent common to two given circles. 19. To construct an isosceles triangle, when one side and one angle are given. 20. If in a right angled triangle one of the acute angles is equal to twice the other, then the Jiypotenuse is equal to twice the shorter leg. 21. Draw a line DE parallel to the base BC of a triangle ABC, so that DE shall be equal to the difference of BD and CE. 22. In a given circle, to inscribe a triangle similar to a given triangle. 23. In a given circle, find the locus of the middle points of those chords which pass through a given point. 24. To describe a circumference tangent to three given equal circumferences, which are tangent to each other. 118 ELEMENTS OF GEOMETRY. 25. If a line bisects an exterior angle of a triangle, it divides the base produced into segments which are proportional to the adjacent sides. That is, if BF bisects the angle ABD, then, CF : FD : : CB : BD. 26. The parts of two parallel lines intercepted by several straight lines which meet at one point, are proportional. The converging lines are also divided in tlie same ratio. 27. Two triangles are similar, when two sides of one are pro- portional to two sides of the other, and the angle opposite to that side which is equal to or greater than the other given side in one, is equal to the homologous angle in the other. 28. The perpendiculars erected upon the several sides of a tri- angle at their centers, meet in one point. 29. The lines which bisect the several angles of a triangle, n)eet in one point. 30. The altitudes of a triangle, that is, the perpendiculars let fall from the several vertices on the opposite s'des, meet in one pomt. 31. The lines which join the several vertices of a triangle with the centers of the opposite sides, meet in one point. 32. Each of the lines last mentioned is divided at the point of meeting into two parts, one of which is twice aa long as the otJver. QUADKILATERALS. 119 CHAPTER VI. QUADRILATERALS. 5538. In a polygon, two angles which immediately succeed each other in going round the figure, are called adjacent angles. The student will distinguish adjacent angles of a polygon from the adjacent angles defined in Article 85. A Diagonal of a polygon is a straight line joining the vertices of any two angles which are not adjacent. Sometimes a diag- onal is exterior, as the diagonal BD of the. figure ABCD. A Convex polygon has all its di- agonals interior. A Concave polygon has at least one diagonal exte- rior, as in the above diagram. Angles, such as BCD, are called reentrant. 339. A Quadrilateral is a polygon of four sides. 340. Corollary. — Every quadrilateral has two diago- nals. 341. Corollary. — An interior diagonal of a quadri- lateral divides the figure into two triangles. EQUAL quadrilaterals. 342. Theorem — Ttvo quadrilaterals are equal when they are each composed of two triangles, which are respect- ively equal, and similarly arranged. 120 ELEMENTS OF GEOxMETRY. For, since the parts are. equal and similarly arranged, the wholes may be made to coincide (40). 31:3. Corollary — Conversely, two equal quadrilaterals may be divided into equal triangles similarly arranged. In every convex quadrilateral this division may be made in either of two ways. 344. Theorem. — Two quadrilaterals are equal when the four sides and a diagonal of one are respectively equal to the four sides and the same diagonal of the other. By the same diagonal is meant the diagonal that has the same position with reference to the equal sides. For, since all their sides are equal, the triangles AEI and BCD are equal, also the triangles AIO and BDF (282). Therefore, the quadrilaterals are equal (342). 345. Theorem Two quadrilaterals are equal when the four sides and an angle of the one are respectively equal to the four sides and the similarly situated angle of the other. By the similarly situated angle is meant the angle included by equal sides. For, if the sides AE, IE, and the included angle E are respectively equal to the side BC, DC, and the included angle C, then the triangles AEI and BCD are equal (284) ; and AI is equal to BD. But since the QUADRILATERALS. 121 three sides of the triangles AIO and BDF are respect- ively equal, the triangles are equal (282). Hence, the quadrilaterals are equal (342). SUM OF THE ANGLES. 346. Theorem. — The sum of the angles of a quadri- lateral is equal to four right angles. For the angles of the two triangles into which every quadrilateral may be divided, are together coincident with the angles of the quadrilateral. Therefore, the sum of the angles of a quadrilateral is twice the sum of the angles of a triangle. Let the student illustrate this w^ith a diagram. In applying this theorem to a concave figure (338), the value of the reentrant angle must be taken on the side toward the polygon, and therefore as amounting to more than two right angles. INSCRIBED QUADRILATERAL. 34T. Problem. — Any four points of a circumference may he joined hy chords, thus making an inscribed quad- rilateral. This is a corollary of Article 47. 348. Theorem — The opposite angles of an inscribed quadrilateral are supplementary. For the angle A is measured by half of the arc EIO (222), and the angle I by half of the arc EAO. Therefore, the two together are measured by half of the whole cir- cumference, and their sum is equal to two right angles (207). (reoni. — 1 1 123 ELEMENTS OF GEOMETRY. TRAPEZOID. 349. If two adjacent angles of a quadrilateral are sup- plemental, the remaining angles are also supple- mental (346). Then, one pair of opposite sides must ^ ^ be parallel (131). A Trapezoid is a quadrilateral Avhich has two sides parallel. The parallel sides are called its bases. 350. Corollary — If the angles adjacent to one base of a trapezoid be equal, those adjacent to the other base must also be equal. For if A and D are equal, their supplements, B and C, must be equal (96). APPLICATION. 351. The figure described in the last corollary is symmetrical. For it can be divided into equal parts by a line joining the middle points of the bases. The symmetrical trapezoid is used in architecture, sometimes for ornament, and sometimes as the form of the stones of an arch. EXERCISES. 352. — 1. To construct a quadrilateral when the four sides and one diagonal are given. For example, the side AB, 2 inches; the side BC, 5; CD, 3; DA, 4; and the diagonal AC, 6 inches. 2. To construct a quadrilateral when the four sides and one angle are given. 3. In a quadrilateral, join any point on one side to each end o'^ the side opposite, and with the figure thus constructed demonstrate the theorem, Article 346. 4. The sum of two opposite sides of any quadrilateral which is PARALLELOGRAMS. 123 circumscribed about a circle, is equal to the sum of the ether two sides. 5. If the two oblique sides of a trapezoid be produced till tliey meet, then the point of meeting, the point of intersection of Hit- two diagonals of the trapezoid, and the middle points of the two bases are all in one straight line. PARALLELOGRAMS. 3«>3. A Parallelogram is a quadrilateral which has its opposite sides parallel. 354. Corollary. — Two adjacent angles of a parallelo- gram are supplementary. The angles A and B, being between the parallels AD and BC, and on one side of the secant AB, are supplementary (126). 355. Corollary. — The opposite angles of a parallelo- gram are equal. For both D and B are supplements of the angle C (96). 356. Theorem. — The opposite sides of a parallelogram are equal. For, joining AC by a diagonal, the triangles thus formed have the side AC common ; the angles ACB and DAC equal, for they are alternate (125) ; and ACD and BAC equal, for the same reason. Therefore (285), the tri- angles are equal, and the side AD ] is equal to BC, and AB to CD. 357. Corollary. — When two systems of parallels cross each other, the parts of one system included between two lines of the other are equal. 124: ELEMENTiS OF (JEOMETRY. 358. Corollary. — A diagonal divides a parallelograTii into two equal triangles. But the diagonal does not divide the figure symmetrical!}, because the position of the sides of the triangles is reversed. 339. Theorem — If the opposite sides of a quadri- lateral are equal, the figure is a parallelogram. Join AC. Then, the triangles ABC and CDA are equal. For the side AD is equal to BO, and DC is equal r ^ j to AB, by hypothesis; and p ^ — q they have the side AC com- mon. Therefore, the angles DAC and BCA are equal. But these angles are alternate Avith reference to the lines AD and BC, and the secant AC. Hence, AD and BC are parallel (130), and, for a similar reason, AB and DC are parallel. Therefore, the figure is a paral- lelogram. 360. Theorem. — If, in a quadrilateral, tivo opposite sides are equal and parallel, the figure is a parallelogram. If AD and BC are both equal and parallel, then AB is parallel to DC. For, joining BD, the trian- ^ B gles thus formed are equal, V ....■-'" \ ■since they have the side BD jp^- q common, the side AD equal to BC, and the angle ADB equal to its alternate DBC (284). Hence, the angle ABD is equal to BDC. But these are alternate with reference to the lines AB and DC, and the secant BD. Therefore, AB and DC are parallel, and the figure is a parallelogram. 361. Theorem. — The diagonals of a parallelogram bi- sect each other. PARALLELOGRAMS. 1*25 The diagonals AC and BD are each divided into eqiu.l parts at 11, the point ^ B of intersection. /'^^^X.- For the triangles ^^. " ^^ ABH and CDH have ^ "^ the sides AB and CD equal (356), the angles ABH and CDH equal (125), and the angles BAH and DCH equal. Therefore, the triangles are equal (285), and AH is equal to CH, and BH to DH. 362. Theorem. — If the diagonals of a quadrilateral bisect each other ^ the figure is a parallelogram. To be demonstrated by the student. RECTANGLE 363. If one angle of a parallelogram is right, the others must be right also (354). A Rectangle is a right angled parallelogram. The rectangle has all the properties of other parallelo- grams, and the following peculiar to itself, which the student may demonstrate. 364. Theorem — The diagonals of a rectangle are equal. RHOMBUS. 365. When two adjacent sides of a parallelogram are equal, all its sides must be equal (356). A Rhombus, or, as sometimes ^-"^T^v^ called, a Lozenge, is a parallelo- ^^-^^ \ ^^>^ gram having all its sides equal. ^""^^-v.^^^ j ^^^-^^^ The rhombus has the follow- ing peculiarities, which may be demonstrated by the student. 126 ELEMENTS OF GEOMETRY. 366. Theorem. — The diagonals of a rhombus are per- pendicular to each other. S67. Theorem. — The diagonals of a rhomhui bisect its angles. SQUARE. 368. A Square is a quadrilateral having its sidc3 equal, and its angles right angles. The square may be shown to have ail the properties of the parallelogram (359), of the rectangle, and of the rhombus. 369. Corollary. — The rectangle and the square are the only parallelograms which can be inscribed in a circle (348). EQUALITY. 370. Theorem. — Two parallelograms are equal when two adjacent sides and the included angle in the one, are respectively equal to those parts in the other. For the remaining sides must be equal (356), and this becomes a case of Article 345. 371. Corollary. — Two rectangles are equal when two adjacent sides of the one, are respectively equal to those parts of the other. 373. Corollary — Two squares are equal when a side of the one is equal to a side of the other. APPLICATIONS. 373. The rectangle is the most frequently used of all quadri- laterals. The walls and floors of our apartments, doors and win- dows, books, paper, and many other articles, have this form. Carpenters make an ingenious use of a geometrical principle in order to make their door and window-frames exactly rectangular. Having made the frame, with its sides equal and its ends equal, PARALLELOGRAMS? 127 they measure tlie two diagonals, and make tlie frame take sucli a shape that these also will be equal. In this operation, what principle is applied? 3*74. A rhombus inscribed in a rectangle is the basis of many orna- ments used in arcliitecture and other work. 315. An instrument called parallel rulers, used in drawin* parallel lines, consists of two ^ rulers, connected by cross pieces a -d ^/ith pins in their ends. The rulers may turn upon the pins, varying their distance. The dis- r tances between the pins along C D tlie rulers, that is, AB and CD, must be equal; also, along the cross pieces, that is, AC and BD. Then the rulers will always be parallel to each other. If one ruler be held fast while the other is moved, lines drawn along the edge of the other ruler, at difi'erent positions, will be parallel to each other. What geometrical principles are involved in the use of this instrument? EXERCISES. 3*76. — 1. State the converse of each theorem that has been given in this chapter, and determine whether each of these coU' verse propositions is true. 2. To construct a parallelogram when two adjacent sides and an angle are given. 3. What parts need be given for the construction of a rect- angle? 4. What must be given for the construction of a square? 5. If the four middle points of the sides of any quadrilateral be joined by straight lines, those lines form a parallelogram. 6. If four points be taken, one in each side of a square, at equal distances from the four vertices, the figure formed by join- ing these successive points is a square. 12S ELE^IENTS OF GEOMETRY. 7. Two parallelograms are similar wlien they have an angle in the one equal to an angle in the other, and these equal angles included between proportional sides. MEASURE OF AREA. 377. The standard figure for the measure of surfaces is a square. That is, the unit of area is a square, the side of which is the unit of length, whatever be the ex- tent of the latter. Other figures might be, and sometimes are, used for" this purpose; but the square has been almost univers- ally adopted, because 1. Its form is regular and simple; 2. The two dimensions of the square, its length and breadth, are the same; and, 3. A plane surface can be entirely covered with equal squares. The truth of the first two reasons is already known to the student : that of the last will appear in the fol- lowing theorem. 378. Any side of a polygon may be taken as the The Altitude of a parallelogram is the distance be- tAveen the base and the opposite side. Hence, the alti- tude of a parallelogram may be taken in either of two ways. AREA OF RECTANGLES. 379. Theovem.-^ The area of a rectangle is measured hy the product of its base by its altitude. That is, if Ave multiply the number of units of length contained in the base, by the number of those units MEASURE OF AREA. 129 contained in the altitude, the product is the number of units of area contained in the surface. Suppose that the base AB and the altitude AD are Jiiultiplesof the same unit of length, for example, four and three. Di- vide AB into four equal parts, and through all the points of divi- sion extend lines parallel to AD. Divide AD into three equal parts, and through the points of division extend lines paral- lel to AB. All the intercepted parts of these two sets of parallels must be equal (357) ; and all the angles, right angles (124). Thus, the whole rectangle is divided into equal squares (372). The number of these squares is equal to the number in one row multiplied by the number of rows ; that is, the number of units of length in the base multiplied by the number in the altitude. In the exam- ple taken, this is three times four, or twelve. The result would be the same, whatever the number of divisions in the base and altitude. If the base and altitude have no common measure, then we may assume the unit of length as small as we please. By taking for the unit a less and less part of the altitude, the base will be made the limit of the lines commensurable with the altitude. Thus, the demonstra-r tion is made general. 380. Corollary — The area of a square is expressed by the second power of the length of its side. An- ciently the principles of arithmetic were taught and il- lustrated by geometry, and we still find the word square in common use for the second power of a number. 381. By the method of infinites (203), the latter part of the above demonstration would consist in supposing 130 ELExMExNTS OF GEOMETRY. the base and altitude of the rectangle divided into infi- nitely small and equal parts ; and then proceeding to form infinitesimal squares, as in the former part of the demonstration. If a straight line move in a direction perpendicular to itself, it describes a rectangle, one of whose dimen^ sions is the given line, and the other is the distance which it has moved. Thus, it appears that the two di^ mensions which every surface has (33), are combined in the simplest manner in the rectangle. A rectangle is said to be coritained by its base and altitude. Thus, also, the area of any figure is called its superficial contents. ^ APPLICATION. 3S2. All enlightened nationB attach great importance to exact and uniform standard measures. In this country the standard of length is a yard measure, carefully preserved by the National Government, at Washington City. By it all the yard measures are regulated. The standards generally used for the measure of surface, are the square described upon a yard, a foot, a mile, or some other cer- tain length; but the acre, one of the most common measures of surface, is an exception. The number of feet, yards, or rods in one side of a square acre, can only be expressed by the aid of a radical sign. The public lands belonging to the United States are divided into square townships, each containing thirty-six square luiles, called sections. AREA OF PARALLELOCxHAMS. 383. The area of a parallelogram is measured hy the product of its base hy its altitude. At the ends of the base AB erect perpen€, com- plete the polygon by drawing the trian- gle DBC from its three known sides (295). Suppose the angles not given were D, C, and F. Then, draw the tri- angles «, e, and ^, and separately, tJie triangle u. Then, having the three sides of the triangle o, it may be drawn, and the poly- gon completed. SIMILAR POLYGONS. 431. Theorem. — Similar polygons are composed of the same number of triangles, respectively similar and simi- larly arranged. Since the figures are similar, every angle in one has 148 ELEMENTS OF GEOMETRY. its corresponding equal angle in the other (303). If, then, diagonals be made to divide one of the polygons into triangles, every angle thus formed may have its corresponding equal angle in the other. Therefore, the triangles of one polygon are respectively similar to those of the other, and are similarly arranged. 43!S. Theorem — If two polygons are composed of the same number of triangles which are respectively similar and are similarly arranged, the polygons are similar. By the hypothesis, all the angles formed by the given lines in one polygon have their corresponding equal angles in the other. It remains to be proved that an- gles formed by any other lines in the one have their corresponding equal angles in the other polygon. This may be shown by reasoning, in the same man- ner as in the case of triangles (304). Let the student make the diagrams and complete the demonstration. 433. Theorem. — Two polygons are similar when the angles formed hy the sides are respectively equal, and there is the same ratio between each side of the one and its homologous side of the other. Let all the diagonals possible extend from a vertex A of one polygon, and the same from the homologous ver- tex B of the other polygon. Now the triangles AEI and BCD are similar, because they have two sides proportional, and the included an- gles equal (317). SIMILAR POLYGONS. 149 Therefore, EI : CD : : AI : BD. But, by hypothesis, EI : CD : : 10 : DF. Then (21), AI : BD : : 10 : DF. Also, if we subtract the equal angles EIA and CDB from the equal angles EIO and CDF, the remainders AIO and BDF are equal. Hence, the triangles AIO and BDF are similar. In the same manner, prove that each of the triangles of the first polygon is similar to its corresponding triangle in the other. Therefore, the figures are similar (432). As in the case of equal polygons (422 and 430), it is only necessary to the hypothesis of this proposition, that all the angles except three in one polygon be equal to the homologous angles in the other. 4S4. Theorem. — In similar polygons the ratio of two homologous lines is the same as of any other two homolo- gous lines. For, since the polygons are similar, the triangles which compose them are also similar, and (309), AE : BC : : EI : CD : : AI : BD : : 10 : DF, etc. This common ratio is the linear ratio of the two figures. Let the student show that the perpendicular let fall from E upon OU, and the homologous line in the other polygon, have the linear ratio of the two figures. (50 ELEMENTS OF GEOMET-RY. 435. Theorem — The perimeters of similar polygons are to each other as any two homologous lines. The student may demonstrate this theorem in the same manner as the corresponding propositions in trian- gles (312). 436. Theorem. — The area of any polygo7i is to the area of a similar polygon, as the square on any line of the first is to the square on the homologous line of the second. Let the polygons BCD, etc., and AEI, etc., be divided into triangles by homologous diag- onals. The trian- gles thus formed in the one are similar to those formed in the other (431). Therefore (391), area BCD : area AEI : : BD : AI : : area BDF : area 2 2 2 2 AIO : : BF : AO : : area BFG : area AOU : : BG : AU : : area BGH : area AUY. Selecting from these equal ratios the triangles, area BCD : area AEI : : area BDF : area AIO : : area BFG : area AOU : : area BGH : area AUY. Therefore (23), area BCDFGHB : area AEIOUYA : : 2 2_ area BCD : area AEI ; or, as BC : AE ; or, as the areas of any other homologous parts; or, as the squares of any other homologous lines. 437. Corollary The superficial ratio of two similar polygons is always the second power of their linear ratio. REGULAR POLYGONS. 151 EXERCISES. 438. — L Compose two polygons of the same number of tri- angles respectively similar, but not similarly arranged. 2. To draw a triangle similar to a given triangle, but with double the area. 3. What is the relation between the areas of the equilateral tri- angles described on the three sides of a right angled triangle? REGULAR POLYGONS. 439. A Regular Polygon is one which has all its sides equal, and all its angles equal. The square and the equilateral triangle are regular polygons. 440. Theorem — Within a regular 'polygon there is a point equally distant from the vertices of all the angles. Let ABCD, etc., be a regular polygon, and let lines bisecting the angles A and B extend till they meet at 0. These lines will meet, for the interior angles which they make with AB are both acute (187). In the triaui^le ABO, the angles at A and B are equal, being halves of the equal angles of the polygon. Therefore, the opposite sides AO and BO are equal (275). Join OC. Now, the triangles ABO and BCO are equal, for they have the side AO of the first equal to BO of the second, the side AB equal to BC, because the polygon is regular, and the included angles OAB and OBC equal, since they are halves of angles of the poly- gon. Hence, BO is equal to OC. Then, the angle OCB is equal to OBC (268), and OC 152 ELEMENTS OF GEOMETRY. bisects the angle BCD, which is equal to ABC. In the same manner, it is proved that OC is equal to OD, and so on. Therefore, the point is equally distant from all the vertices. CIRCUMSCRIBEB AND INSCRIBED. 441. Corollary — Every regular polygon may have a circle circumscribed about it. For, with as a center and OA as a radius, a circumference may be described passing through all the vertices of the polygon (153). 442. Theorem — The point which is equally distant from the vertices is also equally distant from the sides of a regular polygon. The triangles OAB, OBC, etc., are all isosceles. If perpendiculars be let fall from upon the several sides AB, BC, ^ 5 — ; — S etc., these sides will be bisected A/< \ \ ! / ^^D (271). Then, the perpendiculars / '\\\;/ \ will be equal, for they will be "''4'' ^ sides of equal triangles. But they measure the distances from to the several sides of the polygon. Therefore, the point is equally dis- tant from all the sides of the polygon. 443. Corollary. — Every regular polygon may have a circle inscribed in it. For with as a center and OG as a radius, a circumference may be described passing through the feet of all these perpendiculars, and tangent to all the sides of the polygon (178) , and therefore in- scribed in it (253). 444. Corollary. — A regular polygon is a symmetrical figure. 445. The center of the circumscribed or inscribed cir- cle is also called the center of a regidar polygon. The REGULAR POLYGONS. 153 radius of the circumscribed circle is also called the radius of a regular polygon. The Apothem of a regular polygon is the radius of the inscribed circle. 446. Theorem — If the circumference of a circle he di- vided into equal arcs, the chords of those equal arcs will he the sides of a regular polygon. For the sides are all equal, being the chords of equal arcs (185); and the angles are all equal, being inscribed in equal arcs (224). 447. Corollary — An angle formed at the center of a regular polygon by lines from adjacent vertices, is an aliquot part of four right angles, being the quotient of four right angles divided by the number of the sides of the polygon. 448. Theorem — If a circumference he divided into equal arcs, and lines tangent at the several points of divi- sion he produced until they meet, these tangents are the sides of a regular polygon. Let A, B, C, etc., be points of division, and F, D, and E points where the tangents meet. Join GA, AB, and BC. Now, the triangles GAF, ABD, and BCE have the sides GA, AB, and BC equal, as they are chords of equal arcs; and the angles at G, A, B, and C equal, for each is formed by a tangent and chord which inter- cept equal arcs (226). Therefore, these triangles are all isosceles (275), and all equal (285); and the angles F, D, and E are equal. Also, FD and DE, being 154 ELEMENTS OF GEOMETRY. doubles of equals, are equal. In the same manner, it is proved that all the angles of the polygon FDE, etc., are equal, and that all its sides are equal. Therefore, it is a regular polygon. REGULAR POLYGONS SIMILAR. 449. Theorem. — Regular polygons of the same number of sides are similar. Since the polygons have the same number of sides, the sum of all the angles of the one is equal to the sum of all the angles of the other (423). But all the angles of a regular polygon are equal (439). Dividing the equal sums by the number of angles (7), it follows that an angle of the one polygon is equal to an angle of the other. Again : all the sides of a regular polygon are equal. Hence, there is the same ratio between a side of the first and a side of the second, as between any other side of the first and a corresponding side of the second. Therefore, the polygons are similar (433). 450. Corollary — The areas of two regular polygons of the same number of sides are to each other as the squares of their homologous lines (436). 451. Corollary. — The ratio of the radius to the side of a regular polygon of a given number of sides, is a constant quantity. For a radius of one is to a radius of any other, as a side of the one is to a side of the other (434). Then, by alternation (19), the radius is to the side of one regular polygon, as the radius is to the side of any other regular polygon of the same number of sides. 452. Corollary. — The same is true of the apothem and side, or of the apothem and radius. REGULAR POLYGONS. 155 PROBLEMS IN DRAWING. 453. Problem. — To inscribe a square in a given circle. Draw two diameters perpendicular to each other. Join their extremities by chords. These chords form an inscribed square. For the angles at the center are equal by construction (90). Therefore, their intercepted arcs are equal (197), and the chords of those arcs are the sides of a regular polygon (446). 454. Problem. — To inscribe a regular hexagon in a circle. Suppose the problem solved and the figure completed. Join two adjacent angles with the center, making the triangle ABC. Now, the angle C, being measured by one-sixth of the circumference, is equal to one-sixth of four right an- gles, or one-third of two right an- gles. Hence, the sum of the two angles, CAB and CBA, is two-thirds of two right angles (256). But CA and CB are equal, being radii; there- fore, the angles CAB and CBA are equal (268), and each of them must be one-third of two right angles. Then, the triangle ABC, being equiangular, is equilateral (276). Therefore, the side of an inscribed regular hexagon is equal to the radius of the circle. The solution of the problem is now evident — apply the radius to the circumference six times as a chord. 455. Corollary. — Joining the alternate vertices makes an in- scribed equilateral triangle. 456. Problem. — To inscribe a regular decagon in a given circle. Divide the radius CA in extreme and mean ratio, at the point B. \334) BC is equal to the side of a regular inscribed decagon. That is, if we apply BC as a chord, its arc will be one-tenth of the whole circumference. Take AD, making the chord AD equal to BC. Then join DC and DB. Then, by construction, CA : CB : : CB : BA. 15G ELEMENTS OF GEOMETRY. Substituting for CB its equal DA, CA : DA :: DA* BA. Then the triangles CDA and BDA are similar, for they have those sides proportional which include tlie common angle A (317). But the triangle CDA being isosceles, the tri- ano-le BDA is the same. Hence, DB is equal to DA, and also to BC. \ ^ B j Therefore, the angle C is equal to the angle BDC (26S). But it is also equal to BDA. It follows that the angle CDA is twice the angle C. The angle at A being equal to CDA, the angle C must be one-fifth of the sum of these three angles; that is, one-fifth of two right angles (255), or one-tenth of four right angles. Therefore, the arc AD is one-tenth of the circum- ference (207); and the chord AD is equal to the side of an in- scribed regular decagon. 45T, — Corollary, — By joining the alternate vertices of a deca- gon, we may inscribe a regular pentagon. 458. Corollary. — A regular pentedecagon, or polygon of fifteen sides, may be inscribed, by subtracting the arc subtended by the side of a regular decagon from the arc subtended by the side of a regular hexagon. The remainder is one-fifteenth of the circum- ference, for ^ — to = tV 459. Problem. — Given a regular polygon inscribed in a circle^ to inscribe a regular polygon of double the num- ber of sides. Divide eich arc subtended by a given side into two equal parts (194). Join the successive points into which the circumference is divided. Thj figure thus formed is the required polygon. 480. We have now learned how to inscribe regular polygons of 3, 4, 5, and 15 sides, and of any number that may arise from doubling either of these four. The problem, to inscribe a regular pol3'gon in a circle by means of straight lines and arcs of circles, can be solved in only a limited number of cases. It is evident that the solution depends upon the division of the circumference into any number of equal parts; and this depends upon the division of the sum of four right angles into aliquot parts. REGULAR POLYGONS. 157 461. Notice that the regular decagon was drawn by the aid of two isosceles triangles composing a third, one of the two being simi- lar to the whole. Now, if we could combine three isosceles triangles in this manner, we could draw a regu- lar polygon of fourteen, and then one of seven sides. However, this can not be done by means only of straight lines and arcs of circles. The regular polygon of seventeen sides has been drawn in more tlian one way, using only straight lines and arcs of circles. It has also been shown, that by the same means a regular polygon of two hundred and fifty-seven sides may be drawn. No others are known where the number of the sides is a prime number. 403. Problem — Given a regular polygon inscribed in a circle, to circumscribe a similar polygon. The vertices of the given polygon divide the circumference into equal parts. Through these points draw tangents. These tan- gents produced till they meet, form the required polygon (448), EXERCISES. 463. — 1. First in right angles, and then in degrees, express the value of an angle of each regular polygon, from three sides up to twenty. 2. First in right angles, and then in degrees, express the value of an angle at the center, subtended by one side of each of the same polygons. 3. To construct a regular octagon of a given side. 4. To circumscribe a circle about a regular polygon. 5. To inscribe a circle in a regular polygon. 6. Given a regular inscribed polygon, to circumscribe a similar polygon whose sides are parallel to the former. 7. The diagonal of a square is to its side as the square root of 2 is to 1, 1 56 ELEMENTS OF GEOMETRY. A PLANE OF REGULAR POLYGONS. 461. In order that any plane surface may be entirely covered by equal polygons, it is necessary that the fig- ures be such, and such only, that the sum of three or more of their angles is equal to four right angles (92). Hence, to find what regular polygons will fit together so as to cover any plane surface, take them in order according to the number of their sides. Each angle of an equilateral triangle is equal to one-third of two right an- gles. Therefore, six such angles ex- actly make up four right angles; and the equilateral triangle is such a fig- ure as is required. 465. Each angle of the square is a right angle, four of which make four right angles. So that a plane can be covered by equal squares. One angle of a regular pentagon is the fifth part of six right angles. Three of these are less than, and four exceed four right angles; so that the regular pentagon is not such a figure as is required. 466. Each angle of a regular hexagon is one-sixth of eight right angles. Three such make up four right angles. Hence, a plane may be covered with equal regular hexa- gons. This combination is remarkable as being the one adopted by bees in form- ing the honeycomb. 4G7. Since each angle of a regular polygon evi- dently increases when the number of sides increases, and since three angles of a regular hexagon are equal ISOPEKIMETRY. 159 to i(/ r right angles, therefore, three angles of any reg- ular polygon of more than six sides, must exceed four right angles. Hence, no other regular figures exist for the purpose here required, except the equilateral triangle, the square, and the reguliir hsxagon. ISOPEKIMETKY. 46S. Theorem — Of all equivalent polygons of the same number of sides, the one having the least perimeter is reg- ular. Of several equivalent polygons, suppose AB and BO to be two adjacent sides of the one having the least .^J^-- perimeter. It is to be k,^*'^'^^ ^^^^r^ proved, first, that these / A sides are equal. / \ Join AC. Now, if AB / \ and BC were not equal, there could be constructed on the base AC an isosceles triangle equivalent to ABC, whose sides would have less extent (395). Then, this new triangle, with the rest of the polygon, would be equivalent to the given polygon, and have a less perimeter, which is contrary to the hypothesis. It follows that AB and BC must be equal. So of every two adjacent sides. Therefore, the polygon is c({uilateral. It remains to be proved that the polygon will have all its angles equal. Suppose AB, BC, and CD to be adjacent sides. Produce AB and CD till they meet at E. Now the triangle BCE is isosceles. For if EC, for example, were 160 ELEMENTS OF GEOMETRY. longer than EB, we could then take EI equal to EB, and EF equal to EC, and we could join FI, making the ^S two triangles EBC and EIF equal (284). Then, the new polygon, having AFID for part of its perimeter, would be equivalent and isoperimetrical to the given polygon hav- ing ABCD as part of its perimeter. But the given polygon has, by hypothesis, the least possible perimeter, and, as just proved, its sides AB, BC, and CD are equal. If the new polygon has the same area and perime- ter, its sides also, for the same reason, must be equal ; that is, AF, FI, and ID. But this is absurd, for AF is less than AB, and ID is greater than CD. Therefore, the supposition that EC is greater than EB, which sup- position led to this conclusion, is false. Hence, EB and EC must be equal. Therefore, the angles EBC and ECB are equal (268), and their supplements ABC and BCD are equal. Thus, it may be shown that every two adjacent angles are equal. It being proved that the polygon has its sides equal and its angles equal, it is regular. 469. Corollary. — Of all isoperimetrical polygons of the same number of sides, that which is regular has the greatest area. 470. Theorem. — Of all regular equivalent polygons^ that which has the greatest number of sid-es has the least perimeter. It will be sufficient to demonstrate the principle, when one of the equivalent polygons has one side more than ihe other. ISOPERIMETRY. 161 In the polygon having the less number of sides, join the vertex C to any point, as H, of the side BG. Then, 01 CH construct an isosceles triangle, CKH, equivalent to CBH. Then HK and KC are less than HB and BC ; there- fore, the perimeter GHKCDF is less than the perimeter of its equivalent polygon GBCDF. But the perimeter of the regular polygon AO is less than the perimeter of its equivalent irregular polygon of the same number of sides, GHKCDF (468). So much more is it less than the perimeter of GBCDF. 471. Corollary. — Of two regular isoperimetrical poly- gons, the greater is that which has the greater number of sides. EXERCISES. 472. — 1. Find the ratios between the side, the radius, and the {\pothem, of the regular polygons of three, four, five, six, and eiglit sides, 2. If from any point within a given regular polygon, perpen- diculars be let fall on all the sides, the sum of these perpendicu- lars is a constant quantity. 3. If from all the vertices of a regular polygon, perpendiculars be let fall on a straight line which passes through its center, the Geom. — 14 162 ELEMENTS OF GEOMETRY. sum of the perpendiculars on one side of this line is equal to the sum of those on the other. 4. If a regular pentagon, hexagon, and decagon be inscribed in a circle, a triangle having its sides respectively equal to the sides of these three polygons will be right angled. 5. If two diagonals of a regular pentagon cut each other, each is divided in extreme and mean ratio. 6. Three houses are built with walls of the same aggregate length; the first in the shape of a square, the second of a rectan- gle, and the third of a regular octagon. Which has the greatest amount of room, and which the least? 7. Of all triangles having two sides respectively equal to two given lines, the greatest is that where the angle included between the given sides is a right angle. 8. In order to cover a pavement with equal blocks, in the shape of regular polygons of a given area, of what shape must they be that the entire extent of the lines between the blocks shall be a minimum. 9. All the diagonals being formed in a regular pentagon, the figure inclosed by them is a regular pentagon. CIRCLES. 133 CHAPTER VIII. CIRCLES. 473. The properties of the curve which bounds a circle, and of some straight lines connected with it, were discussed in a former chapter. Having now learned the properties of polygons, or rectilinear figures inclos- ing a plane surface, the student is prepared for the study of the circle as a figure inclosing a surface. The circle is the only curvilinear figure treated of in Elementary Geometry. Its discussion will complete this portion of the work. The properties of other curves, such as the ellipse which is the figure of the orbits of the planets, are usually investigated by the application of algebra to geometry. 474. A Segment of a circle is that portion cut oif by a secant or a chord. Thus, ABC and CDE are seg- ments. D A Sector of a circle is that portion included between two radii and the arc intercepted by them. Thus, GHI is a sector. 164 ELEMENTS OF C.KOMETRY. THE LIMIT OF INSCRIBED POLYGONS. 47*5. Theorem — A circle is the limit of the polygons which can be inscribed in it, also of those which can be circumscribed about it. Having a polygon inscribed in a circle, a second poly- gon may be inscribed of double the number of sides. Then, a polygon of double the number of sides of the second may be inscribed, and the process repeated at will. Let the student draw a diagram, beginning with an inscribed square or equilateral triangle. Very soon the many sides of the polygon become confused with the circumference. Suppose we begin with a circumscribed regular polygon; here, also, we may circumscribe a regular polygon of double the number of sides. By repeating the process a few times, the polygon become?: inseparable from the circumference. The mental process is not subject to the same limits that we meet with in drawing the diagrams. We may conceive the number of sides to go on increasing to any number whatever. At each step the inscribed polygon grows larger and the circumscribed grows smaller, both becoming more nearly identical with the circle. Now, it is evident that by the process described, the polygons can be made to approach as nearly as we please to equality with the circle (35 and 36), but can never en- tirely reach it. The circle is therefore the limit of the polygons (198). 476. Corollary. — A circl'e is the limit of all regular polygons whose radii are equal to its radius. It is also the limit of all regular polygons whose apothems are equal to its radius. The circumference is the limit of the perimeters of those polygons. CIRCIES SIMILAR. 165 4T7. By the method of iniinites, the circle is consid- ered as a regular polygon of an infinite number of sides, each side being an infinitesimal straight line. But the method of limits is preferred in this place, be- cause, strictly speaking, the circle is not a polygon, and the circumference is not a broken line. The above theorem establishes only this, that whatever is true of all inscribed, or of all circumscribed polygons, is necessarily true of the circle. 478. Theorem. — A curve is shorter than any other line which joins its ends, and toward which it is convex. For the curve BDC is the limit of those broken lines which have their vertices in it. Then, the curve BDC is less than the line BFC (79). 479. Corollary — The circumference of a circle is shorter than the perimeter of a circumscribed polygon. 480. Corollary. — The circumference of a circle is longer than the perimeter of an inscribed polygon. This is a corollary of the Axiom of Distance (54). 481. Theorem — A circle has a less perimeter than any equivalent polygoyi. For, of equivalent polygons, that has the least perim- eter which is regular (468), and has the greatest number of sides (470). 482. Corollary. — A circle has a greater area than any isoperimetrical figure. CIRCLES SIMILAR. 48S. Theorem — Circles are similar figures. 1 For angles which intercept like parts of a circumfer- ence are equal (207 and 224). Hence, whatever lines i6() ELEMENTS OF GEOMETRY. be made in one circle, homologous lines, making cqu^il angles, may be made in another. This theorem may be otherwise demonstrated, thus: Inscribed regular polygons of the same number of sides are similar. The number of sides may be increased indefinitely, and the polygons will still be similar at eacli successive step. The circles being the limits of the polygons, must also be similar. 484. Theorem. — Ttvo sedors are similar when tlie an- gles made hy their radii are equal. 485. Theorem — Two segments are similar when the angles which are formed hy radii from the ends of their respective arcs are equal. These two theorems are demonstrated by completing the circles of which the given figures form parts. Then the given straight lines in one circle are homologous to those in the other: and any angle in one may have its corresponding equal angle in the other, since the circles are similar. EXERCISE. 486. When the Tyrian Princess stretched the thongs cut from the hide of a bull around the site of Carthage, what course should she have pursued in order to include the greatest extent of terri- tory ? RECTIFICATION OF CIRCUMFERENCE. 487. Theorem. — The ratio of the circumference to its diameter is a constant qiiantity. Two circumferences are to each other in the ratio of their diameters. For the perimeters of similar regular polygons are in the ratio of homologous lines (435); and the circumference is the limit of the perimeters of RECTIFICATION OF CIRCUMFERENCE. 167 regular polygons (476). Then, designating any two circumferences by C and C, and their diameters by D and W, C : C' : : D : D'. Hence, by alternation, C : D : : C : D^ That is, the ratio of a circumference to its diametei is the same as that of any other circumference to its diameter. 488. The ratio of the circumference to the diameter is usually designated by the Greek letter ;r, the initial of 'perimeter. If we can determine this numerical ratio, multiplying any diameter by it will give the circumference, or a straight line of the same extent as the circumference. This is called the rectification of that curve. 489. The number t: is less than 4 and greater than 3. For, if the diameter is 1, the perimeter of the cir- cumscribed square is 4; but this is gi*eater than the circumference (479). And the perimeter of the in- scribed regular hexagon is 3, but this is less than the circumference (480). In order to calculate this number more accurately, let Us first establish these two principles : 400. Theorem. — Given the apothem, radius, and side of a regular polygon ; the apothem of a regular polygon of the same length of perimeter, hut double the number of sides, is half the sum of the given apothem and radius; and the radius of the polygon of double the number of sides, is a mean proportional between its own apothem and the given radius. Let CD be the apothem, CB the radius, and BE the side of a regular polygon. Produce DC to F, making 168 ELEMENTS OF GEOMETRY OF equal to CB. Join BF and EF. From C let the perpendicular CG fall upon BF. Make GH parallel to BE, and join CH and CE. Now, the triangle BCF being isos- celes by construction, the angles CBF and CFB are equal. The sum of these two is equal to the exterior an- gle BCD (261). Hence, the angle BFD is half the angle BCD. Since \ \ i DF is, by hypothesis, perpendicular \ | / to BE at its center, BCE and BFE \j/ are isosceles triangles (108), and the |, angles BCE and BFE are bisected by the line DF (271). Therefore, the angle BFE is half the angle BCE. That is, the angle BFE is equal to the angle at the center of a regular polygon of double the number of sides of the given polygon (447). Since GH is parallel to BE, We have, GH : BE : : GF : BF. Since GF is the half of BF (271), GH is the half of BE. Then GH is equal to the side of a regular poly- gon, with the same length of perimeter as the given polygon, and double the number of sides. Again, FH and FG, being halves of equals, are equal. Also, IF is perpendicular to GH (127). Therefore, we have GH the side, IF the apothem, and GF the radius of the polygon of double the number of sides, with a perimeter equal to that of the given polygon. Now, the similar triangles give, FI : FD : : FG : FB. Therefore, FI is one-half of FD. But FD is, by con- struction, equal to the sum of CD and CB. Therefore, RECTIFICATION OF CIRCUMFERENCE. l(jZawe can have only one common point, unless the line lies wholly in the plane. This is a corollary of Article 58. »I15^ When a line and a plane have only one common point, the line is said to pierce the plane, and the plane CO cut the line. The common point is called the foot of the line in the plane. When a line lies wholly in a plane, the plane is said to pass through the line. PLANE AND LINES, 179 516. Theorem. — The iyiter section of two planes is a straight line. For two planes can not have three points common, unless those points are all in one straight line (59). PEEPENDICULAR LINES. 517. Theorem. — A straight line which is perpendicular to each of two straight lines at their point of intersection, is perpendicular to every other straight line tvhich lies in the plane of the two, and passes through their point of intersection. In the diagram, suppose D, B, and C to be on the plane of the paper, the point A being above, and I below that plane. If the line AB is perpendicu- lar to BC and to BD, it is also perpendicular to every other line lying in the plane of DBC, and passing through the point B ; as, for example, BE. Produce AB, making BI equal to BA, and let any line, as FII, cut the lines BC, BE, and BD, in F, G, and H. Then join AF, AG, AH, and IF, IG, and IH. Now, since BC and BD arc perpendicular to AI at its center, the triangles AFII and IFH have AF equal to IF (108), All equal to III, and FH common. There- fore, they are equal, and the angle AHF is equal to IHF. Then the triangles AHG and IHG are equal (284), and the lines AG and IG are equal. Therefore, the line BG, having two points each equally distant from A and I, is perpendicular to the line AI at its center B (109). ISO ELEMENTS OF GEOMETRY In the same way, prove that any other line through B, in the plane of DBC, is perpendicular to AB. 518. Theorem Conversely, if several straight lines are each perpendicular to a given line at the same point, then these several luus all lie in one plane. Thus, if BA is perpendicular to BC, to BD, and tc BE, then these three all lie in one plane. BD, for instance, must be in the plane CBE. For the intersection of the plane of ABD with the plane of CBE is a straight line (516). This straight intersection is per- pendicular to AB at the point B (517). Therefore, it coincides with BD (103). Thus it may be shown that any other line, perpendicular to AB at the point B, is in the pl?.ne of C, B, D, and E. 510. A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line which passes through its foot in that plane, and the plane is said to be perpendicular to the line. Every line not perpendicular to a plane which cuts it, is called oblique. 520. Corollary. — If a plane cuts a line perpendicu- larly at the middle point of the line, then every point of the plane is equally distant from the two ends of the line (108). 531. Corollary. — If one of two perpendicular lines revolves about the other, the revolving line describes a plane which is perpendicular to the axis. f%22. Corollary Through one point of a straight line there can be only one plane perpendicular to that line. PLANE AND LINES. 181 c M \ \ D B ""E 523. Theorem. — Through a jjoint out of a plane there can he only one straight line perpendicular to the plane. For, if there could be two perpendiculars, then each would be perpendicular to the line in the plane which joins their feet (519). But this is impossible (103). •524. Theorem. — Through a point in a plane there con he only one straight line perpendicular to the plane. Let BA be perpendicular to the plane MN at the point B. Then any other line, BC for example, will be oblique to the plane MN. For, if the plane of ABC be produced, its intersection with the plane MN will be a straight line. Let DE be this intersection. Then AB is perpen- dicular to DE. Hence, BC, being in the plane of A, D, and E, is not perpendicular to DE (103). Therefore, it is not perpendicular to the plane MN (519). 525. Corollary — The direction of a straight line in space is fixed by the fact that it is perpendicular to a given plane. The directions of a plane are fixed by the fact that it is perpendicular to a given line. 526. Corollary — All straight lines which are perpen- dicular to the same plane, have the same direction ; that is, they are parallel to each other. 527. Corollary. — If one of two parallel lines is per- pendicular to a plane, the other is also. 528. The Axis of a circle is the straight line perpen- dicular to the plane of the circle at its center. 182 ELEMENTS OF GEOMETRY. M V B OBLIQUE LINES AND PLANES. 529. Theorem. — If from, a point without a plane, a perpendicular and oblique lines be extended to the plane, then two oblique lines which meet the pla^ie at equal dis^ tances from the foot of the perpendicular, are equal. Let AB be perpendicular, and AC and AD oblique to the plane MN, and the dis- tances BC and BD equal. Then the triangles ABC and ABD are equal (284), and AC is equal to AD. 530. Corollary. — A perpendicular is the shortest line from a point to a plane. Hence, the distance from a point to a plane is measured by a perpendicular line. 531. Corollary. — All points of the circumference of 1 circle are equidistant from any point of its axis. S33. If from all points of a line perpendiculars be let fall upon a plane, the line thus described upon the plane is the projection of the given line upon the given plane. 533. Theorem. — The projection of a straight line upon a plane is a straight line. Let AB be the given line, and MN the given plane. Then, from the points A and B, let the perpen- diculars, AC and BD, fall upon the plane MN. Join CD. AC and BD, being per- pendicular to the same plane, are parallel (526), and lie in cne plane (121), Now, every perpendicular PLANE AND LINES. 183 to MN let fall from a point of AB, must be parallel to BD, and must therefore lie in the plane AD, and meet the plane MN in some point of CD. Hence, the straight line CD is the projection of the straight line AB on the plane MN. There is one exception to this proposition. When the given line is perpendicular to the plane, its projection is a point. 5S4. Corollary. — A straight line and its projection on a plane, both lie in one plane. 535. Theorem — The angle which a straight line makes with its projection on a plane, is smaller than the angle it makes with any other line in the plane. Let AC be the given line, and BC its projection on the plane MN. Then the angle ACB is less than the angle made by AC with any other line in the plane, as CD. With C as a center and BC as a radius, de- scribe a circumference in the plane MN, cut- ting CD at D. Then the triangles ACD and ACB have two sides of che one respectively equal to two sides of the other. But the third side AD is longer than the third side AB (530). Therefore, the angle ACD is greater than the single ACB (294). 536. Corollary — The angle ACE, which a line makes with its projection produced, is larger than the angle made with any other line in the plane. » 537. The angle which a line makes Avith its projec 184 ELEMENTS OF GEOMETRY. tion in a plane, is called the Angle of Inclination of the line and the plane. PARALLEL LINES AND PLANE. 538. Theorem. — If a straight line in a plane is paral- lel to a straight line not in the plane, then the second line and the plane can not have a common point. For if any line is parallel to a given line in a plane, and passes through any point of the plane, it will lie wholly in the plane (121). But, by hypothesis, the sec- ond line does not lie wholly in the plane. Therefore, it can not pass through any point of the plane, to what- ever extent the two may be produced. 539. Such a line and plane, having the same direc- tion, are called parallel. 5i:0. Corollary — If one of two parallel lines is par- allel to a plane, the other is also. 5irl. Corollary — A line which is parallel to a plane is parallel to its projection on that plane. fills. Corollary. — A line parallel to a plane is every- where equally distant from it. APPLICATIONS. 543. Three points, however placed, must always be in the same plane. It is on this principle that stability is more readily obtained by three supports than by a greater number. A three- legged stool must be steady, but if there be four legs, their ends should he in (jne plane, and the floor should be level. Many sur- veying and astronomical instruments are made with three legs. 544. The use of lines perpendicular to planes is very frequent in the mechanic arts. A ready way of constructing a line perpen- dicular to a plane is by the use of two squares (114). Place the angle of each at the foot of the desired perpendicular, one side o^ DlEDllAL ANGLES. 185 each square resting on the plane surface. Bring their perpendic- ular sides together. Their position must tlien be that of a per- pendicular to the plane, for it is perp(!ndicular to two lines in the plane. 545. When a circle revolves round its axis, the figure under- goes no real change of position, each point of the circumference taking successively the position deserted by another point. On this principle is founded the operation of millstones. Two circular stones are placed so as to have tlie same axis, to which their faces are perpendicular, being, therefore, parallel to each other. The lower stone is fixed, while the upper one is made to revolve. The relative position of the faces of the stones under- goes no change during the revolution, and their distance being properly regulated, all the grain which passes between them will be ground with the same degree of fineness. 546. In the turning lathe, the axis round which the body to be turned is made to revolve, is the axis of the circles formed by the cutting tool, which removes the matter projecting beyond a proper distance from the axis. The cross section of every part of the thing turned is a circle, all the circles having the same axis. DIEDRAL ANGLES. 547. A DiEDRAL Angle is formed by two planes meeting at a common line. This figure is also called simply a diedral. The planes are its faces^ and the in- tersection is its edge. In naming a diedral, four letters are used, one in each fr.ce, and two on the edge, the letters on the edge being between the other tAvo. This figure is called a diedral angle^ because it is simi- lar in many respects to an angle formed by two lines. MEASURE OF DIEDRALS. 548. The quantity of a diedral, as is the case with a linear angle, depends on the difference in the directions Oconi. — If) 186 ELEMENTS OF GEOMETRY of the faces from the edge, without regard to the extent of the planes. Hence, two diedrals are equal when they can be so placed that their planes will coincide. 519. Problem. — One diedral may he added to another. In the diagram, AB, AC, and AD represent three planes having the common intersection AE. Evidently the sum of BEAC and OEAD IS equal to BEAD. 550. Corollary Diedrals may be subtracted one from another. A diedral may be bisected or divided in any required ratio by a plane pass- ing through its edge. 551. But there are in each of these planes any num- ber of directions. Hence, it is necessary to determine which of these is properly the direction of the face from the edge. For this purpose, let us first establish the following principle: 552. Theorem. — One diedral is to another as the plane angle, formed in the first hy a line in each face perpen- dicular to the edge, is to the similarly formed angle in the other. Thus, if FO, GO, and HO are each perpendicu- lar to AE, then the die- dral CEAD is to the die- dral BEAD as the angle GO 11 is to the angle FOR. This may be de- monstrated in the same manner as the proposi- tion in Article 197. DIEDRAL ANGLES. 187 553. Corollary. — A diedral is said to be measured by the plane angle formed by a line in each of its faces perpendicular to the edge. 554. Corollary. — Accordingly, a diedral angle may be acute, obtuse, or right. In the last case, the planes are perpendicular to each other. 555. Many of the principles of plane angles may be applied to diedrals, without further demonstration. All right diedral angles are equal (90). When the sum of several diedrals is measured by two right angles, the outer faces form one plane (100). When two planes cut each other, the opposite or ver- tical diedrals are equal (99). PERPENDICULAR PLANES. 556. Theorem — If a line is perpendicular to a plane^ then any plane passing through this line is perpendicular to the other plane. If AB in the plane PQ is perpendicular to the plane MN, then AB must be perpen- dicular to every line in MN which passes through the point B (519); that is, to RQ, the intersection of the two planes, and to BC, which is made perpendicular to the in- tersection RQ. Then, the an- gle ABC measures the inclina- tion of the two planes (553), and is a right angle. There- fore, the planes are perpendicular. 557. Corollary. — Conversely, if a plane is perpen- dicular to another, a straight line, which is perpendicu- p ^ V M R .^ c'"-"^^ Q N 188 ELEMENTS OF GEOMETRY lar to one of them, at some point of their intersection, must lie wholly in the other phine (524). *%^3, Corollary — If two planes are perpendicular to a third, then the intersection of the first two is a line perpendicular to the third plane. OBLIQUE PLANES. 559. Theorem — If from a point within a diedral, per- pendicular lines be made to the two faces, the angle of these lines is supplementary to the angle which measures the diedral. Let M and N be two planes whose intersection is AB, and CF and CE perpendicu- lars let fall upon them from the point C; and DF and DE the intersections of the plane FOE with the two planes M and N. Then the plane FCE mist be perpendicular to each of the planes M and N (556). Hence, the line AB is perpendicular to the plane FCE (558), and the angles ADF and ADE are right angles. Then the angle FDE measures the diedral. But in the quadrilateral FDEC, the two angles F and E are right angles. Therefore, the other two angles at C and D are supplementary. 560. Theorem — Every point of a plane which bisects a diedral is equally distajit from its two faces. Let the plane FC bisect the diedral DBCE. Then it is to be proved that every point of this plane, as A, for example, is equally distant from the planes DC and EC. From A let the perpendiculars AH and AI fall upon the faces DC and EC, and let 10, AO, and IIO be the DIEDKAL ANGLES. 189 intersections of the plane of the angle lAH Avith the three given planes. Then it may be shown, as in the last theorem, that the angle HOA measures the diedral FBCD, and . the angle lOA the y^ diedral FBCE. But z^^^^nT/^ / these diedrals are d^ -/ ^<^ • L equal, by hypothesis. ^v H^^ Ax^f Z^ Therefore, the line AG x^^Axo'"'^ y^ bisects the angle lOH, B E and the point A is equally distant from the lines OH and 01 (113). But the distance of A from these lines is measured by the same perpendiculars, AH and AI, which measure its distance from the two faces DC and EC. Therefore, any point of the bisecting plane is equally distant from the two faces of the given diedral. APPLICATIONS. 561. Articles 548 to 554 are illustrated by a door turning on its hinges. In every position it is perpendicular to the floor and ceiling. As it turns, it changes its inclination to the wall, in which it is constructed, the angle of inclination being that which is formed by the upper edge of the door and the lintel. 562. The theory of diedrals is as important in the study of magnitudes bounded by planes, as is the theory of angles in the study of polygons. This is most striking in the science of crystallography, which teaches us how to classify mineral substances according to their geometrical forms. Crystals of one kind have edges of which the diedral angles measure a certain number of degrees, and crystals of another kind have edges of a diflTerent number of degrees. Crystals of many species may be thus classified, by measuring their diedrals. 563. The i)laiie of the surface of a liquid at rest is called hori- zontal, or the plane of the horizon. The direction of a phnnb- J 90 ELEMENTS OF GEOMETRY. line when the weight is at rest, is a vertical line. The vertical line is perpendicular to the horizon, the positions of both being governed by the same causes. Every line in the plane of the horizon, or parallel to it, is called a horizontal line, and every [)lane passing through a vertical line is called a vertical plane. Every vertical plane is perpendicular to the horizon. Horizontal and vertical planes are in most frequent use. Floors, ceilings, etc., are examples of the former, and walls of the latter. The methods of using the builder's level and plummet to determ- ine the position of these, are among the simplest applications of* geometrical principles. Civil engineers have constantly to observe and calculate the [>osition of horizontal and vertical planes, as all objects are re- ferred to these. The astronomer and the navigator, at every step, refer to the horizon, or to a vertical plane. EXERCISES. 564. — 1. If, from a point without a plane, several equal oblique lines extend to it, they make equal angles with the plane. 2. If a line is perpendicular to a plane, and if from its foot a perpendicular be let fall on some other line which lies in the plane, then this last line is perpendicular to the plane of the other two. 3. What is the locus of those points in space, each of which is equally distant from two given points? PARALLEL PLANES. 565. Two planes which are perpendicular to the same straight line, at different points of it, are both fixed in po- sition (525), and they have the same directions. If the parallel lines AB and CD revolve about the line EF, to which they are both perpendicular, then each of the revolving lines describes a plane. Every direction assumed by one line is the same ns PARALLEL PLANES. 191 that of the other, and, in the course of a complete revo- lution, they take all the possible directions of the two planes. Two planes which have the same directions are called jKiraUel planes. Parallelism consists in having the same direction, whether it be of two lines, of two planes, or of a line and a plane. 560. Corollary. — Two planes parallel to a third are parallel to each other. 567. Corollary. — Two planes perpendicular to the same straight line are parallel to each other. 568. Corollary. — A straight line perpendicular to one of two parallel planes is perpendicular to the other. 569. Corollary. — Every straight line in one of two parallel planes has its parallel line in the other plane. Therefore, every straight line in one of the planes is parallel to the other plane. 570. Corollary. — Since through any point in a plane there may be a line parallel to any line in the same plane (121), therefore, in one of two parallel planes, and through any point of it, there may be a straight line parallel to any straight line in the other plane. 571. Theorem. — Two parallel planes can not meet. For, if they had a common point, being parallel, they would have the same directions from that point, and therefore would coincide throughout, and be only one plane. 572. Theorem — The intersections of two parallel planes hy a third plane are parallel lifies. Let AB and CD be the intersections of the two par- allel planes M and N, with the plane P. Now, if through C there be a line parallel to AB, it 192 ELEMENTS OF GEOMETRY. M/ A 1 B C 1 N/ N / must lie in the plane P (121), and also in the plane N (570). Therefore, it is the in- tersection CD, and the two in- tersections are parallel lines. When two parallel planes are cut by a third plane, eight diedrals are formed, which have properties similar to those of Articles 124 to 128. 573. Theorem — The parts of two parallel lines inter- cepted between parallel planes are equal. For, if the lines AB and CD are parallel, they lie in one plane. Then AC and BD are the intersections of this plane with the two parallel planes M and P. Hence, AC is parallel to BD, and AD is a parallelogram. Therefore, AB is equal to the opposite side CD. 574. Theorem. — Two 'parallel planes are everywhere equally distant. For the shortest distance from any point of one plane to the other, is measured by a perpendicular. But these perpendiculars are all parallel (526), and therefore equal to each other. 5791. Theorem. — In every triedral the sum of tJie three diedral angles is greater than two right angles^ and less than six. Consider the supplementary triedral, with the given one. Now, the sum of the three diedrals of the given triedral, and of the three faces of its supplementary tri- edral, . must be six right angles ; for the sum of each pair is two right angles. But the sum of the faces of the supplementary triedral is less than four right angles (587), and is greater than zero. Subtracting this sum from the former, the remainder, being the sum of the three diedrals of the given triedral, is greater than two and less than six right angles. TRIEDRALS. 203 EQUALITY OF TRIEDRALS. 502. Theorem. — When two triedrals have two faces^ and the included diedral of the one respectively equal to the corresponding p>arfs of the other, then the remaining face and diedrals of the first are respectively equal to tk3 correspondiyig parts of the other. There are tAvo cases to be considered. 1st. Suppose the angles AEO and BCG equal, and the angles AEI and BCD equal, also the included diedrals whose edges are AE and BC. Let the ar- rangement be the same in both, so that, if we go around one triedral in the order 0, A, I, 0, and around the other in the order G, B, D, G, in both cases the triedral will be on the right. Then it may be proved that the two triedrals are equal. Place the angle BCD directly upon its equal, AEI. Since the diedrals are equal, and are on the same side of the plane AEI, the planes BCG and AEO will coin- cide. Since the angles BCG and AEO are equal, the lines CG and EO will coincide. Thus, the angles DCG and lEO coincide, and the two triedrals coincide throughout. 2J. Let the angles AEO and DCG be equal, and the angles AEI and BCD, also the included diedrals, whose edges are AE and DC. But let the arrangement be re- verse ; that is, if we go around one triedral in the order 0, A, I, 0, and around the other in the order G, D, B, G, 204 ELEMENTS OF GEOMETRY. h:^:'-' ,j^._ 7F / in one case the triedral will be to the right, and in the other it will be to the left of us. Then it may be proved that the two triedrals are symmetrical. One of the triedrals can be made to coincide with the symmetrical of the other ; for if the edges BC, GO, and DC be produced beyond C, the triedral CFHK will have two faces and the included diedral respect- ively equal to those parts of the triedral EAOI, and arranged in the same order ; that is, the re- verse of the tri- edral CDGB. Hence, as just shown, the trie- drals CFHK and EAOI are equal. Therefore, EAOI and CDGB are symmetrical triedrals. In both cases, all the parts of each triedral are re- spectively equal to those of the other. 59.^. Theorem. — When hvo triedrals have one face and (he two adjacent diedrals of the one respectively equal to the corresponding parts of the other, then the remaining faces and diedral of the first are respectively equal to the corresponding parts of the other. Suppose that the faces AEI and BCD are equal, that the diedrals whose edges are AE and BC are equal, that the diedrals whose edges are IE and DC are equal, and that these parts are similarly arranged in the two trie- drals. Then the one may coincide with the other. TRIEDRALS. 205 Therefore, it For BCD may coincide with its equal AEI, BC fall- ing on AE. Then the plane of BCG must coincide with that of AEO, since the diedrals are equal ; and the line CG will fall in the plane of AEO. For a similar reason CG will fall on the plane of lEO. must coincide with their intersection EO, and the two triedrals coincide throughout. When the equal parts are in re- verse order in the two triedrals, the arrangement in one must be the same as in the sym- metrical of the other. Therefore, in that case, the two triedrals would be symmetrical. In both cases, all the parts of each triedral are re- spectively equal to those of the other. 594. Theorem. — A71 isosceles triedral and its symmet- rical are equal. Let ABCD be an isosceles triedral, having the faces BAC and DAC equal, and let AEFG be its symmetrical triedral. Now, the faces BAC, DAC, FAG, and FAE, are all equal to each other. The diedrals whose edges are AC and AF being vertical, are also equal. Hence, the faces mentioned being all equal, corresponding equal parts may be taken in the same order in both triedrals ; that is, the face EAF equal to the face BAC, and the face FAG equal to CAD. Therefore, the two triedrals are equal. 206 ELEMENTS OF GEOMETRY. 395« Corollary — In an isosceles triedral, the diedrals opposite the equal faces are equal. For the diedrals whose edges are AB and AD, are eacl: equal to the diedral whose edge is AE. 50l>. Corollary Conversely, if in any triedral two of the diedral angles are equal, then the faces opposite tiiese diedrals are equal, and the triedral is isosceles. For, as in the above theorem, the given triedral can be shown to be equal to its symmetr'iCal. 507. Theorem. — When i.wo triedrals have two faces of the one respectively/ equal to two faces of the other, and the included diedrals unequal, then the third faces are unequal, and that face is greater which is ojyposite the greater diedral. Suppose that the faces CBD and EAi are equal, and that the faces CBF and EAO are also equal, but that the diedral whose edge is CB is greater than the die- dral whose edge is EA. Then the face FBD will be greater than the face OAI. Through the line BC, let a plane GBC pass, making with the plane DBC a diedral equal to that whose edge is AE. In this plane, make the angle CBG equal to EAO. Let the diedral FBCG be bisected by the plane TRIEDRALS. 207 IIBC. BH being the intersection of this plane with the pbine FBD. Then the two triedrals BCDG and AEIO, having two faces and the included diedral in the one equal to the corresponding parts in the other, will have the remain- ing parts equal. Hence, the» faces DBG and lAO aie equal. Again, the two triedrals BCFH and BCGH have the faces CBF and CBG equal, by construction, the face CBH common, and the included diedrals equal, by con- struction. Therefore, the third faces FBH and GEH are equal. To each of these equals add the face HBD, and we have the face FBD equal to the sum of GBH and HBD. But in the triedral BDGH, the face DBG is less thai: the sum of the other two faces, GBH and HBD (586) Hence, the face DBG is less than FBD. Therefore, the face OAI, equal to DBG, is less than FBD. 598. Corollary — Conversely, when two triedrals have two faces of the one respectively equal to two faces of the other, and the third faces are unequal, then the die- dral opposite the greater face is greater than the diedral opposite the less. 599. Theorem. — When two triedrals have their three faces resj)ectively equal, their diedrals will he respectively equal; and the two triedrals are either equal, or they are symmetrical. When two faces of one triedral are respectively equal to those of another, if the included diedrals are une- qual, then the opposite faces are unequal (597). But, by the hypothesis of this theorem, the third faces are equal. Therefore, the diedrals opposite to those faces must be equal. In the same manner, it may be shown that the other 203 ELEMENTS OF GEOMETRY. diedral angles of the one, are equal to the corresponding diedral angles of the other triedral. Therefore, the trie- drals are either equal or symmetrical, according to the arrangement of their parts. 600. Theorem Tivo triedrah which have their die- drals respectively equal, hmoe also their faces resjyectively equal ; and the two triedrals are either equal, or they are symmetrical. Consider the supplementary triedrals of the two given triedrals. These will have their faces respectively equal, because they are the supplements of equal diedral an- gles (589). Since their faces are equal, their diedrals are equal (599). Then the two given triedrals, having ^heir faces the supplements of these equal diedrals, will nave those faces equal ; and the triedrals are either equal or symmetrical, according to the arrangement of their parts. 601. The student may notice, in every other case of equal triedrals, the analogy to a case of equality of tri- angles; but the theorem just demonstrated has nothing analogous in plane geometry. 60!3. Corollary. — All trirectangular triedrals are equal. 603. Corollary. — In all cases where two triedrals are either equal or supplementary, equal faces are opposite equal diedral angles EXERCISES. 694. — 1. In any triedral, the greater of two faces is opposite to the greater diedral angle; and conversely. 2. Demonstrate the principles stated in the last sentence of Article 590. 3. If a triedral have one riglit diedral angle, then an adjacent POLYEDRALS. 209 fac3 and its opposite diedral are either both acute, both right, or Vioth obtuse. POLYEDRALS. 605. A PoLYEDRAL is the figure formed by several planes which meet at one point. Thus, a polyedral is composed of several angles having their vertices at a common point, every edge being a side of two of the angular faces. The triedral is a polyedral of three faces. 600. Problem Any polyedral of more than three faces may he divided into triedrah For a plane may pass through any two edges which are not adjacent. Thus, a polyedral of four faces may be divided into two triedrals ; one of five faces, into three ; and so on. GOT. This is like the division of a polygon into tri- angles. The plane passing through two edges not adja- cent is called a diag- 07ial plane. , A polyedral is //i\ called convex, when // i y^ / every possible diag- /kC ' \\ A''' onal plane lies within / / ^~^\\A / ^ the figure; otherwise ' I ^ \ it is called concave. 60S. Corollary If the plane of one face of a con- vex polyedral be produced, it can not cut the polyedral. 609. Corollary. — A plane may pass through the ver- tex of a convex polyedral, without cutting any face of the polyedral. 610. Corollary — A plane may cut all the edges of a convex polyedral. The section is a convex polygon. Geom.— 18 210 ELEMENTS OF GEOMETRY. Gil. When any figure is cut by a plane, the figure that is defined on the plane by the limits of the figure so cut, is called a plane section. Several properties of triedrals are common to other polyedrals. 612. Theorem — The sum of all the angles of a convex polyedral is less than four right angles. For, suppose the polyedral to be cut by a plane, then the section is a polygon of as many sides as the polye- dral has faces. Let n represent the number of sides of the polygon. The plane cuts off a triangle on each face of the polyedral, making n triangles. Now, the sum of the angles of this polygon is 2n — 4 right angles (424), and the sum of the angles of all these triangles is 2n right angles. Let v right angles represent the sum of the angles at the vertex of the polyedral ; then, 2n right angles being the sum of all the angles of the triangles, 2n — V is the sum of the angles at their bases. Now, at each vertex of the polygon is a triedral hav- ing an angle of the polygon for one fjice, and angles at the bases of the triangles for the other two faces. Then, since two faces of a triedral are greater than the third, the sum of all the angles at the bases of the tri- angles is greater than the sum of the angles of the polygon. That is, 2n — -y > 2n — 4. Adding to both members of this inequality, v + 4, and s ibtracting 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. 211 013. Theorem. — In any convex poJi/edral, 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. EXERCISES. 615. — 1. What is the locus of those points in space, each of which is equally distant from three given points? 2. AVhat is the locns of those points in space, each of which is equally distant from two given planes? 212 ELEMENTS OF GEOMETRY. 3. What is the locus of those points in space, eacli of which is equally distant from three given planes? 4. What is the locus of those poin* 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 the 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. TETKAEDRONS. 213 CHAPTER X. POLYEDRONS. OlO. A PoLYEDRON is a solid, or portion of space, bounded by plane surfaces. Each of these surfaces is a face^ their several intersections are edge^^ 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. TETKAEDRONS. 619. We have seen that three planes can not inclose a space (581). But if any point be taken on each edge of a tricdral, a plane passing through these three points would, with the three faces of / \^4 \ the triedral, cut off a portion ^m^^ '^ \ of space, which would be in- closed by four triangular faces. A Tetraedron is a polyedron having four faces. 214 ELEMENTS OF GEOMETRY. 0!S0. Problem. — Any four points whatever ^ which do not all lie in one plane, may he 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. 031. Either face of the tetraedron may be taken as the base. Then tlie 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). 62-^. Theorem. — There is a point equally distant from the four vertices of any tetraedron. In the plane of the face BCF, suppose a circle w^hose 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. 634. 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. 63«5. 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 // \ diedral angle whose edge is /^ / \ AO, and the other the die- ^y ^ -— /— -^ .\ ^^ dral whose edge is EG. \^ / ^^^.^^^''^ Now, every point of the I first bisecting plane is equally distant from the faces lAO 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 lino 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. G27. Theorem. — Two tetraedrons are equal where three faces of the one art, 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. 028. 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. — Tavo 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. 6S1. 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 TETRAEDRONS. 217 edges, that is, the sides of these triangles, are given. Theiij 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 be united to form a vertex. In this way models of various forms may be made with mor( accuracy than in wood, and the student may derive much hel] 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 — Whe7i 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, (leom. — 19 218 ELEMENTS OF GEOMETRY. When the edges taken in one of the tetraedrons are not fi 'des of one face ; as, AE : BC : : 10 : DF, A then, AE : BC : : IE : CD, as just proved, and 1 : DF : : IE : CD. Therefore, AE : BC : : 10 : 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 10. Join AN, and from H let the perpendicular HG fall upon DF, which is homologous to 10. Join BG. Now, the planes AKN and EIO are perpendicular to each other (556), and the line IN m 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 10 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 BGII. 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. 21U Also, the right angled triangles ANI and BGD are similar, since, by hypothesis, the ap'^les AIN and BDG are equal. Hence, AI : BD : : AN : BG. Therefore, AK : BH : : AI : BD. Thus, by the aid of similar triangles, it may be proved that any two liomologous lines, in two similar tetrae- drons, have the same ratio as two homologous edges. 635. Tliccrem Two tetraedrons are similar ivhen 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 ftices of the one are respectively similar to thos of the other, and they are similarly arranged. For th 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* 320 ELEMENTS OF GEOMETRY. 630. Corollary. — Two tetraedrons are similar when the edges of one are respectively proportional to those of the other, and they are similarly arranged. G40. 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. TETRAEDEONS CUT BY A PLANE. 044. 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. 645. 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 (022). But TEirvAEDRONS. 221 such a plane is parallel to the cutting plane GHP (566). 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 alt3rnation, EIO : CDF : : GHK : LNP. 046. 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. 050. 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 G35 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. 6,^2. If a polyedral is cut by a plane which cuts its several edges, the section is a pol}/gon, and a portion of space is cut oif, 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, pentn; - onal, etc., according to the polygon which forms tic base. The tetraedron is a triangular pyramid. 053. Problem. — Every pyramid can he 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 tvJien the base and two adjacent sides of the 07ie are respectively equal to the corresponding parts of the other, and they are simi- larly arra^iged. 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. G55. Theorem — Tivo similar pyramids are composed i>f tetraedrons respectively similar, ajid similarly arranged ; and, conversely, two pyramids are similar when com^ posed of similar tetraedrons, similarly arranged. 224 ELEMENTS OF GEOMETRY. 0>8. Theorem — When a pyramid is cut hy 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. 657. 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 su n 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. 00!S. 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. 603. The section made by the cutting plane is called the upper base of the frustum. The slant JdgJit 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. 6®e5. 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. 066. 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 OF GEOMETRY. MODEL PYRAMIDS. GGlf* 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. y/o. 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. PRIS3IS. 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 siown by supposing two equal and pirallel 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. rUlSMS. 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 tivo parallel planes pass through a prism, so that each j9?ang cuis 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. 228 ELEMENTS OF GEOMETRY. E f q:::;:.r>: 71 C ,, ^ ,L_..__j^ \ ,,''' 7 HOW DIVISIBLE. 675. Problem. — Every prism can he 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. 673. 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 II, making the intersections BH and CII, in the sides DF and DG. This plane cuts oiF the tet- raedron 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 BDII 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 it be cut by a diagonal plane through the points IT, G, and B. PllISMS. 229 This plane separates two tetraedrons, HBCG and IIBFG. The two faces, HBC and HBG, of the tetrae- dron HBCGj 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 tetracdrons as having their bases BCG and BFG. These are e((ual 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. 677. 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, tlie 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 nmkes himself 230 ELEMENTS OF GEOMETRY. EQUAL PKISMS. OTO. 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. G^O. Corollary — Two right prisms are equal v^licn they have equal bases and the same altitude. GSl, The theory of similar prisms presents nothing diiiicult or peculiar. The same is true of symmetrical prisms, and of symmetrically similar prisms. AREA OF THE SURFACE. O.S2. Theorem. — The area of the lateral surface of a jirism is equal to the p)rodiwt of one of the lateral edges hy the perimeter of a section, made by a plane per pen- dijular to those edges. Since the lateral edges are parallel, the plane TIN, parpendicular 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 PHISMS. 231 of the parallelogram, and one side of the polygon HN as its altitude. Thus, area AG = ABXHP, 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- plieJ by the perimeter of the polygon. 083. 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. ^84. 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 pris,m and the apothem of the base. EXERCISES, 6S5. — 1. A right prism has less surface than any other prism of equal base and equal altitude; and a regular prism lias 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 G83, vvitliout the aid of Article 682. 232 ELEMENTS OF GEOMETRY. MEASURE OF VOLUME. 086. A Parallelopiped is a prism whose bases are parallelograms. Hence, a parallelopiped is a solid in- closed by six parallelograms. 0^7. Theorem. — The opposite sides of a parallelopiped are equal. For example, the faces AI and BD are equal. For 10 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. 68S. Corollary — Any two opposite faces of a paral- lelopiped may be assumed as the bases of the figure. 6^9. A parallelopiped is called right in the same case as any other prism. When the bases also are rectangles, it is called rectangidar. Then all the faces are rectangles. G90. 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 i.s 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, ar.d the straight line among lines. MEASURE OF VOLUME. 233 The cube is taken, therefore, as the unit of measure of volume. That is, Avhatever 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. \ \ \ \ \ v\ \ \ \-^ 1 \ \ \ \ VOLUME OF PARALLELOPIPEDS. 691. Theorem. — The volume of a rectangular paral- lelopiped 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 parallele- 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 Avould we measure the whole of the prism. But the remainder of botli 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 parallelepiped is meas- ured by the product of its length, breadth, and altitude. 69!S. Theorem — The volume of any parallelopiiyed is equal to the product of its length, hreadth, and altitude. Inasmuch as this has just been demonstrated for the rectangular parallelepiped, it will be sufficient to show that any parallelepiped 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 parallelepipeds 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 cc incidence of equal figures, as follows ; Let AF be any oblique parallelopiped. It may be shown to be equivalent to the parallelopiped AL, which has a rectangular base, AH, since the prism LHEO is equal to the prism DGAI. But the poTallelopipeds AF and AL have the same length, breadth, and altitude. 236 ELEMENTS OF GEOMETRY. By similar reasoning, the prism AL may be shoAvn 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. 69S. 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 fisure, fold at the dividing lines, and glue the edges. It is well to have at least eight of one size. CI05. Corollary — The volume of any parallelopiped is equal to the product of its base by its altitude. 696. Corollary. — The volumes of any two parallelo- pipeds are to each other as the products of their threj dimeusiond. VOLUME OF PRISMS. 697. Theorem. — The volume of any triangular pnf^m 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- epiped ; 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 TETRAEDRONS. 700. Theorem. — Two tetraedrons of equivalent bases and of the same altitude are equivalent. Suppose the bases of the two tetraedrons to be in the E C F 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 o'f 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 tAvo 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 SIMILAIl POLYEDRONS. 229 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. tOG, — 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 wliich 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. TOT. 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. — An^ two similar polyedrons may he divided into the same number of similar tetraedrons, which shall be respectively siinilar, and similarly arranged. For, after dividing one into tetraedrons, the construe 240 ELEMENTS OF GEOMETRY. tiori 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 poly edroyis 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 : : EF : CF^ ; : AH^ : BG^ )^y the proportionality of homologous lines, (634), J AH : i BG : : EI : CF : : AH : BG. Multiplying these proportions (701), we have AEIO : BCFD : : EF : CF'^ : : AH^ : BG^ 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. Ill, What is the ratio between the edges of two cubes, one of wliich lias 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 desirec? degree of approximation. REGULAR POLYEDRONS. 712. A Regular Polyedron is one whose faces arc- 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 ocfaedron has eight, the dodecaedron twelve, and the icosaedron twenty faces. Geom.— 21 £42 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 thos3 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 KEGULATt POLYEDKONS. ')'14. 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 ih*? regular tetraedron and the cube have already been given. REGULAR POLYEDRONS. 213 For the octaedron, draw eight equal regular trian- gles, as ill the diagram. For the dodecaedron, draw twelve equal regular penta- gons, as in the diagram. For the icosacdron, draw twenty equal regular trian- gles, as in the diagram. There are many crystals, which, though not regular, in iIk 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 wliich 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 parallelepiped 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 tlie opposite faces bi- 244 ELEMENTS OF GEOMETRY. sect each o^htr; 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. It' 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 f^^e square feet, of the upper base one and four-fifths square feet, and the altitude iu two feet; required the volume. SOLIDS OF REVOLUTION. 245 CHAPTER XI SOLIDS OF REVOLUTION 710. 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 Mglit 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 surftice. The plane surfaces of a cylinder are called its bases ^ 246 ELEMENTS OF GEOMETllY. 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. TIO, 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 w^hen 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 diff'erent 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. •ySl. 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. 73!S. 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 also 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 (Q66). 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 higlit by the circumference of a section midway between the two bases (665). TSS. 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. '7^9, — 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. TSO. 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. T35. 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. '7SG, — 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 eenter, radius, and diame- ier of the sphere are the same as those of the generating circle. The spherical surface is described by the circumference. SPHERES. 251 •ySS. 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. T39. 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. •742. 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 ^^p^— - -=-^ other point of the plane, they are ^^' oblique and longer than the radius, ^fc- 1/^^^ which is perpendicular (530). There- ^^fc^^W fore, every point of the plane except ^B^^^^ 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. 2r)2 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 diifer- 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 JS and S are the poles of both the sections in the last diagram. TSO. 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 (he surface of a sphere, is the arc of a great circle, passing through the ttvo poiyits. 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 ELEMENTS 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 Avhich joins its extremities. 75^. 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 passes through the center of a sphere (749). 753. CDrollary — 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. •755. The student of geography will recognize the equator as a great circle of tlie 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 tliat art. A vessel cross- ing liie ocean Ironi 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. T56. The curvature of tlie 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, upoi\ 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. 75S. 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. 7S0. 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. T61« 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. 76ti. A Spherical 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 ha,ve 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. 761. 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. 7S5. 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. 708. 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 •^58 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. y^ ^^^^^^>\ Of these seven spherical tri- / F/ / ;\ angles, that which lies vertically \C^^--^J "^ /^^/ \ opposite the given triangle, as \ / 7^--,^ \ GKH to FDB, has its sides \ / 7-.. '> / ""'/b respectively equal to the sides \ I / /^ "y of the given triangle, but they g'^< l ^ ^^ are arranged in reverse order ; for the corresponding triedrals are symmetrical. Such spherical triangles are called symmetrical. *767. Corollary. — If two spherical triangles are equal, their corresponding triedrals are also equal ; and if two spherical triangles are symmetrical, their corresponding triedrals are symmetrical. 7*68. Corollary. — On the same sphere, or on equal spheres, equal triedrals at the center have equal corre- sponding spherical triangles ; and symmetrical triedrnls at the center have symmetrical corresponding spherical triangles. 789. Corollary — The three sides and the three an- gles of a spherical triangle are respectively the measures of the three faces and the three diedrals of the triedral at the center. 770. Corollary Spherical triangles are isosceles, equilateral, rectangular, birectangular, and trirectangu- iar, according to their triedrals. 771. Corollary. — The sum of the angles of a spher- ical triangle is greater than two, and less than six right angles (591). 772. Corollary An isosceles spherical triangle is SPHERES. 259 equal to its symmetrical, and has equal angles oppo- site the equal sides (594). 773. Corollary The radius being the same, two spherical triangles are equal, 1st. When they have two sides and the included an- gle of the one respectively equal to those parts of the other, and similarly arranged; 2d. When they have one side and the adjacent angles of the one respectively equal to those parts of the other, and similarly arranged; 3d. When the three sides are respectively equal, and similarly arranged; 4th. When the three angles are respectively equal, and similarly arranged. 774. Corollary. — In each of the four cases just given, when the arrangement of the parts is reversed, the tri- angles are symmetrical. POLAR TRIANGLES. 775. If at the vertex of a triedral, a perpendicular bo erected to each face, these lines form the edges of a supplementary triedral (590). If the given vertex is at the center of a sphere, then there are two spherical tri- angles corresponding to these two triedrals, and they have all those relations which have been demonstrated concerning supplementary triedrals. Since each edge of one triedral is perpendicular to the opposite face of the other, it follows that the vertex of each angle of one triangle is the pole of the opposite side of the other. Hence, such triangles are called polar triangles, though sometimes supplementary. 776. Theorem. — If with the several vertices of a spher- ical triangle as poles, arcs of great circles be made, then a 260 ELEMENTS OF GEOMETRI. second trianglt is formed whose vertices are also poles of the first. 777. Theorem — Uach angle of a spherical triangle is the supplement of the opposite side of its polar triangle. Let ABC be the given triangle, and EF, DF, and DE be arcs of great circles, whose poles are respectively A, B, andC. Then ABC and DEF are polar or supplementary- triangles. These two theorems are corollaries of Article 589, but they can be demonstrated by the student, with the aid of the above diagram, without reference to the triedrals. 778. The student w^ill derive much assistance from drawing the diagrams on a globe. Draw the polar tri- angle of each of the following : a birectangular triangle, a trirectangular triangle, and a triangle with one side longer than a quadrant and the adjacent angles very acute. INSCRIBED AND CIECUMSCR I BE D. 779. A sphere is said to be inscribed in a polyedron when the faces are tangent to the curved surface, in which case the polyedron is circumscribed about the sphere. A sphere is circumscribed about a polyedron when the ver- tices all lie in the curved surface, in which case the poly- edron is inscribed in the sphere. 780. Problem — Any tetraedron may have a sphere inscribed in it; also, one circumscribed about it. For within any tetraedron, there is a point equally distant from all the faces (625), which may ^^ 'he cen- SPHERICAL AREAS. 261 ter of the inscribed sphere, the radius beiiig the perpen- dicular distance from this center to either face. There is also a point equally distant from all the vertices of any tetraedron (623), which may be the center of the circumscribed sphere, the radius being the distance from this point to either vertex. 781. Corollary. — A spherical surface may be made to pass through any four points not in the same plane. EXERCISES. 782. — 1. In a spherical triangle, the greater side is opposite the greater angle; and conversely. 2. If a plane be tangent to a sphere, at a point on the circum- ference of a section made by a second plane, then the intersection of these planes is a tangent to that circumference. 3. When two spherical surfaces intersect each other, the lin of intersection is a circumference of a circle; and the straight line which joins the centers of the spheres is the axis of that circle. SPHERICAL AREAS. 783. Let AHF be a right angled triangle and BFD a semicircle, the hypotenuse AF be- ing a secant, and the vertex F in the circumference. From E, the point where AF cuts the arc, let a perpendicular EG fall upon AB. Suppose the whole of this figure to revolve about AD as an axis. The triangle AFH describes a cone, the trapezoid EGHF describes the frustum of a cone, and the semicir- cle describes a sphere. The points E and F describe the circumferences of 262 ELEMENTS OF GEOMETRY. the bases of tke frustum ; and these circumferences lie in the surface of the sphere. A frustum of a cone is said to be inscribed in a sphere, when the circumferences of its bases lie in the surface of the sphere. 784. Theorem. — The area of the curved surface of an inscribed frustum of a cone, is equal to the product of the altitude of the frustum by the circumference of a circle whose radius is the perpendiciilar let fall from the center of the sphere upon the slant hirjht of the frustum. Let AEFD be the semicircle which describes the given sphere, and EBHF the trape- zoid Avhich describes the frustum. ^ Let IC be the perpendicular let fall /ff from the center of the sphere upon yiX the slant hight EF. ^/ kX Then the circumference of a circle 1 of this radius would be tt times twice \ CI, or 2;rCI ; and it is to be proved \ that the area of the curved surface ^--^ of the frustum is equal to the prod- uct of BH by 2.tCL The chord EF is bisected at the point I (187). From this point, let a perpendicular IG fall upon the axis AD. The point I in its revolution describes the circumference of the section midway between the two bases of the frustum. GI is the radius of this circumference, which is therefore 27rGI. The area of the curved surface of the frustum is equal to the product of the slant hight by this circumference (727); that is, EF by 2-GI. Now from E, let fall the perpendicular EK upon FH. The triangles EFK and IGC, having their sides respect- ively perpendicular to each other, are similar. Therefore, EF : EK : : CI : GI. Substituting for the second term SPHERICAL AREA&. 263 EK its 'equal BH, and for the second ratio its equi- multiple 2;rCI : 2;:GI, we have EF : BH : : 2;rCI : 2;rGI. Bj multiplying the means and the extremes, EFx2;ria-:BHX2;rIC. But the first member of this equation has been shown to be equal to the area of the curved surface of the frustum. Therefore, the second is equal to the same area. 785. Corollary. — If the vertex of the cone we're at the point A, the cone itself would be inscribed in the sphere; and there would be the same similarity of tri- angles, and the same reasoning as above. It may be shown that the curved surface of an inscribed cone is equal to the product of its altitude by the circumfer- ence of a circle whose radius is a perpendicular let fall from the center of the sphere upon the slant hight. •786. Theorem — The area of the surface of a sphere is equal to the product of the diameter hy the circumfer- ence of a great circle. Let ADEFGB be the semicircle by which the sphere is described, having inscribed in it half of a regular polygon which may ^ be supposed to revolve with it about // "^^ the common diameter AB. /^ "Of. Then, the surface described by the / side AD is equal to 2;rCI by AH. 1 The surface described by DE is \ equal to 2;rCI by HK, for the per- \^— pendicular let fall upon DE is equal ^^^^^Ub to CI; and so on. If one of the sides, as EF, is parallel to the axis, the measure is the same, for the surface is cylindrical. Adding these sev- 264 ELEMENTS OF GEOMETRY. eral equations together, we find that the entire surface described by the revolution of the regular polygon about its diameter, is equal to the product of the circumfer- ence whose radius is CI, by the diameter AB. This being true as to the surface described by the perimeter of any regular polygon, it is therefore true of the surface described by the circumference of a cir- cle. But this surface is that of a sphere, and the radius CI then becomes the radius of the sphere. Therefore, the area of the surface of a sphere is equal to the product of the diameter by the circumference of a great circle. 787. Corollary. — The area of the surface of a sphere is four times the area of a great circle. For the area of a circle is equal to the product of its circumference by one-fourth of the diameter. 788. Corollary — The area of the _,-_ surface of a sphere is equal to the inT^SIillS^^ area of the curved surface of a cir- |ii||iliii'P^^^^^ cumscribing cylinder; that is, a cyl- llijjr^.... jti inder whose bases are tangent to the iiiir~~hl^^^ surfa.ce of the sphere. PliiSiiill AREAS OF ZONES. 789. A Zone is a part of the surface of a sphere included between two parallel planes. That portion of the sphere itself, so inclosed, is called a segment. The circular sections are the bases of the segment, and the distance between the parallel planes is the altitude of the zone or segment. One of the parallel planes may be a tangent, in which case the segment has one base. SPHERICAL AREAS. 205 TOO. Theorem — The area of a zone is equal to the product of its altitude hy the circumference of a great circle. This is a corollary of the last demonstration (786). The area of the zone described by the arc AD, is equal to the product of AH by the circumference whose ra- dius is the radius of the sphere. AREAS OF LUXES. TOl. Theorem. — Tlie area of a lune is to the area of the whole spherical surface as the angle of the lune is to four right angles. It has already been shown that the angle of the lune is measured by the arc of a great circle whose pole is at the vertex. Thus, if AB is the axis of the arc DE, then DE measures the angle DAE, which is equal to the angle DCE. But evidently the lune varies exactly with the angle DCE or DAE. This may be rigorously demonstrated in the same manner as the principle that angles at the center have the same ratio as their intercepted arcs. Therefore, the area of the lune has the same ratio to the whole surface as its angle has to the whole of four right angles. TRIRECTANGULAR TRIANGLE. T02. If the planes of two great circles are perpen- dicular to each other, they divide the surface into four equal lunes. If a third circle be perpendicular to these Geom.— 23 266 ELEMENTS OF GEOMETRY. two, each of the four lunes is divided into two equal triangles, which have their angles all right angles and their sides all quadrants. Hence, this is sometimes called the quadrantal triangle. This triangle is the eighth part of the whole surface, as just shown. Its area, therefore, is one-half that of a great circle (787). Since the area of the circle is ;r times the square of the radius, the area of a trirectangu- lar triangle may be expressed by J;rR^. The area of the trirectangular triangle is frequently assumed as the unit of spherical areas. AREAS OF SPHERICAL TRIANGLES. 793. Theorem. — Two symmetncal spherical trianales are equivalent. Let the angle A be equal to B, E to C, and I to B. Then it is known that the other parts of the ^ triangle are respect- ively equal, but not superposable ; and it is to be proved that the triangles are equiv- alent. Let a plane pass through the three points A, E, and I ; also, one through B, C, and D. The sections thus made are small circles, which are equal; since the distances between the given points are equal chords, and circles described about equal triangles must be equal. Let be that pole of the first circle which is on the same gide of \\io, S2)hcre as the triangle, and F the corre- SPHERICAL AREAS. 2G7 spending pole of the second small circle. Let be joined bj arcs of great circles OA, OE, and 01, to the several vertices of t\e first triangle ; and, in the same way, join FB, FC, and FD. Now, the triangles AOI and BFD are isosceles, and mutually equilateral ; for AO, 10, BF, and DF are equal arcs (753). Hence, these triangles are equal (772). For a similar reason, the triangles lOE and CFD are equal ; also, the triangles AOE and BFC. Therefore, the triangles AEI and BCD, being composed of ecual parts, are equivalent. The pole of the small circle may be outside of the given triangle, in which case the demonstration would be by subtracting one of the isosceles triangles from the sum of the other two. •704. It has been show^n that the sum of the angles of a spherical triangle is greater than the sum of the angles of a plane triangle (771). Since any spherical polygon can be divided into triangles in the same man- ner as a plane polygon, it follows that the sum of the angles of any spherical polygon is greater than the sum of the angles of a plane polygon of the same number of sides. The difference betw^een the sum of the angles of a spherical triangle, or other polygon, and the sum of the angles of a plane polygon of the same number of sides, is called the spherical excess. 795. Theorem. — The area of a spherical triangle is equal to the area of a trirect angular triangle, multiplied hy the ratio of the spherical excess of the given triangle to one right angle. That is, the area of the given triangle is to that of the trirectangular triangle, as the spherical excess of the given triangle is to one right angle. 268 ELEMEJ^TS OF GEOMETRY. Let AEI be any spherical triangle, and let DHBCGF be any great circle, on one side of which is the given triangle. Then, comsidering this circle as the plane of reference of the figure, produce the sides of the trian- gle AEI around the sphere. Now, let the several angles of the given triangle be represented by a, e, and i; that is, taking a right an- gle for the unit, the angle EAI is equal to a right angles, etc. Then, the area of the lune AEBOCI is to the whole surface as a is to 4 (791). But if the tri- rectangular triangle, which is one-eighth of the spher- ical surface, be taken as the unit of area, then the area of this lune is 2a. But the triangle BOC, which is a part of this lune, is equivalent to its opposite and symmetrical triangle DAF. Substituting this latter, the area of the two triangles ABC and DAF is 2a times the unit of area. In the same way, show that the area of the tw^o tri- angles IDH and IGC is 2z, and that the area of the two triangles EFG and EHB is 2e times the unit of area. These equations may be wTitten thus : area (ABO + ADF) = 2a times the trirectangular tri- angle ; area ( IDH + IGC ) = 2i times the trirectangular tri- angle ; area (EFG + EHB) = 2e times the trirectangular tri- angle. In adding these equations together, take notice that the triangles mentioned include the given triangle AEI SPHERICAL AREAS. 269 three times, and all the rest of the surface of the hem- isphere above the plane of reference once ; also, that the area of this hemispherical surface is four times that of the trirectangular triangle. Then, by addition of tlio equations : area 4 trirect. tri. + 2 area AEI = 2 («-f e-{-i) trir. Iri. Transposing the first term, and dividing by 2. area AEl = (a -]- e -\r i — 2) trir. tri. But (a-i-e-i-i — 2) is the spherical excess of the given triangle, taking a right angle as a unit ; that is, it is the ratio of the spherical excess of the given trian- gle to one right angle. 796. Corollary. — If the square of the radius be taken as the unit of area, then the area of any spherical tri- angle may be expressed (792), J(a + e + ^ — 2};rR2. AREAS OF SPHERICAL POLYGONS. TOy. Theorem — The area of any spherical polygon is equal to the area of (he trirectangular triangle multiplied hy the ratio of the spherical excess of the polygon to one right angle. For the spherical excess of the polygon is evidently the sum of the spherical excess of the trian- gles which compose it; and its area is the sum of their areas. EXERCISES. 198. — 1. What is the area of the earth's surface, supposing it to be in the shape of a sphere, with a diameter of 7912 miles? 270 ELEMENTS OF GEOMETllY. 2. Upon the same hypothesis, what portion of the whole sur- face is between the equator and the parallel of 30° north latitude? 3. Upon the same hypothesis, what portion of the whole sur- face is between two meridians which are ten degrees apart? 4. What is the area of a triangle described on a globe of 13 inches diameter, the angles being 100°, 45°, and 53°? VOLUME OF THE SPHEEE. •700. Theorem. — The vohijne of any poly edr on in which a sphere can be inscribed^ is equal to one-third of the prod- uct of the entire surface of the polyedron by the radius of the inscribed sphere. For, if a plane pass through each edge of the poly- edron, and extend to the center of the sphere, these planes will divide the polyedron into as many pyramids as the figure has faces. The faces of the polyedron are the bases of the pyramids. The altitude of each is the radius of the sphere, for the radius which extends to the point of tangency is perpendicular to the tangent plane (742). But the vol- ume of each pyramid is one-third of its base by its altitude. Therefore, the volume of the Avhole polyedron IS one-third the sum of the bases by the common alti- tude, or radius. 800, Theorem. — The volume of a sphere is equal to one-third of the product of the surface by the radius. For, the surface of a sphere may be approached as nearly as Ave choose, by increasing the number of faeej of the circumscribing polyedron, until it is evident that the sphere is the limit of the polyedrons in which it is inscribed. Then, this theorem becomes merely a corol- lary of the preceding. 801. Corollary. — The volume of a spherical pyramid, SPHERICAL VOLUMES. 271 or of a spherical wedge, is equal ^;o one-third of the product of its spherical surface by the radius. 802. A spherical Sector is that portion of a sphere which is described by the rev- olution of a circular sector about a diameter of the circle. It may have two or three curved surfaces. Thus, if AB is the axis, and the generating sector is AEC, the sector has one spherical and one conical surface ; but if, with the same axis, the gener- ating sector is FCG, then the sector has one spherical and two conical surfaces. 803. Corollary. — The volume of a spherical sector is equal to one-third of the product of its spherical surface by the radius. 804. The volume of a spherical segment of one base is found by subtracting the volume of a cone from that of a sector. For the sector ABCD is composed of the segment ABC and the cone ACD. The volume of a spherical segment of two bases is the difference of the volumes of two segments each of one base. Thus the segment AEFC is equal to the segment ABC less EBF. 805. All spheres are similar, since they are gener- ated by circles which are similar figures. Hence, we might at once infer that their surfaces, as well as their volumes, have the same ratios as in other similar solids. These principles may be demonstrated as follows : 272 ELEMENTS OF GEOMETRY. 800. Theorem. — The areas of the surfaces of two spheres are to each other as the squares of their diameters; and their volumes are as the cubes of their diameters, or other homologous lines. For the superficial area of any sphere is equal to t: times the diameter multiplied by the diameter (786); that is ;rD'^. But ;r is a certain or constant factor. Therefore, the areas vary as the squares of the diame- ters. The volume is equal to the product of the surface by one-sixth of the diameter (800) ; that is, TtW by JD, or ^TiW. But Jtt is a constant numeral. Therefore, the volumes vary as the cubes of the diameters. USEFUL FORMULAS. SOT. Represent the radius of a circle or a sphere, or that of the base of a cone or cylinder, by R ; repre- sent the diameter by D, the altitude by A, and tha slant hight by H. Circumference of a circle . = ttD = 27rR, Area of a circle = JttD^ = ;rR'^, Curved surface of a cone = J;rDH = ttRH, Entire surface of a cone = ;rR(H-}-R), Volume of a cone = j^^ttD^A = J;rR^A, Curved surface of a cylinder = ;rDA = 2;rRA, Entire surface of a cylinder = 27rR(A + Rj. Volume of a cylinder = J;rD-A = ;rR^A, Surface of a sphere =;rD-=4;rR^, Volume of a sphere = iyTD'^=|;rR^, ;: = 3.1415926535. EXERCISES FOR -GENERAL REVIEW. 273 EXERCISES. SOS, — 1. What is the locus of those points in space which are at the same distance from a given point? 2. What is the locus of those points in space which are at the samo distance from a given straight line? 3. Wliat is the locus of those points in space, such that the distance of each from a gi\^en straight line, has a constant ratio to its distance from a given point of that line? EXERCISES FOR GENERAL REVIEW. 809. — I. Take some principle of general application, and state all its consequences which are found in the chapter under review; arranging as the first class those which are immediately deduced from the given principle; then, those which are derived from these, and so on, 2. Reversirng the above operation, take some theorem in the latter part of a chapter, state all the principles upon which its proof immediately depends; then, all upon which these depend; and so on, back to the elements of the science. 3. Given the proportion, a : h :'. c : d, to show that c — a : d — h : : a : b ; also, that a-\-c : a — c : : b-^d : b — d. 4. Form other proportions by combining the same terms. 5. What is the greatest number of points in which seven straight lines can cut each other, three of them being parallel; and what is the least number, all the lines being in one plane? 6. If two opposite sides of a parallelogram be bisected, straight lines from the points of bisection to the opposite vertices will tri- sect the diagonal. 7. In any triangle ABC, if BE and CF be perpendiculars to any line through A, and if D be the middle of BC, then DE is equal to DF. 8. If, from the vertex of the right angle of a triangle, there extend two lines, one bisecting the base, and the other perpen- 274 ELEMENTS OF GEOMETRY. dicular to it, the angle of these two lines is equal to the differ- ence of the two acute angles of the triangle. 9. In the base of a triangle, find the point from which lines extending to the sides, and parallel to them, will be equal. 10. To construct a square, having a given diagonal. 11. Two triangles having an angle in the one equal to an angle in the other, have their areas in the ratio of the products of the sides including the equal angles. 12. If, of the four triangles into which the diagonals divide a quadrilateral, two opposite ones are equivalent, the quadrilateral has two opposite sides parallel. 13. Two quadrilaterals are equivalent when their diagonals are respectively equal, and form equal angles. 14. Lines joining the middle points of the opposite sides of any quadrilateral, bisect each other. 15. Is there a point in every triangle, such that any straight line through it divides the triangle into equivalent parts? 16. To construct a parallelogram having the diagonals and one side given. 17. The diagonal and side of a square have no common meas- ure, nor common multiple. Demonstrate this, without using the algebraic theory of radical numbers. 18. To construct a triangle when the three altitudes are given. 19. To construct a triangle, when the altitude, the line bisect- ing the vertical angle, and the line from the vertex to the middle of the base, are given. 20. If from the three vertices of any triangle, straight lines be extended to the points where the inscribed circle touches the Bides, these lines cut each other m one point. 21. What is the area of the sector whose arc is 50°, and whose radius is 10 inches? 22. To construct a square equivalent to the sum, or to the diii ference of two given squares. 23. To divide a given straight line in the ratio of the areas of two given squares. 24. If all tlie sides of a polygon except one be given, its area will be greatest when the excepted side is made the diameter of a circle which circumscribes the polygon. EXERCISES FOR GENERAL REVIEW. 275 25. Find the locus of those points in a plane, such that the sum of the squares of the distances of each from two given points, shall be equivalent to the square of a given line. 26. Find the locus of those points in a plane, such that the difference of the squares of the distances of each from two given points, shall be equivalent to the square of a given line. 27. If the triangle DEF be inscribed in the triangle ABC, the circumferences of the circles circumscribed about the three trian- gles AEF, BFD, CDE, will pass through the same point. 28. The three points of meeting mentioned in Exercises 28, 29, and 30, Article 337, are in the same straight Jine. 29. If, on the sides of a given plane triangle, equilateral tri- angles be constructed, the triangle formed by joining the centers of these three triangles is also equilateral; and the lines joining their vertices to the opposite vertices of the given triangle are equal, and intersect in one point. 30. The feet of the three altitudes of a triangle and the cen- ters of the three sides, all lie in one circumference. The circle thus described is known as " The Six Points Circle." 31. Four circles being described, eacli of which shall touch the three sides of a triangle, or those sides produced ; if six lines be made, joining the centers of those circles, two and two, then the middle points of these six lines are in the circumference of the circle circumscribing the given triangle. 32. If two Imes, one being in each of two intersecting planes, are parallel to each other, then both are parallel to the intersec- tion ff the planes. 33. If a line is perpendicular to one of two perpendicular planes, it is parallel to the other; and, conversely, if a line is par- allel to one and perpendicular to another of two planes, then the planes aro perpendicular to each other. 34. How mny a pyramid be cut by a plane parallel to the base, ■.0 as to make the area or the volume of the part cut off have a £;iven ratio to the area or the volume of the whole pyramid? 35. Any regular polyedron may have a sphere inscribed in it; also, one circumscribed about it 36 In any polyedron, the sum of the number of vertices and the number of faces exceeds by two the number of edges. 276 ELEMENTJS OF GEOMETRY. 37. How many spheres can be made tangent to three given planes? 38. Apply to spheres the principle of Article 331; also, of Article 191, substitutmg circles for chords. 39. Discuss the possible relative positions of two spheres. 40. What is the locus of those points in space, such that the sum of the .squares of the distances of each from two given points, is equivalent to a given square? 41. What is the locus of those points in space, such that the difference of the squares of the distances of each from two given points, is equivalent to a given square ? 42. A frustum of a pyramid is equivalent to tlie s.um of three pyramids all having the same altitude as the frustum, and having lor their bases the lower base of the frustum, the upper base, and a mean proportional between them. 43. The surface of a sphere can be completely covered with tlie surfaces either of 4, or of 8, or of 20 equilateral spherical tri- angles. 44. The volume of a cone is equal to the product of its whole surface by one-third the radius of the inscribed sphere. 45. If, about a sphere, a cylinder be circumscribed, also a cone whose slant hight is equal to the diameter of its base, tlien the area and volume of the sphere are two-thirds of the area and volume of the cylinder; and the area and volume of the cylinder are two-thirds of the area and volume of the cone. TRIGONOMETRY. CHAPTER XII. PLANE TRIGONOMETRY. 811. Trigonometry is the science in which the rehi- tions subsisting between the angles, sides, and area of any triangle are investigated. The science was origi- nally called Plane Trigonometry or Spherical Trigonom- etry, according as the triangle was plane or spherical. Plane Trigonometry has now a wider meaning, com- prising algebraic investigations concerning angles and their functions, and the methods of calculating these functions. MEASURE OF ANGLES. 812. In Elementary Geometry, the unit for the meas- ure of angles is usually the right angle. The frequent fractions which the use of this unit gives rise to, render it inconvenient for calculation. It has been divided into degrees, minutes, and seconds (208). This sexagesimal division of angles has been in use since the second century. Efforts have been made to substitute for it the centesimal division, making the right angle contain one hundred grades^ each grade one hun- dred minutes^ and so on ; but this plan has never been generallv in use. (277) 278 PLANE trigonoml:tiiy. 81S. There is another unit which has been called the circular measure of an angle. It is used in trigonometri- cal investigation, and is also called the analytical unit* It is that angle at tiie center of a circle whose intercepted arc has the same lin- ear extent as the radius. Thus, if the arc AB has the same linear extent as the radius AC, then the angle C is the unit of circular measure. Hence, this unit of measure is equal to 180° 57°. 29578— =57° 17' 44'^ 8+, Also, 1°=tt7q=.017453 times the circular measure. 814. Various instruments are used for the measure of angles. A protractor is used to measure the angle of two lines in a drawing. It is usually shaped like a semi- circumference with its diameter, the arc being marked with the degrees from to 180. Let it be required to measure the angle ABC. Place the center of the straight edge, which is marked by a notch on the instrument, at the vertex B; let the edge lie along one side of the angle, as BC ; then read the degree marked where the other side BA passes the arc of the instrument. This gives the size of the angle. The same instrument is used for drawing angles of a known size. One side of the angle being drawn, place the center of the protractor at the point which is to be the vertex; then the required number of degrees, on the FUNCTIONS OF ANGLES. 279 edge of the arc, ^vill indicate a point on the other side of the angle. Connect this point with the vertex to com- plete the angle. The student should be provided with a protractor, a six-inch scale, and a pair of dividers. Large protractors, made of wood, pasteboard, or tin-plate, are useful for blackboard work. EXERCISES. 815. 1. Find the circular measure of an angle of 3° 4' 5^\ 2. Draw a triangle having one side two inches, another three inches, and the included angle 100°. Find the other angles and side by measurement. 3. Draw a triangle with the sides three, four, and five inches in length. Find the angles by measurement. These exercises may be extended and varied, referring to Articles 295 to 301 inclusive. FUNCTIONS OF ANGLES. 816. When two quantities are so related that any variation in one causes a variation in the other, each is a function of the other. Thus, »Jx is a function of x; the area of a circle is a function of its radius (500). A quantity may be a function of several others. Thus, x^ y^ is a function of x and y ; the area of a triangle is a function of its base and altitude (386). The angles of a triangle are functions of the ratios of the sides (316); and tlie ratios of the sides of a triangle are functions of tiie angles (309). For example, if the lengths of the sides be as the num- bers 3, 4, and 5, then the angle opposite the longest side is a right angle (413); and each of the acute angles is also a function of the numbers 3, 4, and 5. 200 PLANE TRIGONOMETRY. For another example; if the tri- angle ACD has its angles 30°, 60°, and 90°, then it may be shown that AC , CD_ AC .. AD=^' AD-^^''''^OD=^^^- A D B Let the student now solve the 1st Exercise of Art. 472. 817. Theorem — If from any point in one side of an ayigle, a perpendicular fall upon the other side, a right angled triangle is formed, and the ratios of the sides of this triangle are functions of the given angle. For, if any number of tri- angles were thus formed with a given angle, all of these tri- angles would be similar (306), and their sides would have the same ratios (309). When the given angle is greater than a right angle, one side may be produced to meet the perpendicular. 818. If from any point on one side of a given angle a perpendicular fall on the other side as a base, then The Sine of the given angle is the ratio of the perpen- dicular to the hypotenuse of the right angled triangle thus formed. The Tangent of the angle is the ratio of the perpen- dicular to the base. The Secant of the angle is the ratio of the hypotenuse to the base. The Cosine of the angle is the ratio of the base to the hypotenuse. FUNCTIONS OF ANGLES 281 The Cotangent of the angle is the ratio of the base to the perpendicular. The Cosecant of the angle is the ratio of the hypote- nuse to the perpendicular. The abbreviations sin., tan., sec, cos., cot., and cosec. are used respectively for these six functions. Thus, the sine of the angle A is written sin. A. These six are all the ratios that can be formed by the simple combination of the sides of the triangle. They are called, therefore, the simple functions of an angle. Other functions have been formed by composition and by division. Of these, the following is used at the present day: The Versed sine is the ratio of the excess of the hypotenuse over the base, to the hypotenuse. Hence, vers. sin. A=l — cos. A. 819. A table of sines of every degree from to 90° may be made by drawing and measurement. Draw a right angled triangle, with an angle at the base equal to the angle whose sine w^e wish to find. It will simplify the work to make the hypotenuse the length of a certain unit. Divide the length of the perpendicular by that of the hy- potenuse. The quotient is the sine. By careful drawing and measurement, a table of sines may be made that shall be true to two places of decimals. A table of tangents may be formed in a similar manner, making the base the length of a certain unit. By calculation, the functions may all be found to any required degree of accuracy. 820. The etymology of sine, tangent, and secant ap- pears from the method which was formerly used to define these terms, which was as follows : Trij:.-24. 282 TLANE TRIGONOMETRY. If with any radius an arc be de- scribed about the vertex C as a cen- ter, and if from B, one extremity of the intercepted arc, a perpendicular BD fall upon the side CA, then BD is called the sine of the arc BA, or of the angle C. If a perpendicular to AC be produced to meet CB at E, then AE is the tangent and CE the secant of the arc AB, or of the an- gleC. The student readily perceives that if the radius is taken as the unit of length, then the lengths of BD, AE, and CE are respectively the sine, tangent, and secant of the angle C. The names tangent and secant are taken from the geometrical tangent and secant. Arc, chord, and sine are derived from the fancied resemblance of the figure to the bow of the archer. The curve BAF is the bow or arc^ the chord BE joins its ends, and BD touches the breast or sinus^ of the archer. So also DA has been called the sagitta or arrow. When used now, it is called the versed sine. The oldest work on Trigonometry now extant is the Almagest of Ptolemy,- written in the second century. He divides the radius into sixty parts, also the arc whose chord is equal to the radius into the same number of parts. This mode of measuring arcs, and consequently angles, remains in use, but the sexagesimal division of lines was long since abandoned. The use of sines was introduced by the Arabian mathematicians about the eighth or ninth century. Napier, a Scotch Baron, who lived in the early part of the seventeenth century, has done more for the science of Trigonometry than any other one man. FUNCTIONS OF ANGLES. 283 EXERCISES. 821.— 1. Demonstrate tan. 45°=1; also, sin. 60°=^i/3, 2. Construct* an angle whose sine is f ; one whose tangent is 4; one whose secant is |. SIGNS OF ANGLES AND OF THEIR FUNCTIONS. 822. An angle may be conceived to be generated by the revolution of a line about a point. Thus, the line AB beginning at AX, may take the positions AB,AB', AB'', AB''', AX, and so on indefi- nitely, repeating at each revolution all the positions of the first. In Trigonometry, the amount of this revolution is con- sidered as an angle, so that an angle may be greater than the sum of two or of four right angles. In the strict geometrical definition, an angle being the difference of two directions, can not be greater than two right angles. Quantities conceived to exist in a certain direction or mode are called positive, and are designated by the sign -|-; while the quantities in the opposite direction are called negative, and are designated by the sign — . In the present investigation, the angle is supposed to be generated by the motion of the line AB up from AX. Angles so formed are positive, and when estimated in the opposite direction they are negative. Thus, if BAG is an acute angle, it is positive. If it is negative, it is greater than the sum of three right angles. The com- 284 PLANE TRIGONOMETRY. plement of an angle greater than a right angle must be negative, and the same is true of the supplement of an angle greater than two right angles. The directions to the right of YY' and those upwards from XX' are positive. Then the directions to the left from YY' and those downward from XX' are negative. Thus, AC, CB, and C'B' are positive, while AC, C"B'', and C'B''' are negative. 823. Theorem — The functions of any acute angle are 'positive. For when the revolving line is in the first quarter of its revolution, that is, between AX and AY, all the sides of the triangle ABC are positive. The same is true of the functions of any angle which is equal to 4n right angles -|- an acute angle, n being any entire number positive or negative. 824. Theorem — The tangent, secant, cosine, and co- tangent of obtuse angles are negative, while the sine and cosecant of obtuse angles are positive. For, when the revolving line is in the second quarter of its revolution, that is, between AY and AX', the side AC of the triangle AB'C is the only negative term. Hence, the functions of which it forms one term are neg- ative. The same is true of the respective functions of any angle which is equal to an obtuse angle ± 4n right angles. 82«5. In this manner, the signs of the functions may be found, and arranged according to the quarter of the revolving line AB. The following table exhibits the signs of the functions of all angles whatsoever: FUNCTIONS OF ANGLES. 285 REVOLVING LINE IN First quarter, Second quarter, Third quarter, Fourth quarter. NE & COSEC. COS. & SEC. TAN. A COT. + + + + + + 836. Corollary. — The functions of two angles are the same, when one of the angles is greater than the other by four right angles. EXERCISES. 827. — 1. Demonstrate the following equations : sec. 120°= — 2; COS. 135°= — V2- 2. The ratio of one straight line to its projection upon another is what function of their angle ? 3. Construct an angle whose tangent is — 1 ; one whose sine is — ^. 4. Construct an angle whose cosine is — f. ANGLES OF A GIVEN FUNCTION. 828. Theorem. — Any given simple function, when taken irrespective of its algebraic sign, belongs to four different angles within each revolution. If BAG is the acute angle of a given function, the revolving line AB will, at some point in each quarter of its revolution, form an acute angle with XX', equal to the angle BAG. 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 BAG. 820. Corollary. — Any simple function of an angle is 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 anoxic and of its ne";ative 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"-|-xV) = -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 7i-180°+(— 1)^A includes all angles which have the same sine as A; The formula n'360°±A includes all the angles which have the same cosine as A; and The formula ?i-180°-(-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 auizle. 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/S; cot. 405°= 1. 3. Write a formula containing all the values of A when tan. A=l. LIMITS OF FUNCTIONS. S32, Theorem. — Hie sine of any angle can not he greater titan 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 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, Avhen the sine is — 1. Then again it increases to the end of the revolu- tion, where the sine is 0. The cosii.e 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 : 0° 90° 180° 270° 360° 1 00 1 00 1 00 00 1 0—10 00—1 00 1 00 00—1 1 00 1 00 The versed sine increases from to 2 as the angle increases from 0° to 180°, and decreases from 2 to through the other two quarters. EXERCISES. 835. — 1. Trace the value of this expression: cos. A — sin. A. as A varies from 0° to 360°. 2. What are the sine 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 ^: ono whose tangent is ^; one whose cosine is j,. 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, h the base, and c 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 h are either positive or nega- tive. Then, sin. A=-^ C COS. A=*-, c tan. ^'l cot. -4 sec. '"V coscc. A=f. a 837. Corollary — The sine and cosecant of air 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 Tris— 25. 290 PLANE TRIGONOMETRY. of each of the other simple functions in terms of the sine of the same angle. Beginning with the equation, and dividing both members by c^, That is, the sum of the squares of the sine and cosine of any angle is equal to unity. Hence, sin. A = Vl — cos.^ A ; also, cos. A = Vl — sin."^ 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 h - = 7 X-- CDC That is, sin. A = tan. A cos. A. ^^ . sin. A sin. A Hence, tan. A = -= , COS. A vl — sin.'^A Since the tangent and cotangent are reciprocals, COS. A Vl — sin.^ A cot. A = — r- = — I • sin. A sm. A Since the secant and cosine are reciprocals, . ^ . 1 Vsec.^^A — 1 sm. A = \ I —r — r ^ sec.^A sec. A FUxNCTiONS OF ANGLES. 291 EXERCISES S40. — 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. This valuable exercise should be continued throughout the work. 3. Given 2 sin. A = tan. A, to find A. Ans. 0°, 60°, 120°, 180°, 240°, or 300°. 4. If sin. A = |, what is the value of cos. A ? 5. If sin. A = ^, what is the value of tan. A? 6. Demonstrate sin.\ 18° ^^(/5 — 1). Notice that 18° is the angle made by the apothegm and radius of a regular decagon. FUNCTIONS OF (90° rt A). 841. Theorem. — The cosine of an angle is the sine of its complement. That is, COS. A = sin. (90° r— 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 TLA NE TRIG (JxN ( )M KTRY. 843. Theorem — Sin. (90'' + .4) = cos. A, and cos. (90° + A) = — sin. A. 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. |/(10 + 2y/5) 4. Demonstrate tan. 72° 1/5 — 1. FUNCTIONS OF TWO ANGLES. 845. 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 foil FD and FK respect- ively perpendicular to CD and GIL 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. FUN{ TIONS OF ANGLES. 293 Likewise GK = GFXcos. A, and GF=CGXsin. B; hence, GK=CGXcos. A sin. B. Also, GK+DF=GK+KH=GH=CGXsin. (A+B); therefore. sin . (A+B)=sin. A cos. B-f-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 i# 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.(90°+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, (rr.) 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, (iii.) COS. (A — B) = cos. A cos. B + 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 : . . j^-r>s _ sin. (A-f-B) sin. A cos. B -\- cos. A sin. B cos. (A-j-B) cos. A cos. B — sin. A sin. B ' Dividing both terms of the fraction by cos. A cos. B, /A I -ON tan. A -1- tan. B , . tan. (A + B) = \ , . (v.) ^ ^ 1 — tan. A tan. B ^ ^ ^. ., , ,. .^. tan. A — tan. B , , am.lariy, tan. (A - B) = ^^^-^-^-^-^ , . (vi.) EXERCISES. 846. — 1. Demonstrate formula ii 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° = |-(730 + 6 v/5 + i/5 — 1.) FUNCTIONS OF 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. 2 A = cos.2 A — sin/^ A, . . . (n.) 2 tan. A tan.2A = j _^^^,^ , .... (m.) By substituting (n — 1)A for B in the original formulas, sin. ?^A, cos. nA, 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.^ A + sin.^ A = 1 (838), we have cos . 2A = 1 — 2 sin.*^ A ; also, cos. 2A = 2 cos.' A — 1. These formulas being true for all angles, J A may be substituted for A. Then, transposing, 2 sin.' i^ = 1 — cos. A, and 2 cos.' JA = 1 + cos. A. Therefore, sin. iA= -J I (1 — cos. A), COS. i 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. «-^ 1 Tx A sec. A — 1 848, — 1. Demonstrate tan. -;r- = — : : — . 2 tan. A 2. What is the value of sin. 15°; cos. 3°; sin. 1° 3CK? FORMULAS FOR LOGARITHMIC USE. 849. In order to render a formula fit for logarithmic calculation, products and quotients must be substituted for sums and diiferences. 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), 29G PLANE TRIGONOMETRY. sin. (A+B) + sin. (A— B)=2sin. Acos.B, (i.) sin. (A + B) — sin. (A — B) = 2 cos. A sin. B, (ii.) 1 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 -j- B = C, and A — B = D ; whence, A= i(C + D), and B= l(C-D). Then, sin. C+sin. D=2 sin. 1{G +T)) cos. h{^ — J)), (v.) sin. C — sin. D = 2 cos. 1{C + D) sin. J(C — D), (vi.) cos.C+cos.D = 2 cos. J(C + D) cos.J(C— D), (vii.) COS. D — COS. C = 2 sin. i{G + J)) sin. |(C — D), (viii.) By dividing v by vi, sin. C+sin. D wn i t\\ *. ^ /n t\\ tan. J(C+D) -, — -J—, — - = tan. J (C+D) cot. i(C— D)= ^)^ ' ^^; . sm.C— sin.D ^v i / 2\ i tan. i(C— D) Hence, sin. C+sin.D : sin. C— sin. D : : tan. K^+D) : tan. i(C— D). (ix.) EXERCISES. 850. — 1. Demonstrate sin. 5A=:5sin. A — 20sin.'A+16 sin.^A. 2. Demonstrate sin. (A + B) sin. (A — B) = sin.^A — sin.-^B. TRIGONOMETRICAL 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 J, and the sine of 18° is ^(-^5 — 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. (A+2A) = sin. Acos.2A+ cos. A sin. 2A. Substituting the values of cos. 2 A and sin. 2 A (847), sin. 3A = 3 cos.^ A sin. A — sin.^ A. Hence (838), sin. 3A = 3 sin. A — 4 sin.^ A. Put 1° for A ; then, knowing the value of sin. 3°, and representing the unknown sin. 1° by x, 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 V 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. •29S PLANE TRIGONOMETRY. 852. The Table of Natural Sines and Tangents gives these functions to six places of figures for every 10' from 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 84° 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 rehed 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. S53. Before proceeding to the study of this article, the student should understand the use of the tables of loiirarithms 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 acteristic of every logarithmic trigonometric function is increased by 10. For example, sin. 30° = I ; log. 1 = 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 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 lY. 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 V^ by the number of seconds; and ndd 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.I" =2.5, ... 2.5x14 .. = 35, Therefore, . . tab. log. sin. 40° 13' 14" = 9.810052. To find the tabular logarithmic cosine of 75° 40' 21'^ tab. log. COS. 75° 40' = 9.393685, P. P. 1" = 8.23, . . 8.23 X 21 . . = 173, Therefore, . . 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 tanojents of small anojles 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, 8.120623. Subtract as in Table V, 13, tab. log. sin. 45' 23" = 8.120610. 856. 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 end minutes. If the given function is not in the table, take that which is next less; subtract it fi-om 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. 1"= 4.38, . . . . 217^4.38=50. 302 PLANE TRIGONOMETRY. Therefore, 53° 2' 10'' is the angle Avhose logarithmic cotanirent is 9.876543. o 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. RIGHT ANGLED TRIANGLES. 857. The principles have now been established, by which, whenever certain parts of a triangle are knowri, 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. 859. Problem. — Given one leg and one angle, to find the other parts. The quotient of one leg divided by the sine of tlie opposite angle is the hypotenuse. The product of oi.e 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 legsi 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. Since 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, Omittinor 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° 30^. Find the other angle and the two legs. Solve this both with and without loga- rithms. •504 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 lino 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 tlie sun r* 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. One angle of a triangle being the supplement of the sum of the other two, when two are known the third may be found by subtraction. Also, the sine bf either angle is 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. 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), a-'=b''+e' — 2b'A'D, and in the second figure (412), a'=b'+c'+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 a-= b--\- e~ — 2bc cos. A. By similar reasoning, it may be shown that b^ = a^ -{- c^ — 2ac cos. B, and c^=: a^+ b^— 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 : TriiT.— 26. 30G PLANE TRIGONOMETRY. 866. Theorem — Expressing the sum of the sides of any triangle hy p, then sin. — = ^^^ lAM-. ^. For, by the formula just demonstrated, COS. A = !-— . 2bc Hence (847, iv). .n4 = ,/,/ yp {yp - a) Similarly, find the cosine and tangent of JB and of JC. PLANE TRIANGLES. 307 867. Theorem — The sides of any triangle are pro- portional to the sines of the opposite angles. That is, a :h :: sin. A : sin. B. For, whether A is acute or obtuse, BD = ABsin. A, and BD = BC-sin. C. Therefore, c sin. A = a sin. C, and a : c :: sin. A : sin. C. Similarly, a : b :: sin. A : sin. B. 868. Theorem. — One side of a triangle is equal to the sum of the products found by multiplying each of the other sides by the cosine of the angle which it forms with the first side. For, AC = CD ± DA = BC-cos. C + BA-cos. A. That is, b = a cos. C -\- c cos. A. 869. Theorem. — The sum of any two sides of a tri- angle is to their difference as the tangent of half the sum ■of the tivo opposite angles is to the tangent of half their difference. By Art. 867, a : b :: sin. A : sin. B. By composition and division, a-{-b : a — b :: sin. A + sin. B : sin. A — sin. B. Hence (849, ix), a+b : a — b :: tan. J(A. + B) : tan. J (A — B.) 308 PLANE TRIGONOMETRY. 8T0. Problem. — Given the sides of a triangle, to find the angles. This rule is derived from the formula for the sine of half an angle (866). From half the sum of the sides, subtract each of the sides adjacent to the required angle ; multiply together these remainders; divide this product by the product of the two adjacent sides, and extract the square root of the quotient. This root is the sine of half the angle sought. The student may write rules for the solution of this problam from the formulas for cos. JA, and tan. JA, and cos. A. For example, the given sides are a = 3457, b = 4209, and c = 6030.4. For finding all the angles, the formula for the tangent of half an angle is the best, because the same numbers are used for every angle. To find the angle C, i^p =6848.2 Jj? -5 =2639.2 .i.p_« = 3391.2 lp — c= 817.8 log. (Ip — a) . = 3.530353 log. ihp—b) . = 3.421472 a.cAog. Ip . . = 6.164424 a.cAog. {yp—c)= 7.087353 tab. log. tan.2 JC = 20.203602 tab. log. tan. JC = 10.101801, ' which is the tab. log. tan. 51° 39' 16'^4. Therefore, the angle C is 103° 18' 33''. In the above calculation, the sum of the logarithms exceeds by 20 the sum required, on account of the arithmetical complement twice used ; but the tabular logarithm of tan.^ JC being also 20 more than the true logarithm of tan.'-* .^C, no correction is necessary. PLANE TRIANGLES. 309 Find in a similar manner the other two angles, and test the result by comparing the sum with 180°. There is another method of solving this problem. By dividing any triangle into two right angled triangles, if the sides are known, the altitude and the segments of the base may be found (328). Then the angles may be cal- culated as in the solutions of right angled triangles. 8T1. Problem — Given two angles and a side, to find the other angle and sides. Find the third angle by subtracting the sum of the given two from 180°. Then find the remaining sides by the formula (867), sin. A : sin. B : : a : J. 8T2. Problem. — Given two sides and an angle opposite one of them, to find the other angles and side. Find the angle opposite the other given side by the formula, a : b : : sin. A : sin. B. Find the third angle by subtraction, and the third side by the formula, sin. A : sin. C : : a : e. When the side opposite the given angle is equal to or greater than the other given side, there can be only one solution (287). When it is less than the other given side, there may be two solutions (291 and 300). This is' called the ambiguous case. The result is indicated by the trigonometrical formula, for the angle is found by its sine ; and for a given sine there are two angles, one acute and one obtuse. 310 ■ PLANE TRIGONOMETRY. The side opposite the given angle may be so small as to make the triangle impossible (300.) This result is also indicated by the trigonometrical solution, for the sine of the angle sought is found to be greater than unity, which is impossible. 873. Problem — Given two sides and the included angle, to find the othef' angles and side. Find the sum of the other angles by subtraction, and the difference of those angles by the formula (869), a + b : a — b :: tan. i(A+B) : tan. i(A — B). Knowing half the sum and half the difference of the two required angles, take the sum of these two quantities for the greater and their difference for the less of the angles. The third side is found as in the preceding problems. This problem may be solved, without logarithms, by the formula (865), a^ =b^^ c^ — 2bc COS. A. AREAS. 874. Theorem — The area of a triangle is equal to half the product of any two sides multiplied by the sine of the included angle. Thus, the area of triangle ABC = ^ be sin. A. For the altitude BD (see last figure) is the product of the side c by the sine of the angle A. The student may now review Art. 390. PLANE TRIANGLES. 311 Take some point C, from which APPLICATIONS. 875. 1. To measure the distance from one point to another, when the line between them can not be passed over with the measuring chain or rod. Let A and B be the two points, both A and B are visible, and such that the lines AC and BC can be measured with the rod or chain. Measure these and the angle C. Then, in the triangle ABC, two sides and the included angle are known, from which the third side AB can be calculated (873). If A and B are visible from each other, as when the obstacle between them is open Water^ then the angles A and B may be ob- served. In that case it is necessary to measure only one of the sides AC or BC ; for, knowing sides may be calculated (871) one side and the angles, the other 2. To find the height and distance of an inaccessible object. Let P be the top of all object, whose distance from, and height above, the point A are required. At A observe the angle PAC, that is, the angle of inclination of the line AP with the plane of the horizon (537 and 563). Then, measure any length AB, on a horizontal line directly towards the object, and at B observe the angle PBC. In the triangle APB, the side AB and the angle A are known ; also the angle ABP, since it is the supplement of PBC; hence, AP can be calculated. Then PC = AP-sin. A, and AC = AP-cos. A; thus determining the height and distance of the object. The angle A is called the angular elevation of the point P as seen at A, the angle PBC being the elevation of the same point as seen at B. If P were below the level of A, the angle thus ob- served would be the angular depressioyi of the object. 312 PLANE TRIGONOMETRY When, as is generally the case, it is inconvenient to measure the line AB "on a horizontal line directly toward the object," meas- ure any length AB in any conve- nient direction; at A, observe the angle PAB, and the elevation PAC; and at B, observe the angle PBA. Then, in the triangle APB, the side AB and the adjacent angles being known, the side AP may be found, and the height and distance of P calculated as before. 3. To find the distance between two visible but inaccessi- ble objects. Let P and N be the objects, C and B two accessible points from vrhich both the objects are visi- ble. At C observe the angles PCN and NCB, and if C, B, N, P are not all in the same plane, ob- serve also the angle PCB. At B observe the angles PBC and NBC. Measure CB. In the triangle PCB, the side CB and its adjacent angles being known, the side CP can be found. In the triangle NCB, the side CB and its adjacent angles being known, the side CN may be found. Then, in the triangle PCN, the sides CP and CN and their included angle being known, the side PN may be found. 4. To find the ividth of a river without an instrument for observing angles. Let P be a visible point on the further bank, and A a point opposite to it on this side. Take B, C, and D, any convenient ac- cessible points, such that B, A, and P are in a straight line, and C, D, and P are in a straight line ; and measure AB, AC, AD, BD, and CD. PLANE TRIANGLES. 313 All the sides of the triangles ABD and ACD being known, the angles BAD and ADC may be found, and hence their supplements DAP and ADP. Then, from the side AD and the two adjacent angles of the triangle ADP, the side AP may be calculated. EXERCISES. 876. — 1. The sides of a triangle being 70, 80, and 100, what are the angles? 2. Two angles of a triangle are 76° 30^ 23^^ and 54° IV 5V', and the side opposite the latter is 40.451.; find the other sides. 3. Two sides of a triangle are 243.775 and 907.961, and the angle opposite the former is 15° 16^ 17^^; find the other parts. 4. Two sides of a triangle are 196.96 and 173.215, and the in- cluded angle 40°; find the other angles and side. 5. From a station B, at the base of a mountain, its summit A ia seen at an elevation of 60°; after walking one mile towards the summit, up a plane, making an angle of 30° with the horizon to another station C, the angle BCA is observed to be 135°. Find the height of the mountain. 6. Two sides of a parallelogram are 25 and 17.101, and one of its diagonals 38.302; find the other diagonal. 7. A person observing the elevation of a spire to be 35°, advances 80 yards nearer to it, and then finds the elevation is 70° ; required the height of the spire. 8. From the top of a tower whose height is 124 feet, the angles of depression of two objects, lying in the same horizontal plane with the base of the tower and in 48°; what is their distance apart? Trio:. —27, 314 SPHERICAL TRIGONOMETRY CHAPTER XIII. SPHERICAL TRIGONOMETRY. 877. Spherical Trigonometry is the investigation of the relations which exist between the sides and angles of spherical triangles. Each side of a spherical triangle being an arc, is the measure of an angle. It has the same ratio to the whole circumference that its angle has to four right angles. It may be measured by degrees, minutes, and seconds, as an angle is measured. It has its sine, tangent, and other trigonometrical functions ; it being understood that the sine, etc., of an arc are the sine, etc., of the angle at the center which that arc subtends. The propositions Avhich express the relations between the sides and angles of a spherical triangle, apply equally well to the faces and diedral angles of a triedral (766 and seq.). If the investigation were made from this point of view, as it well might be, the proper title of the subject would be Trigonometry in Space. THREE SIDES AND AN ANGLE. 878. Theorem — The cosine of any side of a spherical triangle is equal to the product of the cosines of the other two sides, increased by the product of the sines of those sides and the cosine of their included angle. SPHERICAL ARCS AND ANGLES. 315 Let ABC be a spherical triangle, the center of the sphere, AD and AE tangents re- spectively to the arcs AB and AC. Thus, the angle EAD is the angle A of the spherical triangle ; the angle EOD is measured by the side a, and so on. From the triangles EOD and EAD (865), DE'= 0D'+ OE — 20D0E cos. a, DE'= AD'+ AE — 2AD- AE cos. A. By subtraction, the triangles OAE and OAD being right angled, = 20T+ 2AD-AE cos. A — 20D-0E cos. a; rru r OA OA , AE AD . Therefore, cos. a = -^ . ^^+ -^-- . ^ cos. A; that is. cos. a = COS. b COS. c + sin. h sin. e cos. A. In the above construction, the sides which contain the angle A are supposed less than quadrants, since the tan- gents at A meet OB and OC produced. That the for- mula just demonstrated is true when these sides are not less than quadrants, is shown thus : 316 SPHERICAL TRIGONOMETRY. Suppose one of the sides greater than a quadrant, for example, AB. Produce BA and BC to B', and repre- sent AB^ and CB' by d and a' respectively. Then, in the triangle AB'C, as just demonstrated, cos. a'= cos. h cos. c' -\- sin. h sin. c' cos. B'AC. Now, a', c\ and B'AC are respectively supplements of a, c, and BAG. Hence, cos. a = COS. h cos. c -\- sin. b sin. c cos. A. When botJh the sides -which contain the angle A arc greater than quadrants, produce them to form the aux- iliary triangle, and the demonstration is similar to the last. Suppose that one of the sides b and c is a quadrant, for example, e. On AC, produced if necessary, take AD equal to a quadrant, and join BD. Now A is a pole of the arc BD (754), and therefore that arc measures the angle A (760). Then, from the triangle BCD, cos* a = cos. CD cos. BD + sin. CD sin. BD cos. CDB ; but — CD is the complement of b, BD measures A, SPHERICAL ARCS AND ANGLES. 317 and CDB is a right angle. Hence, this equation be- comes, COS. a = sin. h cos. A, and the formula to be demonstrated reduces to this, when c is a quadrant. The proposition having been demonstrated for any angle of any spherical triangle, cos. h = COS. a COS. c+ sin. a sin. c cos. B, cos. c = COS. a COS. h + sin. a sin. h cos. C. These have been called the fundamental equations of Spherical Trigonometry. By their aid, when any three of the elements of a spherical triangle are known, the others may be calculated. A SIDE AND THE THREE ANGLES. 879. Since the formulas just demonstrated are true of all spherical triangles, they apply to the polar triangle of any given triangle. Therefore, denoting the sides and angles of the polar triangle, by accenting the letters of their corresponding parts in the given triangle, COS. a' = cos. h' cos. c' -\- sin. h' sin. c' cos. A', but a'=180° — A, 5^=180° — B, and A' = 180°— «, etc. (777). Substituting these values of a', h', etc., COS. (180°— A) = COS. (180° — B) cos. (180°— C) + sin. (180° — B) sin. (180°— C) cos. (180° -«). 318 SPHERICAL TRIGONOMETRY. Reducing (829), and changing the signs, COS. A = — COS. B COS. C + sin. B sin. C cos. a. Similarly, cos. B = — cos. A COS. C + sin. A sin. C cos. 6, cos. C = — COS. A cos. B + sin. A sin. B cos. e. None of the above formulas is suited for logarithmic calculation. FORMULAS FOR LOGARITHMIC USE. 880. Let p represent the perimeter, that is, p == a+b+c. By transposing and dividing the fundamental formula (878), . COS. a cos. 6 cos. C rT^^ o fOAr \ cos. A = : — Therefore (845, iv), sin. sin. c ^ . sin.6 sin.c+cos.5 cos.c — cos.a cos.(6— c)— cos.a 1— cos.A= ; ^ . = ;^ — -A sm.osin. c. sm. osin.c. Substituting for this numerator its value (849, viii), and dividing by 2, J-(l - COS. A) = si"- K^+ ^ — 0) sin. ),{a-h-\-c) ^ sin. b sin. c. Substituting p for its value, and extracting the root (847, IV), Sin ■^ _ pin. (ij? — 6)sin. (Vp — c) 2 ^ sin. h sin. c. SPHERICAL ARCS AND ANGLES. 319 To find tbe value of the cosine of half the angle, sin. 6 sin.c?— C0S.6 cos.c-fcos.a cos. a — cos.(6+6') l-fcos.A=z- s'm.b sin. c. sin. b sin. c. Hence, cos.^ = >"" ^-^ f -^^^'"^ 2 ^ sin. 6 sin. c. Dividing sin. ^ A by cos. J A, tan. ^ = J^^'^l^"" ^^ ^'"' ^^P~''^ " ^ sin 1- Q sin. Ip sin. (Ip — a) Find the analogous formulas for the sine, cosine, and tangent of JB and of JC. 881. Let E represent the spherical excess, that is, E=A + B + C-180°. By reasoning upon the polar triangle as in the pre- ceding article, the formula for the sine of half an angle becomes 180°— a _^ / sin. A(lcS()°— A+B— C) sin. ^(180°— A— B+C) ^ sin. (180'— B) sin. (180° — C) r—a Isin. I but sin. i(180°— a) = sin. (90°— »«) = cos. },a, and sin. J(180° — A + B — C) = sin. (B - JE), etc. Therefore, cos. ^ ^ J^^"' (^- ^^) ^'"- ^^- ^^) . 2 ^^ sin. B sin. C Similarly, from the formula for the cosine of half the angle. sin - = J sin.|Esin.(A-^-E) '2 ^Z sin. B sin. C 320 SPHERECAL TRIGONOMJ]TRY. Hence, tan. sin. JE sin. (A — JE) \sin. (B— IE) sm. (C — ^E) * Since E must be less than 360° (771), sin. JE is pos- itive; and since sin. ^a is a real quantity, sin. (A — ?,E) 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 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. c sin. B. Similarly, PD = OP • sin. b sin. C ; therefore, OP * sin. c sin. B = OP • sin. 6 sin. C. sin. B sin. b sin. C sin. e or sin. B : sin. C : : sin. b : sin. c SPHERICAL ARCS AND ANGLES. s 321 The figure supposes h, 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 hy 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 hy the cotangent of the angle opposite the first side. We have (878 and 882), COS. a = COS. h COS. c+ sin. h sin. c cos. A, cos. c ==. COS. a cos. b -}- sin. a sin. b cos. C, sin. a sin. C sm. c = sin. A Eliminate c by substituting these values of cos. c and sin. c in the first equation, , , , . . , ^,, , , sin.asin.icoa.Asin.C COS. a = (cos.a C0S.6 + sin.a sin.6 cos.C ) cos.o H -. -. ; ^ ^ sm. A transposing and reducing, since 1 — cos.^5 = sin.^ 6, cos.asin.^6=sin.a sin.^cos.i cos.C+sin.a sin.^cot.Asin.C ; dividing by sin. a sin. b, cot. a sin. b = cos. b cos. C + cot. A sin. C. 322 SPHERICAL TRTGONOMETRY. The demonstration being general, may be applied to other angles and sides, making these five additional formulas : cot. h sin. a = cos. a cos. C + cot. B sin. C, cot. h sin. c = cos. c cos. A + cot. B sin. A, cot. c sin. h = cos. b cos. A+ cot. C sin. A, cot. e sin. a = cos. a cos. B + cot. C sin. B, cot. a sin. c = cos. e cos. B -|- cot. A sin. B. FORMULAS OF DELAMBRE. 884. Putting J A and JB for A and B respectively, in formula I, Art. 845, sin. J (A + B) = sin. JA cos. JB + cos. ^A sin. ^B. Substitute the values of the factors of the second mem- ber, as found in Art. 880, . A+B sin.(Jp— a)+sin.(Jjt?— 5) k'm.lp sin. {}^p~c) ^ sm. — - — = ; \— : ; J ; 2 sm. c ' sm. a sm. o but, . N . /, IN . /^ a—h\ . Ic a—h\ (849,1), .... =2sin. ic cos. |(a_5), and (847, i), sin. c=2sin. Jc cos. ^c. Substituting these values, also cos. \Q> for the radi- cal (880), . A + B cos. Ua—h) ,^ sm. ^- — = -^ COS. ?,C, 2 cos.-Jc sm or. sin. J(A + B) _ COS. l{a—h) COS. JC COS. \c SPHERICAL ARCS AND ANGLES. 323 Similarly, by beginning with formulas ii, iii, and iv of Art. 845, we find, sin. |(A — B) _ sin, ^(a — h) COS. JC ~~ sin. ^c COS. |(A + B) _ COS. Ija + h) sin. JC COS. Ic cos.|(A — B) sin. ^(a+ 5) sin. JC 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. l(a — b) cot. JC COS. 2(^+ ^) tan. |(A — B) _ sin. » (a —b) cot. AC sin. i(a + 5) tan. l{a+b) __ cos. i(A — B) tan. ic ~" COS. i(A + B) ' tan. ^(a — b) __ sin. |(A — B) tan. |c """ sin. J(A + 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. l(a — h) and cot. |C are necessarily positive; hence, tan. J(A-j- B) and COS. J(«+ b) are of the same sign; thus, ^(A+B) and li^"^^) ^^^ either both less or both greater than ninety degrees. In the second of the above equations, sin. Jl'^ ~)~ ^) and cot. jC are positive; hence, tan. J(A — B) and sin. l{a — b) have the same sign ; thus, ^(A — B) and l(a — h) 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. 8Sf. — 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. EIGHT 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. 878, 1st formula, cos. a = cos. b cos. e, . (i.) Art. 879, 1st formula, cos. a = cot. B cot. C, . (ii.) Art. 882, sin. b = sin. a sin. B 1 / v. " " sin. a = sin. a sin. C ^ Art. 883, 1st formula, tan. b = tan. a cos. CI / >. " " 6th formula, tan. c = tan. a cos. B J RIGHT ANGLED TRIANGLES. 325 Art. 883, 3rd formula, tan. 6 = sin. c tan. B 1 , s " " 4th formula, tan, c = sin. b tan. C J Art. 879, 2nd formula, cos. B = sin. C cos. b\ , . " ^' 3rd formula, cos. C = sin. B cos. c j In deducing li, 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. 800. 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 TRIG0N0MI:T11Y. 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 ojblique angle^ to find the other angle and the sides. Find the other oblique angle by formula ii, the side opposite the given angle by iii, and the adjacent side by 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. 102. 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 ayid the adjacent ob- lique angle, to find the other sides and angle. Find the hypotenuse by IV, the other side by v, and the otlier 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 III. 895. Problem — Given the tivo oblique angles, to find the three sides. Find the hypotenuse by li, and the other sides by VI. 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 Give7i a side and its opposite angle, to find the other sides and angle. Find the hypotenuse by ill, the other side by V, and the other nngle by vi. 328 SPHERICAL TRIGONOMETRY. Here the triangle is ambiguous, as all the parts are found by their sines. Sup- pose BAG 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 rigtit 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 b and c, the sides a and c may be produced to B', making the isosceles triangle 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 supplementary. 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° 5F 8^^ and 35° 23^ 30^^; find the hypotenuse and the angles. 3. The hypotenuse is 71° W ZT' 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 bo 90°, and conversely. 6. The sides are 90°, 76° 49' 55^', 41° 45M6''; 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. — Giveri 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 Triir._28. 330 SPHERICAL 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 Delambre. For example, given the side a = 76° 35' 36'', h = 50° 10' 80", and the angle C = 34° 15' 3". By the 1st analogy, tan.KA+B)=cot.^c ;;;;^g-g . tab. log. cot. i-C . . . = 10.511272 tab. log. cos. J(a — 6) . = 9.988355 a. c. tab. log. cos. \{a-\-h) = 0.348717 tab. log. tan. |(A + B) = 10.848344 . • . J(A + B) = 81° 55' 47" By the 2nd analogy, ^sin. l(a — h) tan. KA — B) = cot. ?,C^. f, , '• tab. log. cot. JC . . . = 10.511272 tab. log. sin. J (a — ^^) . = 9.358899 a, e. tab. log. sin. l{a+h) = 0.048648 tab. log. tan. i(A— B) = 9.918819 .•.J(A— B) =39° 40' 38" Hence, A = 121° 86' 20", and * B = 42° 15' 14". SPHERICAL TRIANGLES. 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. J(A+B) tan. 3(?=tan. |(a — h) sin. i(A-B) tab. log. tan. J(a — 6) . = 9.370544 tab. log. sin. |(A + B) . = 9.995677 a, c, tab. log. sin. » (A — B) = .194877 tab. log. tan. Jc . . . = 9.561098 002. 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. 003. Problem — Given two sides and an angle opposite one of them, to find the other 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 is 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 h are given, h being greater than <2, 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 h 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 thf.t the triangle is impossible. OOir. Problem. — Given two ayigles and a side opposite one of them, to find 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. ^33 EXERCISES. 905.— 1. The sides are 60° 4^ 54^^, 135° 49^ 2(r^, and 146° SV \y^] 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^ V, 36° 45^ 28^^ and 130° y 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^ y, 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° 42^ 43^^, 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 TRlGONOxMETRY. 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 CoxMMON 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, " " " 1000 " 3, etc. Again, the logarithm of 1 is 0, U ii ii ii " Jo or .1 " -1, " xJoOr.Ol "-2, etc. 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 = 1.301080 = — 1 + .301030, log .08 = 2.903090 = — 2 + .908090. If any number is between 1 and 10, its logarithm is between 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 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 hoiv 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 OF LOGARITHMS. 909. Let c represent the characteristic and d the mantissa of any logarithm, and let N represent the number. By the definition, 10^+^ = N. Multiplying by 10, 10« + i-J-^= ION. That is, if (?-}" ^ ^s th® logarithm of N, c-f- 1 + ^ 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 muLiplied 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 tlie log- arithm of every number from 1000 to 11000. It follows LOG Alii iHMS. 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 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 o^ 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 OF A GIVEN NUMBER. OlO. 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 ; then prefix the charac- Triff.— 29. 838 TRIGONOMETRY. teristic 3; and the logarithm of 1245 is 3.095169. Sim- ilarly, log. 124500 = 5.095169, 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 difiierence. 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 difierence 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 LOGAKIiHuS. 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 TRIGONOMETRY. Expressing the fraction |J 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 grea'ter 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 a; and y represent the logarithms of M and N respectively. By the definition, 10"^ = M. Similarly, 10^ =N. Multiplying the first by the second, 10^ + ^= MXN. . Dividing the first by the second, 10^-^=M-f-N. That is, x-\-y is the logarithm of the product of M multiplied by N, and x — ^ is the logarithm of the quo- tient of M divided by N. Hence, the following rules for multipHcation and division by logarithms : To multiply, add the logarithms of the factors. The sum is the logarithm of the product. 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 = .301030, log. .000314 = 4.496930, The remainder, 3.804100 is the logarithm of 6369.43, the quotient, true 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. 342 TRIGONOMETRY. 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 . . . n 3456 X 89123 For example, to find the value of Q7f^Q w 409-1 • log. 3456 = 3.538574, log. 89123 = 4.949990, a. clog. 9753 =6.010862, a. clog. 4321 =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. Then, 10^ = N. Raising both members to the x^^ power, 10^^=N=^. Taking the x^^ root of both members, 10^= v^N. LOGARITHMS. 343 That is, xy is the logarithm of the x^^ power of N, and I is the logarithm of the x^^ root of N. Hence, these rules for involution and evolution by logarithms : To raise a number to a required power, multiply its logarithm hy 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 aflfected 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 J. 015. 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 1^3 and |/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 onlj 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. 916. — 1. Calculate the value of these expressions: l/8932 X .045726, -|/7609 -r-^Io; y^l3^ X 14«-f- 1.25«. 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? Additional exercises may be made upon the formulas of Art. 807. TABLES 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. ■ 1 Num. 100, Log. 000. TABLE I.— LOGARITHMS N. 100 1 2 3 4 5 6 7 8 9 D. 000000 0434 0868 1301 1734 2166 2598 3029 3461 3891 432 101 4321 4751 5181 5609 6038 &466 6894 7321 7748 8174 428 102 8600 9026 9451 9876 .0300 .0724 .1147 .1570 .1993 .2415 424 103 012837 3259 3680 4100 4521 4940 5m 5779 6197 6616 420 104 7033 7451 7868 8284 8700 9116 9532 9947 .0361 .0775 416 105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 107 9384 9789 .0195 .0600 .1004 .1408 .1812 .2210 .2619 .3021 404 108 033421 3826 4227 4628 5029 5430 58,'M) 6230 6629 7028 401 109 7426 7825 8223 8620 9017 9414 9811 .0207 .0602 .0998 397 110 041393 1787 2182 2.576 2969 3362 3755 4148 4.540 4932 393 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 390 112 9218 9606 9993 .0380 .0766 .1153 .1538 .1924 .2309 .2694 386 113 053078 3463 3846 42:30 4613 4996 5378 5760 6142 6524 382 114 6905 7286 7666 8046 8426 8805 9185 9563 9942 .0320 379 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 117 8186 8557 8928 9298 9668 .0038 .0407 .0776 .1145 .1514 369 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 5182 367 119 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 364 120 079181 9543 9904 .0266 .0626 .0987 .1347 .1707 .2067 .2426 360 121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 358 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 356 123 9905 .0258 .0611 .0963 .1315 .1667 .2018 .2370 .2721 .3071 352 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 349 125 096910 7257 7604 7951 8298 8644 8990 9335 9681 .0026 346 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 344 127 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 341 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 .0253 338 129 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 130 113943 4277 4611 4944 5278 5611 5943 6276 6608 6940 333 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 .0245 330 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 328 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 325 134 7105 7429 T753 8076 8399 8722 9«5 9368 9690 .0012 323 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 136 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 137 6721 7037 7354 7671 7987 a303 8618 8934 9249 9564 310 138 9879 .0194 .0508 .0822 .1136 .1450 .1763 .2076 .2389 .2702 313 139 143015 3327 3639 3951 4263 4574 4885 5196 5507 6818 311 140 146128 6438 6748 7058 7367 7676 7986 8294 8603 8911 309 141 9219 9527 9835 .0142 .0449 .0756 .1063 .1370 .1676 .1982 307 142 152288 2594 2900 3205 3510 3815 4120 4424 4728 ,503'? 305 143 5336 5(>40 5943 6246 6549 6852 7154 7457 7759 8061 303 144 8362 8664 8965 9266 9567 9868 .0168 .0469 .0769 .1068 301 145 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 147 7317 7613 7908 S'Mi 8497 8792 9086 9380 9674 9968 294 148 17Qg62 0555 0848 1141 14;^ 1726 2019 2311 2603 2895 293 149 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 N. 1 2 3 4 5 6 7 8 9 D. 346 OF NUMBERS. Num. 199, Log. 300. N. 1 2 3 4 5 6 7 8 9 D. 150 176091 6381 6670 69.59 7248 7.536 7825 8113 8401 8689 288 151 8977 9264 ft552 9839 .0126 .0413 .0699 .0986 .1272 .1558 287 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 153 4691 4975 52.59 5542 5825 6108 ft391 6674 6956 72;^9 283 154 7521 7803 8084 8366 8647 8928 9209 9490 9771 .0051 281 155 19a33^ 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 156 31^ 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 157 5900 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 1.58 8657 8932 9206 9481 9755 .0029 .0303 .0577 .0850 .1124 274 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 272 160 204120 4391 4663 4934 5204 5475 5746 6016 6286 6556 270 161 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 162 9515 9783 .0051 .0319 .a586 .0853 .1121 .1388 .1654 .1921 267 163 212188 2454 2720 2986 32.52 3518 3783 4049 4314 4579 266 164 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 165 217484 7747 8010 8273 8536 8798 9060 9323 9585 9816 263 166 220108 o;?70 0631 0892 11.53 1414 1675 1936 2196 2456 261 167 2716 2976 32;^ 3496 3755 4015 4274 4533 4792 5051 259 168 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 .0193 256 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 255 171 2996 3250 .3.504 37.57 4011 4264 4517 4770 5023 5276 253 172 5528 5781 60.« 628.5 6,537 6789 7041 7292 7514 7795 252 173 8046 8297 &548 8799 9049 9299 9550 9800 .0050 .0300 250 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 175 243038 3286 3534 3782 4a30 4277 4525 4772 5019 5266 248 176 5513 5759 6006 62.52 6499 6745 6991 7237 7482 7728 246 177 7973 8219 8464 8709 8954 9198 9443 9687 9932 .0176 245 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 243 179 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 255273 5.514 57,>5 5996 6237 &477 6718 6958 7198 7439 241 181 7679 7918 81.58 8398 8637 8877 9116 9355 9594 9m 239 182 260071 0310 0.548 0787 1025 1263 1501 1739 1976 2214 238 im 2451 2688 292;) 3162 3399 3636 3873 4109 4346 4582 237 m 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 185 267172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 186 9513 9746 9980 .0213 .0446 .0679 .0912 .1144 .1377 .1609 233 187 271842 2074 2306 2.538 2770 3001 3233 3464 3696 3927 232 188 4158 4389 4620 48,50 5081 .5311 5542 5772 6002 6232 2JJ0 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 190 278754 8982 9211 9439 9667 9895 .0123 .0351 .0578 .0806 228 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 192 3301 3527 37,53 3979 4205 4431 4656 4882 5107 5332 228 193 5557 5782 6007 62;^2 6456 6681 6905 7130 7354 7578 225 194 7802 8026 8249 »473 8696 8920 9143 9366 9589 9812 223 195 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 222 196 2256 2478 2699 2920 3141 3363 3584 3804 . 4025 4246 221 197 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 198 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 199 8853 9071 9289 9507 9725 9943 .0161 .0378 .0595 .0813 218 N. 1 2 3 4 5 6 7 8 9 D. ■ ■ ■ ■ , ..J 347 Num. 200, Log. 301. TABLE T.— LOGARITHMS j N. 1 2 3 4 5 6 1 1 7 8 9 D. 200 30io;» 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 201 3196 3412 3628 3844 4a59 4275 j 4491 4706 4921 51,36 216 202i 5aji 5566 5781 5996 6211 6425 &m 6854 7068 7282 215 203 7496 7710 7924 8137 83,51 8564 8778 8991 9204 9417 213 204 9630 9843 .00.56 .0268 .0481 .0693 .0906 .1118 .1330 .1542 212 205 3117&4 1966 2177 2389 1^600 2812 3023 3284 3445 3656 211 206 3867 4078 4289 4499 4710 4920 5l;30 5340 5551 5760 210 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 20f) 208 8063 8272 8481 8689 8898 9106 9314 9522 9730 99;^ 208 209 320146 0354 0562 0769 0977 1184 1391 1598 1805 2012 207 210 3t^l9 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 2a5 212 6336 6541 6745 6950 7155 7a59 7563 7767 7972 8176 204 213 83801 8583 8787 8991 9194 9398 9601 9805 .0008 .0211 203 214 380414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 332438 2640 2842 3044 3246 3447 3649 8850 40,51 4253 202 216 4454 46.55 4856 50.57 5257 5458 5658 5859 6059 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 8a58 8257 200 218 8456 8656 88.55 9054 9253 9451 96,50 9849 .0047 .0246 199 219 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 198 220 342423 2620 2817 3014 3212 a409 3606 3802 3999 4196 197 221 4392 4589 4785 4981 5178 5374 5570 5766 5962 61,57 196 222 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 223 8305 8500 8694 8889 9083 9278 9472 9666 9860 .0054 194 224 350248 0442 0636 0829 1023 1216 1410 1603 1796 1989 194 225 352183 2375 2568 2761 29.54 3147 aas9 a582 3724 3916 193 226 4108 4301 4493 46&5 4876 5068 5260 5452 5643 5834 192 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 228 79a5 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 229 9835 .0025 .0215 .0404 .0593 .0783 .0972 .1161 .ia50 .1539 189 230 361728 1917 2105 2294 2482 2671 2859 8048 3236 3424 188 231 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 232 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 234 9216 9401 9587 9772 99.58 .0143 .0828 .0513 .0698 .0883 185 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 184 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 %^1 4748 4932 5115 5298 5481 56ft4 5846 6029 6212 6394 183 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 .0030 181 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 181 241 2017 2197 23/V 2557 2737 2917 3097 ;3277 a4,56 36^36 180 242 3815 3995 4174 4,3,5;} 4,533 4712 4891 5070 5249 ,5428 179 213 5606 5785 5964 6142 ft^l 6499 6677 6856 7oai 7212 178 244 7390 7568 7746 7923 8101 8279 8456 8634 8811 8989 178 245 389166 9343 9.520 9698 9875 .00,51 .0228 .0405 .0582 .0759 177 246 3909a5 1112 1288 1464 1641 1817 1993 2169 2345 2.521 176 247 2697 2873 3048 3224 aioo a575 3751 3926 4101 4277 176 248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 175 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 N. 1 2 3 4 5 6 7 8 9 D. 348 OF NUMBERS. Num. 299, Log .476. N. 1 2 3 4 5 6 7 8 9 D. 250 397940 8114 8287 8461 8634 8808 8981 91.54 9328 9501 173 251 9674 9847 .0020 .0192 .0365 .a538 .0711 .0883 .1056 .1228 173 2.52 401401 1573 1745 1917 2089 2261 24:33 2t>05 2777 2949 172 253 3121 3292 3464 3G3.5 3807 3978 4149 4320 4492 46&S 171 251 4834 5005 5176 5346 5,517 5688 5858 6029 6199 6370 171 255 406.540 6710 6881 7a51 7221 7,391 7561 7731 7901 8070 170 256 8240 8410 &579 8749 8918 9087 9257 9126 9.5a5 9764 169 257 9933 .0102 .0271 .0440 .0()09 .0777 .0946 .1114 .1283 .1451 169 258 411620 1788 19.56 2124 2293 2461 2629 2796 2964 81,32 168 259 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 167 260 414973 5140 5307 5474 5641 5808 5974 6141 6308 6474 167 261 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 263 99.56 .0121 .0286 .0451 .0616 .0781 .0^ .1110 .1275 .14:39 165 264 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 1&4 265 423246 3410 a574 3737 3901 4065 4228 4392 4555 4718 1&4 286 4882 5045 5208 5371 om 5697 5860 6023 6186 6349 163 267 6511 6674 68;^ 6999 7161 7321 7486 7648 7811 7973 162 268 813.5 8297 8459 8621 8783 8944 9106 9268 9429 9591 162 269 9752 9914 .0075 .0236 .0.398 .0559 .0720 .0881 .1012 .1203 161 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 161 271 2969 3130 3290 3450 3610 3770 3930 4090 4249 4409 160 272 4569 4729 4888 5048 5207 5,367 5526 5685 5844 6004 159 273 6163 6322 6481 6640 6799 69.57 7116 7275 74.33 7592 159 274 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 275 439333 9491 9648 9806 9964 .0122 .0279 .0437 .0594 .0752 1.58 276 440909 1066 1224 1381 1.538 1695 1852 2009 2166 2323 1.57 277 2480 2637 2793 29.50 3106 32<)3 »119 a576 3732 :3889 1.57 278 4045 4201 4357 4513 4669 4825 4981 5137 .5293 5149 156 279 5604 5760 5915 6071 6226 6382 6,537 6692 6818 7003 155 280 447158 7313 7468 7623 7778 7933 8088 8242 8397 S>52 155 281 8706 8861 9015 9170 9,324 9178 9633 9787 9941 .0095 1-A 282 450249 0403 0557 0711 0865 1018 1172 1326 1479 1633 154 283 1786 1940 2093 2247 2400 25,53 2706 2859 3012 3165 153 ay 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 2a5 454845 4997 51.50 5302 5454 5606 5758 5910 6062 6214 152 286 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 287 7882 8033 8184 83;56 8487 86:38 8789 8940 9091 9212 151 288 9392 9;^3 9694 9845 999.5 .0146 .0296 .0447 .0,597 .0748 151 289 460898 1018 1198 1348 1499 1649 1799 1948 2098 2218 150 290 462398 2548 2697 2847 2997 3146 3296 m5 3594 3744 150 291 3893 4042 4191 4;^0 4490 46:39 4788 4936 5085 5234 149 292 5383 5532 5680 5829 5977 6126 6274 ftl23 6571 6719 149 293 6868 7016 7164 7312 7460 760cS 7756 7904 8052 8200 148 m 8^47 849,5 8643 8790 8938 9085 923:3 9380 9527 9675 148 29.5 469S22 9969 .0116 .02ft3 .0410 .a5.57 .0704 .0851 .0998 .1145 147 290 471292 1488 1.585 1732 1878 2025 2171 2318 2464 2610 146 297 27.56 290:3 3049 3195 3341 3487 3633 3779 3925 4071 146 298 4216 4332 4508 4&53 4799 4944 5090 5235 5381 5526 146 299 5671 5816 5902 6107 6252 6397 6542 6687 6832 6976 145 N. 1 2 3 4 5 6 7 8 9 D. 349 Num. 300, Log. 477. TABLE I.— LOGARITHMS N. 1 2 3 4 5 6 7 8 9 D. 300 477121 7266 7411 7555 7700 7Mi 7989 8133 8278 8422 145 301 8566 8711 8855 8999 9143 9287 9431 9575 9719 986:3 144 302 480007 0151 0294 0438 0582 072.5 0869 1012 1156 1299 144 303 1443 1586 1729 1872 2016 21.59 2302 2445 2588 2731 143 304 2874 3016 3159 3302 3445 a587 3730 3872 4015 4157 143 305 484300 4442 45a5 4727 4869 5011 51.53 5295 5437 5579 142 306 5721 5863 6005 6147 6289 6430 a572 6714 68,55 6997 142 307 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 308 8551 8692 88.33 8974 9114 9255 9396 9537 9677 9818 141 309 9958 .0099 .0239 .0380 .0520 .0661 .0801 .0941 .1081 .1222 140 310 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 140 311 2760 2900 3040 3179 3319 3458 3.597 3737 3876 4015 139 312 41.5.5 4294 4433 4.572 4711 48,50 4989 5128 5267 5406 1,39 313 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 139 314 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 315 498311 8448 8586 8724 8862 8999 9137 9275 9412 9550 138 316 9687 9824 9962 .0099 .0236 .0374 .0511 .0648 .078,5 .0922 137 317 501059 1196 ims 1470 1607 1744 1880 2017 2154 2291 137 318 2427 2.564 2700 2837 2973 3109 3246 3382 3518 3655 136 319 3791 3927 4063 4199 43a5 4471 4607 4743 4878 6014 136 320 5051.50 5286 5421 55.57 5693 5828 5964 6099 6234 6370 136 321 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 135 322 7856 7991 8126 8260 8395 8530 8664 8799 89:34 9068 136 323 9203 9337 9471 9606 9740 9874 .0009 .0143 .0277 .0411 134 324 510545 0679 0813 0947 1081 1215 1349 1482 1616 1750 134 32.5 511883 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 320 3218 3351 3484 3617 3750 38m 4016 4149 4282 4415 133 327 4.S48 4681 4813 4946 5079 5211 5344 5476 5609 6741 133 328 5874 6006 6139 6271 6403 6535 6668 6800 69,32 7064 132 329 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 518514 8646 8777 8909 9040 9171 9303 9434 9566 9697 131 331 9828 9959 .0090 .0221 .0353 .0484 .0615 .0745 .0876 .1007 131 332 5211.38 1269 1400 1.530 1661 1792 1922 2053 2183 2314 131 m 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 334 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 130 335 52.5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 129 336 6a39 6469 6598 6727 6856 698.5 7114 7243 7372 7501 129 337 76;^ 77,59 7888 8016 8145 8274 8402 85:31 8660 8788 129 338 8917 9045 9174 9302 9430 9559 9687 9815 9943 .0072 128 339 530200 0328 0456 0584 0712 0840 0968 1096 1223 1361 128 340 531479 1607 1734 1862 1990 2117 2245 2372 2500 2627 128 341 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 127 342 4026 4153 4280 4407 45^ 4661 4787 4914 5041 6167 127 343 6294 5421 5547 5674 5800 .5927 605:3 6180 6306 64:32 126 344 6558 6685 6811 69;^ 7063 7189 7315 7441 7667 7693 126 345 537819 7945 8071 8197 a322 8448 8574 8699 8825 8951 126 346 9076 9202 9327 9452 9578 9703 9829 9954 .0079 .0204 125 347 540329 04.55 orm 07a5 om 0955 1080 1205 1330 1454 125 348 1579 1704 1829 1953 2078 ^W0.'{ 2327 24.52 2576 2701 125 349 2825 2950 3074 3199 3323 3447 a571 3696 3820 3944 124 N. 1 2 3 4 5 6 7 8 9 D. 1 350 OF NUMBERS. Nnm. 399, Log. 601. N. 1 2 3 4 5 6 7 8 9 D. a50 544068 4192 4316 4440 4564 4688 4812 4936 5060 5183 124 351 5307 5431 55.55 5678 5802 5925 6019 6172 6296 6419 124 a52 6543 6666 6789 6913 70.36 7159 7282 7405 7529 7652 123 353 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 123 m 9003 9126 9249 9371 9494 9616 9739 9861 9984 .0106 123 355 550228 oa5i 0473 0595 0717 0840 0962 1084 1206 1328 122 356 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 a57 2668 2790 2911 30.33 3155 3276 3398 3519 3640 3762 121 a58 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 359 5094 5215 5336 54.57 5578 5699 5820 5940 6061 6182 121 360 556303 6423 6.544 6664 6785 69a5 7026 7146 7267 7387 120 361 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 120 362 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 363 9907 .0026 .0146 .0265 .038.5 .a5W .0624 .0743 .0863 .0982 119 364 561101 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 365 FA9fm 2412 2.531 2650 2769 2887 3006 3125 8244 8362 119 366 3481 3600 3718 3837 39.55 4074 4192 4311 4429 4548 119 367 4666 4784 4903 5021 5139 52.57 5;37(i 5494 5612 5730 118 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 369 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 568202 8319 84.36 8.5.54 8671 8788 8905 9023 9140 9257 117 371 9374 W91 9608 972.5 9842 99.59 .0076 .0193 .0309 .0426 117 372 57a543 0660 0776 0893 1010 1126 1243 ia59 1476 1592 117 378 1709 1825 1942 20.58 2174 2291 2407 2523 2639 2755 116 374 2872 2988 3104 3220 3336 ^452 3568 3684 3800 3915 116 375 574031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 376 5188 5303 5419 5oM 5650 576,5 5880 5996 6111 6226 115 377 6341 64.57 6.572 6687 6802 6917 7032 7147 7262 7377 115 378 7492 7607 7722 7&« 7951 8066 8181 8295 8410 8525 115 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 579784 9898 .0012 .0126 .0241 .0355 .0469 .0583 .0697 .0811 114 381 58092.5 1039 1153 1267 1381 1495 1608 1722 1836 1950 114 382 2063 2177 2291 2404 2518 2631 2745 28o8 2972 3085 114 38:3 3199 asi2 3426 3539 36.52 3765 3879 3992 4105 4218 113 381 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 385 5&5461 5.574 5B86 5799 5912 6024 61.37 6250 6362 6475 113 386 6587 6700 6812 692.5 70;37 7149 72(i2 7374 7486 7599 112 V 387 7711 7823 79.3,5 8017 8160 8272 8384 8496 8608 8720 112 as8 8832 8944 90;56 9167 9279 9391 9.J03 9615 9726 9838 112 389 9950 .0061 .0173 .0284 .0396 .0507 .0619 .0730 .0842 .0953 112 390 59106.5 1176 1287 1399 1510 1621 1732 1843 1955 2066 111 391 2177 2288 2399 2510 2621 2732 2843 2954 8064 3175 111 392 3286 3397 3508 3618 3729 3840 39.50 4061 4171 4282 111 393 ms 4.503 4614 4724 4834 4945 5055 5165 5276 5386 110 394 5496 5606 5717 5827 5937 6047 6157 6267 6377 6487 110 395 596597 6707 6817 6927 7037 7146 7256 7366 7476 7586 110 396 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 110 397 8791 SJKX) 9009 9119 9228 9.337 9446 9556 9665 9774 109 398 9883 9992 .0101 .0210 .0319 .0428 .0537 .0646 .0755 .0864 109 399 600973 1082 1191 1299 1408 1517 1625 1734 1843 1951 109 N. 1 2 3 4 5 6 7 8 9 D. 351 - Num. 400, Log. 602. TABLE I.— LOGARITHMS N. 400 1 2 3 4 5 6 7 8 9 D. 602060 2169 2277 2,386 2494 2603 2711 2819 2928 30:36 108 401 3144 32.53 .3361 3469 3577 3686 3794 3902 4010 4118 108 402 4226 4334 4442 45.50 4658 4766 4874 4982 5089 5197 108 403 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 108 404 6381 6489 6.596 6704 6811 6919 7026 7133 7241 7348 107 405 607455 7562 7669 7777 7884 7991 8098 8205 a3i2 8419 107 406 8.526 863:3 8740 8847 8954 9061 9167 9274 9,381 9488 107 407 9594 9701 9808 9914 .0021 .0128 .02,34 .0341 .0447 .0.554 107 408 610660 0767 0873 0979 1086 1192 1298 1405 1511 1617 106 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 612784 2890 2996 3102 3207 a313 3419 352,5 3a30 Sim 106 411 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 106 412 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 105 413 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 lft5 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 105 415 618048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 416 9093 9198 9:302 9406 9511 9615 9719 9824 9928 .0032 104 417 620136 0240 0;344 0448 0.5,52 0656 0760 0864 09f)8 1072 104 418 1176 1280 1384 1488 1,592 1695 1799 1903 2007 2110 104 419 2214 2318 2421 2525 2628 2732 2835 .2939 3042 3146 104 420 623249 3353 34,56 3559 36ft3 3766 3869 3973 4076 4179 103 421 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 103 422 5312 5415 5,518 5(521 5724 5827 5929 60:32 6135 6238 103 423 6340 6443 6546 6648 6751 6853 6956 7a58 7161 7263 103 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 102 425 628389 8491 &593 8695 8797 8900 9002 9104 9206 9308 102 426 WIO 9512 9613 9715 9817 9919 .0021 .0123 .0224 .0326 102 427 630428 0.5:30 06:31 07:33 08:3.5 09:36 1088 1139 1241 1342 102 428 1444 1.545 1647 1748 1849 1951 2052 21,5:3 22,55 2356 101 429 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367 101 430 6a3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 101 431 4477 4.578 4679 4779 4880 4981 5081 5182 5283 5383 101 432 5484 5,584 508,5 5785 5886 5986 6087 6187 6287 6388 100 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 100 434 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 100 435 638489 a589 8689 8789 8888 8988 9088 9188 9287 9387 100 436 9486 9586 9686 978.5 9885 9984 .0084 .0183 .028:3 .0382 99 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 99 4:i8 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 439 2465 2563 2662 2761 2860 2959 3058 3156 3255 33*4 99 440 643453 3551 3650 3749 3847 3946 4044 4143 4242 4340 99 441 4439 4.5:37 4636 4734 4.S:32 4931 5029 6127 5226 5.324 98 442 &422 5,521 5619 5717 5815 5913 6011 6110 6208 asoo 98 443 6404 6.502 6600 6698 6796 6894 6992 7089 7187 7285 98 444 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 98 445 648360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 446 93a5 9432 9530 9627 9724 9821 9919 .0016 .0113 .0210 97 447 6,50308 0405 0502 0599 0696 0793 0890 0987 1084 1181 97 448 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 97 449 2246 2343 2440 2536 2633 2730 2826 2923 3019 3116 97 N. 1 2 8 4 6 6 7 8 9 D. 352 OF NUMBERS. Num. 499, Log .698. N. 1 2 3 4 5 6 7 8 9 D. 4;50 653213 3309 3405 3502 3598 3695 3791 3888 3984 4080 96 451 4177 4273 4369 446;5 4.562 4658 4751 48.50 4M6 5042 96 452 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 96 453 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 4r>5 658011 8107 8202 8298 8393 8488 8584 8679 8774 8870 95 456 89&5 9060 91.55 92.50 9346 9441 9,536 9631 9726 9821 95 457 9916 .0011 .0106 .0201 .0296 .0391 .0486 .0581 .0676 .0771 95 458 66086.5 0960 ia5.5 11.50 1245 1339 14,34 1.529 1623 1718 95 459 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 95 460 662758 2852 2917 3041 31.35 3230 3324 3418 3512 3607 94 461 3701 3795 3889 398;^ 4078 4172 4266 4360 4454 4548 m 462 4642 47.36 48.30 4924 5018 5112 5206 5299 5393 5487 M 463 5581 5675 5769 58()2 5ft56 6O50 6143 6237 6:331 6424 M 464 6.518 6612 6705 6799 6892 6986 7079 7173 7266 7360 M 4a5 6674.53 7.>46 7640 77.33 7826 7920 8013 8106 8199 82a3 93 466 8386 8479 8ij72 8665 8759 8852 8iM5 9038 9131 9224 93 467 9317 9410 9,503 9.596 9689 9782 9875 9967 .0060 .0153 93 468 670246 0339 0431 orm 0617 0710 0802 0895 0988 1080 93 469 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 93 470 672098 2190 2283 2;375 2467 2.560 26,52 2744 2836 2929 92 471 3021 3113 320-5 3297 3390 3482 ;3.574 3666 3758 3850 92 472 3942 4034 4126 4218 4310 4402 4494 4.586 4677 4769 92 473 4861 4953 5045 5137 5228 5320 &412 5503 5595 5687 92 474 5778 5870 5962 6053 6145 6236 6328 fr419 6511 6602 92 475 676694 6785 6876 6968 7059 7151 7242 7333 7424 7516 91 476 7607 7698 7789 7881 7972 8063 81M 8245 8336 &427 91 477 8.518 8609 8700 8791 8882 8f)73 9004 9155 9246 9337 91 478 9428 9519 9610 9700 9791 9882 9973 .0063 .0154 .0245 91 479 680.336 0426 0517 0607 0698 0789 0879 0970 1060 1151 91 480 681211 ia32 1422 1513 1603 1693 17*4 1874 1964 2055 90 481 2145 223*5 2;326 2416 2.506 2596 2686 2777 2867 2957 90 482 3047 31;^ 3227 mn 3407 3497 3587 3677 3767 3857 90 48;^ 3947 4a37 4127 4217 4:307 4396 4486 4576 4666 4756 90 484 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 90 485 685742 5831 5921 6010 6100 6189 6279 6368 6458 6547 89 4m 66.'i6 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 487 7.529 7618 7707 7796 788(5 7975 8004 8153 8242 8331 89 488 8420 8;309 8.598 8687 8776 8865 89.53 9042 9131 9220 89 489 9309 9398 9486 9575 9664 9753 9841 9930 .0019 .0107 89 490 690196 02&5 0373 0462 0550 0039 0728 0816 0905 0993 89 491 1081 1170 12.58 1347 14:3.5 1524 1612 1700 1789 1877 88 1 492 1965 20.53 2142 22m 2318 2406 2494 2583 2671 2759 88 493 2847 29a5 302:3 3111 3199 3287 3375 3463 3551 3639 88 494 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 49.5 694605 4693 4781 4868 4956 5044 5131 5219 5307 5394 88 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 87 497 6356 &444 6531 6618 6706 6793 6880 6968 7055 7142 87 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 499 8101 8188 8275 8362 8449 8535 8622 8709 8796 8883 87 N. 1 2 3 4 5 6 7 8 9 D. Trig.— 30. 353 Num. 500, Log. 698. TABLE I.— LOGARITHMS 1 N. 1 2 3 4 5 6 7 8 9 D. 500 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 87 501 9838 9924 .0011 .0098 .0184 .0271 .oa58 .0444 .0531 .0617 87 602 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 86 503 1568 1654 1741 1827 1913 1999 2086 2172 2268 2344 86 504 2431 2517 2603 2689 2775 2861 2947 3033 3119 3206 86 505 703291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 506 4151 4236 4322 4408 4494 4679 4665 4761 4837 4922 86 507 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 86 608 5864 5949 6035 6120 6206 6291 6376 6462 6647 6632 85 509 6718 6803 6888 6974 7059 7144 7229 7315 7400 7486 85 510 707570 7655 7740 7826 7911 7996 8081 8166 8261 8336 86 511 8421 8506 8591 8676 8761 8846 8931 9015 9100 918.5 85 512 9270 9355 9440 9524 9609 9694 9779 9863 9948 .0033 85 513 710117 0202 0287 0371 0456 0540 0626 0710 0794 0879 85 514 0963 1048 1132 1217 1301 1386 1470 1554 1639 1723 84 515 711807 1892 1976 2060 2144 2229 2313 2397 2481 2666 84 516 2650 2734 2818 2902 2986 3070 3164 3238 3323 3407 84 517 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 84 518 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 M 519 5167 5251 5335 5418 5502 6586 5669 5753 5836 5920 84 520 716003 6087 6170 62.54 6337 6421 6504 6588 6671 6754 83 521 6838 6921 7004 7088 7171 7254 7338 7421 7604 7,687 83 522 7671 7754 7837 7920 8003 8086 8169 8263 8336 8419 83 523 8502 8585 8668 8751 8834 8917 9000 9083 9166 9248 83 524 9331 9414 9497 9580 9663 9745 9828 9911 9994 .0077 83 525 720159 0242 0325 0407 0490 0673 0656 0738 0821 0903 83 526 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 82 527 1811 1893 1975 2058 2140 ^?^^ 2306 2387 2469 2652 82 528 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 529 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 724276 4358 4440 4522 4604 4685 4767 4849 4931 5013 82 531 5095 5176 5258 5340 5422 5603 5585 5667 5748 5830 82 532 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 82 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 534 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 535 728354 8435 8516 8697 8678 8759 8841 8922 9003 9084 81 536 9165 9246 9327 9408 9489 9670 9651 9732 9813 9893 81 537 9974 .0055 .0136 .0217 .0298 .0378 .0469 .0540 .0621 .0702 81 538 730782 0863 0944 1024 1105 1186 1266 1347 1428 1608 81 539 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 540 732394 2474 25;55 2635 2716 2796 2876 2956 3037 3117 80 541 3197 3278 3358 3438 3518 3698 3679 3759 38^^0 3919 80 542 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 543 4800 4880 4960 5040 5120 5200 5279 5369 5439 5519 80 644 5,599 5679 5759 5838 5918 5998 6078 6157 6237 6317 80 645 736397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 546 7193 7272 73*52 7431 7511 7690 7670 7749 7829 7908 79 547 7987 8067 8146 8225 8306 8384 8463 8543 8622 8701 79 548 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 549 9572 9651 9731 9810 9889 9968 .0047 .0126 .0205 .0284 79 N. 1 2 3 4 6 6 7 8 9 D. 354 1 1 OF NUMBERS. Nnm. 599, Log. 778. N. 1 2 3 4 5 6 7 8 9 D. 5.50 740363 0442 0521 0600 0678 0757 0836 0915 0994 1073 79 551 1152 1230 1.309 1388 1467 1,546 1624 1703 1782 1860 79 552 1939 2018 2096 2175 22,54 2332 2411 2489 2568 2647 79 553 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78 554 3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 78 555 744293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 51.53 5231 5309 5387 546.5 5543 5621 5699 5777 78 657 5855 5933 6011 6089 6167 6245 6323 6401 6479 6556 78 558 6634 6712 6790 ,6868 6945 702;3 7101 7179 7256 7334 78 559 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 8266 &343 8421 8498 8576 8653 8731 8808 8885 77 561 8963 9040 9118 9195 9272 9a50 9427 9504 9582 9659 . 77 562 9736 9814 9891 9968 .004.5 .0123 .0200 .0277 .0364 .0431 77 563 7.50.508 0586 0663 0740 0817 0894 0971 1048 1125 1202 77 564 1279 1356 1433 1510 1.587 1664 1741 1818 1895 1972 77 565 752048 2125 2202 2279 2;i56 24.33 2.509 2586 2663 2740 77 566 2816 2893 2970 3047 312;] 3200 3277 3353 3430 3506 77 567 a583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 568 4348 4425 4.501 4578 46.54 4730 4807 4883 4960 5036 76 569 5112 5189 5265 5341 5417 5494 5570 5646 6722 5799 76 570 755875 5951 6027 6103 6180 62,56 6332 ft408 6484 6560 76 571 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 7396 7472 7.S48 7624 7700 VVVo 7851 7927 8003 8079 76 573 81.55 sm 8306 8382 sm &533 8609 8685 8761 8836 76 574 8912 8988 9063 9139 9214 9290 9366 9441 9617 9592 76 575 759668 9743 9819 9894 9970 .004.5 .0121 .0196 .0272 .0347 75 576 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 75 577 1176 12.51 1.326 1402 1477 1,5.52 1627 1702 1778 1853 75 578 1928 2003 2078 21.53 2228 2;«3 2378 2453 2529 2604 76 579 2679 2754 2829 2904 2978 3ft53 3128 3203 3278 3363 75 580 763428 3503 a578 3653 3727 3802 3877 3952 4027 4101 75 581 4176 42.51 4326 4400 4475 45^50 4624 4699 4774 4848 75 582 4923 4998 5072 5147 5221 5296 5370 6445 5520 5594 75 5m 5669 6743 5818 5892 5966 6041 6115 6190 6264 6338 74 584 e413 6487 6562 6636 6710 6785 6859 6933 7007 7082 74 585 767156 7230 7.S04 7379 74.53 7527 7601 7675 7749 7823 74 586 7898 7972 8046 8120 8194 8268 8.342 »416 8490 8564 74 587 8638 8712 8786 88(«) 89.34 9008 9082 91.56 9230 9303 74 588 9377 9451 952.5 9599 9673 9746 9820 9894 9968 .0042 74 589 770115 0189 0263 0336 0410 0484 0557 0631 0705 0778 74 590 770852 0926 0999 1073 1146 1220 1293 1367 1440 1614 74 591 1587 1G61 1734 1808 1881 19.55 2028 2102 2175 2248 73 592 ?^?9, 2.395 2468 2542 2615 2688 2762 2835 2908 2981 73 593 3a55 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 591 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 774517 4590 4663 4736 4809 4882 49.55 5028 5100 6173 73 596 5246 5319 6392 6465 5538 5610 5683 5756 6829 6902 73 597 5974 6047 6120 6193 6265 6338 6411 6483 &566 6629 73 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 72 N. 1 2 3 4 5 6 7 8 9 D. 355 Num 600, Log. 778. TABLE I.— LOGARITHMS N. 1 2 3 4 5 6 7 8 9 D. 778151 8224 8296 8308 8441 8.513 85.S.5 8658 8730 8802 72 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 eo2 £5)6 9669 9741 9813 9885 99,57 .0029 .0101 .0173 .0245 72 608 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 72 604 1037 1109 1181 1253 1324 1396 1468 1540 1612 1681 72 60-) 781755 1827 1899 1971 2042 2114 2186 22.58 2329 2401 72 60i) 2473 2.544 2616 2688 27,59 2831 2902 2974 3046 3117 72 607 31K9 3260 3:W2 3403 3475 3546 3618 3689 3761 3832 71 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 610 785330 5401 5472 5543 5615 5686 5757 5828 5899 5970 71 611 6041 6112 6183 6254 6325 6396 6467 6,538 6609 6680 71 612 6751 6822 6893 6964 70a5 7106 7177 7248 7319 7390 71 613 7460 75:^1 7602 7673 7744 7815 7885 79,56 8027 8098 71 614 8108 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 615 788875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 616 9581 9651 9722 9792 9863 9933 .0004 .0074 .0144 .0215 70 617 790285 0356 0126 0496 0567 0637 0707 0778 0848 0918 70 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 619 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 2462 2.532 2602 2672 2742 2812 2882 2952 3022 70 621 3092 3162 3231 3301 3371 3441 3511 3581 3651 3721 70 622 3790 3860 39,30 4000 4070 4139 4209 4279 4349 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 70 624 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 62;j 795880 5949 6019 6088 6158 6227 6297 6366 6436 6505 69 626 6574 6644 6713 6782 68.52 6921 6990 7060 712',. 7198 69 627 7268 7.337 7406 7475 7.545 7614 7683 7752 7821 7890 69 628 7960 8029 8098 8167 82.36 8305 8374 8443 8513 8582 69 629 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 799341 9409 9478 9,547 9616 9685 9754 9823 9892 9961 69 631 800029 0098 0167 0236 0305 0373 0442 0511 0580 0648 69 632 0717 0786 0854 0923 0992 1061 1129 1198 1266 133.5 69 633 1404 1472 1,541 1609 1678 1747 1815 1884 1952 2021 69 631 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 69 635 802774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 636 3457 3525 a594 3662 3730 3798 3867 39:35 4003 4071 68 637 4139 4208 4276 4344 4412 4480 4,548 4616 4685 47.53 68 638 4821 4889 49.57 5025 5093 5161 5229 5297 5365 54;33 68 639 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 68 640 806180 6248 6316 6384 6451 6,519 &587 6655 6723 6790 68 641 68.58 6926 6994 7061 7129 7197 7264 7.332 7400 7467 68 6i2 7.5a5 7603 7670 7738 7806 7873 7941 8008 8076 8143 68 643 8211 8279 8;^46 8414 8481 8,549 8616 8684 8751 8818 67 644 8886 8953 9021 9088 91.56 922;3 9290 9358 9425 9492 67 &15 809560 9627 9694 9762 9829 9896 9964 .0031 .0098 .0165 67 646 8102;B o.%o 0367 0434 0501 0rj69 0636 0703 0770 0837 67 647 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 648 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 649 2245 2312 2379 2445 2512 2579 2W6 2713 2780 2847 67 N. 1 2 3 4 5 6 7 8 9 D. 356 OF NUMBERS. Num. 699, Log .845. N. 1 2 3 4 5 6 7 8 9 3). 650 812913 2980 3047 3114 3181 3247 3314 3381 3448 3514 67 651 3581 3648 3714 3781 3848 3914 3981 4048 . 4114 4181 67 652 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 653 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 66 654 5578 5614 5711 5777 5843 5910 5976 6042 6109 6175 66 655 816241 6308 6374 6440 6506 6573 6639 6705 6771 6838 66 656 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 657 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 658 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 659 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819.544 9610 9676 9741 9807 9873 9939 .0004 .0070 .0136 66 661 820201 0267 oa33 0399 0464 0530 0595 0661 0727 0792 66 662 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 663 1514 1579 1615 1710 1775 1841 1906 1972 2037 2103 65 664 2168 2233 2299 2364 2430 2495 2560 ?fiW 2691 2756 65 665 822822 2887 2952 3018 3083 3148 3213 3279 3344 ai09 65 666 3474 3539 3605 3670 3735 3800 3865 3980 3996 4061 65 667 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 65 668 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 65 669 6426 5491 5556 5621 5686 5751 5815 5880 5945 6010 65 670 826075 6140 6204 6269 em 6399 0404 6528 6593 66.58 65 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 65 672 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 673 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 675 829304 9368 9432 9497 9561 9625 9690 97,54 9818 9882 64 676 9947 .0011 .0075 .0139 .0204 .0208 .0332 .0396 .0460 .0525 64 677 830589 0653 0717 0781 Ofy45 0909 0973 1037 1102 1166 64 678 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 64 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 ^45 64 680 832509 2573 2637 2700 2704 2828 2892 2956 3020 3083 64 681 3147 3211 3275 aS38 3402 3466 3530 3593 3657 8721 64 682 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 683 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 684 5056 5120 5183 5^7 5310 5373 5437 5500 5564 5627 63 685 835691 5754 5817 5881 6m 6007 6071 6134 6197 6261 63 686 6324 6387 6451 6514 6577 6041 6704 6767 6830 6894 63 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 688 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 838&49 8912 8975 9038 9101 9164 9227 9289 9352 9415 63 691 9478 9541 9604 9667 9729 9792 9855 9918 9981 .0043 63 692 840106 0169 02132 0294 0357 0420 0482 0545 0608 0671 63 693 0733 0796 0&39 0921 0984 1046 1109 1172 1234 1297 63 694 1359 1422 1485 1547 1610 1672 17aj 1797 1860 1922 63 695 -841985 2047 2110 2172 2235 2297 2360 2422 2484 2547 62 690 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 62 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 698 3855 3918 3980 4042 4104 4166 4229 4291 4a53 4415 62 699 4477 4X39 4601 4664 4726 4788 4850 4912 4974 5036 62 N. 1 2 3 4 5 6 7 8 9 D. 357 ^ Num. 700, Log. 845. TABLE I.— LOGARITHMS N. 700 1 2 3 4 5 6 7 8 9 D. 845098 5160 5222 5284 5346 5-108 5470 5532 5594 5656 62 701 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 7W 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 705 848189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 707 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 708 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 61 709 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 710 8512-58 1320 1381 1442 1503 1.564 1625 1686 1747 1809 61 711 1870 1931 1992 2053 2114 2175 2236 2297 2-358 2419 61 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 3090 3150 3211 3272 3333 3394 34.55 3516 3577 3637 61 714 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 61 715 854306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 716 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 717 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 718 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 719 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 720 857332 7393 7453 7513 7574 7634 7694 7755 7815 7875 60 721 7935 7995 80.56 8116 8176 8236 8297 8,357 8417 8477 60 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 723 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 724 9739 9799 9859 9918 9978 .0038 .0098 .0158 .0218 .0278 60 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 60 726 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 727 1534 1594 1654 1714 1773 183:3 1893 1952 2012 2072 60 728 2131 2191 2251 2310 2370 2430 2489 2-549 2608 ^668 60 729 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 86a323 3382 3442 3501 3561 3620 3680 3739 3799 3858 59 731 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 732 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 733 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 59 734 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 735 866287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 786 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 59 738 8056 8115 8174 8233 8292 8350 8409 ms 8527 8586 59 739 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 869232 9290 9349 9408 9466 9525 9584 9642 9701 9760 59 741 9818 9877 9935 9994 .0053 .0111 .0170 .0228 .0287 .0345 59 742 870404 0462 0521 0579 0638 0696 0755 0813 0872 0930 58 743 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 744 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 58 745 872156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 74€ 2739 2797 2855 2913 2972 30:30 3088 3146 3204 3262 58 747 3321 a379 3437 3495 3.353 3611 3669 3727 3785 3844 58 74^ 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 74i 4482 4540 4598 4656 4714 • 4772 4830 4888 4945 5003 58 N. 1 2 3 4 5 6 7 8 9 D. 358 OF NUMBERS. Num. 799, Log. 903. N. 1 2 3 4 5 6 7 8 9 D. 750 875061 5119 5177 5235 5293 5351 5409 5466 6524 5682 58 751 5640 5698 5756 5813 5871 .5929 5987 6045 6102 6160 58 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 7.53 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 58 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 755 877947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 756 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 757 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 67 758 9669 9726 9784 9841 9898 9956 .0013 .0070 .0127 .0185 57 759 880242 0299 0356 0413 0471 0528 0585 0642 0699 0756 57 760 880814 0871 0928 09&5 1042 1099 1156 1213 1271 1328 57 761 138.5 1442 1499 1.5.56 1613 1670 1727 IIM 1841 1898 57 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 67 763 2525 2.581 2638 2695 2752 2809 2866 2923 2980 3037 57 764 3093 3150 3207 3264 3321 3377 34S4 »491 3548 3605 57 765 883661 3718 3775 3832 8888 3^5 4002 4059 4115 4172 57 766 4229 4285 4342 4399 4455 4512 4.569 4625 4682 4739 57 767 47a5 4852 4909 4965 5022 5078 5135 5192 5248 5305 57 768 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 57 769 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 56 770 886491 6547 6604 6660 6716 6773 6829 6885 6942 6998 56 771 7054 7111 7167 7223 7280 7.336 7392 7449 7505 7561 56 772 7617 7674 7730 7786 7812 7898 7955 8011 8067 8123 56 773 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 66 774 8741 8797 8853 8909 8f)65 9021 yovv 9ia4 9190 9246 56 775 889302 9358 9414 9470 9.526 9582 9638 9694 9760 9806 56 776 9862 9918 9974 .oo;^j .0086 .0141 .0197 .0253 .0309 .ft365 56 777 890421 0477 0533 0.589 0645 0700 0756 0812 0868 0924 56 778 0980 1035 1091 1147 1203 12.59 1314 i;^o 1426 1482 66 779 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 56 780 892095 2150 2206 2262 2317 2373 2429 2484 2540 2595 56 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 66 782 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 56 783 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 55 784 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 65 785 894870 492.5 4980 5036 5091 5146 5201 62,57 5312 6367 55 786 5423 5478 5533 5588 6644 5699 6754 6809 68&4 6920 55 787 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 55 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 789 7077 7132 7187 7242 72^ 7352 7407 7462 7517 7572 65 790 897627 7682 7737 7792 7847 7902 7957 8012 8067 8122 55 791 8176 8231 8286 8^41 8396 8451 8.506 8561 8615 8670 65 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 55 794 9821 9875 9930 9985 .0039 .0094 .0149 .0203 .0258 .0312 65 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 55 796 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 65 797 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 54 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 N. 1 2 3 4 5 6 7 8 9 D. 359 Num. 800, Log. 903. TABLE L— LOGARITHMS N. 1 2 3 4 5 6 7 8 9 D. 800 903090 3144 3199 3253 3307 3361 3416 3470 3524 3578 54 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 802 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 803 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 804 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 805 905796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 806 6335 6389 6443 6497 6551 6604 66.58 6712 6766 6820 54 807 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 54 808 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 54 811 9021 9074 9128 9181 9235 9289 9342 9;396 9449 9503 54 812 9556 9610 9663 9716 9770 9823 9877 9930 9984 .0037 53 813 910091 0144 0197 0251 0304 oa58 0411 0464 0518 0571 53 814 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 53 815 911158 1211 1264 1317 1371 1421 1477 1.530 1584 1637 53 816 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 53 817 2222 2275 2328 2381 2435 2188 2541 2594 2647 2700 53 818 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 53 819 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 820 913814 3867 3920 3973 4026 4079 4132 4184 4237 4290 53 821 4343 4396 4449 4502 4.555 4608 4660 4713 4766 4819 53 822 4872 4925 4977 5030 5083 6136 5189 5241 5294 5347 53 823 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 53 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 53 825 916454 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 826 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 53 827 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 52 828 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 52 829 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 52 830 919078 9130 9183 9235 9287 9340 9892 9444 9496 9549 52 831 9601 9653 9706 9758 9810 9862 9914 9967 .0019 .0071 52 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 52 833 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 52 834 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 835 921686 1738 1790 1842 1894 1946 1998 2050 2102 2154 52 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 52 837 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 52 838 3244 3296 3348 3399 S451 3503 &555 3607 3658 3710 52 839 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 52 840 924279 4331 4383 4434 4486 4.538 4589 4641 4693 4744 52 841 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 52 842 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 52 843 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 51 844 6342 6394 6445 6497 6518 6600 6651 6702 6754 6805 51 845 926857 6908 6959 7011 7062 7114 7165 7216 7268 7319 51 846 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 51 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 51 848 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 51 849 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 51 1 , 1 2 3 4 5 6 7 8 9 D. 360 1 ' OF NUMBERS. Nxun. 899, Log .954. N. 850 1 2 3 4 5 6 7 8 9 9879 D. 61 929419 9470 9521 9572 9623 9674 972,5 9776 9827 8.51 9930 9981 .0032 .0083 .01,34 .0186 .0236 .0287 .0338 .0.389 51 852 930440 0491 0542 0592 0643 0094 0746 0796 0847 0898 51 8.>3 0949 1000 10.51 1102 11.53 12W ia>4 1305 13,56 1407 51 8.54 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 8oo 931966 2017 2068 2118 2169 2220 2271 2322 2372 2423 51 856 2474 2524 2576 2626 2677 2727 2778 2829 2879 2930 51 &57 2981 3031 3082 3133 3183 3234 3286 3335 3386 &137 51 858 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 51 859 3993 4044 4094 4145 4195 4216 4296 4347 4397 4448 51 860 934498 4549 4.599 46.50 4700 4751 4801 4852 4902 4953 50 861 5003 5061 5104 6154 6206 6256 5306 6356 5106 5467 60 862 6507 5558 5608 66.58 6709 6769 6809 6860 5910 5960 50 863 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 50 864 6514 6564 6614 6665 6715 6766 6815 6865 6916 6966 60 865 937016 7066 7117 7167 7217 7267 7317 7367 7418 7468 50 866 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 50 868 8520 mo 8620 8670 8720 8770 8820 8870 8920 8970 50 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 50 870 939519 9569 9619 9669 9719 9769 9819 9869 9918 9968 50 871 940018 0068 0118 0168 0218 0267 0317 0367 0117 0467 50 872 0516 a566 0616 0666 0716 0766 0815 0865 0915 1 0964 50 873 1014 1064 1114 1163 1213 1263 1313 1362 1412 1 1462 50 874 1511 1561 1611 1660 1710 1760 1809 1859 1909 1968 60 875 942008 20.58 2107 2167 2207 2266 2306 2355 24a5 2455 50 876 2.501 2.5.54 2603 2653 2702 2762 2801 2861 2901 2950 50 877 3000 3049 3099 3148 3198 3217 3297 3346 3396 3445 49 878 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 49 879 3989 4038 4088 4137 4186 4236 4286 4335 4384 4433 49 880 944483 4532 4681 4631 4680 4729 4779 4828 4877 4927 49 881 4976 502.5 6074 6124 5173 6222 5272 6321 5370 5419 49 882 5469 5518 6567 5616 5666 5715 57M 6813 5862 5912 49 883 5961 6010 6059 6108 61.57 6207 0266 6305 6364 64as 49 8M 6452 6501 6651 6600 6649 6698 6747 6796 6845 6894 49 88,5 946943 6992 7041 7090 7140 7189 72.38 7287 7336 7385 49 886 74;^ 7483 7532 7581 7630 7679 7728 WW 7826 7875 49 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 888 8413 8462 8.511 8560 8(309 86.57 8706 8765 8804 8863 49 889 8902 8961 8999 9048 9097 9146 9196 9214 9292 9341 49 890 949390 9439 9488 9536 958.5 96,34 9683 9731 9780 9829 49 891 9878 9926 9975 .0024 .0073 .0121 .0170 .021iJ .0267 .0316 49 892 950365 0414 0462 Oijll 0(560 0608 0(557 0706 ! 0764 0803 49 893 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 49 894 1338 1386 1435 1483 1632 1580 1629 1677 1726 1776 49 895 961823 1872 1920 1969 2017 2066 2114 2163 2211 2260 48 896 2308 2356 240.5 2453 2.502 2560 2599 2647 2696 2744 48 897 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 48 898 3276 3325 3373 3421 3470 3518 3666 3615 3663 3711 48 899 N. . 3760 3808 3856 3906 3 3953 4001 5 4049 4098 4146 4194 48 D. 1 2 4 6 7 8 9 Trig.— 31. 361 Num. 900, Log. 954. TABLE L— LOGARITHMS N. 1 2 3 4 5 6. 7 8 9 D. 900 954243 4291 4339 4387 4435 4484 4r>32 4580 4628 4677 48 901 4725 4773 4821 4869 4918 4966 5014 5062 5110 6158 48 902 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 903 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 904 6168 6216 6265 6313 6361 6409 6457 6505 6558 6601 48 905 956649 6697 6745 6793 6840 6888 6986 6984 7032 7080 48 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 75,59 48 907 7607 7655 7703 7751 7799 7847 7894 7942 7990 8088 48 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 909 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 9089 9137 9185 9232 9280 9828 9875 9423 9471 48 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 912 9995 .0042 .0090 .0138 .0185 .0233 .0280 .0328 .0376 .0423 48 913 960471 0518 0566 0613 0061 0709 0756 0804 0851 0899 48 914 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 47 915 961421 1469 1516 1563 1611 1658 1706 1753 1801 1848 47 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 917 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 918 2843 ^890 2937 2985 8032 3079 8126 3174 3221 3268 47 919 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 47 920 963788 38a5 3882 3929 3977 4024 4071 4118 4165 4212 47 921 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 922 4731 4778 4825 4872 4919 4966 5018 5061 5108 5155 47 923 5202 5249 5296 5343 5390 5437 &484 5531 5578 5625 47 924 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 47 925 966142 6189 6236 6283 6829 6876 6428 6470 6517 65&4 47 926 6611 6658 6705 6752 0799 6845 6892 6939 6986 7033 47 927 7080 7127 7173 7220 7267 7314 7301 7408 7454 7601 47 928 7548 7595 7642 7688 77:35 7782 7829 7875 7922 7969 47 929 8016 8062 8109 8156 8203 8249 8296 8343 8390 8486 47 930 968483 8530 8576 8623 8670 8716 8768 8810 8856 8903 47 931 8950 8996 9043 9090 9136 9183 9229 9276 9328 9369 47 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 am 9882 9928 9975 .0021 .0068 .0114 .0161 .0207 .0254 .0300 47 934 970347 0393 0440 0486 0533 0579 0626 0672 0719 0765 46 935 970812 0&58 0904 0951 0997 1044 1090 1137 1188 1229 46 936 1276 1322 1369 1415 1461 1508 15,>4 1601 1647 1693 46 937 1740 1786 1832 1879 1925 1971 2018 20(34 2110 2167 46 938 2203 2249 2295 2342 2388 2434 2481 2527 2573 2()19 46 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 940 973128 3174 3220 3266 3313 3359 8405 3451 3497 3543 46 941 3590 3636 3682 3728 3774 3820 8866 3913 3a59 4005 46 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 913 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 944 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 945 975432 5478 5.524 5570 5616 5662 5707 5753 5799 5845 46 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 mn 46 947 6a50 6396 6442 6488 6538 6579 6625 6671 6717 6763 46 94e 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 im > 7266 731ii 7358 7403 7449 7495 7541 7586 7632 7678 46 N. 1 2 3 4 5 6 7 8 9 D. 362 [ OF NUMBERS. Hum. 999, Log. 9S9. N. 1 2 3 4 5 6 7 8 9 D. 950 977724 7769 7815 7861 7906 79.52 7998 8043 8089 8ia5 46 951 8181 8226 8272 8317 8363 8409 84,54 8500 8546 8591 46 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 953 9093 9138 9184 9230 9275 9:521 9366 9412 94.57 9;503 46 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 9o5 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 45 9,56 04.58 0.503 0549 0594 0640 0685 07:^ 0776 0821 0867 45 9-57 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 9.58 1366 1411 14,56 1501 1,>47 1592 1637 1683 1728 1773 45 959 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 45 960 982271 2316 2.362 2407 2452 2497 2.543 2588 2633 2678 45 961 2723 2769 2814 28,59 2904 2949 2994 3040 3085 3130 45 962 3175 3220 3265 3;^io a'i56 3401 3446 3491 3536 a581 45 96;^ 3626 3671 3716 3762 3807 38,52 3897 3942 3987 4032 45 964 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 965 984,527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 mi 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 967 5426 5471 5516 5561 5606 5651 5696 5741 5786 58:30 45 968 5875 5920 596.5 6010 6055 6100 6144 6189 6234 6279 45 969 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 986772 6817 6861 6906 6951 6996 7040 7085 71.30 7175 45 971 7219 7261 7309 7;^ 7398 7443 7488 75:^ 7577 7622 45 972 7666 7711 7756 7800 7845 7890 im 7979 8024 8068 45 973 8113 81,57 8202 8247 8291 83:36 8,-i81 8425 8470 8514 45 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 975 9890a5 9049 9094 9138 9183 9227 9272 9316 9361 9405 45 976 94,50 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 9895 9939 9983 .0028 .0072 .0117 .0161 .0206 .0250 .0294 44 978 990339 om 0428 0472 0,516 0561 0605 06.50 0694 0738 44 979 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 44 980 991226 1270 1315 1359 1403 1448 1492 1536 1580 1625 44 981 1669 1713 1758 1802 1846 1890 19,35 1979 2023 2067 44 982 2111 21,56 2200 2244 2288 2333 23-77 2421 2465 2509 44 988 2,554 2598 2642 2686 2730 2774 2819 2863 2907 2951 . 44 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 985 99,3436 3480 3524 a568 3613 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 988 47,57 4801 4845 4889 49:« 4977 5021 5065 5108 5152 44 989 5196 5240 52i« 5328 5:^72 5416 5460 5504 5547 5591 44 990 9956.3.5 5679 ,5723 5767 5811 5854 5898 5942 5986 6030 44 991 6074 6117 6161 62a5 6249 6293 6337 6380 6424 6468 44 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 994 7386 7430 7474 7517 7561 7605 7648 7692 7736 7779 44 995 997823 7867 7910 7954 7998 8041 8085 8129 8172 8216 44 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 997 8695 8739 8782 8826 8869 8913 8956 9000 9043 9087 44 998 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 43 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 N. 1 2 3 4 5 6 7 8 9 D. 363 Num. 1000, Log. 000. TABLE I.— LOGARITHMS N. 1 2 3 4 5 6 7 8 9 D. 1000 000000 0043 0087 0130 0174 0217 0260 0304 0347 0391 43 1001 0434 0477 0521 0564 0608 0(i51 0694 0738 0781 0824 43 1002 0868 0911 0954 0998 1041 1084 1128 1171 1214 1268 43 1003 1301 1344 1388 1431 1474 1517 1561 1604 1647 1690 43 1004 1734 1777 1820 1863 1907 1960 1993 2036 2080 2123 43 1005 002166 2209 22.52 2296 2339 2382 2425 2468 2612 2555 43 1006 2598 2641 2684 2727 2771 2814 2867 2900 2943 2986 43 1007 3029 3073 3116 3159 3202 3245 3288 3331 3374 3417 43 1008 3461 3504 3547 3590 3633 3676 3719 3762 3806 3848 43 1009 3891 3934 3977 4020 4063 4106 4149 4192 4236 4278 43 1010 004321 4364 4407 4450 4493 4536 4579 4622 4665 4708 43 1011 4751 4794 4837 4880 4923 4966 5009 5052 5096 5138 43 1012 5181 5223 5266 5309 5352 5395 5438 6481 5524 5567 43 1013 5609 5652 5695 5738 5781 5824 5867 6909 6952 5995 43 1014 6038 6081 6124 6166 6209 6252 6295 6338 6380 6423 43 1015 006466 6509 6552 6594 6637 6680 6723 6765 6808 6851 43 1016 6894 6936 6979 7022 7065 7107 7150 7193 7236 7278 43 1017 7321 7364 7406 7449 7492 7534 7677 7620 7662 7705 43 1018 7748 7790 7833 7876 7918 7961 8004 8046 8089 8132 43 1019 8174 8217 8259 8302 8345 8387 8430 8472 8515 8558 43 1020 008600 8643 8685 8728 8770 8813 8856 8898 8941 8983 43 1021 9026 9068 9111 9153 9196 9238 9281 9323 9366 9408 42 1022 9451 9493 9536 9578 9621 9663 9706 9748 9791 9833 42 1023 9876 9918 9961 .0003 .0045 .0088 .0130 .0173 .0215 .0258 42 1024 010300 0342 0385 0427 0470 0512 0564 0597 0639 0681 42 1025 010724 0766 0809 0851 0893 0936 0978 1020 1063 1105 42 1026 1147 1190 1232 1274 1317 ia59 1401 1444 1486 1528 42 1027 1570 1613 1655 1697 1740 1782 1824 1866 1909 1951 42 1028 1993 2035 2078 2120 2162 2204 2247 2289 2331 2373 42 1029 2415 2458 2500 2542 2584 2626 2669 2711 2753 2795 42 1030 012837 2879 2922 2964 3006 3048 3090 3132 3174 3217 42 1031 3259 3301 3343 3385 3427 3469 3511 3553 3596 3638 42 1032 3680 3722 3764 3806 3848 3890 3932 3974 4016 4058 42 1033 4100 4142 4184 4226 4268 4310 4353 4395 4437 4479 42 1034 4521 4563 4605 4647 4689 4730 4772 4814 4856 4898 42 1035 014940 4982 5024 5066 5108 5150 5192 52.34 5276 5318 42 1036 5360 5402 5444 5485 5,527 6.569 6611 6663 5696 5737 42 1037 5779 5821 5863 5904 5946 5988 6030 6072 6114 6156 42 1038 6197 6239 6281 6323 6365 6407 6448 6490 6532 6574 42 1039 6616 6657 6699 6741 6783 6824 6866 6908 6950 6992 42 1040 017033 7075 7117 7159 7200 7242 7284 7.326 7367 7409 42 1041 7451 7492 7534 7576 7618 7ft59 7701 7743 7784 7826 42 1042 7868 7909 7951 7993 8034 8076 8118 8159 8201 8243 42 1043 8284 8326 8368 8409 8451 8492 8634 8576 8617 8659 42 1044 8700 8742 8784 8825 8867 8908 8950 8992 9033 9075 42 1045 019116 9158 9199 9241 9282 9324 9366 9407 9449 9490 42 1046 9,53? 9573 9615 9656 9698 9739 9781 9822 9864 9905 42 1047 9947 9988 .0030 .0071 .0113 .01.54 .0195 .0237 .0278 .0320 41 1048 020361 0403 0444 0486 0527 0568 0610 0661 0693 0734 41 1049 0775 0817 0858 0900 0941 0982 1024 1065 1107 1148 41 N. 1 2 3 4 5 6 7 8 9 D. 364 OF NUMBERS. Num. 1099, Log .041. N. .1 2 3 4 5 6 7 8 9 D. ia50 021189 1231 1272 1313 1355 1396 14.37 1479 1520 1561 41 lOol 1603 1644 1685 1727 1768 1809 18,51 1892 1933 1974 41 m2 2016 2a57 2098 2140 2181 2222 226;^ 2305 2346 2387 41 1053 2128 2470 2.511 2552 9593 2635 2676 2717 2758 2799 41 1054 2841 2882 2923 2964 3005 3047 3088 3129 3170 3211 41 1055 023252 3294 333.5 3376 3417 3458 3499 3541 3582 3623 41 ia56 3664 3705 3746 3787 3828 3870 3911 3952 3993 4034 41 1057 4075 4116 41.57 4198 4239 4280 4321 4363 4404 4445 41 ia58 4486 4527 4568 4609 4&50 4691 4732 4773 4814 4855 41 1059 4896 4937 4978 5019 5060 5101 5142 5183 5224 5265 41 1060 025306 5347 5388 5429 5470 5511 5552 5593 5634 5674 41 1001 5715 5756 5797 5&38 5879 5920 5961 6002 6043 6084 41 1062 6125 6165 6206 6247 6288 6329 6370 6411 6452 6492 41 1063 6533 6574 6615 66,56 6697 6737 6778 6819 6860 6901 41 1064 6942 6982 7023 7064 7105 7146 7180 7227 7268 7309 41 1065 027a50 7390 7431 7472 7513 7^i 7594 7635 7676 7716 41 1066 7757 7798 7839 7879 7920 7961 8002 8042 8083 8124 41 1067 8164 8205 8246 8287 8327 8368 8409 8449 »490 8531 41 1068 8.571 8612 8653 8693 8734 8775 8815 8856 8896 8937 41 1069 8978 9018 9059 9100 9140 9181 9221 9262 9303 9343 41 1070 029384 9424 9465 9506 9.546 9587 9627 9668 9708 9749 41 1071 9789 9830 9871 9911 99.52 9992 .003;3 .0073 .0114 .0154 41 1072 030195 0235 0276 0316 0357 0397 (M38 0478 0519 0559 40 1073 0600 0640 0681 0721 0762 0802 0^3 om 0923 0964 40 1074 1004 1045 1085 1126 1166 1206 1247 1287 1328 1368 40 1075 031408 1449 1489 1530 1570 1610 1651 1691 1732 1772 40 1076 1812 1853 1893 1933 1974 2014 2ft->4 2095 2135 2175 40 1077 2216 2256 2296 2337 2377 2417 24.58 2498 2538 2578 40 1078 2619 2659 2699 2740 2780 2820 2800 2901 2941 2981 40 1079 3021 3062 3102 3142 3182 322;3 3263 3303 3343 3384 40 1080 033424 3464 3504 3544 3585 3025 3665 3705 3745 3786 40 1081 3826 3866 3906 3946 3986 4027 4067 4107 4147 4187 40 1082 4227 4267 4308 4348 4388 4428 4408 4508 4548 4588 40 1083 4628 4669 4709 4749 4789 4829 4869 4909 4949 4989 40 1084 5029 5069 5109 5149 5190 5230 5270 5310 6350 5390 40 1085 035430 5470 5510 5550 5590 5630 5670 5710 5750 5790 40 1086 5830 5870 5910 5950 5990 6030 6070 6110 6150 6190 40 1087 6230 6269 6309 6349 6389 6429 6469 6509 6549 6589 40 1088 6629 6669 6709 6749 6789 6828 6868 6908 6948 6988 40 1089 7028 7068 7108 7148 7187 7227 7267 7307 7347 7387 40 1090 037426 7466 7506 7546 7586 7626 7665 7705 7745 7785 40 1091 782) 7865 7904 7944 7984 8024 8064 8103 8143 8183 40 1092 8223 8262 8302 8342 8382 &421 8461 8501 8541 8580 40 1093 8620 8660 8700 8739 8779 8819 8859 8898 8938 8978 40 1094 9017 9057 9097 9136 9176 9216 9255 9295 9335 9374 40 1095 039414 9454 9493 9533 9573 9612 9652 9692 9731 9771 40 1096 9811 9850 9890 9929 9969 .0009 .0048 .0088 .0127 .0167 40 1097 040207 0246 0286 0325 0365 0405 0444 0484 0523 0563 40 1098 0602 0642 0681 0721 0761 0800 0840 0879 0919 0958 40 1099 0998 1037 1077 1116 1156 1195 1235 1274 1314 1353 39 N. 1 2 8 4 6 6 7 8 9 D. 365 TABLE IL— LOGARITHMS OF PRIME j N. Logarithm. N. Logarithm. N. Logarithm, 2 30102 99956 63981 238 367;i5 59210 26019 547 73798 73263 33431 3 47712 12.547 19662 239 37839 79009 48138 557 74585 51951 73729 5 69897 00043 36019 241 38201 70425 74868 563 75050 83948 51346 7 84509 80400 14257 251 39967 37214 81038 569 75511 22663 95071 11 04139 26851 58225 257 40993 31233 31295 571 75663 61082 45848 13 11394 33523 06837 263 41995 57484 89758 577 76117 58131 55731 17 23044 89213 78274 269 42975 22800 02408 587 76863 81012 47614 19 27875 36009 52829 271 43296 92908 74406 593 77305 469.33 64263 23 36172 78360 17593 277 44247 97690 64449 599 77742 68223 89311 29 46239 79978 98956 281 44870 63199 05080 601 77887 44720 02740 31 49136 16938 34273 283 45178 64355 24290 607 78318 86910 75258 37 56820 17210 66995 293 46686 76203 54109 613 78746 04745 18415 41 61278 38567 19735 307 48713 837.54 77186 617 79028 51640 33242 43 63346 84555 79587 311 49276 03890 26838 619 79169 06490 20118 47 67209 78579 35717 313 49554 43375 46448 631 80002 93592 44134 53 72427 58696 00789 317 50ia5 92622 17751 641 80685 80295 18817 59 77085 20116 42144 331 51982 79937 75719 643 80821 09729 24222 61 78532 98850 10767 337 52762 99008 71339 647 81090 42806 68700 67 82607 48027 00826 847 54032 94747 90874 653 81491 31 812 75074 71 85125 83487 19075 349 54282 54269 59180 659 81888 54145 94010 73 86332 28601 20456 353 54777 470.53 87823 661 82020 14594 &5640 79 89762 70912 90441 359 55509 44485 78319 673 82801 50642 23977 83 91907 80923 76074 367 56466 60642 52089 677 83058 86686 85144 89 94939 00066 44913 373 57170 88318 08688 683 83442 07036 81533 97 98677 17342 66245 379 57863 92099 68072 691 83947 80473 74198 101 00432 13737 82643 383 58319 87739 68623 701 84571 80179 66659 103 01283 72247 05172 389 58994 96013 25708 709 85064 62351 83067 107 02938 37776 85210 397 59879 05067 63115 719 85672 88903 82883 109 03742 64979 40624 401 60314 43726 20182 727 86153 44108 59038 113 05307 84434 83420 409 61172 33080 07342 733 86510 39746 41128 127 10380 37209 55957 419 62221 40229 66295 739 86864 44383 94826 131 11727 12956 55764 421 62428 20958 35668 743 87098 88137 60575 137 13672 05671 56407 431 63447 72701 60732 751 87563 99370 04168 139 14301 48002 54095 433 63648 78963 53365 757 87909 58795 00073 149 17318 62684 12274 439 64246 45202 42121 761 88138 46567 70573 151 17897 69472 93169 443 64640 37262 23070 769 88592 63398 01431 157 19589 9ft524 09234 449 65224 63410 03323 773 88817 94939 18325 163 21218 76044 03958 457 65991 62000 69850 787 89597 47323 59065 167 22271 64711 47583 461 66370 092,53 89648 797 90145 83213 96112 173 23804 61031 28795 463 66558 09910 17953 809 90794 85216 12272 179 25285 30309 79893 467 66931 68805 66112 811 90902 0a542 111.56 181 25767 85748 69185 479 68033 55134 14563 821 91434 31571 19441. 191 28103 33672 47728 487 68752 89612 14634 823 91539 mm 12270 193 28555 73090 07774 491 69108 14921 22968 827 91750 55095 52547 197 29446 62261 61593 499 69810 05456 23390 829 91855 45305 50274 199 29885 30764 09707 503 70156 79850 55927 839 92376 19608 28700 211 32428 24.552 97693 509 70671 77823 36759 853 93094 90311 67523 223 34830 48630 48161 521 71683 77232 99524 857 93298 08219 23198 227 35602 58571 93123 523 71850 16888 67274 859 93399 316;i8 31242 229 35983 54823 39888 541 73319 72651 06569 863 93601 07957 15210 366 NUMBERS LESS THAN 1000. 877 881 883 887 907 911 Logarithm. N. 94299 959a3 66041 919 94497 59084 12048 929 94596 070a5 77r)69 937 94792 36198 31726 941 95760 72870 60095 947 95951 83769 72998 9.53 Logarithm. TS. mSSl 5.5113 861 11 967 96801 57139 93642 971 97173 95908 87778 977 97^58 962;M 272.57 983 976;^ 99790 03273 991 97909 29006 38326 997 Logarithm. 98,542 64740 .83002 98721 92299 08005 98989 456;S7 18773 992.5.5 35178 .32136 99607 36.544 8.5275 99869 51583 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 83G43; 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; log. n = log. (n-l) + 2M (.y^^ + 1 3(2 n — ly + &C Omitting the second fraction in the parenthesis, the logarithm will be found correct to three times as many figures as there are in the number n. Using this term gives the result true to five times as many figures as there are in n. For example, to find the logarithm of 1013, log. 1012 = 2 log. 2 + log. 11 +log. 23 = 3.00518 05125 03780 2M-J-2025 = .00042 89328 21633 2 M -i- 3(2025)3 = .00000 00000 34867 log. 1013 =3.00560 94453 60280 For some large numbers it may be necessary to repeat the operation. When one of the prime factors oi n — 1 is greater than 1000, it may be better to find the loga- rithm of n + 1, and then log. n by subtracting the differ- ence. For example, log. 2027 can be found more readily from log. 2028 than from log. 2026. 367 TABLE IIL- -NATURAL SINES. Deg. 0' 10' 20' 30' 40' 50' 60' Deg. 000000 002909 00.5818 008727 011635 014.544 017452 89 1 017452 020361 02)269 020177 02908.5 031992 034899 88 2 031899 037806 040713 043619 04ft325 049431 052336 87 3 052;336 055211 a58145 061049 0639.32 066854 069750 86 4 069756 072658 075559 078459 081359 084258 0871.50 85 5 087156 090053 0929.50 095846 098741 101635 104528 84 6 104528 107421 110313 113203 110093 118982 121809 83 7 121869 124756 127642 130526 133410 136292 139173 82 8 139173 142053 144932 147809 150686 L53561 156434 81 9 156434 159307 162178 165048 167916 170783 173648 80 10 173648 176512 179375 182236 185095 187953 190809 79 11 190809 193661 19&"j17 199368 202218 205065 207912 78 12 207912 210756 21^3599 216440 219279 222116 224951 77 13 221951 227784 230616 233445 236273 239098 241922 76 14 241922 244743 247563 250380 253195 256008 258819 75 15 258819 261628 264434 267238 270040 272840 27.5637 74 16 275637 278432 281225 284015 286803 289589 292372 73 17 292372 295152 297930 300706 303479 306249 309017 72 18 309017 311782 314">45 317305 320062 322816 325568 71 19 325.568 328317 331063 333807 336547 339285 342020 70 20 342020 344752 347481 350207 352931 355651 358368 69 21 a58368 361082 363793 366501 3(59206 371908 374(,07 68 22 374607 377302 379994 382683 385369 388052 3007.31 67 23 390731 393407 39G080 398749 401415 404078 400737 66 24 406737 409392 412045 414693 417338 419980 422618 65 25 422618 42525:3 427884 430511 4331.35 435755 438371 64 26 438371 440984 443593 44C198 448799 451397 453990 63 27 453990 456i580 459166 461749 464327 466901 469472 62 28 469172 472038 474600 477159 479713 482263 484810 61 29 484810 487^52 489890 492424 494953 497479 500000 60 30 500000 502517 505030 507538 510043 512543 5150.38 69 31 51.3038 517529 520016 522499 524977 527450 529919 58 32 529919 5.32384 534844 537.300 539751 5-12197 544639 57 ,33 &44639 517076 519.509 551937 &54360 556779 559193 56 34 559193 561602 564007 566406 668801 571191 573576 55 a5 573576 5759o7 5783.32 580703 583069 585429 687785 .54 36 5877&5 590136 592482 594823 597159 599489 601815 53 37 601815 604136 606451 608761 611067 613367 615661 52 38 61.5661 617951 62023;) 622515 624789 627057 629320 51 39 629320 631578 633831 636078 638320 640557 642788 50 40 642788 645013 647233 649448 6.51657 65^3861 656059 49 41 6.560.59 ft58252 6604.39 6626^0 664796 666966 669131 48 42 669131 671289 673143 67.S590 677732 679868 681998 47 43 681998 681123 686242 688:355 690462 692563 6^4658 46 44 694658 696748 698832 700909 702981 705047 707107 45 Deg. 60' 50' 40' 30' 20' 10' 0' Deg. NATUI lAL CO SINES. 368 TABLE III.— NATURAL TANGENTS. 1 1 Deg. 0' icy 2^ 30' 40' 50' 60' Deg. ! 000000 002909 0a5818 008727 0116.36 014545 017455 89 1 0174.5.5 020:36.5 023275 026186 029097 032009 034921 88 2 034921 037834 040747 04:3661 046576 019491 052408 87 3 052108 a55325 058243 061163 064083 067004 0()9927 86 4 069927 072851 075775 078702 081629 084558 087489 85 5 087489 090421 0933.54 096289 099226 102104 105104 84 6 105104 108046 110990 113936 116883 119833 122785 83 7 122785 12.5738 128694 131652 134613 137.576 140541 82 8 140.541 143508 146478 149451 152426 155404 158384 81 9 158384 161368 164354 167343 170331 173329 176327 80 10 176.327 179328 18^3^ 185339 188^49 191.363 194380 79 11 194380 197401 200425 20ai52 206483 209518 212557 78 12 212.557 215599 218645 221695 224748 227806 230868 77 13 2.30868 2339.34 237004 240079 243157 246241 249328 76 14 249328 252420 255516 258618 261723 204834 267949 75 15 267949 271069 274194 277325 280460 283600 286745 74 16 286745 289896 293052 296213 299380 302553 305731 73 17 305731 308914 312104 315299 3ia500 321707 324920 72 18 324920 328139 331.364 331595 a37833 »41077 ^44328 71 19 344328 34758.5 350848 354119 357396 360679 363970 70 20 363970 367268 370573 373885 877204 380530 383804 69 21 383864 3^205 390.554 393910 397275 400046 401026 68 22 404026 407414 410810 414214 417626 421046 424475 67 23 424475 427912 43i:i58 4»4812 438276 441748 445229 66 24 445229 448719 452218 455726 459244 462771 466308 65 2.5 466308 469854 473410 476976 480551 484137 487733 64 26 48773;^ 491:339 4949.5i5 498582 502219 505867 509525 63 27 50952.5 513195 516875 620.567 624270 627984 531709 62 28 5;J1709 6:35146 539195 5429.56 646728 550513 554309 61 29 5M309 558118 561939 665773 669619 573478 577350 60 30 577*50 581235 685134 589045 592970 696908 600861 59 31 600861 604827 608807 612801 616809 620832 624869 68 32 024869 628921 632988 637070 041167 645280 649408 67 m 649408 6.53.551 657710 661886 666077 670284 674509 66 34 674.509 678749 683007 687281 691572 695881 700208 65 a5 700208 7045.51 708913 713293 717691 722108 726543 64 36 720.543 730996 7.35469 739961 744472 749003 753554 53 37 7,53551 75812-5 762716 767327 771959 T76612 781286 52 38 781286 785981 790697 795436 800196 804979 809784 61 39 809784 ' 814612 819463 824:336 829234 834155 839100 50 40 839100 844069 849062 8.54081 859124 864193 809287 49 41 869287 874407 879.553 884725 889924 895151 900404 48 42 900101 905685 910994 916331 921697 927091 932515 47 43 9.32515 937968 943451 948965 951508 960083 965689 46 44 965689 971326 976996 982697 988432 994199 1.000000 45 Deg. 60' 50' 40' 30' 20' 10' O' Deg. N ATURA L COTA NGENT 3. 369 1 j TABLE III.- -NATURAL SINES. 1 f Deg. 0' 10' 20' 30' 40' 50' 60' Deg. 45 707107 709161 711209 713250 715286 717316 719340 44 46 719340 721.3.57 723369 725374 727374 729307 731354 43 47 7313.54 733334 7^5309 737277 739239 741195 743145 42 48 743145 745088 747025 748956 750880 752798 754710 41 i 49 754710 756615 758514 760406 762292 764171 766044 40 1 50 766044 767911 769771 771625 773472 775312 777146 39 : 51 777146 778973 780794 782608 784416 786217 788011 38 1 52 788011 789798 791579 793a53 795121 796882 798636 37 ! 53 798636 800383 802123 8038.57 805584 807304 809017 36 54 809017 810723 812423 814116 815801 817480 819152 35 55 819152 820817 822475 824126 825770 827407 829038 34 56 829038 830661 832277 833886 8a5488 837083 838671 33 57 838671 840251 841825 843391 844951 846503 848048 32 58 848048 849586 851117 852640 854156 855665 857167 31 59 857167 858662 860149 861629 863102 864567 866025 30 60 866025 867476 868920 870356 871784 873206 874620 29 61 " 874620 876026 877425 878817 880201 881578 882948 28 62 882948 8S4309 88.5664 887011 888350 889682 891007 27 63 891007 892323 893633 894934 896229 897515 898794 26 64 898794 900065 901329 902585 903834 905075 906308 25 65 906308 907533 908751 909961 911164 912358 913545 24 66 913545 914725 915896 917060 918216 919364 920505 23 67 920505 921638 922762 923880 924989 926090 927184 22 68 927184 928270 929348 930418 931480 932534 933580 21 69 933580 934619 935650 936672 937687 938694 939693 20 70 939693 9406S4 941666 942641 943609 944568 945519 19 71 945^319 946462 947397 948324 949243 950154 951057 18 72 951057 951951 952838 953717 954.588 955450 956305 17 73 9.56305 9.57151 957990 958820 959642 960456 961262 16 74 961262 962059 962849 963630 964404 965169 965926 15 75 965926 966675 967415 968148 968872 969588 970296 14 76 970296 970995 971687 972370 973045 973712 974370 13 77 974370 975020 975662 976296 976921 977539 978148 12 78 978148 978748 979341 97992.5 98a500 981068 981627 11 79 981627 982178 982721 983255 983781 984298 984808 10 80 984808 985309 985801 986286 986762 987229 987688 9 81 987688 988139 988582 989016 989442 989859 990268 8 82 990268 990669 991061 991445 991820 992187 992546 7 83 992546 992896 993238 993572 993897 994214 994522 6 84 994522 994822 995113 995396 995671 995937 996195 5 85 996195 996444 9966g5 996917 997141 997357 997564 4 86 997564 99776;^ 997953 998135 998308 998473 998630 3 87 998630 998778 998917 999048 999171 999285 999391 2 88 999.391 999488 999577 999657 999729 999793 999848 1 89 999848 999894 999932 999962 999983 999996 1.000000 Deg. 60' 50' 40' 30' 20' 10' 0' Deg. NATUB .AL CO SINES. 370 Fj TABLE IIL— NATURAL TANGENTS. Deg. 0' 10' 20' 30' 40' 50' 60' Deg. 45 1.000000 1.005835 1.011704 1.017607 1.023546 1.029.520 1.0a>530 44 46 1.035530 1.041577 1.047060 1.0.53780 1.059938 l.W)6134 1.072369 43 47 1.072:369 1.078642 1.0849.55 1.091309 1.0977021 1.1041:37 1.110612 42 48 1.110612 1.117131 1.123691 l.l;%294 1.1:36941 1.14:3633 1.1.50.368 41 49 1.150368 1.157149 1.163976 1.170850 1.177770 1.184738 1.191754 40 50 1.191754 1.198818 1.205933 1.213097 1.220312 1.227579 1.234897 39 51 1.231897 1.212269 1.249693 1.2.57172 1.261706 1.272296 1.279W2 38 52 1.279912 1.287645 1.29.5106 1.303225 1.311105 1.319014 1.. 327045 37 53 1.327045 l.;«5108 1.34.3233 l.a51422 l.a59676 1.367996 1.376:382 36 54 1.376382 1.3»1835 1.393357 1.401948 1.410610 1.419343 1.428148 35 55 1.428148 1.437027 1.44.5980 1.455009 1.464115 1.473298 1.482561 34 56 1.482561 1.491904 1.501328 1.510835 1.. 520426 1..530102 1.539865 33 57 1.539805 1.549716 1..5;396.>5 1.569686 1.579808 l.,590024 1.600335 32 58 1.600a3.j 1.610742 1.621247 l.&3ia52 1.642.558 1.65.3366 1.664279 31 59 1.664279 1.675299 1.686426 1.69766^3 1.709012 1.720474 1.732051 30 60 1.732051 1.743745 1.75.55.59 1.767494 1.7795.52 1.791736 1.804048 29 61 1.804048 1.816189 1.829063 1.^1771 1.854616 1.867600 1.880726 28 62 1.880726 1.893997 1.907415 1.920982 1.9.S4702 1.948577 1.962611 27 63 1.962611 1.976805 1.991164 2.005690 2.020:380 2.0.3-52-56 2.050304 26 64 2.050304 2.063532 2.080944 2.096544 2.1123:35 2.128321 2.144507 25 65 2.144507 2.160896 2.177492 2.194300 2.211323 2.228.568 2.246037 24 66 2.246037 2.263736 2.281609 2. 29984:3 2.318261 2.336929 2.a55852 23 67 2.35.5852 2.37i5a37 2.394489 2.414214 2.4,34217 2.454,506 2.475087 22 68 2.475087 2.495966 2.517151 2.-5:38648 2. .560465 2.582609 2.C05089 21 69 2.605089 2.627912 2.651087 2.674621 2.698525 2.722808 2.747477 20 70 2.747477 2.772515 2.798020 2.823913 2.850235 2.876997 2.904211 19 71 2.904211 2.931888 2.960042 2.9886a5 3.017830 3.047492 3.077684 18 72 3.077684 3.108421 3.139719 3.171.5a5 3.204064 3.237144 3.27085:3 17 73 3.2708.>3 3.305209 3.3402:53 3.375943 3.4123«j:3 3.449512 3.487414 16 74 3.487414 3.526094 3.565575 3.6058&1 3.647047 3.689093 3.732061 15 75 3.732051 3.775952 3.820828 3.866713 3.913642 3.961652 4.010781 14 76 4.010781 4.061070 4.112561 4.165300 4.219a32 4.274707 4.331476 13 77 4.331476 4.389694 4.449418 4.510709 4..57:3629 4.638246 4.704630 12 78 4.704630 4.7728.57 4.843005 4.9151.57 4.989403 6. 0(5.58:35 5.144554 11 79 5.144554 5.225665 6.309279 6.395517 6.484505 6.576379 5.671282 10 80 5.671282 5.769369 5.870804 5.975764 6.0844.38 6.197028 6.313752 9 81 6.313732 6.434843 6.. 560.554 6.69115() 6.820944 6.968234 7.11.5.370 8 82 7.115370 7.26872.5 7.428706 7.5957.54 7.770:351 7.9.53022 8.144346 7 83 8.144346 8.344956 8.-55.5547 8.77(5887 9.009826 9.255.304 9..ai.%4 6 84 9.514364 9.788173 10.07803 10.3854C 10.71191 11.05943 11.43005 6 85 11.43005 11.82617 12.2.50,51 12.70620 13.19688 13.72674 14.30067 4 86 14.30067 14.92442 15.60478 16.34986 17.16934 18.07498 19.08114 3 87 19.08114 20.20.355 21.47040 22.90377 24.54176 26.43160 28.6362.5 2 88 28.63625 31.241.58 34.36777 38.18846 42.96408 49.10:388 57.28996 1 89 57.28996 68.75009 85.93979 114.5887 171.8864 343.7737 00 Deg. 60' 50' 40' 30' 20' 10' 0' Deg. NATURAL COTANGENTS. 371 M. 1 2 3 4 5 6 . 7 8 9 10 11 12 13 U 15 16 17 18 19 20 21 22 23 24 2-3 26 27 28 29 30 31 32 33 34 So 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 \~m7 89 '^ TABLE IV.— LOGARITHMIC 6.463726 764756 910847 7.065786 162696 241877 366816 417968 463726 7.505118 542906 577668 609853 639816 667845 694173 718997 742478 764754 7.785943 806146 825451 843934 861662 895085 910879 92(3119 940842 7.95.5082 968870 982233 995198 8.007787 020021 031919 W3301 0.54781 065776 8.076500 08C9 097183 107107 110926 126471 i;35810 144953 irMQiJ 162681 8.171280 179713 187985 196102 204070 211895 219581 227134 234557 24185.5 5017 2934 2082 1615 1319 1115 5 852.5 762.6 689.8 629.8 579.3 536.4 499.3 467.1 438.8 413.7 391.3 371.2 353.1 336.7 321.7 308.0 295.4 283.9 273.2 263.2 254.0 245.3 237.3 229.8 222.7 216.1 209.8 203.9 198.3 193.0 188.0 ia3.2 178.7 174.4 170.3 166.4 162.6 159.1 155.6 152.4 149.2 146.2 143.3 140.5 137.8 135.3 132.8 130.4 128.1 125.9 123.7 121.6 Tang. PPl" M^ 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 6.463726 764756 940847 7.065786 162696 241878 30882;5 366817 417970 463727 7.505120 542909 577672 609857 639820 667849 694179 719003 742484 764761 7.7859,51 8061.5.5 82.5460 843944 861674 878708 910894 926134 940858 7.955100 9822.53 995219 8.007809 020044 031945 01:3.527 051809 085800 8.076531 086997 097217 107203 110963 126510 1358.51 144996 153952 162727 8.171328 179763 188036 1961.56 204126 211ft53 219641 227195 234621 241921 5017 293.5 2082 1615 1320 1116 966.5 852.5 762.6 689.9 629.8 579.4 536.4 499.4 467.1 438.8 413.7 391.3 371.2 353.2 336.7 321.7 308.0 295.5 283.9 273.2 .2 2.54.0 245.4 237.3 229.8 222.7 216.1 209.8 203.9 198.3 193.0 188.0 183.3 178.7 174.4 170.3 166.4 162.7 1.59.1 1.55.7 152.4 149.3 146.2 143.3 140.6 137.9 135.3 132.8 130.4 128.1 125.9 123.8 121.7 Cosine. PPl" (lotans. PPl" M Sin-. 8.241855 249033 256094 263042 276614 283243 289773 296207 302546 308794 8.314954 321027 327016 332924 338753 344504 350181 355783 361815 366777 8.372171 377499 3S2762 387962 393101 398179 403199 408161 41,3068 417919 8.42271 427462 432156 436800 441394 445941 450440 454893 459301 463665 8.467985 472263 476498 480693 484848 488963 493040 497078 501080 505045 8.508974 512867 516726 52a551 524343 528102 531828 535523 539186 512819 Cosinf PIM" 119.(; 117.7 115.8 114.0 112.2 110.5 108.8 107.2 105.6 104.1 102.7 101.2 99.82 98.47 97.14 95.86 94.60 93.38 92.19 91.03 89.90 88.80 87.72 86.6 85.64 84. 83. 82.71 81.77 80. 79. 79.09 78.23 77.40 76.57 75.77 74.99 74.22 73.46 72.73 72.00 71.29 70.60 69.91 69.24 68.59 67.94 67.31 66.69 66.08 65.48 64. 64.31 63.75 63.19 62.64 62.11 61.58 61.06 60.55 PPl' Tang. 8.241921 219102 a561ft-) 263115 2699,56 276691 283323 289856 296292 3026J34 308884 8.315046 321122 327114 33302,5 3388,50 344610 3502891 3558951 3614301 PPl" 8.3722t)2 377622 382889 388092 893234 398315 403338 408304 413213 418068 8.422869 427618 432315 436962 441.560 446110 450613 455070 459481 463849 8.468172 472454 476693 480892 4&5050 489170 493250 497293 501298 505267 8.509200 513098 516961 520790 524586 528349 532080 535779 539447 543084 119.7 117.7 115.8 114.0 112.2 110.5 108.9 107.2 10,5.7 104.2 102.7 101.3 99.87 98.51 97.19 195.90 94.65 93.43 92.24 91.08 89.95 88.85 87.77 86.72 85.70 84.70 83.71 82.76 81.82 80.91 80.02 79.14 78.29 77.45 76.63 75.83 75.05 74.28 73.52 72.79 72. 71.35 70.66 69.98 69.31 68.65 68.01 67.38 66.76 66.15 65.55 64.96 64.39 03.82 63.26 62.72 62.18 61.65 61.13 60.62 Cotiniff. PPl" H72 HS^ 2" SINES AND TANGENTS. 3« 87" Sine. IPPl" Tang. FPl" M. 8.512819 516422 M9995 553.389 557054 560540 563999 567431 574214 577566 60.04 59. .55 59.06 58.58 .58.11 .57.65 .57.19 .56.74 56.30 55.87 8.580892 f.-^i 584193 " 587469 590721 5971.52 600332 609734 8.612823 615891 618937 621962 624965 627948 630911 63:^854 636776 639680 8.642563 64.5428 648274 651102 653911 65G702 659475 662230 664968 6671 8.670.393 673080 675751 678405 6S1043 683665 686272 688803 691438 8.698.543 699073 701589 70;090 706.577 709049 711.507 7139.52 716:}8;3 718800 ,54.60 ;54.19 53.79 .5:^.39 53.00 52.61 52.23 51.86 .51.49 ,51.12 50.76 50.41 ,50.06 49.72 49.-38 49.04 48.71 48.39 48.06 47.75 47.43 47.12 40.82 46.. 52 46.22 4;5.92 4.5.6:3 45. a5 45.06 44.79 U.'A 44.24 4.3.97 43.70 43.44 43.18 42.92 42.67 42.42 42.17 41.92 11.68 41.44 41.21 40.97 40.74 40.51 40.29 Cosinf PPI 8..54;»84 .546691 550268 55,3817 557a3() 560828 564291 567727 571l;37 574.520 577877 8.581208 584514 587795 59ia51 594283 597492 600677 610094 8.613189 616262 619313 622.343 62.5a52 628340 631308 6342.56 637184 640093 8.642982 6458-53 648704 651.537 654352 657149 659928 ()6543;3 668160 .670870 67:3563 6702:39 678900 681.544 684172 686784 69196:3 694529 8.697081 699617 702139 704^40 707140 709618 71208:3 7145^4 716972 719396 Cotang. 60.12 59.62 59.14 58.66 58.19 57.73 57.27 56.82 56.38 .55.95 55.52 55.10 .54.68 .54.27 .53.87 53.47 53.08 52.70 52.32 51.94 51.58 51.21 50.85 50.50 .50.15 49.81 49.47 49.13 48.80 48.48 48.16 47.^4 47..53 47.22 46.91 46.61 46.31 46.02 4.5.73 45.44 45.16 44.88 44.61 44., 34 44.07 43.80 43.54 43.28 43.a3 42.77 42.52 42.28 42.03 41.79 41.55 41. 41. 40.85 40.62 40.40 Sine. PPI" Tans. 8.718800 721204 72:3.595 725972 728:337 730688 7^3027 7X5354 737667 739969 742259 8.744.536 746802 7490,55 751297 7,5a52S 75574 7579,55 760151 762337 764511 8.766675 768828 770970 773101 77.522:3 777,333 779434 781.524 7a%05 78.567,5 8.7877.36 789787 791828 79:38.59 795881 797894 799897 801892 80:3876 80.Wj2 8.807819 809777 811726 81.3667 81,5599 817522 8194:36 8213^13 82:3240 82:130 8.827011 828884 8:30749 832G07 8:344.56 830297 8381:30 8399.56 841774 843585 40.06 39.84 J9.62 39.41 39.19 .38.98 ;38.77 .38.57 .38.36 ,38.16 37.96 :37.76 ;37.,56 ,37.37 .37.17 .36.98 .36.80 .36.61 36.42 ,36.24 .36.06 a5.88 a5.70 35.53 ,3.5. a5 35.18 .35.01 a4.84 :34.67 ,a4..51 :34.:i5 .34.18 ai.02 .a3.8G .'3:3.70 :3;3.,54 :i3.39 :3;3.23 :33.08 .32.93 .32.78 32.63 .32.49 32.34 32.19 32.05 :31.91 31.77 :31.63 :31.49 31. a5 31.22 ,31.08 .30.95 30.82 30.69 .30.56 30.43 ,30.30 30.17 Cosinf PPI" PPl' M 8.719396 721806 724204 726588 728959 731317 7a3663 735996 738317 740626 742922 8.74.5207 747479 749740 751989 754227 756453 7.58668 760872 7(5.3065 765246 8.767417 709578 771727 77:3866 775995 778114 780222 782,320 784408 78(5486 8.788.554 790613 792602 794701 79(5731 798752 800763 8027(5.5 8047.58 80(5742 8.808717 810(]8.3 812m 1 814.589 816529 818461 820:»1 822298 824205 826103 8.827992 829874 8:31748 83361:? 8a>471 8.37321 &39163 840998 842825 844644 40.17 39.95 39.74 .39.52 :39.30 39.09 :38.89 38.68 .38.48 38.27 :38.07 37.87 ,37.68 :37.49 37.29 37.10 36.92 36.73 36..55 .m .18 :36.00 .a5.83 .a5.65 .a5.48 ,a5.3i .a5.i4 :34.J>7 31.80 34.64 :34.47 .34.;31 :34.15 .a3.99 :a}.,s;3 :5:3.68 .a3..52 :}3.37 ,3:3.22 :33.07 ,32.92 ;32, :32.62 ;32.48 :32.a3 32.19 :52.05 ,31.91 .31.77! :31.6:3 31.-50 31.. 36 31.23 .31.10 30.96 :30.83 .30.70 30.57 30.45 30.32 (^otang. PPI" 378 »«" 1!_ "m. 1 2 3 4 5 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 85" TABLE IV.— LOGARITHMIC 50 Sino 8.843585 845387 847183 848971 8.50751 852525 854291 856049 857801 85954G 861283 8.863014 864738 866455 868165 871565 873255 874938 876615 87828.5 8.879949 881607 883258 884903 886542 888174 891421 893035 894643 8.896246 897842 901017 902.596 904169 905736 907297 908853 910404 8.911949 913488 915022 916550 918073 919591 921103 922610 924112 92;5609 8.927100 928587 930068 931544 983015 934481 935942 937398 9-38850 940296 Cosine. PPl' .30.05 29.92 29.80 29.67 29.-55 29.43 29.31 29.19 29.07 28.96 28.84 28.73 28.61 28.50 28. 28.28 28.17 28.08 27.95 27.84 27.73 27.63 27.52 27.42 27.31 27.21 27.11 27.00 26.90 26.80 26.70 26.60 26.51 26.41 26.31 26.22 26.12 26.03 25.93 25.84 25.75 25.66 25.56 25.47 25.38 25.29 25.20 25.12 25.03 24.94 24.86 24.77 24. 24.60 24.52 24.43 24.35 24.27 24.19 24.11 PPl' Tang. PPl" M 8.844644 846455 848260 850a57 851846 855403 857171 862433 8.864173 865906 867632 871064 872770 874469 876162 877849 879529 8.881202 8845.30 886185 887833 889476 891112 892742 894366 8.897596 899203 900803 902398 905570 90714" 908719 910285 911846 8.913401 914951 91649.5 918034 919568 921096 922619 924136 925649 927156 8.928658 9301-5;5 931647 93;31.34 934616 9:36093 937-565 939032 940494 941952 Cotanar. 30.19 30.07 29.95 29.82 29.70 29., 58 29.46 29. a5 29.23 29.11 29.00 28.88 28.77 28.66 28.54 28.43 28.32 28.21 28.11 28.00 27.8 27.7 27.68 27.58 27.47 27.37 27.27 27.17 27.07 26.97 26.87 26.77 26.67 26.58 26.48 26.38 26.29 26.20 26.10 26.01 25.92 25.83 25.74 25.65 25.56 25.47 25.38 25.30 25.21 25.12 25.03 24.95 24.87 24.78 24.70 24.62 24.53 24.45 24.37 PPl M. Sir 8.940296 941738 943174 944606 946034 947456 948874 950287 951696 953100 954499 8.95.5894 957284 958670 960052 961429 962801 964170 965534 968249 8.969600 97094' 972289 973628 974962 976293 977619 978941 980259 981573 8.9828&:} 984189 985491 986' 988083 989374 990660 991943 993222 994497 8.995768 997036 999560 9.000816 002069 . 003318 004563 005805 007044 9.008278 009510 010737 011962 013182 014400 01-5()13 016824 018031 019235 CJosine. PJ'l" 24.03 23.94 23.87 23.79 23.71 23.63 23.55 23.48 23.40 23.32 23.25 23.17 23.10 23.02 22.95 22.88 22.80 22.73 22.66 22. 22.52 22.45 22.38 22.31 22.24 22.1 22.10 22.03 21.97 21. 21.83 21.77 21.70 21.6:3 21.57 21.50 21.44 21.38 21.31 21.25 21.19 21.12 21.06 21.00 20.94 20.88 20.82 20.76 20.70 20.64 20.58 20.52 20.46 20.40 20.:34 20.29 20.2:3 20.1 20.12 20.06 PPl' Tang. PI* 8.941952 943404 944852 946295 947734 949168 950597 952021 953441 954856 956267 8.957674 959075 960473 961866 963255 964639 966019 967394 968766 970133 8.971496 972855 974209 975560 976906 978248 979586 980921 982251 983577 8.984899 986217 987532 988842 990149 991451 992750 994045 995337 996624 8.997908 999188 9.000465 001738 003007 004272 005534 006792 008047 009298 9.010546 011790 013031 014268 015502 016732 017959 01918:3 020403 021620 24.21 24.13 24.05 23.97 23.90 23.82 23.74 23.66 23.60 23.51 23.44 23.37 23.29 23.22 23.14 23.07 23.00 22.93 22.86 22.79 22.71 22.65 22.57 22.51 22.44 22.37 22,30 22.23 22.17 22.10 22.04 21.97 21.91 21.84 21.78 21.71 21.65 21.68 21.52 21.46 21.40 21.34 21.27 21.21 21.15 21.09 21.03 20. 20.91 20.85 20.80 20.74 20.68 20.62 20.56 20.51 20.45 20.40 20.33 20, ("otang. I'PI' 374 84« 6« SINES AND TANGENTS. 1^0 Siiu!. 9.01923.5 020435 021G32 022825 02401G 023203 02(J386 027o(>7 028744 029918 031089 9-0322.")7 0:^3421 034582 03)741 03(5896 038048 039197 040342 01148.') 012625 9.013762 044895 016026 0171.54 048279 049400 0.50519 05ia3.5 052749 0538.59 9.054966 056071 057172 058271 059337 0:)0460 061.551 0626:39 063724 064806 9.0()o885 066962 068036 069107 070176 071242 072306 07:3366 074124 075480 9.07()533 077.583 078631 079676 080719 0^1759 082797 08:38:32 084864 O.S;5894 Pl'l" 20.00 19.95 19.89 19.a4 19.78 19.73 19.67 19.62 19.57 19. .51 19.46 19.41 19.36 19.30 19.2.5 19.20 19.15 19.10 19.05 18.99 18.94 18.89 18.84 18.80 18.75 18.70 18.65 18.60 18.;55 18.. 50 18.45 18.41 18.:36 18.31 18.27 18.22 18.17 18.13 18.08 18.04 17.99 17.94 17.90 17.86 17.81 17.77 17.72 17.68 17.63 17.. 59 17.5.5 17.50 17.46 17.42 17.38 17.33 17.29 17.2.5 17.21 17.17 Cosine. Tan J PPl" .021620 0228;34 024044 02.52.51 026455 027655 0288.52 0:30046 0.31237 03242.5 a3:3609 .034791 ft3.5969 0:37144 0a8316 0394,S5 040651 041813 042973 0441.30 045281 .0464:34 047.582 048727 049869 051008 0,52144 05,3277 054407 0555:3.5 0566.59 .057781 0.58900 060016 0611:30 062240 063348 0644.53 06.555(5 0666.55 0677.52 1.068846 PP 071027 0721131 0731971 0742781 07.53.56 07&132 077.505 078.576 ).079644 080710 081773 082833 083891 084947 086000 087050 088098 089144 20.23 20.17 20-11 20.06 20.00 19.95 19.90 19.85 19.79 19.74 19.69 19.64 19.58 19.5:3 19.48 19.43 19.38 19.-33 19.28 19.23 19.18 19.13 19.08 19.03 18.98 18.93 18.89 18.84 18.79 18.74 18.70 18.6.5 18.60 18.55 18.51 18.46 18.42 18.:37 18.33 18.28 18.24 18.19 18.15 18,10 18.06 18.02 17.97 17, 17.89 17.84 17.80 17.76 17.72 17.67 17.63 17.59 17.55 17., 51 17.47 17.43 Cotang. I PPl" M I PPl" 9.085894 086922 087947 088970 089990 091008 092024 093037 094047 09.5056 096062 9.097065 098066 099065 100062 1010^56 102048 1030:37 104025 105010 10.5992 9.106973 1079.51 108927 109901 110873 111842 112809 113774 114737 11.5698 9. 116(5.56 117613 118567 119519 120469 121417 122362 123306 124248 125187 9.126125 127000 127993 128925 129854 130781 131706 132630 133551 1.34470 9.13.5,387 136303 1,37216 138128 1,39037 139944 140850 1417.54 142655 143555 17.13 17.09 17.04 17.00 16.96 16.92 16.88 16.84 16.80 16.76 16.73 16.68 16.65 16.61 16.57 16.53 16.49 16.45 16.41 16.38 16.31 16.30 16.27 16.2:3 16.19 16.16 16.12 16.0! 16. o; 16.01 15.97 15.94 15.90 15.87 15.83 15.80 15.76 15.73 15. ()9 15.66 15.62 15..59| 15.56! 15.. 52 15.491 15.451 15.42! 15.39 15.35 15. .32 15.29 15.25 15.22 1.5.19 15.16 15.12 15.09 15.06 15.03 15.00 'lai .089144 090187 091228 092266 093302 094336 095367 097422 098446 099468 ). 100487 101,504 102519 103532 104542 105550 106556 107559 108,560 109.559 ).110556 111.551 112.543 113533 114521 115507 116491 117472 118452 119429 9.120404 121377 122M8 123317 124284 12.5249 126211 1271 1281.30 129087 9.1:30041 130994 131944 1:32893 133839 134784 ia5726 1136667 13760.5 1.38542 9.13947 140409 14iai0 142269 143196 144121 145044 145966 146885 147803 PP Cotang. 17.:38 17.34 17.30 17.27 17.22 17.19 17.15 17.11 17.07 17.03 16.99 16.95 16.91 16.87 16.84 16.80 16.76 16.72 16.69 16.65 16.61 16.58 16.54 16.50 16.46 16.43 16.39 16.36 16.:32 16.29 16.25 16.22 16.18 16.15 16.11 16.07 16.04 16.01 15.97 1.5.94 15.91 15.87 15.84 15.81 15.77 15.74 15.71 15.67 15.64 15.61 15.58 15.55 15.51 15.48 15.45 15.42 15.39 15. a5 15.32 15.29 IM. PPl' 83" 375 M. M. 1 SI' TABLE IV.— LOGARITHMIC 9« iSinc. PPl" TauK. PPl" BL_ 60 59 58 57 56 55 54 53 52 51 50 9.143555 144453 145349 146243 147136 148026 148915 149802 150686 151569 152451 9.153330 154208 155083 155957 156830 157700 158569 159435 160301 161164 9.162025 162885 163743 164600 1&5454 166307 167159 168008 168856 169702 9.170547 171389 172230 173070 173908 174744 175578 176411 177242 178072 9.178900 179728 180r»l 181374 182196 183016 183834 184651 185466 180280 9.187092 187903 188712 189519 19032.5 191130 191933 192734 193534 194332 14.96 14.93 14.90 14.87 14.84 14.81 14.78 14.75 14.72 14.69 14.66 14.63 14.60 14.57 14.54 14.51 14.48 14.45 14.42 14.39 14.36 14.33 14.30 14.27 14.24 14.22 14.19 14.16 14.13 14.10 14.07 14.05 14.02 13.99 13.96 13.94 13.91 13.88 13.86 13.83 13.80 13.77 13.74 13.72 13.69 13.06 13.64 13.61 13.59 13.56 13.53 13.51 13.48 13.46 13.43 13.41 13.C 13.^ 13.^ 13.30 Cosine. PPV 9.147803 148718 149632 150544 151454 152363 153269 154174 155077 155978 156877 9.157775 158671 159565 100457 101347 162236 163123 164008 164892 105774 9.166654 167532 168409 169284 170157 171029 171899 172767 173634 174499 9.175362 176224 177084 177942 178799 179655 180508 181360 182211 183059 9.18390 184752 18559 186439 187280 188120 188958 189794 190629 191462 9.192294 193124 193953 194780 195606 196430 197253 198074 198894 199713 15.26 15.23 15.20 15.17 15.14 15.11 15.08 15.05 15.02 14.99 14.96 14.93 14.90 14.87 14.84 14.81 14.79 14.76 14.73 14.70 14.67 14.64 14.61 14.58 14.55 14.53 14.50 14.4 14.44 14.42 14. 14. 14.33 14.31 14.28 14.25 14.23 14.20 14.17 14.15 14.12 14. 14.07 14.04 14.02 13.99 13.96 13.93 13.91 13.89 13.86 13.84 13.81 13.79 13.76 13.74 13.71 13.69 13.66 13.64 ('Otana. PPl' Sine. 9.194332 195129 195925 19071?) 197511 198302 199091 199879 200066 201451 202234 9.203017 203797 204577 205354 206131 206906 20767! 208452 209222 209992 9.210760 211526 212291 213055 213818 214579 215338 216097 216854 217609 9.218363 219116 219868 220618 221367 222115 222861 223606 224349 225092 9.225833 226ij73 227311 228048 228784 229518 230252 2;30984 231714 232444 9.233172 233899 2M625 235^49 236073 PPl' 237515 238235 239670 13.28 13.26 13.23 13.21 13.18 13.16 13.13 13.11 13.08 13.06 13.04 13.01 12.99 12.96 12.94 12.92 12.89 12.87 12. 12.82 12.80 12.78 12.75 12.73 12, 12.68 12.66 12.64 12.62 12.59 12.57 12.55 12.53 12.50 12.48 12.46 12.44 12.42 12.39 12.37 12.35 12.33 12.81 12.28 12.26 12.24 12.22 12.20 12.18 12.16 12.14 12.12 12.09 12.07 12.05 12.03 12.01 11.99 11.97 11.95 Tail-. PPl" JNf. 9.199713 200529 201345 202159 202971 203782 204592 2a5400 200207 207013 207817 9.208619 209420 210220 211018 211815 212611 213405 214198 214989 215780 9.210568 217356 218142 218926 219710 220492 221272 222052 222830 223607 9.224382 225156 22.5929 226700 227471 228239 229007 229773 230539 231.302 9.232065 232826 233586 234345 2,35103 235859 236614 237368 238120 238872 9.2^9622 240371 241118 241865 242610 243354 244097 244839 245579 246319 13.61 13.59 13.57 13.54 13.52 13.49 13.47 13.45 13.42 13.40 13.38 13. a5 13.33 13.31 13.28 13.26 13.24 13.21 13.19 13.17 13.15 13.12 13.10 13.08 13.06 13.03 13,01 12.99 12.97 12.94 12.92 12.90 12.88 12.86 12. &4 12.81 12.79 12.77 12.75 12.73 12.71 12.69 12.67 12.65 12.62 12.60 12.58 12.56 12.54 12.52 12.50 12.48 12.46 12.44 12.42 12.40 12.38 12.36 12.34 12.32 Cosine. I'Pl" Cotang. PPl" M. 376 80° io< SINES AND TANGENTS. 11 « 30 31 32 33 31 a3 m 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.239670 240386 241101 211814 212.526 2mS7 213947 214656 215:363 2K30)9 216775 9.217478 218181 2188.^3 249583 250282 2509.S0 251677 252373 253087 2.5:3761 9. 214453 255144 2.>58:31 256.52-3 257211 2-57898 258;5a3 259268 259951 2606.33 9.261314 261994 262673 26;i3.51 231027 264703 265377 266a51 286723 287395 9.26806.' 268734 269102 27003! 2707*5 271400 272011 272726 273;3S8 274019 9.274708 27.5337 276025 276681 277,337 277991 278645 279297 279948 280599 I'I'l' 11.93 11.91 11.89 11.87 11.85 11.83 11.81 11.79 11.77 11.75 11.73 11.71 11.69 11.67 11.65 11.63 11.61 11.59 11..58 11.56 11.54 ll.,52 11.50 11.48 11.46 11.44 11.42 11.41 11.39 11.37 11. a5 11.33 11.31 11.30 11.28 11.26 11.24 11.22 11.20 11.19 11.17 11.15 11.13 11.12 11.10 11.08 11.06 11.05 11.03 11.01 10.99 10.98 10.96 10.94 10.92 10.91 10.89 10.87 10.86 10.84 Tuner. 9.246319 247a57 247794 248530 249264 249998 250730 2.51461 2.52191 2.52920 253648 9.2.51.374 2-55100 255824 2.56517 2)7269 257990 258710 259429 260146 26086:3 9.2(31578 262292 26.30a5 263717 264428 265138 265847 2665.55 267261 267967 9.268671 269375 270077 270779 271479 272178 272876 273573 274269 274964 9.275658 276351 277043 277734 27&121 279113 279801 280488 ■ 281174 281858 9.282542 283225 28390' 284588 28.5268 285947 286624 287301 287977 288652 PP 12.30 12.28 12-26 12.24 12.22 12.20 12.18 12.17 12.15 12.13 12.11 12.09 12.07 12.0.5 12.03 12,01 12.00 11.98 11.96 11.94 11.92 11.90 11.89 11.87 11.85 11.83 11.81 11.79 11.78 11.76 11.74 11.72 11.70 11.69 11.67 11.65 11.64 11.62 11. 11.58 11.57 11..55 11.53 11.51 11.50 11.48 11.47 11.45 11.43 11.41 11.40 11.38 11.36 11.35 11.33 11 11.30 11.28 11.26 11.25 M. M. SiiK 9.280599 281218 281897 282544 283190 2»4480 285124 28.5766 286408 287048 2889&1 289600 290236 290870 291504 292137 292768 Cosine. PPl" Cotang. PPl" M. M. Cosine. PPl" Cotang. PPl" M. 9.294029 294658 295286 2ft5913 296,539 2971frl 297788 298412 2990ai 299<}55 9.300276 300895 301514 302132 302748 3033ty 303979 301593 305207 305819 9.306430 307041 307650 308259 308867 309474 310080 310685 311289 311893 9.312495 31:3097 313698 314297 314897 315195 316092 316689 317284 317879 PPl" 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 10.61 10.59 10.58 10.,56 10.54 10.. 53 10.51 10.. 50 10.48 10.46 10.45 10.43 10.42 10.40 10.39 10.37 10.36 10. ai 10.-32 10.-31 10.29 10.28 10. 2« 10.25 10.23 10.22 10.20 10.19 10.17 10.16 10.14 10.13 10.11 10.10 10.08 10.07 10.06 10.04 10.03 10.01 10.00 9.98 9.97 9.96 9.04 9.93 9.91 .288652 289326 290671 291342 292013 292682 293350 294017 2946H4 295349 .296013 296677 297339 298001 298662 299322 2fi9980 300638 301295 301951 .302607 30.3261 303914 301.567 305218 805869 306519 30716K 307816 30846:3 .5309109 309751 310399 311012 311685 312327 312968 313608 31424 314885 .315,523 316159 316795 317430 318064 318697 319330 319961 320592 321222 .321851 322479 323106 324358 32560: 326231 326853 327475 PPl' 11.23 11.22 11.20 11.18 11.17 11.15 11.14 11.12 11.11 11.09 11.07 11.06 11.01 11.03 11.01 11.00 10.98 10.96 10.95 10.93 10.92 10.90 10.89 10.87 10.86 lO.gl 10.83 10.81 10.80 10.78 10.77 10.75 10.74 10.73 10.71 10.70 10.68 10.67 10.65 10.64 10.62 10.61 10.60 10.58 10.5' 10.55 10.51 10.53 10.. 51 10.50 10.48 10.47 10.45 10.44 10.43 10.41 10.40 10 10.37 10.36 79' Trio:.— 32. 377 7H" 1*J0 TABLE IV.— LOGARITHMIC 13* Sine. .817879 318473 319066 319658 320249 320840 321430 322019 5226071 323194 323780 .324366 324950 325534 326117 326700 327281 328442 329021 329599 .330176 330753 331329 331903 332478 333051 333624 334195 334767 335387 .335906 336475 337043 337610 338176 338742 389307 389871 340434 340996 .341558 ^42119 342679 343239 343797 344355 344912 345469 346024 34&579 .347134 347687 348240 348792 349343 350443 350992 351540 352088 Cosine. PPl 9.87 9.86 9.84 9.83 9.82 9.80 .79 9.77 9.76 9.75 9.73 9.72 9.70 9.69 9.68 9.66 9.65 9.64 9.62 9.61 9.60 9.58 9.57 9.56 9.54 9.53 9.52 9.50 9.49 9.48 9.46 9.45 9.44 9.43 9.41 9.40 9.89 9.87 9.36 9.85 9.34 9.82 9.31 9.30 9.29 9.27 9.26 9.25 9.24 9.22 9.21 9.20 9.19 9.17 9.16 9.15 9.14 9.13 rang. PPl" M 9.327475 328095 328715 329884 329953 330570 3:51187 3318a3 332418 33;?083 3&8646 9.384259 834871 335482 336098 386702 387311 387919 338527 339138 a39739 9.340344 340948 341552 342155 342757 348858 348958 344;558 345157 345755 9.846358 346949 347.545 348141 348735 349829 349922 a50514 351106 351097 9.352287 a32876 353465 354053 354640 355227 a55813 356398 357566 9.358149 a">8731 a-^9318 3.39898 360474 miorjs 361632 362210 362787 363364 10.85 10.33 10.32 10.80 10.29 10.28 10.26 10.25 10.24 10.28 10.21 10.20 10.19 10.17 10.16 10.15 10.13 10.12 10.11 10.10 10.08 10.07 10.06 10.04 10.08 10.02 10.00 9.99 9.98 9.97 9.96 9.94 9.93 9.92 9.91 9.90 9.88 9.87 9.86 9.85 9.88 9.82 9.81 9.80 9.79 .75 9. 9. 9.73 9.71 9.70 9.69 9.68 9.67 9.66 9.65 9.63 9.62 9.61 Cotang. PPl" M M, 9.332088 352685 a53181 a33720 a34271 354815 a55a38 355901 356448 356984 357524 9.a38064 a38603 359141 359678 360215 360752 361287 361822 362356 362889 9.368422 368954 36448.3 365016 365546 366075 366604 367181 3676.59 368185 9.368711 809236 369761 370285 370808 371380 371852 372378 372894 373414 9.373988 374452 374970 375487 370008 376519 377035 377549 378063 378577 9.379089 379601 380113 380624 381134 381643 382152 382661 383168 383675 PPl' M. Cosine. PPl" 9.11 9.10 9.09 9.08 9.07 9.a3 9.04 9.08 9.02 9.01 8.99 8.98 8.97 8.96 8.95 8.98 8.92 8.91 8.90 8.89 8.88 8.87 8.85 8.84 8.88 8.82 8.81 8.80 8.79 8.77 8.76 8.75 8.74 8.73 8.72 8.71 8.70 8.69 8.67 8.66 8.65 8.64 8.03 8.62 8.61 8.60 8.59 8.58 8.57 8.56 8.54 8.53 8.52 8.51 8.50 8.49 8.48 8.47 8.46 8.45 Tans. .868364 368940 364515 365090 365664 366237 366810 367382 367953 36&524 369094 .369668 370282 370799 871367 371938 372499 3730(M 373629 374193 374756 .375319 375881 376442 377003 377563 378122 378 379239 379797 380854 .380910 381466 382020 382575 383129 383082 384234 384786 385337 385888 .386438 386987 387586 388084 388031 389178 889724 390270 390815 391360 .391903 392447 392989 39a>]l 394078 394614 3951.54 395694 396233 396771 9.00 9.69 9.58 9.57 9.55 9.54 9.53 9.52 9.51 9.50 9.46 9.45 9.44 9.43 9.42 9.41 9.40 9.39 9.38 9.37 9.35 9.34 9.38 9.32 9.31 9.30 9.29 9.28 9.27 9.26 9.25 9.24 9.23 9.22 9.21 9.20 9.19 9.18 9.17 9.15 9.14 9.18 9.12 9.11 9.10 9.09 9.08 9.07 9.06 9.ft5 9.04 9.03 9.02 9.01 9.00 8.99 8.98 8.97 Cotang. PPl" M 7^' 378 •76^ 140 SINES AND TANGENTS. 150 31. 8ii 9. 1 2 3 4 5 6 7 399575 400062 400549 4010a5 401520 402005 402489 402972 4034.55 .403938 404420 404901 40.5382 405802 406341 406820 407299 407777 408254 .408731 409207 409682 410157 4106.32 411106 411.579 4120.52 412.521 412996 8.43 8.42 8.41 8.40 .383675 L ,. 3841821""'^ 384687 385192 385697 386201 386704 387207 387709 3.S8210 388711 389211 389711 390210 390708 391206 391703 392199 392695 393191 39;}685 394179 394673 395166 395658 3961.50 39<3641 397132 397621 398111 398600 8.37 8.36 8.35 8.34 8M 8.32 8.31 8.30 8.28 8.27 8.26 8.25 8.24 8.2;^ 8.22 8.21 8.20 8.20 8.18 8.17 8.17 8.16 8.15 8.14 8.13 8.12 8.11 8.10 8.09 8.08 8.07 8.08 8.05 8.04 8.03 8.02 8.01 8.00 7.99 7.98 7.97 7.96 7.95 7.94 7.94 7.93 7.92 7.91 7.90 7.89 7.88 7.87 7.80 PPl' 9.39.')771 17846 398919 3994.55 40a524 4010.58 401591 402124 9.402ft50 403187 403718 404249 404778 40.5:^08 4ft58;36 406364 406892 407419 9.407945 408471 409.521 410045 4ia569 411092 411615 412137 412658 9.413179 413699 414219 414738 4152.57 415775 416293 416810 417326 417842 9.4183.58 418873 419387 419901 420415 420927 421440 421952 422463 422974 9.423484 423993 424.503 425011 425519 426027 426;534 427041 427547 428052 PPl" M 8.96 8.95 8.94 8.93 8.92 8.91 8.90 8.89 8.8.S 8.87 8.86 8.a5 8.84 8.83 8.82 8.81 8.80 8.79 8.78 8.77 8.76 8.75 8.74 8.74 8.73 8.72 8.71 8.70 8.69 8.68 8.67 8.66 8.6.5 8.64 8.64 8.63 8.62 8.61 8.60 8.59 8..58 8.-57 8. ,56 8.55 8.55 8..54 8.53 8.52 8.51 8.,50 8.49 8.48 8.48 8.47 8.46 8.45 8.44 8.43 8.43 M. Sine. I PPl" 9. 412996 413467 413938 414408 414878 41.5347 415815 416283 416751 417217 4176»4 418150 418615 419079 4ia>44 420007 420470 421395 421857 422318 ). 422778 423238 423697 424156 424615 42507 425530 425987 426443 426899 ). 427354 427809 428263 428717 429170 429623 430075 430527 430978 431429 5.431879 432329 432778 433226 433675 4^^122 434569 4;i3016 43.5462 4a5908 ). 436353 436798 437242 437686 438129 438572 439014 439456 439897 440338 ^50 rotang. PPl" m. M. Cosi 379 7.85 7.84 7.83 7.83 7.82 7.81 7.80 7.79 7.78 7.77 7.76 7.75 7.74 7.73 7.73 7.72 7.71 7.70 7.67 7.67 7.66 7.65 7.64 7.63 7.62 7.61 7.60 7.60 7.59 7.58 7.57 7.56 7.55 7.54 7.53 7.53 7.52 7.51 7.50 7.50 7.49 7.48 7.47 7.46 7.45 7.45 7.44 7.43 7.42 7.41 7.40 7.40 7.39 7.38 7.37 7.. 36 7.35 7.35 Tang. ).428a52 428.558 429062 429566 430070 4a573 431075 431577 432079 432580 .43a580 434080 434579 435078 435576 436073 436,570 4370()7 437.56;j 9.438554 439048 43a543 44003*) 440529 441022 441514 442006 442497 442988 9.443479 443968 444458 444947 44^35 445923 446411 447384 447870 9.448356 448841 449326 449810 450294 450777 451260 451743 452225 452706 9.453187 453668 454148 454628 455107 455586 456064 456542 457019 457496 PPl" Cotaiip. PPl" M. -540 16" TABLE IV.— LOGARITHMIC 17° Sine. 9.440338 440778 441218 44ia58 442096 4425*5 442973 443410 44:3847 444284 444720 9.44515.5 44.5590 446025 4464.59 447326 447759 448191 448623 4490.54 9.4494a5 449915 450345 4,50775 451204 451632 4.52060 452488 452915 453342 9.453768 454194 454619 4.55014 455469 455893 4.56316 456739 457162 457584 9.458006 458427 458848 459268 4.59688 460108 460527 460946 461364 461782 9.462199 462616 46.3032 463448 463864 464279 464694 465108 465522 465935 PPi" 7.34 7. as 7.32 7.31 7.31 7.30 7.29 7.28 7.27 7.27 26 7.25 7.24 23 7.23 7.22 7.21 7.20 7.20 7.19 7.18 7.17 7.17 7.16 7.15 7.14 7.13 7.13 7.12 7.11 7.10 7.10 7.09 7.08 7.07 7.07 7.08 7.05 7.05 7.04 7.03 7.02 7.01 7.00 7.00 6.99 6.98 6.98 6.97 6.96 6.95 6.95 6-94 6.93 6.93 6.92 6.91 6.90 Cosino. PPI Tans. IPPl 9.4.57496 457973 458449 45892.5 459400 459875 460349 460823 461297 461770 462242 9.462715 463186 463&58 464128 464599 4&5069 46.>539 466008 466477 9.467413 467880 468347 468814 469746 470211 470676 471141 471605 9.472069 472532 472995 473457 473919 474381 474842 475303 >175763 476223 9.476683 477142 477601 478059 478517 478975 479432 479889 480345 480801 9.481257 481712 482167 482021 483075 48a529 483982 484435 484887 485339 7.94 7.93 7.93 7.92 7.91 7.90 7.90 7.89 7.88 7.88 7.87 7.86 7.85 7.8.5 7.84 7.83 7.83 7.82 7.81 7.80 7.80 7.79 7.78 7.78 7.77 7.76 7.75 7.75 7.74 7.73 7.73 7.72 7.71 7.70 7.70 7.70 7.69 7.68 7.67 7.67 7.66 7.65 7.65 7.64 7.63 7.63 7.02 7.61 7.60 7.60 7.60 7.59 7.58 7.57 7.57 7.56 7.55 7.55 7.54 7.58 M. M. 60 Cotans. PPI" M iMiie. I'Fl' Tang. PPI" M. 466348 466761 467173 467585 467996 468407 468817 469227 469637 470046 9.470455 470863 471271 471679 472086 472492 472898 473304 473710 474115 9.474519 474923 475327 475730 476133 476536 476938 477340 477741 478142 9.478542 478942 479342 479741 480140 480539 480937 481334 481731 482128 9.482525 482921 483316 483712 484107 484501 484895 485289 485682 486075 9.486467 486860 487251 487643 488034 488424 488814 489201 6.88 6.88 6.87 6.86 6.85 6.85 6.84 6.83 6.83 6.82 6.81 6.80 6.80 6.79 6.78 6.78 6.77 6.77 6.76 6.75 6.74 6.74 6.73 6.72 6.72 6.71 6.70 6.70 6.69 6.68 6.67 6.67 6.66 6.65 6.65 6.64 6.63 6.63 6.62 6.62 6.61 6.60 6.59 6.59 6.58 6.57 6.57 6.56 6.55 6.55 6.54 6.53 6.53 6.52 6.51 6.50 6.50 6.50 6.49 6.48 485791 486242 486693 487143 487593 488043 488492 488941 489390 489&S8 9.490286 490733 491180 491627 492073 492.519 492965 493410 493854 494299 9.494743 495186 495630 496073 496515 497399 497841 498282 498722 9.499163 499603 500042 500481 500920 501359 601797 502235 502672 503109 9.503546 603982 504418 501854 605289 605724 606159 606593 607027 507460 9.5078a3 508326 608759 509191 609622 610064 510485 610916 611346 511776 7.53 7.52 7.51 7.51 7.50 7.49 7.49 7.48 7.47 7.47 7.46 7.46 7,45 7.44 7.44 7.43 7.43 7.42 7.41 7.40 7.40 7.40 7.39 7.38 7.37 7.37 7.36 60 7.35 7.34 7.34 7.33 7.33 7.32 7.31 7.31 7.30 7.30 7.29 7.28 7.28 7.27 7.27 7.26 7.25 7.25 7.24 7.24 7.23 7.22 7.22 7.22 7.21 7.20 7.19 7.19 7.18 7.18 7.17 7.17 tS^ 880 Cosino. PPI" Cotang. PPI" M. IS' SINES AND TANGENTS. 19^ Sill. .489982 490:?71 490759 491147 4915^5 491922 492308 493081 493466 4938.51 .4942-36 494621 495005 49.5388 49.5772 4961.>4 496.5:^7 497:301 497682 9.498064 498444 498825 499204 499584 4999^3 500342 500721 501099 501476 9.. 501851 502231 502607 502984 50:«()0 503735 504110 504485 504860 50.52;i4 9.. 50.5608 505981 506354 506727 5070[)9 507471 507843 508214 50a585 508956 9.509326 509696 510065 510134 510803 511172 511540 511907 512275 512&42 M. Cosine. PPl 6.48 6.47 .47 .46 .45 .45 .44 6.43 .43 .42 6.42 6.41 .40 .40 .38 6.38 6.37 6.37 6.36 6.36 6.:i5 .34 6.33 6.33 6.32 .32 6.31 6.30 6.:30 6.29 6.28 6.28 6.28 6.27 6.26 6.26 6.2.5 .2.5 .24 6.23 6.22 .22 6.22 6.20 6.20 6.20 6.19 6.18 6.18 6.17 6.17 6.16 6.15 6.15 6.15 6.14 6.13 6.13 6.12 9.511776 512206 512635 513064 513493 513921 514349 514777 515204 51.5631 516057 9.516484 516910 517;S3.5 517761 518186 518610 5190.34 5194.58 519882 520.305 9.520728 521151 521.573 52199.5 522417 522838 52:5259 52.3680 524100 524.520 9.524910 5253.59 52.5778 526197 526615 5270:33 527451 527868 52828.5 528702 9.529119 529.535 529951 530366 530781 531196 531611 53202-5 532439 532853 9.533266 53:3679 5:34092 5^4504 5:^916 5a5:328 535739 536150 536.561 5369' PPl' .16 7.16 7.15 7.14 7.14 7.13 7.13 7.12 7.12 7.11 7.10 7.10 7.09 7.09 7.08 7.08 7.07 7.06 7.06 7.a5 .0.5 7.04 7.03 7.03 7.03 7.02 7.02 7.01 7.01 7.00 6.99 .99 6.97 6.97 6.96 6.96 .95 6.95 6.94 6.93 6.93 6.93 6.92 6.91 6.91 6.90 6.90 6.88 6.87 6.87 6.86 6.86 6.85 6.85 6.84 M. M 9.512642 513009 513375 513741 514107 514472 514837 515202 51.5566 515930 51629-4 9.516657 517020 517382 517745 518107 518468 5iaS29 519190 519.551 519911 9.520271 5204;50 645708 645962 646218 64(5474 616729 646981 .617240 647494 647749 648001 648258 618512 648766 649020 649274 649527 .619781 050034 6^30287 650539 650792 651014 651297 651519 651800 652052 .652301 652555 652806 653057 653308 65a558 653808 &54059 654309 654558 1.654808 655058 655307 655556 655805 656054 I' PI" Tans. I'1>I" M 656551 656799 6.57047 31 4.31 30 4.30 4.;30 4.-30 4.30 4.29 4.29 4.29 4.28 4,28 4.28 4.27 4.27 4.27 4.26 4.26 4.26 4.2-5 4.25 4.'M 4.24 4.24 4.23 4.23 4.23 4.23 4.23 4.22 4.22 4.22 4.22 4.21 4.21 4.21 4.20 4.20 4.20 4.19 4.19 4.18 4.18 1.18 4.18 4.17 4.17 4.17 4.17 4.16 4.16 4.16 4.16 4.15 4.15 4.15 4.14 4.14 4.14 4.13 Cosino, 9.688182 688.502 688823 689143 689463 690103 690423 690742 691062 691381 .691700 692019 6923:38 692&56 693293 69,3612 6939:30 694248 694.566 .694883 69.5201 69.5518 6961.53 696470 696787 697103 697420 697736 .6980-5:3 699001 699316 6996:32 699947 7002(33 700-578 700893 9.701208 701523 701837 702152 702466 702781 70:3095 703409 70-3722 704036 9.704:3-50 70466:5 704976 705290 70^5603 705916 706228 706541 7068,54 707166 5.34 5.:^ 5.-34 -5.33 5.33 -5.a3 .5.33 5.33 5.32 5.32 5.32 5.31 5.-31 5.31 5.31 5.31 .30 ,5.30 .5.30 5., 30 .5.29 .29 5.29 ,5.29 Siiip. 9.6,57047 657295 657,542 657790 6580,37 658284 658531 6,58778 659025 659271 ft59517 9.ft597()3 660009 6602-55 66a501 660746 660991 6612:3() 661481 661726 661970 9.6()2214 662459 66270:3 I'Pr 663133 663(577 663920 6641(53 664406 9. 664(548 664891 6651:3:3 66,5;{75 66.5617 66;58,59 066100 666:342 666583 666824 9. 6(57065 067:305 667,546 667786 668027 6(582(57 66&506 668746 668986 6(5922-5 9.669464 669703 669942 670181 670419 670658 670896 671134 671372 671609 4.13 4.13 4.12 4.12 4.12 4.12 4.11 4.11 4.11 4.10 4.10 4.10 4.09 4.09 4.09 1.09 4.08 4.08 4.08 4.07 4.07 4.07 4.07 4.06 4.06 4.06 4.a5 4.05 4.a5 4.05 4.04 4.04 4.04 4.03 4.03 4.03 4.02 4.02 4.02 4.02 4.01 4.01 4.01 4.01 4.00 4.00 4.00 3.99 3.99 3.99 3.99 3.98 3.98 3.98 3.97 3.97 3.97 3.97 3.96 3.96 PP1"| Cotans. PPl" M. M. Cosino. PPl" Cotang. PPl" M ins. 9.7071(56 707478 -• 707790 '^ 708102 708414 708726 7090:37 709:349 709660 709971 710282 9.710-593 710904 711215 711.52.5 7118:36 712146 7121.56 712766 71:3076 71:3:38(5 9.71-369(5 714005 714:314 714624 714933 715242 715551 71,5860 716168 716477 9.71(578-5 717093 717401 717709 718017 718:32-5 71863:3 718940 719248 719-555 9.719862 720169 720476 72078:3 721089 721:39(5 721702 722009 722315 722621 9.722927 723232 72:3538 72:3844 724149 724451 724760 72,5065 725370 725674 63" Trig.— 33. 385 6*i« 28 « TABLE IV.— LOGARITHMIC 29« 1 2 3 4 5 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 .Sine 9.671609 671847 672084 672321 672558 672795 673032 673268 673505 673741 673977 9.674213 674448 674684 674919 675155 675390 675624 675859 676094 676328 9.676562 676796 677030 677264 677498 677731 677964 678197 678430 678663 Pin' 679128 679360 679592 679824 680056 680288 680519 680750 9.681213 681443 681674 681905 682135 682365 682,595 682825 683055 683284 9.683514 683743 683972 684201 684430 684658 684887 68.5115 68.5343 685571 (JositK 3.96 3.95 3.95 3.95 3.95 3.94 3.94 3.94 3.94 3.93 3.93 3.93 3.92 3.92 3.92 3.92 3.91 3.91 3.91 3.91 3.90 3.90 3.90 3.90 3.89 3.89 3.89 3.88 3.88 3.88 3.88 3.87 3.87 3.87 3.87 3.86 3.86 3.86 3.85 3.a5 3.85 3.85 3.84 .84 3.84 3.84 3.83 3.83 3.83 3.83 3.82 3.81 3.81 .81 3.80 3.80 3.80 TaiiK. m 9.725674 725979 726284 726588 726892 727197 727501 727805 728109 728412 728716 9.729020 729323 729626 729929 730233 7305a5 730838 731141 731444 731746 9.732048 732351 7.32653 732955 73.3257 733558 734162 7.34463 734764 9.735066 735367 735668 735969 736570 736870 737171 737471 737771 9.738071 738371 738671 738971 739271 739570 739870 740169 740468 740767 9.741066 741365 741664 741962 742261 742559 742858 743156 743454 743752 PPl" CotariK. PPl" M 5.08 5.07 5.07 5.07 5.07 5.07 5.06 5.06 5.06 5.06 5.06 5.05 5.05 5.05 5.05 5.05 5.04 5.04 5.04 5.04 5.04 5.03 5.03 5.03 5.03 5.03 5.02 5.02 .5.02 5.02 5.02 5.02 5.01 5.01 5.01 5.01 5.01 5.00 5.00 5.00 5.00 5.00 4.99 4.99 4.99 4.99 4.99 4.99 4.98 4.98 4.98 4.98 4.98 4.97 4.97 4.97 4.97 4.97 4.97 31. M 60 59 58 57 56 55 54 53 52 61 50 49 48 47 4Q 45 44 43 42 41 40 9.685,571 685799 686027 686254 686482 686709 686936 687163 687616 687843 688521 688972 689198 689423 689648 M. 690098 9.690323 690548 690772 690996 691220 691444 691668 691892 692115 692339 9.692562 692785 693008 69,3231 693453 693676 693898 694120 694;S42 694564 9.694786 695007 695229 695450 695671 696113 696334 696554 696775 9.696995 697215 6974,35 697654 697874 PPl' 698313 698532 698751 3.80 3.79 3.79 3.79 3.79 3.78 3.78 3.78 3.78 3.77 3.77 3.77 3.77 3.76 3.76 3.76 8.76 3.75 3.75 3.75 3.75 3.74 3.74 3.74 3.74 3.73 3.73 3.73 3.73 3.72 3.72 3.72 3.71 3.71 3.71 3.71 3.70 3.70 3.70 3.70 3.69 3.69 3.69 3.69 3.68 3.6» 3.68 3.68 3.67 3.67 3.67 .67 3.66 3.66 3.66 3.66 3.65 3.65 3.65 3.65 ang. 9.743752 7440,50 744348 744645 744943 745240 74,55.38 7458a5 746132 746429 746726 9.747023 747319 747616 747913 748209 748505 748801 749097 749393 749689 9.749985 750281 750576 750872 751167 751462 751757 752052 752347 752642 9.752937 753231 753526 753820 754115 754409 754703 754997 755291 755585 9.7.55878 756172 756465 756759 7.57052 757345 757638 757931 758224 758517 ). 758810 759102 759395 759687 759979 760272 760564 7()0856 761148 761439 PPl 4.96 4.96 4.96 4.96 4.96 4.96 4.95 4.95 4.95 4.9(5 4.95 4.94 4.94 4.94 4.94 4.94 4.93 4.93 4.93 4.93 4.93 4.93 4.92 4.92 4.92 4.92 4.92 4.92 4.91 4.91 4.91 4.91 4.91 4.91 4.90 4.90 4.90 4.90 4.90 4.90 4.89 4.89 4.89 4.89 4.89 4.89 4.88 4.88 4.88 4.88 4.88 4.88 4.87 4.87 4.87 4.87 4.87 4.87 4.86 Cosine. PPl"| Cotantr. PPl" M ev 386 60< 30' SINES AND TANGENTS. 31< .698970 699626 699814 700062 700280 700498 700716 700933 701151 9.701368 70158.5 701802 702019 7022:36 702452 702669 70288.5 703101 703317 9.70a5*i 703749 703964 704179 704395 704610 704825 70.5040 70.5254 705469 9.70.56aS 705898 706112 706326 7065:39 7067.53 7069(37 707180 707393 707606 9.707819 7080:?2 708245 7084.58 708670 708882 709094 709.:]06 709518 7097:30 9.709941 71015:3 710364 710,575 710786 710997 711208 711419 711629 711839 CosiiK PPI' 3.65 3.64 3.64 3.&1 3.63 3.63 3.63 3.63 3.63 3.62 3.62 3.62 3.62 3.62 3.62 3.61 3.61 3.60 3.60 .3.60 3.60 3.60 3.59 3.59 3.59 3.58 3.58 :3.58 3.58 3.58 3.57 3.57 3.57 3.57 3.56 3.56 3.56 3.55 ;3.55 :3.55 3.55 3.55 ;3.55 3.54 3.54 3.53 ;3.5;3 .3..53 3.53 :3..53 3..52 3.52 3.52 3.52 3.52 3.52 3.51 3.. 51 3.50 3.50 PPl" Taiu 9.761439 761731 762023 762314 762897 763188 763479 763770 764061 764:3.52 9.764643 7649:33 76.5224 765.514 76.5805 766095 766:385 766675 76696.5 76725.5 9.767545 767834 768124 768414 768703 769.571 770148 9.7704.37 770726 771015 771303 771.592 771880 772168 772457 772745 773033 9.773321 773608 773896 774184 774471 774759 775ai6 775333 775621 775908 9.77619.5 776482 776768 777055 777342 777628 777915 778201 778488 778774 Cotaiu PPI" M 4.86 4.86 4.86 4.86 4.85 4.85 4.85 4.8.5 4.85 4.85 4.85 4.84 4.84 4.84 4.84 4.84 4.83 4.83 4.83 4.83 4.8:3 4.83 4.83 4.82 4.82 4.82 4.82 4.82 4.82 4.81 4.81 4.81 4.81 4.81 4.81 4.80 4.80 4.80 4.80 4.80 4.80 4.80 4.79 4.79 4.79 4.79 4.79 4.79 4.79 4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.77 4.77 4.77 4.77 PPI" I M. M. 60 9.7 .71 712050 712260 IVMiQ 712679 713098 713308 713517 71:3726 71:39a5 .714144 714:3.-)2 714.5IU 714769 714978 715186 71.5:394 7ir)602 715809 71601 .716224 71f>432 7166:39 716)^6 717a5:3 7172.59 71740() 717( 717879 71808.5 .718291 718197 718703 718909 719114 719:320 71ft52.5 7197:30 7199.35 720140 .720:345 720.549 720754 7209.58 721162 721366 721570 721774 721978 722181 ,722385 722588 722791 722994 723197 72:3400 723603 723805 724007 724210 PPI' 3.50 3.50 3.50 3.50 3.50 3.49 3.49 3.49 3.48 3.48 3.48 3.48 3.47 :3.47 3.47 3.47 3.47 3.46 :3.46 3.46 3.46 3.45 3.45 3.45 3.45 3.45 3.44 3.44 3.44 3.43 3.43 3.43 ;3.43 3.43 3.43 3.42 :3.42 13.42 3.42 3.41 3.41 3.41 :3.41 3.40 ;3.40 .40 3.40 3.40 3.40 3.:39 3.39 3.39 3.38 3.38 3.38 3.37 3.37, 37 Cosino. |PPl" Cotang laiiK. .778774 779060 779346 779632 779918 780203 780489 780775 781060 78iai6 781&31 .781916 782201 782486 782771 783056 783341 PPI" M. 783910 784195 781479 '.7847(>1 785048 785332 785616 785900 786184 786468 786752 787036 787319 .78760:3 787886 788170 7884,53 789302 789585 789868 790151 .790434 790710 790999 791281 79156:3 791846 792128 792410 792692 792974 .793256 793538 793819 794101 794383 794664 794946 79.5227 79.5508 79.5789 69 59" 387 5S< S'Z' TABLE IV.— LOGARITHMIC 33" 51° Tautr. IPr 9.724210 724412 724614 724816 725017 725219 725420 725622 725823 726024 726225 9.726426 726626 726827 727027 727228 727428 727628 727828 728027 728227 9.728427 728626 72882.5 729024 729223 729422 729621 729820 730018 730217 9.730415 730613 7;^811 731009 731206 731404 731602 1 7317991 731996! 732193' 9.732390 732587 732781 732980 733177 733373 73a569 73;^65 73;S961 7341571 9.734353!., „. 734.549 ^-^ 7;34744 i'^. 734939 ff. 7.S5ia5 ' • - 7a5525 l'^ 7357191^-^ 7:«9i4i:-^ 736109 ^-"^^ 3.;^ 3.37 3.37 3.36 3.36 3.36 3.35 3.35 3.a5 3.35 3.35 3.34 3.34 3.34 3.34 3.33 3.33 3.33 3.33 3,33 3.33 3.32 3.32 3..S2 3.32 3.32 3.31 3.31 3.31 3.30 3.30 3.. 30 3.30 3.;30 3.29 3.29 3.29 3.29 3.29 3.28 3.28 3.28 3.28 3.28 3.27 3.27 3.27 3.27 3.27 3.26 3.26 9. 795789 j 796070 7963.51 796632 796913 797194 797474 797755 7980.36 798316 9.798877 799157 7994.37 799717 799997 800277 800,557 800836 801116 801.396 9.801675 8019.55 802234 802.513 802792 803072 803351 803909 804187 9.804466 804745 80502:3 805302 80.5580 805859 806137 806415 80(5971 ). 807249 807.527 807805 808361 808638 809193 809471 809748 9.810025 810302 810580 81085' 811134 811410 811687 811964 • 812241 812,517 4.68 4.()8 4.68 4.68 4.68 4.68 4.68 4.68 4.67 4.67 4.67 4.67 4,67 4.67 4.07 4.67 4.66 4.66 4.66 4.66 4.66 4.66 4.66 4.6.5 4.65 4.65 4.65 4.65 4.65 iM 4.60 4,65 4.64 4.64 4.64 4.64 4.64 4.64 4,63 4,63 4.63 4.63 4.63 4.63 4.63 4.63 4.6:3 4.62 4.62 4.62 4.62 4.62 4.62 4.62 4.62 4.60 4.61 4.61 4.61 4.61 CoHine. I l'JM"| Cotang. ! PIM M. 9.736109 736:303 736498 73()692 736886 737080 7:37274 737467 737661 737855 738048 9.738241 738434 738627 738820 739013 739206 739590 739783 739975 9.740167 740359 740550 740742 740934 741125 741,316 741508 741699 741 9.742080 742271 742462 742652 742842 743033 74.3223 743413 IM'i" 3.24 3.24 5.24 3.23 3.23 3.23 3.23 3.23 3.22 3.22 3.22 3.22 3.22 3.22 3.21 3.21 3.21 3.21 3.20 3.20 3.20 3.20 3.20 3.19 3.19 3.19 :3.19 3.19 3.18 3.18 3.18 3.18 3.18 3.17 3.17 3.17 3.17 3.17 74.3602^-J^ 74:3792!^-^ 74.qt|8'2 ^'^^ 744171 744361 744.350 744739 744928 745117 745306 745494 745683 9.745871! :3.16 3.16 3.15 3.15 3.15 3.15 3.15 3.14 3.14 3.14 3 14 7460601 J ;* 7462481^ ,. J 746436! ^-f^ 7466241^;^ 746812if}^ 746999 !*^-;5 /4/187 Q ,„ 747374 i^-J^ 747562p^ osiiic. PIM' Tni.K. 9.812.517 812794 813070 8i;]347 813623 813899 814176 814452 814728 815001 815280 9.815555 81.58:31 816107 816382 8166.58 816933 817209 817484 817759 818035 9.818310 818585 818860 819135 819410 819684 819959 8202,34 820508 820783 9.8210.57 821332 821606 821880 8221.54 822429 822703 822977 823251 823524 9.823798 824072 824:345 824619 824893 825166 825439 825713 825986 826259 9.826532 826805 827078 827351 827624 827897 828170 828442 828715 828987 Cotang. PPl" M 388 56' 34° SINES AND TANGENTS. 35° Siiif .747.^2 747749 747936 74812 J 748;U0 748497 7486a3 748870 749056 749213 749429 .749815 749.S01 749987 750172 7.50.3.58 750543 7.50729 7.50914 751099 751281 .751469 751654 751839 752023 752208 752392 752576 752760 752944 753128 .7,53:312 753495 75.3679 753862 754046 754229 754412 7.54.595 754778 754960 .755143 75.5326 75.5.508 75.5690 75.5872 756051 756236 756418 7.56600 756782 .756963 757144 757326 757507 757688 757869 7580.50 7,58230 75^11 758591 Cosine. PPl FPl' 3.12 3.12 3.12 3.12 3.11 3.11 3.11 3.11 ^.11 3.10 3.10 3.10 3.10 3.09 3.09 3.09 3.09 3.09 3.08 3.08 3.08 3.08 3.08 3.08 3.07 3.07 3.07 3.07 3.07 3.07 3.08 3.08 3.06 3.08 3.05 .3.05 3.05 3.05 3.05 3.01 3.01 3.01 3.01 3.01 3.01 3.03 3.03 3.03 3.03 3.03 3.02 3.02 3.02 3.02 3.02 3.02 3.01 3.01 3.01 8.01 ). 828987 829260 829.5:32 8298a5 8:30077 8:30.349 8.30621 8:30893 a3116.:> 8314:37 831709 J. 8:31981 832253 8:32525 832796 8:33038 8:3:3.3.39 8:3:3611 83-3882 &311.54 a34425 8:34967 8:352:38 8135.509 83.5780 8.36051 836.322 8;36593 836861 837131 1.837405 8:37675 8.37946 8:38216 8:38487 838757 839027 839297 8:39568 8:39838 1.810108 840378 840818 810917 811187 8114,57 841727 841996 842266 842ai5 ).842.S05 &4:3074 843343 84,3612 843882 844151 844420 844689 844958 845227 FPl" M 4, 1, 4 4 4 4 4 4 4 4 4 4 4 4 4.48 4.48 Cotang. PPl" M ,7.58.591 7,58772 7,58952 759132 759312 759492 759672 759852 760031 760211 760:390 .76ft569 760748 7C0927 761 10<) 76128.5 761464 761W2 761821 761999 762177 .762:3.56 7(i2rm 762712 762889 76:3067 763245 76,3422 76:3600 763777 76:3954 .7&1131 7(>i;308 764485 764662 7(H8;38 765015 765191 765367 765544 765720 .765896 766072 766247 766423 766598 766774 760919 767124 767:300 767475 .767649 7<)7824 767999 768173 768348 768522 768697 768871 769045 769219 I'l' 3.01 :3.00 :3.00 :3.00 3.00 ,3.00 3.00 2.99 2.99 2.99 2.99 2.98 2.98 2.98 2.98 2.98 2.98 2.97 2.97 2.97 2.97 2.97 2.97 2.96 2.96 2.96 2.96 2.96 2.9.5 2.95 2.95 2.95 2.95 2.9.5 2.94 2.M 2.94 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 2.91 2.91 2.90 2.90 2.90 2.90 2.90 Tan 9.84.5227 845496 845764 8460:33 846:302 846570 846839 847108 847376 847644 847913 9.848181 848449 818717 848986 849254 849522 849790 850057 850325 85059:3 9.850861 851129 851.396 851664 851931 852199 852466 8527:38 85:3001 85:32(58 9.853585 85:3802 854069 8iy3:3(i 854<)03 8^870 855187 85^04 855()71 8559:58 9.856204 856471 856737 857004 857270 857.537 85780:3 858069 8583:36 858602 9.858868 859iai 859400 859932 860198 8604&4 860730 861261 55 « 389 Cosine. PPl" Cotang. PPl" M. __ 36° TABLE IV.— LOGARITHMIC 3^0 1 2 3 4 5 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 41 45 46 47 48 49 50 51 52 53 54 55 56 57 58 M. r69219 769740 770087 770260 770433 770606 770779 770952 9.771125 771298 771470 771643 771815 771987 772159 772331 772503 772675 9.772847 773018 773190 773361 773533 773704 77;«75 774046 774217 774388 9.774558 774729 774899 775070 775240 775410 775580 775750 775920 776090 9.776259 776429 776598 776768 77693- 777106 777275 777444 777613 777781 9.777950 778119 77828- 778455 778624 778792 T78960 779128 779295 779463 PPl' C(>sin< 2.90 2.89 2.89 2.89 2.89 2.89 2.88 2.88 2.88 2.88 2.88 2.88 2.87 2.87 2.87 2.87 2.87 2.87 2.87 2.86 2.86 2.86 2.86 2.86 2.85 2.85 2.85 2.a5 2,85 2.85 2.84 2.84 2.84 2.84 2.84 2.84 2.m 2.83 2.83 2.83 2.83 2.83 2.82 2.82 2.82 2.82 2.82 2.82 2.81 2.81 2.81 2.81 2.81 2.80 2.80 2.80 2.80 2.80 2.80 •J'a J. 861261 861527 861792 862a58 862323 862589 8628W 863119 86338.5 863650 863915 ). 864180 864445 864710 864975 865240 865505 865770 8660:35 866564 867094 867358 867623 867887 868152 868416 868945 869209 9.869473 869737 870001 870265 870529 870793 871057 871321 871585 871849 ). 872112 872376 872640 872903 873167 873430 873694 873957 874220 874484 ). 874747 875010 875273 875537 875800 876063 876326 876589 876852 877114 ■IM" 4.43 4.43 4.43 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.41 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.39 4.39 4.39 4.39 4.39 4.39 4.39 39 4.39 39 4.39 39 4.39 4.38 4.38 38 4.38 4.38 PPl"l Cotang. IPIM" M bine. 9.779463 7796;31 779798 780133 780300 780467 780634 780801 780968 781134 9.7»1301 781468 781634 781800 781966 782132 782298 782464 782680 782796 9.782961 783127 783292 783458 783623 783788 784118 784282 784447 ). 784612 784776 784941 785105 785269 7854:33 785597 785761 785925 786089 1.786252 786416 786579 786742 786906 787069 787232 787395 787557 787720 '.787883 788045 788208 788370 788532 788694 788856 789018 789180 789342 PPl 2.79 2.79 2.79 2.79 2.79 2.78 2.78 2.78 2.78 2.78 2.78 2.78 2.77 2.77 2.77 2.77 2.77 2.77 2.76 2.76 2.76 2.76 2.76 2.75 2.75 2.75 2.75 2.75 2.75 2.75 2.74 2.74 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.73 2.73 2.73 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 Tai 9.877114 877377 877640 877903 878165 878428 878691 878953 879216 879478 879741 9.880003 880265 880528 880790 881052 881314 881577 8818:39 882101 882363 9.882625 882887 883148 883410 883672 883934 884196 88445' 884719 884980 9. 88.5241 885504 885765 886026 886288 886549 886811 887072 887333 887594 9.887855 888116 888378 4.38 4.38 4.38 4.38 4.38 4.38 4.38 4.38 4.37 4.37 4.37 4.:37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 35 4.35 890204 ^'f. 9.890465|j J 890725 ^-^ 889161 889421 889682 889943 PPl" 891247 891507 891768 892028 892289 892.549 892810 4.34 4.34 4.34 4.34 4.34 4.34 4.34 53' 390 PPl" Cotaiig. PPl^^ M. 38" M. ! SINES AND TANGENTS. 39" ppi' .789342 789.304 789827 790149 790310 790471 790632 790793 7909.>4 .791115 791275 791436 791596 791757 791917 792077 792237 792:397 792-W7 .792716 792876 793035 793195 7933.54 79a514 79.3673 7938.32 793991 794150 .794.308 794467 794626 794784 794942 795101 7952.59 79;>417 795575 7957*3 .795891 796049 796206 796;364 79i;521 796679 7968:36 796993 797150 797307 .797464 797621 797777 797934 798247 798403 798560 798716 798872 2.69 2.69 2.69 2.69 2.69 2.69 2.68 2.68 2.68 2.68 2.68 2.68 2.68 2.67 2.67 2.67 2.67 2.67 2.67 2.66 2.66 2.66 2.66 2.66 2.65 2.65 2.65 2.65 2.65 2.65 2.64 2.64 2.64 2.64 2.64 2.64 2.64 2.63 2.63 2.63 2.63 2.63 2.63 2.63 2.62 2.62 2.62 2.£2 2.62 2.62 2.62 2.61 2.61 2.61 2.61 2.61 2.61 2.60 2.60 2.60 PiM' Tanjr ). 892810 89:3070 89:33:31 89:38.51 894111 894:372 894632 894892 8951.52 89.5412 .895672 89.5932 896192 896452 896712 896971 897231 897491 897751 9.898270 899827 900087 900346 900605 9.900864 901124 901383 901642 901901 902160 902420 902679 902938 903197 9.903456 90.3714 90:3973 9042.32 904491 904750 90.5008 905267 905526 90.5785 9.90. PPl" J.798872 799028 799184 799339 79949.5 799651 799806 799962 800117 800272 800427 J. 800582 800737 801047 801201 801356 801511 801665 801819 801973 .802128 802282 802436 802743 802897 8a3a50 803204 80;3357 803511 9.80:3664 803817 803970 804123 804276 804428 804581 804734 801886 8050:39 9.805191 805:^3 8054S)t5 805647 805799 803951 806103 806254 806406 806557 9.806709 806860 807011 807163 807314 807465 807615 807766 807917 808067 Cosine. 2.60 2.60 2.60 2.60 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 2M 2.&4 2.54 2.54 2,53 2.53 2.53 2.53 2.-53 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 PPl' Tai 9.908.369 90S628 908886 909144 909402 909660 909918 910177 910435 910951 9.911209 911467 911725 911982 912240 912498 912756 913014 913271 91:3529 9.913787 914044 914302 914.560 914817 915075 915332 91.5590 915t«7 916101 9.916362 916619 916877 917134 917891 917648 917906 918163 91H420 918677 9.9189;i4 919191 919448 919705 919962 920219 920476 920733 920990 921247 9.921503 921760 922017 922274 922530 922787 923044 923300 923557 923814 Cotang. PPl' 61^ 391 50* 40^ TABLE IV.— LOGAUITHMIC 41 .808067 808218 808368 808519 808819 809119 809269 809419 809569 1.809718 809868 810017 810167 810316 810465 810614 810763 810912 811061 1.811210 811358 811507 811655 811804 811952 812100 812248 812396 812544 1.812692 812840 812988 813ia5 813283 813430 81*578 81372.5 813872 814019 >.814166 814313 814460 814607 814753 814900 8ir)(>46 815193 815:339 81548.5 ). 815632 815778 815924 816215 816:361 816507 8166.52 816798 816943 M. Cosine. I'l' 2.51 2.51 2.51 2.50 2.. 50 2..50 2..50 2. .50 2.-50 2.50 2.49 2.49 2.49 2.49 2.49 2.48 2.48 2.48 2.48 2.48 2.48 2.48 2.48 2.47 2.47 2.47 2.47 2.47 47 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.45 2.45 2.45 2.4.5 2.45 2.45 2.45 2.44 2.44 2.44 2.44 2.44 2.44 2.43 2.43 2.43 2.43 2.43 2.43 2.43 2.42 2.42 2.42 »P1' rang. PPl" JM ). 92:3814 924070 924327 92458:3 924840 925096 9253,52 925609 925865 926122 926378 ). 926634 926890 927147 927403 927659 927915 928171 928427 928684 928940 ). 929196 929452 929708 929964 930220 930475 930731 930987 931243 931499 ).9317;55 932010 9:32266 932522 932778 933033 933289 9:33545 933800 934056 ). 934311 934r}67 934822 935078 935333 935589 9:35844 936100 936*55 9:36611 9.9:3686<» 93712 L 9373". 7 937632 937887 938142 938:398 9386.53 939163 Cotang. I PPl" M M. }>V 9.816943 817088 817233 817379 817524 817668 817813 817958 818103 818247 818392 9.818536 818681 81882') 818969 819113 819257 819401 819545 819689 819832 9.819976 820120 820263 820406 8205,50 820693 820836 820979 821122 821265 9.821407 821550 821693 821835 821977 822120 822262 822404 822546 822688 9.822830 822972 823114 82:3255 82:3397 823539 823680 823821 2.42 2.42 2.42 2.42 2.41 2.41 2.41 2.41 2.41 2.41 2.41 2.41 2.40 2.40 2.40 2.40 2.40 2.40 2.40 2.39 2.39 2.39 2.39 2.39 2.39 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.38 2.37 2.37 2.37 2.37 2.37 2.37 2.37 2.37 2.36 2.:36 2.36 2.36 2.36 2.36 2.35 12.35 2.a5 2.35 8241041 9.824245 824,386 824,527 824668 824808 824949 825090 82,5230 825371 82-5511; 2.;35 2.34 2.34 2.34 2.34 2.34 2.34 Cosine. PPl' Tiuig. 9.9:39163 939418 9:39673 939928 94018:3 940439 940694 940949 941204 94145fe 941713 9.941968 94222:3 9424' 94273:3 942988 943243 943498 94:3752 944007 944262 9.944517 944771 945026 945281 945535 945790 946045 946299 946554 946808 9.947063 947318 947572 947827 948081 948335 948590 948844 949099 949353 9.949608 949862 950116 950.371 950025 950879 9511:3:3 "951388 951642 951896 9.952150 952405 952659 952913 953167 953421 95.3675 953929 954183 ft544;37 Cotang. J. PI, 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.2.5 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.23 4.23 4.23 4.23 4.28 •PI" 49' 392 48* 42" SINES AND TANGENTS. 43" Sine. 31 m 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 9. 825.511 8256.51 82-5791 825931 82tj071 826211 8263.51 826491 826631 826770 82<}910 827049 827189 827328 827467 827606 827745 827884 828023 828162 828301 828439 828.578 828716 82.S&5.5 828993 829131 829269 829407 829.545 82968;i 829821 8299.59 s;mm 830234 8m372 830.509 8;}0646 830781 830921 8.31058 831195 8513.32 83U69 8.3160J 831742 831879 8.32015 832152 8;^2288 832425 .832;561 832697 8.328.33 8.32969 833105 83,3241 833377 83,5512 833648 PPl"i Tang. 2.34 2.m 2.m 2.3:3 2.33 2.a3 2.33 2.a3 2.33 2.-3.3 2.32 2.32 2.32 2.32 2.-32 2.32 .-32 2.32 2.31 2.31 2.31 2.. 31 2.. 31 2.31 2.30 2.30 2.30 2.30 2.;30 2.30 2.30 2.30 2.29 2.29 2.29 2.29 2.29 2.29 2.29 2.2S 2,2,S 2.2.S 2.28 2.2S 2.28 2.28 2.28 2.28 2.27 2.27 2.27 2.27 2.27 2.27 2.27 2.26 2.26 2.26 2.26 2.26 9.9,54437 9-54691 9.54946 955200 9.554-54 9.55708 95.5961 956215 9-56469 9.5672;3 956977 9.9-57231 95748-5 957739 9-57993 958247 958500 9587-54 9-59516 9.9-59769 960023 960277 960.530 960784 9610:?8 961292 961545 981799 9620.52 9.962306 962.560 962813 96:5067 96.3520 963574 96:5828 961081 96433.5 964.588 9.904842 96-5095 96;5;349 96i5602 9658.55 9(56109 966:362 966616 966869 96712:3 9.967:376 967629 967883 968136 968389 968643 969149 969403 969656 PF M. Sine. PPl" 9.83;3783 833919 8.340.54 8:34189 &3432.5 834460 834595 834730 8348&5 834999 835134 9.835269 83540:3 835538 835672 835807 835941 836075 836209 8:36:343 8:36477 9.836611 836745 836878 837012 837146 837279 837412 837->46 8:37679 8:37812 9.837945 838078 838211 8:38344 8:38477 838610 838742 838875 839007 839140 9.839272 8:59404 8:39-536 839668 839800 8:39932 840064 {340196 840:528 840459 9.840591 840722 840854 840985 841116 841247 841378 841509 841640 841771 2.26 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 2.25 12.24 '2.24 2.24 2.24 2.24 2.24 2.24 2.23 2.2:3 2.23 2.23 2.23 2.23 2.23 :2.22 2.22 2.22 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 2.18 2.18 2.18 2.18 4'y( Cosine. PPl" Cotang. PPl" M. M. Cosine. PP 393 Tang. 9.9()96.56 970162 970416 970922 971175 971429 971682 9719.35 972188 9.972441 972695 972948 973201 9734.54 97:3707 97.3960 974213 974466 974720 9.974973 975226 975479 9757:32 975985 976238 976491 976744 976997 977250 9.977.503 977756 978262 978515 978768 979021 979274 979527 979780 9.980033 980286 980,538 980791 981044 981297 981550 981803 982056 982309 i 9.982562 982814 983067 983320 983573 983826 984079 984332 984584 984837 Cotang. PPl" M 31 4«'^ 394 46' SINES AND TANGENTS. ^-yc ISilK: PFl" 9.&56934 857056 8,57178 &57300 857422 8,57543 8.5766.5 857786 857908 858029 8.58151 9.858272 858393 8,58514 858635 858756 8.58877 8,58998 859119 8,592:39 8.59360 9.859480 859601 859721 8.59842 859962 8()(X)82 860202 8()0322 860442 860562 9.8534 02.5787 026040 026293 026546 026799 027052 027305 027559 027812 10.028065 028318 02.^571 028825 029078 029:331 0295m 029838 030091 030344 PPI"1 M. 4.21 4.21 4.21 4.21 4.21 4.21 21 4.21 4.21 4.21 21 4.21 4.21 21 4.21 4.22 4.22 4.22 22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 (Cosine. PPl" Cotane. PPl" M |PPr'i Tang. I PFl" 9.8(>4127j 864245 864:363 i 864481 864.598 864716 8&483:3 864950 865068 865185 865302 9.86,'5419 865536 865(>53 865770 86.5887 866004 866120 866237 866:35:3 8(56470 9.866586 866703 866819 867a51 867167 867283 867399 867515 8676:31 9.867747 867862 867978 868093 868209 868324 868440 868555 868670 868785 9.868960 869015 869130 869245 869:360 869474 869.589 869704 869818 86993:3 9.870047 870161 870276 870:390 870504 870618 870732 870846 870960 871073 1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.95 1.95 1.95 1.95 1.95 1.95 95 l.ft5 1.95 1.94 1.94 1.94 1.94 i.m 1.94 1.94 1.94 1.93 1.93 1.93 1.93 1.93 1.93 1.93 1.9:3 1.93 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.92 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.91 1.90 1.90 1.90 1.90 1.90 1.90 1.90 1.90 1.90 10.030:344 0:50597 0:50851 031104 031:3,57 031611 031864 032117 032,371 032624 032877 10.0:33131 03:3384 0:336:38 033891 034145 0.'y:398 0:34651 034905 0a5158 03,>412 10.035665 035919 036172 0.36426 036680 0:3693:3 037187 037440 037694 037948 10.038201 038455 038708 038962 039216 039470 039723 0:39977 040231 040484 10.0107:38 040992 041246 041500 041753 042007 042261 042515 042769 043023 10.043277 04*531 04;37a5 044039 044292 044546 044800 045054 04.5309 045563 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 .23 4.23 4.2:3 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.2:3 4.23 4.23 4.23 4.23 4.23 4.2:3 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.2:3 4.2:3 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 Cosine. PPl" Cotana. PPl" M. 4a' 395 42« 4S» TABLE IV.— LO(iARlTHMlC 49' M. 4fF 9.87107;} 871187 871301 871414 871528 871641 871755 871868 871981 872095 872208 9.872-321 872434 872547 872659 872772 872885 872998 873110 873223 ■ 873335 9.873448 873560 873672 873784 873896 874009 874121 874232 874344 874456 9.874568 874680 874791 874903 87.5014 875126 8752;^7 875348 875459 875571 9.875682 875793 875904 876014 876125 8762;56 876347 876457 876.568 876678 9.876789 876899 877010 877120 877230 877340 877450 877560 877670 877780 I'Pl' Tans. |1M»1"! M. 1.90 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.89 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.88 1.87 1.87 1.87 1.87 1.87 1.87 1.87 1.87 1.87 1.87 1.86 1.86 1.86 1.86 1.86 il.86 1.86 1.85 1.8.5 11.85 1.85 1.&5 1.85 1.85 1.85 l.S;3 l.») 1.&5 1.84 1.84 1.84 1.84 1.84 1.84 1.84 1.84 1.83 1.83 1.8;^ 1.83 1.83 Cosine. 10.04.>563 045817 04(«)71 04(532,5 046579 04683;^ 047087 047341 047595 047850 048104 10.048358 048612 0488(57 049121 049,375 049629 0198^1 050i;38 0,50392 050647 10.0-50901 051156 051410 051665 051919 052173 a52428 052682 052937 0-53192 10.053446 053701 0-539-55 054210 0544a5 0.54719 054974 0-5.5229 05,518;3 0-557,38 10.0-55993 056248 056.502 056757 057012 0.5726' 057522 0.57 0-58032 058287 10.058541 058796 059051 059;}06 059,561 059817 060072 060,327 060582 060837 4.23 4.23 4.23 4.23 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.24 4.25 4.2-5 4.25 4.25 4:2,5 4.2-5 4.25 4.23 4.25 4.a5 4.25 4.25 4.2,5 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.25 PIM" Cotang. PPl" M. I'lM' i.as 1.83 1..S3 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 1.79 1.79 1.79 1.79 1.79 1.78 1.78 1.78 1.78 8829/ ' 1 --0 883084;, '^'2 883191 1 ;*,1° 9.88,32971 J'^ 883404!!'^ 883;j10 9.877780 877890 877999 878109 878219 878:528 8784;:58 87&547 878656 8787(56 878875 9.878981 879093 879202 879311 879420 879529 879637 879746 879855 879963 9.880072 880180 880289 880397 880505 880613 880722 880830 880938 881046 9.881153 881201 881369 881477 881584 881692 881799 881907 882014 882121 9.882229 882^«6 882443 8825i')0 882&17 882764 88:5617 883723 88.3829 883936 884042 881148 884254 M. Cosine. PPl 1.77 1.77 1.77 1.77 1.77 1.77 1.77 Tiin g. 10.()()08;37 061092 0(;i;547 061(i02|'' 061K58!^' 062113 062368 062023 062879 063134 063,389 10.063645 06:3900 0641.56 064411 0646(57 064922 065178 06543:3 065689 06,5944 10.066200 0664rj5 0(5(5711 0(56967 067222 067478 0677,34 067990 068245 0(58501 10.068757 069013 069269 069,525 069780 070036 070292 070548 070804 071060 10.071:316 071573 071829 072085 072341 072597 07285:3 073110 07:3:366 07:3()22! 10.073878 0741:35 074:591 074648i^ 074{)04i^' 075160 1 J 075417!;*' 07567;' ■* 07,59:30 076186 Cotang. PPl H«)() 40" 50'' SINES AND TANGENTS. 51 » 8*^360' 88446« 884572 884677 884783 884889 884994 885100 885205 88.5311 .885416 88.5522 88-5627 885732 88.5837 885942 886152 886571 886676 887093 887198 887302 887406 9.887510 887614 887718 887822 887926 888134 888237 888.341 888444 9.888548 888651 88875.- 888858 8889J1 8.S90ol 889168 889271 889374 889477 9.889579 889382 88978.J 88J888 889990 89a093 890400 890503 1.76 1.76 1.76 1.76 1.76, 1.76 1.76 1.76 1.76 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 Cosine. Tang. 10.076186 076443 076700 0769-56 077213 077470 077726 077983 078240 078497 078753 10.079010 079267 079.524 079781 080038 080295 08a552 081066 081.323 10.081.580 0818:J7 082352 PPl' 082866 08:3123 083381 08:^638 08;i896 10.0841.53 084} 10 OS 4668 0S4925 0*5183 085440 085698 0859.>() 086213 086471 10.036729 0SJ986 087244 087.')02 0S7760 088018 0.S8275 088533 0S8791 089049 10.089;»7 089565 089823 090082 090340 090-598 090856 091114 091372 091631 PPl" 4.27 4.28 4.28 4.28 1.28 4.28 4.28 28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 Cotang. PIM M. 9.89a503 89060.5 890707 PI' 890911 891013 891115 891217 891319 891421 89152:3 9.891624 891726 891827 892132 89223:3 892334 8924:35 892536 9.892638 892739 893041 893142 893243 893343 893444 9.893645 89:3745 89:5846 894046 894146 894246 894346 894446 894546 9.894646 894746 894846 894945 , 895045 895145 895244 895:343 895443 895542 9.895641 895741 895840 8959:39 896038 896137 896236 896335 896433 896532 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.69 i;69 1.69 1.69 1.69 1.69 1.69 1.69 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1-67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.65 1.65 1.65 l.a5 1.6;5 1.65 1.65 1.65 1.65 1.65 1.65 Tans. Cosine 10.091631 091889 092147 092406 092664 092923 093181 093440 09:3698 093957 094215 10.094474 09473:3 094992 095250 095509 095768 096027 096286 096544 10.097062 097321 097580 097!!M0 098099 098358 098 TABLE IV.— LOGARITHMIC 53« bine. 37 J. 896532 896631 896729 896828 896926 897025 897123 897222 897320 897418 897516 9.897614 897712 897810 897908 898006 898104 898202 l»l' 898397 898494 898787 898884 899078 899176 899273 899370 899467 9.899564 899660 899757 899854 899951 900047 900144 900240 900;«7 900433 9.900529 900626 900722 900818 900914 901010 901106 901202 901298 901394 9.901490 90158.5 901681 901776 901872 901967 902063 902158 902253 902349 Cosint 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.63 1.63 1.63 1.63 1.63 1.63 1,63 1.63 1.63 1.63 1.63 1.63 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 1.59 'I'aiig. IPPl" M. 10.107190 107451 107711 107972 108232 108493 108753 109014 109275 1095:3.5 109796 10.110057 110318 110579 110839 111100 111361 111622 111884 112145 112406 10.112667 112928 113189 113451 113712 113974 1142.35 114496 114758 115020 10.115281 115543 115804 116066 116328 116590 116852 117113 117375 117637 10.117899 118161 118423 118686 118948 119210 119472 1197a5 120259 10.120522 1207R4 121047 121809 121.372 1218.35 122097 122360 122623 122886 PPl" Cotaiig. PPl .36 M. 9.902349 902444 902;339 902634 902729 902824 902919 903014 903108 903203 903298 9.903:592 903487 90;3581 903676 903770 903864 904053 904147 904241 9.904335 904429 904523 904617 904711 904804 904898 904992 905085 905179 9.90.5272 905366 905459 905552 905645 9067 905832 905925 90C018 906111 9.906204 906296 906389 906482 906575 90()667 906760 906852 906945 9070,37 9.907129 907222 907314 907406 907498 907590 907682 907774 907866 907958 P Pl" 59 1.59 1.58 l.,58 1.58 1.58 1..58 1..58 1.58 1.58 1.58 1.58 1..57 1..57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.55 1.55 1.55 1.55 1.55 1.55 L55 1.55 1.55 1.55 1.55 1.55 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.53 1.53 1.53 1.53 1.53 1.53 1.53 Tang. 10.122886 123148 123411 123674 123937 124200 124463 124727 124990 125253 125510 10.125780 126043 126306 126570 126833 127097 127360 127624 127888 128151 10.128415 128679 128943 129207 129471 129735 129999 130263 130527 130791 10.131055 131320 131584 131848 132113 132377 132642 132906 133171 133436 10.133700 133965 1M2S0 134495 134760 1:35025 135290 ia5555 135820 136085 10.13()a50 1:36615 136881 137146 137411 137677 137942 138208 138473 1:38739 CoKinf. PPl" (totalis 38 39 3-8 • :-Ji)8 PPl" M. 36* 54' SINES AND TANGENTS. 55' fSiiie IP Pi" 35" D. 9079.58 j 9080i9 :*.„ 908141 9082;J8 908;B:il 908416 908307 908599 908781 908873 9.9089134 909055 9(J9146 9092.37 909328 909419 909510 909G01 909091 909782 9.909873 9099()3 9100.34 910144 9102^5 91032.5 910415 910.50() 91059(5 9.910776 910.S66 9109.56 911046 911136 911226 911315 91140) 911495 9115H1 9.911674 911763 91185:^ 911942 9120;^1 912121 912210 912299 912388 912477 9.912.566 91265;5 912744 91283;^ 912922 91.3010 913099 913187 913276 918365 1.53 1..53 1..53 1.53 1.52 1..52 1..52 1.52 1..52 1.-52 1..52 1.-52 1.52 1..52 1..52 1..51 1.-51 1.51 1..51 1.-51 1..51 1..51 1.51 1.-51 1.51 1.50 1.-50 1..50 1..50 1.-50 1.-50 1.-50 1.-50 1..50 1.-50 1.-50 1.50 1.49 1.49 1.49 1.49 1.49 1.49 1.49 1.49 1.49 1.49 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.48 1.47 Tans. PPr 10.1-38739 l,S90a5 139270 139=536 139802 1400(38 140334 140600 141132 141398 10.141(j(>4 141931 142197 142463 142730 142996 143263 14*529 14:3796 144032 10.141329 144-596 144863 145l:W 145397 145661 145931 146198 146165 148732 10.146999 147267 147r>i4 147801 148069 1483;J6 148604 148871 149139 149107 10.149J75 149943 1.50210 1-50478 ir30746 151014 151283 151551 151819 1-52087 10.152356 152624 1-52892 1-53161 ir,3430 1-53698 1.53967 154236 154504 154773 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.43 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.44 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.47 4.47 4.47 4.47 4.47 4.47 4.47 4.47 4.47 4.47 4.47 4.47 4.48 4.48 4.48 4.48 4.48 4.48 Cosine. PPl" Cotaiigr. PPl" M M. feme. PPl' laiitr. 9.913365 9134.53 913541 913630 913718 913806 913894 913982 914070 914158 914246 9.914334 914422 914510 914598 914685 914773 914860 914^8 9150a5 915123 9.91,5210 915297 91.538.5 915472 91-55.59 91.5646 9157:« 915820 915907 91-59{M 9.916081 916167 916254 916341 91W27 916514 916600 916687 916773 916859 9.916946 91703ii 917118 917201 917290 917376 917462 917548 917634 917719 9.917805 917891 917976 918062 918147 918233 918318 918404 918489 918574 1.47 1.47 1.47 1.47 1.47 1.47 .47 1.47 1.47 1.47 1.47 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1.46 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.44 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.42 1.42 1.42 1.42 1.42 1.42 1.42 10.154773 155042 155311 155580 155849 156118 156388 156657 156926 157195 157465 10.157734 158004 158273 158543 158813 159083 159352 159622 159892 160162 10.160432 160703 160973 161243 161513 1617i« 1620;54 162325 162595 162866 10.163i;36 16;i407 163678 163949 164220 164491 164762 165033 165304 165575 10.165846 166118 166389 166661 166932 167204 16747 16774' 168019 168291 10.168563 168835 16910 169371 169651 169923 170195 170468 170740 171013 Cosine. PPl" Cot-nng. PPl" M PPl" 49 49 50 50 50 50 50 50 50 50 50 50 51 51 ,51 ,51 ,61 ,51 ,51 ,51 ,51 ,52 ,52 ,52 ,52 ,52 ,52 ,52 ,52 ,52 ,52 .52 .53 .53 .53 53 53 53 53 53 53 53 54 54 54 54 391) 34' 56' TABLE IV.— LOGARITHMIC 5^0 Sine. 9.918574 918Ho9 918745 918830 918915 919000 91908.5 919169 9192.54 9193;?9 919424 9.919-508 919593 919677 919762 919846 919931 920015 920099 920184 920268 9.920352 920436 920520 920004 92068S 920772 920a56 920939 921023 921107 9.92119U 921274 921357 921441 921524 921607 921691 921774 9218.57 921940 9.922023 922103 922189 922272 9223.55 922438 922.520 922603 PP 1.42 1.42 1.42 1.42 1.42 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.-39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.38 1.38 1.38 1.38 1.38 1.38 1.38 1.38 l.;38 1.38 1.38 1.38 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1..37 1.37 PPl" Taii«. IP PI" 10.171013 171285 171558 171830 172103 172376 172649 172922 173195 17M68 173741 10.174014 174287 174561 174834 175107 17.5381 17565.5 175928 176202 176476 10.176749 177023 177297 177571 177846 178120 178394 178668 178943 179217 10.179492 179766 180041 180316 180')90 180865 181140 181415 181690 181965 10.182241 182-516 182791 183067 ia3342 18,3618 183893 184169 184445 184720 10.184996 185272 185548 1&5824 186101 186377 186653 186930 187206 187483 PPl' M. 9. 923591 92.3673 9237.55 92;S837 923919 924001 9240&^ 924164 924246 924.328 924409 924491 924,572 924654 9247a5 924816 924897 924979 925060 925141 925222 925,303 925384 925465 925545 925626 92570' 925788 925868 925949 92€029 .926110 926190 926270 926,351 926431 926511 926591 926671 926751 926831 .926911 926991 927071 927151 927231 927310 927390 927470 927549 927G29 .927708 927787 927867 927M6 928025 928104 928183 928342 928420 PPl 1.37 1.37 l.,37 l.,36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 l.,35 1.35 1.35 i.a5 i.a5 1.35 l.,35 1.35 1.3-5 1.35 1.35 1.34 1.34 1.34 1.34 1..34 1,34 1.34 1.34 1.34 1.34 l.,34 1.33 1.33 1.33 l.,33 i.SS 1.33 1.33 1.33 i.;33 1.33 1.33 1.33 1.32 1.32 1.32 1.82 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 Taiie. iPPl 10.1874831 1877591 4.61 188036 188:^13 188,5901 188866! 1891431 1894201 i4.61 4.61 189698 189975 4.61 4.61 4.62 4.62 4.62 4.62 ^o:4.62 190252 10.190529^-^; 191084 • 191,362 191639 19191' 192195 192473 192751 193029 10.193307 19:3585 193863 194141 194420 194698 194977 19525,5 1955^34 195813 10.196091 196370 196649 196928 197208 197487 197766 198045 198:325 198604 10.198884 199164 199443 199723 200003 200283 200563 200843 20112:3 201404 10.201684 201964 202245 202526 202806 203087 203368 203649 204211 PP1"| Cotaiie. PPl" 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.64 4.64 4.64 4.64 4.64 4.64 4.65 4.65 4.65 4.65 4.65 4.65 4.65 4.65 4.65 4.66 4.66 4.66 4.66 4.66 4.66 4.66 4.67 4.67 4.67 4.67 4.67 4.67 4.67 4.67 4.68 4.68 4.68 4.68 4.68 4.68 4.68 4.68 3»' 400 3-^" 58 « SINES AND TANGENTS. Sine. ). 928420 928499 928578 928657 92S7m 928815 928893 928972 929050 929129 929207 9.92928a 92i);i61 929142 929521 929599 929677 92975.5 9298*^ 929911 929989 9.9300t)7 9;30145. 93022;^ 930300 930378 930456 930.533 930611 930688 I' PI" 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1..S1 1..31 1.30 i.m \ i.m i.;jo i.;3o 1..30 l.:30 1.30 1.30 1.30 tl.30 1.30 1.29 1.29 1.29 1.29 1.29 9307661 }-?2 .9308431^*'^ 930921 9;W998 931075 9311.52 931229 931306 931383 931460 931.537 .931611 931691 931768 931845 931921 1.29 1.29 1.29 1.29 1.29 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 931998 i 932075;, r; 9321.51 !-f! 932218 •£; 932.301 j,:' 11.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 932457 932533 932609 932685 932762 932838 932914 932990 933066 Cosinf 10.204211 204492 204773 20.5054 205336 20.5617 206181 206462 206744 207026 10.207308 207590 207872 2081.54 208437 208719 209001 209284 209.566 209849 10.210132 210115 210698 210981 211264 211.547 21ia30 212114 212:^97 212681 10.212964 213248 2l;i532 213816 214100 214384 214668 2149.52 21.5236 21.>521 10.21.5805 216090 216.374 216t).59 210944 217229 217514 217799 218084 2mm 10.2186.54 218940 219225 219511 219797 220082 220368 220654 220940 221226 PP 4.68 4.69 4.69 4.69 4.69 4.69 4.69 4.69 4.70 4.70 4.70 4.70 4.70 4.70 4.70 4.70 4.71 4.71 4.71 4.71 4.71 4.71 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.74 4.74 4.74 4.74 4.74 1.75 4.75 4.75 4.75 4.75 4.75 He 4.7o 4.76 4.76 4.76 4.76 4.76 4.76 4.77 4.77 4.77 PPl" Cotang. PPl ). 9,3:^066 9a3141 93.3217 933293 9.3X369 93:^45 933520 9.33596 933671 933747 933822 933973 9^4048 93412:5 9^4199 934274 934349 934424 9344J)9 9^4.574 9.9»4649 9^472;^ 934798 934873 934948 93.5022 9a-;097 935171 935246 9;i5;i20 9.9a5395 9;i>469 9*5543 9a5618 9a5692 9li5766 9a5840 9a5914 935988 9360«)2 9.9:i6i:^i 936210 936284 936a57 93<)431 936505 936578 93()652 93(572.5 936799 9.936872 936946 937019 937092 937165 9372;38 937312 93738.5 937458 937531 Cosine. PPl IM' 59 » I PPl"; 31. 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.25 1.25 1.25 1.2.5 1.2.5 1.25 1.25 1.25 1.25 1.2.5 1.25 1.2.5 1.25 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.23 1.2;} 1.23 1.23 1.23 1.2;? 1.23 1.23 1.2;3 1.23 1.23 1.23 1.23 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 10.221226 221512 221799 222085 222372 222^58 222945 22.3232 22a518 223805 224092 10.224379 2246(>7 224954 225241 225529 22.5816 22t3104 226392 226679 226967 10.22725.5 227543 227832 228120 228408 229274 229.563 229852 10.230140 2:J042J) 230719 231008 231297 231.586 231876 2.32166 232455 23274.5 10.2;i'J0;iJ 2^m2r, 2;i3 234195 2a4486 2a4776 2a5067 235357 235648 10.235939 2:^6521 236812 237103 237394 237686 237977 238561 86 Cotang. PPl" M 31» Trii -34. 401 3a< 60< TABLE IV.— LOGARITHMIC 61° ). 937581 937604 937676 937749 937822 93789.3 937967 938040 938113 938185 9382.58 ). 938330 938402 938475 938547 938763 ). 939052 939123 939195 939267 939339 939410 939482 939554 939625 939697 ). 939768 939840 939911 940054 940125 940196 940267 940338 940409 ), 940480 940551 940622 940693 940763 940834 9401)05 940975 941046 941117 9.941187 941258 941328 941398 941469 941539 941609 941679 941749 941819 PPl" 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 rang. PPl^M 3L 10.238561 238852 239144 239436 239728 240021 240313 240605 240898 241190 24148;^ 10.241776 242069 242362 2426,55 242948 243241 243535 243828 244122 244415 10.244709 245003 245297 245591 245885 246180 246474 246769 247063 247358 10.247^53 247948 248243 248538 248833 249128 249424 249719 250015 250311 10. 25060' 250903 251199 251495 251791 25208 252;:}84 252681 252977 253274 10.2r}a571 2,53868 2r;4165 254462 254760 255057 255355 255652 255950 256248 89 M. 9 1 2 3 4 5 6 7 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 ,941819 941889 9419,59 942029 942099 942169 942239 942308 9423: 942448 942517 .942587 942656 942726 942795 9428&4 942934 943003 943072 943141 943210 .943279 943348 943417 943486 943555 943624 943693 943761 943830 943899 1.943967 944036 944104 9441 944241 944309 944377 944446 944514 944582 1.944650 944718 944786 944854 944922 944i)90 94,5058 945125 945193 D45261 1.945328 945396 945464 945531 945598 945666 945733 945800 945868 PPl" 1.17 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1-13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.12 1.12 1.12 1.12 1.12 1.12 1.12 M. Cosine. PPl" I Ootaiig. PPl" M. M. I Cosine. PPl" Cotang. PPl" M Tang. 10.256248 256546 256844 257142 2.57441 257739 258038 258336 258635 258^34 259233 10.259,5.32 259831 260130 260430 260729 261029 261329 261629 261929 262229 10.262529 262829 263130 263430 26.3731 264031 264332 26463:3 264934 265236 10.265537 265838 266140 266442 266743 267045 267347 267649 267952 268254 10.208556 268859 269162 269465 26976' 270071 270374 2706' 270980 271284 10.271588 271891 272195 272499 272803 273108 273412 273716 274021 274326 •PI" 4.97 4.97 4.97 4.97 4.97 4.97 4.98 4.98 4.98 4.98 4.98 4.99 4.99 4.99 4.99 4.99 4.99 5.00 5.00 5.00 5.00 5.00 ,5.01 ,3.01 5.01 5.01 5.01 5.02 5.02 5.02 5.02 5.02 5.02 5.03 5.03 5.03 5.03 5.03 5.04 5.04 5.04 5.04 5.04 5.05 5.05 5.05 5.05 5.05 5.06 5.06 5.06 5.06 5.06 5.07 5.07 5.07 5.07 5.07 5.08 5.08 29" 402 2S^ 62" SLNES AND TANGENTS. 63° iM. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2-5 26 27 28 29 30 31 32 m 34 aj 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 51 52 53 54 55 56 57 58 59 60 M. Siiie. PPl" Tiiiiii. iPP 9. 9459^5 940002 946069 946136 946203 946270 946337 946401 946471 916.538 946604 9.946671 9467;38 946804 946871 • 946937 947001 947070 947136 94720-3 947269 9.9473;ij 947401 947467 9475;« 947600 94766.") 947731 947797 94786:3 947929 9.94799') 948060 948126 948192 9482.^ 94832:3 948:388 9484,54 948.519 948.384 9.9486.50 948715 948780 948845 948975 949040 949105 949170 9492.^) ). 949:300 949:364 949429 949494 949-558 949623 949688 949752 949816 949881 1.12 1.12 1.12 1.12 1.12 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 i.U 1.11 1.11 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.07 1.07 1.07 10.274326; 274&30i^^^ 27493.5|'^-"^ iv- oinK)«Oo 27.O240; 275.546 ?-X„ 27,58.51 2761-56 2764621"; 276768 277073 277379 10.27768.5 277991 278298 278604 278911 279217 279.524 2798:31 2801:38 280445 10.2807,52 2810(30 281:367 5.( 5.09 5.10 5.10 5.10 .5.10 5.10 ,5.11 .5.11 5.11 .5.11 .5.11 .5.12 ,5.12 .5.12 .5.12 5.12 13 281675: ,o 2819*3 '^ 282291 282599 282907 2*3215 28:3.523 10.2838.32 284140 284449 284758 28,5067 28,5376 28,5686 28.5995 286:304 286614 10.286924 287234 287544 2878.54 288164 288475 288785 289096 28940' 289718 10.290029 290:540 2906.51 2909()3 291274 291586 291898 292210 292.522 292834 iiie. PPl"! (' .13 .13 5.13 5.14 5.14 j.l4 5.14 5.14 5.15 5.15 5.15 5.15 .16 .5.16 5.16 5.16 5.16 .5.17 5.17 5.17 5.17 5.18 5.18 5.18 5.18 .5.18 5.19 5.19 5.19 5.19 5.19 5.20 ,5.20 .5.20 5.20 ppT" M. M. 60 59 1 58 2 57 3 56 4 55 5 54 6 53 7 52 8 51 9 50 10 49 11 48 12 47 13 46 14 45 15 44 16 43 17 42 18 41 19 40 20 39 21 38 22 37 23 36 24 a5 25 34 26 33 27 32 28 31 29 30 30 29 31 28 32 27 a3 26 34 25 35 24 36 23 37 22 38 21 39 20 40 19 41 18 42 17 43 16 44 15 45 14 46 13 47 12 48 11 49 10 50 9 51 8 52 7 53 6 54 5 55 4 56 3 57 2 58 1 59 117 60 M. iPPl" Tanir. PPl" 9.9498811 , 949945:!'"' 950010, "' 9.50074 }•"' a5oi3s:J-"' 950202 1 ;•"' 950266 \'^ 950330 , ^' 950:394 ;•"' 9504.58 !'": 950522 1 !•"' 9.950586!, "^ l.Ob 1.06 1.06 1.06 1.06 1.06 1.06 1.06 .06 .06 1.06 1.06 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 950650 950714 9.50778 9.50841 950905 950968 95103'2 951096 951159 9.951222 951286 951:349 951412 951476 951539 951602 9516(55 951728 951791 9.951854 951917 9,51980 952043 952106 952168 952231 952294 952356 952419 9.952481 952544 952606 952669 952731 952793 952855 952918 952980 953042 9.953104 953166 9,53228 953290 9,5:3,3.52 953413 953475 953,5.37 95:3599 9.53660 Cosine. PI'l" 10.2928:34 293146 293459 293772 294081 2943.97 294710 2950241 5.21 5.21 5.21 5.21 .5.21 5.22 -5,22 295:337 r^'l 295650 '^ff 295964!^-g 10.296278 "^'^ 296.591 296905 297219 297534 297848 298163 5.23 5.23 5.23 5.24 5.24 5.24 5.24 298792 ?-^f 299107 i^-f"? 10.29W22|:?-r? 2997371^-2^ 3000531--^^ 300:UJ8 ^'^^ 3004 - .,« 10.305752 ^-^^ 306070 306:388 306707 307025 S07344 307662 307981 308300 308619 309258 309577 310217 310537 310857 311177 311498 311818 5.30 5.30 5.30 5.31 5.31 5.31 5.31 5.31 5.32 5.32 5.32 5.33 5.33 5.33 5.33 5.33 5.34 5.34 5.34 403 20 « 64» TABLE IV.— LOGARITHMIC 65° 40 41 42 43 -44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.9.53(560 953722 95878;^ 9.5;«4r 953901" Tail!?. 1.03 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.00 1.00 1.00 1.00 1. 00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .99 .98 .98 .98 .98 10.311818 312139 312460 312781 313102 313423 31,3745 314066 314,388 314710 31.5032 10.31.5354 315676 31599f) 316321 316644 316967 317290 317613 317937 318260 10.318584 318908 319232 319556 319880 320205 320,529 320854 321179 321.504 10.321829 322154 322480 322806 32^131 323457 323783 324110 324436 324763 10.325089 325416 325743 326071 326398 326726 327053 327381 327709 328037 10.328365 328694 32902;^ 329;i51 329680 3:30009 a30339 330668 3^30998 31^327 PPI'M (^ot I' I' 5.34 .5.-3^5 5.35 5.35 5,35 5.36 5.36 .5.36 5.36 5.37 5.37 5.37 .5.37 5.38 5..38 5.38 5.38 5.39 5.39 5.39 5.39 5.40 5.40 5.40 5.40 5.41 -5.41 -5.41 5.41 5.42 5.42 ,5.42 5.42 5.43 5.43 .5.43 5.43 5.44 .5.44 ,5.44 5.44 5.45 5.45 5.45 5.46 5.46 5.46 5.46 5.47 5.47 5.47 5.47 5.48 5.48 5.48 5.48 5.49 5.49 5.49 5.50 PP1' Hinc ). 9.57276 95733.5 957393 9.57452 9,57511 957.570 957628 957687 957746 957804 957863 ). 957921 957979 958038 958096 958154 958213 958271 958329 958387 958445 ).958503 958561 958619 958677 958734 958792 958965 959023 .959080 959138 959195 9592.53 959310 959368 9.59425 959482 959,539 959-596 .9596.54 959711 959768 959825 959882 959938 959995 960052 9t)0109 960165 .960222 960279 9603a5 9(i()392 960448 96a50.5 9(50561 960618 960674 960730 PP PPl" Tang. PPl" M. 10. ). 331327 3316.57 ^''c: 331987 'I'l^ 332318 ^-.y a32648 "^•?{ 332979 J'^l 33;3309 .'f 333640!?"^; 33.3971 1^-^^ 3.34302^-J^ 334634?*?^ ).331965?*^ 3352971^-^; 335629 ^-e^ 335961 l'^ as6293 rrj 33662.5 :^-3 336958 :?•:! 337291 ^•?! 337624 ^f 337ft57 r*;^ ).338290 ?*^ 338623 iT^ 339290?'^ 5.57 5.57 5.57 5.58 5.58 5.58 5.58 5..59 5.59 5.59 5.59 5.60 5.60 .5.60 5.61 5.61 5.61 5.61 5.62 5.62 5.62 5.63 5.63 5.63 5.63 .5.64 5.64 5.64 5.65 5.65 5.65 339958 ai0292 340627 340961 341296 1.341631 ..66 404 «4^ I 06« SINKS AND TANGENTS. 67 23^ Siup. >. 960730 960786 960843 960899 96095.5 961011 961067 961123 961179 961235 961290 ). 961346 961402 9614,58 961513 961.5t39 961624 9617;i5 961791 901816 9.961902 9619.57 962012 962067 962123 962178 9622:i3 962288 962343 902308 9.9624-')3 9<52508 962562 962617 962672 962727 962781 9628;36 962890 962945 9.902999 963054 963108 96:3163 963217 963271 963325 96;«79 96:3434 963488 9.963:542 96:3.593 96:3a50 96:5701 90375 963811 96386,5 963919 963972 964026 PPI' .93 .93 .93 .93 .93 .93 .93 .93 .92 .92 .92 .92 .92 .92 .92 .92 .92 .92 .92 .92 .92 .92 .91 .91 .91 .91 .91 .91 .91 .91 .91 .91 .91 .91 .91 .90 .90 .90 .90 .90 .90 .90 .90 .90 .90 .90 .90 .90 .90 I'Fl l«f;ii!|5.67 a52097 352438 3.52778 a53119 3-53460 a5;3801 a54143 354484 a54826 10.,a>5168 a>5.510 3558.52 a56194 3.5a5:37 ,356880 a57223 a57566 a57909 3.582.5:3 10.358.596 a58940 359284 a59629 a59973 .360318 36066:3 361008 361:3.53 361698 10.362014 362:389 3627.a5 363081 S(m2S 363774 364121 364468 364815 365162 10.365510 36.5857 366205 .366.553 366901 3672.50 367598 367947 368296 368645 10.:368995 309344 369694 370044 370394 370745 371095 371446 371797 372148 Cosine. PPl' 5.67 5.68 5.68 .5.68 5.69 5.69 5.69 5.70 5.70 5.70 5.71 5.71 5.71 5.72 5.72 5.72 5.72 5.73 5.73 5.73 5.74 5.74 5.74 5.75 5.75 5.75 5.76 5.76 5.76 5.77 ,5.77 5.77 5.77 .5.78 5.78 5.78 5.79 5.79 5.79 5.80 5.80 5.80 ,5.81 5.81 5.81 5.82 5.82 5.82 5.83 5.83 ,5.83 5.83 .5.84 5.84 5.85 5.85 .5.85 Cotang. M. PPl" 9.964026 964080 964133 9^4187 964240 964294 964347 964400 964454 964.507 964560 9.964613 964666 964720 964773 9W826 964879 9649:31 9<>4984 965037 9ft5090 9.965143 96;5ia5 965248 965:301 9ft5a53 96.540(3 96;54.58 96i5511 96.5.503 965615 9.965668 965720 965' 965824 965876 965929 965981 9660:33 966085 966136 9.966188 966240 966292 966344 906:395 9()6447 966499 9665;50 966602 966653 9.9667ft5 966756 966808 966859 966910 966961 967013 9670(34 967115 9671(30 89 : 10. 372148 372499 372851 373203 37a55,5 373907 374259 374612 374964 37,5317 37.5670 10.370024 .376377 376731 37708.5 3774:39 377793 378148 378.503 378858 379213 10.379.568 379924 3802S0 3806.36 380992 .381.348 3817ft5 382061 382418 382776 10..38;31.3;3 38:3491 38;3849 38420' aS4.56;5 38492:3 38.5282 385(>11 386000 386:3,59 10..386719 387079 387439 aS7799 3881.59 aS8520 388880 389241 389603 10.390320 390088 391050 391412 391775 392137 392.500 393227 393590 5.85 5.86 5.86 5.86 5.87 5.87 5.87 ,5.88 5.88 5.88 5.89 5.89 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 5.97 5.97 5.97 5.98 5.98 5.98 0.00 6.00 6.00 6.01 6.01 6.01 0.02 0.02 6.02 6.03 6.03 6.03 6.04 6.04 6.04 6.05 6.05 6.06 6.06 Cosino. PPl" I (?otang. PPl' 405 22' 68° TABLE IV.— LOGARITHMIC 69 < Sino. 9.967166 967217 967268 967319 967370 967421 967471 967522 967573 967624 967674 9.967725 967775 967826 967876 967927 967977 968027 968078 968128 968178 9.968228 968278 968;329 968379 968429 9684 968528 968578 968628 968678 9.968728 968777 968827 968877 968926 968976 969025 969075 969124 969173 9.969223 969272 969321 969370 969420 969518 96956' 969665 ). 969714 969762 969811 970006 9700.55 970103 970152 10.393.390 393954 394318 394683 39504 395412 395777 396142 396507 396873 397239 10.397605 ^'^^ Taiu I'l' 6.06 6.07 6.07 :::6.o7 6.08 6.08 6.09 6.09 5.09 5.10 397971 398337 398704 399071 400173 400541 400909 10.401278 401646 402015 402384 402753 403122 403492 403862 404232 404602 10.404973 405344 405715 406086 406458 406829 407201 407574 407946 408319 ^E 10.408692 409065 409438 409812 410186 410560 ^"^ 6.10 6.11 6.11 6.11 6.12 6.12 6.13 6.13 6.13 6.14 6.14 6.15 6.15 6.15 6.16 6.16 6.16 6.17 6.17 6.18 6.18 6.18 6.18 6.19 6.19 6.20 6.20 6.21 6.22 6.22 6.22 6.23 6.23 410934 411309 411684 412059 10.412434 412810 41318.5 4ia561 414314 414691 415068 415445 415823 6.24 6.24 6.2.5 6.25 6.25 6.26 6.26 6.27 6.27 6.27 6.28 6.28 6.29 6.29 M. ill.'. PPi 9.970152 970200 970249 970297 970345 970394 970442 970490 970,538 970586 ■9706a5 9.970683 970731 970779 970827 970874 970922 970970 971018 971066 971113 9.971161 971208 971256 971303 971351 971398 971446 971493 971540 971588 9.971635 971682 971729 97r 971823 971870 971917 971964 972011 972058 9.972105 972151 972198 972245 972291 972338 97238.5 972431 972478 972524 9.972570 972617 972663 972709 972755 972802 972848 972894 972940 972980 M. Cosine. PPl" Cotang. I PPl" 31. M. | Cosine. PPl" CotHUg. |PPl" M .79 1 .7: rang. PPl" 31. 10.415823 416200 416578 417335 417714 418472 418851 419231 419611 10.419991 420371 420752 421133 421514 421896 42227' 422659 423041 423424 10.423807 424190 424573 424956 425340 425724 426108 426493 426877 427262 10.427648 428033 428419 428805 429191 429578 429965 430;%2 430739 431127 10.431514 4319021 432291 432680 433068 433458 433847 434237 434627 435017 10.435407 435798 436189 436581 436972 437364 437756 438149 438541 438934 29 6.30 6.30 6.31 6.31 6.32 6.32 6.32 6.33 6.a3 6.34 6.34 6.34 6.35 6.36 6., 36 6.36 6., 37 6.37 6.38 6.38 6.39 6.39 6.39 6.40 6.40 6.41 6.41 6.42 6.42 6.42 6.43 6.43 6.44 6.44 6.45 16.45 6.45 6.46 6.46 16.47 6.47 6.48 6.48 6.49 6.49 6.49 6.50 6.50 6.51 6.51 6.52 6.52 6.53 6.53 6.53 6.54 6.54 6.55 21" 406 20' 70° SINES AND TANGENTS. nr M. Si lie 9.972980 97;«)82 973078 973124 973169 973215 973261 973:307 973;352 973398 973144 9.973489 97.35;35 973-580 97362.5 973671 973716 973761 973807 973852 97;3897 9.973942 973987 974032 974077 974122 974167 974212 974257 974302 974347 9.974391 974436 974481 97452.5 974.570 974614 9746.59 974703 974748 974792 9.974836 974880 97492.5 974969 97,5013 97.5057 975101 975145 975189 9752;« 9.975277 975321 975365 975J08 975452 97M96 975539 975583 975627 975670 M. Cosine. PPl" .77 .76 .76 .76 .76 .75 .75 .75 .75 .75 .75 .75 .75 .75 .75 .75 .75 .75 .75 .75 .74 .74 .74 .74 .74 .74 .74 .74 .74 .74 .74 .74 .74 .74 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 .73 Tang. PPl" M. 10.4.38934 439327 439721 440115 440509 440903 441297 441692 442087 442483 442879 10.443275 443671 444067 444464 444861 44.52.59 44.5656 4460.51 4464.52 4468.51 10.4472.50 447649 448018 448418 418847 449218 449648 4;50049 4.504.50 4.508,51 10.4512.53 4516.55 4.520.57 4.52460 4.52862 4,53265 4.53669 454072 454476 4.54881 10.45528.5 4,5.5690 45(5095 4,56501 45690() 457312 4,57719 4,5812.5 458,5:32 4.58939 10.4.59347 459755 460163 460.57 460980 461389 461798 462208 462618 463028 6.55 6.,56 6.56 6.-57 6.57 ..58 .58 6.59 6.59 .59 .60 6.60 6.61 6.61 6.62 6.62 6.63 6.63 6.64 6.64 6.6.5 6.65 6.6,5 6.66 6.66 6.67 6.67 6.68 6.68 6.69 6.69 6.70 0.70 6.71 6.71 6.72 6.72 6.73 .73 6.74 6.74 6.75 6.75 6.76 6.76 6.77 6.77 6.78 6.78 6.79 6.79 6.80 6.80 6.81 6.81 6.82 6.82 6.83 6.83 6.84 5 4 3 2 1 Cotang. PPl" M. PPl' 9.975670 975714 9757,57 975800 975844 975887 975930 975974 976017 976060 976103 9.976146 976189 9762:32 976275 976318 976361 97{>104 976446 976489 976532 9.976.574 976617 9766(i0 976702 976745 976787 9768:30 976872 976914 976957 9.976999 977041 977083 97712.5 977167 977209 977251 9772i<3 9773:35 977377 9.977419 977461 977503 977544 977586 977628 977669 977711 977752 977794 9.977835 977877 977918 977959 978001 978042 978083 978124 978165 978206 Cosine .72 .72 .72 .72 .72 .72 .72 .72 .72 .72 .72 .72 .72 .71 .71 .71 .71 .71 .71 .71 .71 .71 .71 .71 .71 .71 .71 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .69 .69 .69 .68 .68 Inns. PPl" 31. 10.463028 46:3439 463850 464261 464672 465084 465496 465908! 466321 466734 467147 10.467561 467975 468:389 468804 469219 4696:34 47'X)49 470465 470881 471298 10.471715 4721:32 472,519 472967 473385 473803 474222 474641 475060 475480 10.475900 476320 476741 477162 47758^3 4780a5 478427 478849 479272 479695 10.480118 48a542 48096(5 481:390 481814 4822.39 48266.5 483090 483516 483943 10.484369 484796 485223 485651 486079 486507 PPl" 487365 487794 488224 6.84 6.85 6.85 6.86 6.86 6.87 6.87 6.88 6.88 .90 6.90 6.91 6.91 6.92 6.93 .93 6.93 6.94 6.95 6.a5 6.96 6.96 97 6.97 6.98 6.98 .99 6.99 7.00 7.01 7.01 7.02 7.02 7.03 7.03 7.03 7.04 7.05 7.05 7.06 7.06 7.07 7.08 7.08 7.09 7.09 7.10 7.10 7.11 7.12 7.12 7.13 7.13 7.14 7.14 7.15 7.16 7.16 19^ 407 Cotang. PPl"! M. __ 12° TABLE IV.— LOGARITHMIC 73" .978206 978247 978288 978329 978370 978411 978452 978493 978533 978.574 978615 .978655 978696 978737 978777 978817 978858 978898 978939 978979 979019 ,979059 979100 979140 979180 979220 979260 979300 979340 979380 979420 ,979459 979499 979539 979579 979618 979658 979697 979737 979776 979816 ,979&55 979895 979934 979973 980012 980052 980091 980130 980169 980208 ,980247 980286 98032;5 980364 980403 980442 980519 980.558 980596 M. OosiiiH. PPi" Thus. riM" M^ 60 59 58 57 56 5.5 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 10.488224 4886.54 4«9081 489515 489946 490378 490809 491241 491674 492107 492540 10.492973 493407 49.3841 494276 494711 495146 495.582 496018 4964,54 496891 10.497328 4977a5 498203 498641 499080 499519 499958 500397 500837 501278 10.501718 ,502159 502601 503043 503485 503927 504370 504814 5052,57 50.5701 10.506146 506590 507035 507481 507927 508373 508820 509267 509714 .510162 10.510610 5110;)9 ,511,508 5119.'.7 512407 5128.57 5i;«07 5137.58 514209 514661 Cotiuiir. Pl'l' &iti( ). 980596 98063.5 9806' 980712 9807.50 980789 980827 980866 980904 980942 9.981019 98105 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 9.982146 98218^3 982220 982257 982294 982a31 982367 982404 982441 982477 9.982514 982.551 982.587 982624 982()(i(l 982{)9<) 982733 9827(i9 982805 982842 Cosine. PPI 'Pi" Tang. 10.514661 515113 51,556.5 516018 516471 516925 517379 517833 518288 618743 519199 10.519655 520111 520568 521025 521483 521941 522399 5228.58 523317 523777 10.524237 524697 525158 525619 526081 526543 527005 527468 627931 628395 10.528859 529324 529789 630254 530720 531186 531653 532120 532587 533065 10.53352,3 533992 534461 634931 635401 636872 636;S42 636814 637285 637758 10.5,38230 a38703 639177 639651 640125 510600 541075 541551 642027 542504 Cotang. 11< 408 PPl'^ M. 16^ 74' SINES AND TANGENTS. 750 Si I 9.982842 982878 982914 9829.30 983022 983038 983094 983130 983166 983202 .983238 983273 983309 983345 983381 983416 983452 983487 983523 9835^38 .983594 983629 983664 983700 9837a3 983770 983840 983875 984015 9840.30 98408.3 984120 984155 984190 984224 9842.39 9.981294 984328 984363 984397 984432 984466 984500 984535 981569 984603 9.984638 984672 984706 984740 984774 984808 984842 984876 984910 984944 Cosii ..59 .59 Tang. 10.542.504 542981 5431.58 543936 544414 544893 545372 545852 546332 546813 547294 10.547775 548257 548740 549223 549706 550190 550674 551159 551644 552130 10.552616 533102 653589 554077 554.365 535053 53.3.342 536032 556521 557012 10.557503 557994 558486 558978 559471 559964 560457 560952 561446 561941 10.562437 563430 563927 564424 561922 56.3421 565920 566420 566920 ,567420 567921 568423 568925 569427 5699,30 570434 5709;« 571442 571948 15 Trig.— 35. ppi' 7.95 7.96 7.96 7.97 7.98 7.99 7.99 8.00 8.01 8.02 8.02 8.03 8.04 8.05 8.08 8.06 8.07 8.08 8.09 8.09 8.10 8.11 8.12 8.12 8.13 8.14 8.15 8.16 8.16 8.17 .18 8.19 8.19 8.20 8.21 8.22 8.23 8.23 8.24 8.25 8.26 8.27 8.28 8.28 8.29 8.30 8.31 8.32 8.32 8.3:3 8.34 8.35 8.36 8.37 8.38 8.38 8.39 8.40 8.41 8.42 M. I'T'l' Sine. iPPl" .9849441 984978 985011 985045 985079 985113 985146 985180 985213 985247 9.985314 985^47 985381 985414 985447 985480 985514 983547 985580 985613 9.985646 985679 985712 985745 985778 985811 985876 985909 985942 9.98.3974 986007 986039 986072 986104 98(5137 986169 986202 986234 986266 9.986299 986331 986363 986427 986459 986491 986523 986555 986587 9.986619 986651 986683 986714 986746 986778 98t)809 986841 986873 'I'ang. 10.571948 572453 572959 573466 573973 574481 574989 575497 576007 576516 577026 10.577537 578048 578500 579073 579585 580099 580613 581127 581642 582158 10.582674 583190 583707 584225 584743 585262 585781 I'Pl" 586821 587342 10.5878()3 588385 588908 5^9431 589955 590479 591004 591529 592055 592581 10.593108 593636 694164 694692 595222 595751 596282 596813 597344 597876 10.598409 598942 599476 600010 600515 601081 601617 6021.34 602691 603229 8.43 8.43 .44 8.45 8.46 8.47 8.48 8.48 8.49 8.50 8.51 8.52 8.53 8.54 8.55 8.55 8.56 8.57 8.58 8.59 8.60 8.61 8.62 8.63 8.64 8.64 8.65 8.66 8.67 8.68 8.69 8.70 8.71 8.72 8.73 8.74 8.74 8.75 8.76 8.77 8.78 8.79 8.80 8.81 8.82 8.83 8.84 8.85 8.86 8.87 8.88 8.89 8.90 8.91 8.92 8.93 8.94 8.95 8.96 8.96 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 a3 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 18 12 11 10 9 8 7 ti 5 4 3 2 1 PPI' 409 14< •760 TABLE IV.— LOGARITHMIC .y^c 9.98G904 986936 986967 986998 987030 987061 987092 987124 9S7155 987186 987217 9.987248 987279 987310 987341 987372 987403 987434 987465 987496 987526 9.987557 987618 987649 987679 987710 987740 987771 987801 987832 9.987862 987892 987922 987953 987983 988073 988103 988133 .988163 988193 988252 988312 988342 988371 988401 .988460 988489 988519 988724 i'an^. PPl" M 10.003229 603707 G04306 C04846 G05386 C05027 G0G409 607011 G07553 C0S097 C08640 10.G091S5 609730 610276 610822 C11309 611916 612464 613013 613562 614112 10.614663 615214 615766 616318 616871 617425 617980 618;534 619090 619646 10.620203 620761 621319 621878 622437 623558 624119 624681 625244 10.62.5807 626371 626936 627501 628067 6286a3 629201 629768 630337 10.631476 632047 632618 633190 633763 634336 634910 635485 636060 rPl" CotanK. PPl 8.97 8.98 8.99 9.00 9.01 9.02 9.03 9.04 9.05 9.06 9.07 9.08 9.09 9.10 9.11 9.12 9.13 9.14 9.15 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25 9.26 9.27 9.28 9.29 9.30 9.31 9.32 9.33 9.34 9.35 9.37 .40 9.41 9.42 9.43 9.44 9.45 9.46 9.48 9.49 9.50 9.51 9.52 9.53 9.54 9.55 9.57 9.58 9.59 9.60 51 M. Sine. .988724 988753 9S8782 988811 988840 9SSS98 9SSC27 9SS950 9S8985 9S9014 9.989042 989071 9S9100 9S9128 989157 9891SG 989214 989243 989271 989300 9.989328 989356 989385 989413 989441 989409 989497 989525 989553 989582 9.989610 989637 PPl" 989721 989749 989777 989804 1 989832 i 989860! 9.9898871 989915 989942 989970 .49 .49 .48 .48 .48 .48 .48 .48 .48 .48 .48 .48 .48 .48 .48 .48 .48 .47 .47 .47 .47 .47 .47 .47 .47 .47 .47 .47 .47 I .47 I .47 .47 .47 .46 .46 .46 ! .46 .46 990025 990052 990079 99010 990134 ). 990161 990188 990215 990243 990270 990297 990324 990351 .46 .46 .46 .46 .46 .46 .46 .46 .45 .45 .45 .45 .45 .45 .45 .45 .45 .45 .45 Tariff. PPl" M 10.636636 637213 637790 6388G8 638947 639520 640107 640687 6412G9 641851 642434 10.643018 643602 644187 644773 645360 645947 646535 647124 647713 648303 10.648894 649486 650078 650671 6512&5 651859 652455 653051 653647 654245 10.654843 655442 656042 656642 6572431 6578451 6584481 659052| 6596561 660261 10.660867 661473 662081 6626891 6632981 663907! 664518 665129 665741 666354 10.666967 667582 668197 668813 669430 670047 670666 671285 671905 672525 9.61 9.62 9.63 9.65 9.6G 9.C7 9.68 9. CD 9.70 9.71 9.73 9.74 9.75 9.70 9.77 9.79 9.80 9.81 9.82 9.83 9.85 9.87 9.88 9.90 9.91 9.92 9.93 9.96 9.97 9.98 9.99 10.00 10.02 10.03 10.04 10.06 10.07 10.081 10. 10 1 lO.llj 10.121 10.13 - 10.15 ^ 59 58 57 56 55 51 I 53 ro 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 18 17 PPl" CotariK. PPl" ]M 10. 1() 10.17 10.19 10.20 10.21 10.23 10.24 10.25 10.26 10.28 10.29 10.30 10.32 10.33 10.35 13' 410 12* TS" SINES AND TANGENTS. 19' .990404 990431 990458 990483 990511 990538 990565 990591 990(318 990645 990671 990724 990750 990777 990829 990a5.5 990882 990908 990934 .990960 991012 991038 991064 991090 991115 991141 991167 991193 .991218 991244 991270 991295 991321 991346 991372 991397 991422 991448 .991473 991498 991524 991549 991574 901599 991624 991649 991674 .991724 991749 901 991799 991823 901848 901873 99189 991922 991947 PPl" Tan!?. PPl" M. 10.672525 673147 673769 674393 675017 675642 676267 676894 677521 678149 678778 10.679408 681303 682570 683205 683841 684477 685115 10.685753 .n. Co.=i 687032 687673 688315 688958 689601 690246 690891 691537 10.692184 692832 693481 694131 694782 69.5433 696086 696739 10.698705 699362 700020 700678 701338 701999 702601 703323 703987 704651 10.705316 705983 706650 707318 707987 708058 709329 710001 710674 711348 10.36 10.37 10.39 10.40 10.41 10.43 10.44 10.45 10.47 10.48 10.50 10.51 10.53 10.54 10.55 10.57 10.58 10.60 10.61 10.62 10.64 10.6.5 10.67 10.68 10.70 10.71 10.73 10.74 10.75 10.77 10.78 10.80 10.81 10.83 10.84 10.86 10.87 10.89 10.90 10.92 10.93 10.95 10. 10.98 11.00 11.01 11.03 11.04 11.06 11.07 11.09 11.11 11.12 11.14 11.15 11.1 11.18 11.20 11.22 11.23 SliiP. I PPl' .991947 991971 991996 992020 992044 992093 992118 992142 992166 992190 .992214 992239 992263 992287 992311 992335 992406 992430 .992454 992478 992501 992525 992549 992572 992619 992643 992666 .992690 992713 992736 992759 992783 992806 992829 992852 992875 .992921 992944 99296' 992990 993013 993036 993059 993081 993104 993127 .993149 9931 993195 993217 993240 993262 993284 2 1 PPl" rotans. PPl" IM. M. Co!»ine. PPl" Cotang. PPl" M. 993329 993351 Tan-. 10.711348 712023 712699 713376 714053 714732 715412 716093 716775 717458 718142 10.718826 719512 720199 720887 721576 722266 722957 723649 724342 725036 10.725731 726427 727124 727822 728521 729221 729923 730625 731329 732033 10.732739 733445 73415:3 734862 735572 736283 736995 737708 738422 739137 10.739854 740571 741290 742010 742731 743453 744176 744900 745626 746352 10.747080 747809 748539 749270 750002 750736 751470 752206 752943 753681 PPl" I m. 60 11.25 11.26 11.28 11.30 11.31 11.33 11.35 11.36 11.38 11.40 11.41 11.43 11.45 11.47 11.48 11.50 11.51 11.53 11.55 11.57 11.58 11.60 11.62 11.64 11.65 11.67 11.69 11.70 11.72 11.74 11.76 11.78 11.79 11.81 11.83 11.85 11.87 11. 11.90 11.92 11.94 11.96 11.98 12.00 12.01 13.03 12.05 12.07 12.09 12.11 12.13 12.15 12.17 12.18 12.20 12.22 12.24 12.26 12.28 12.30 11« 4ii lO' so< TABLE IV.— LOGARITHMIC 81< SiJie. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2.5 20 27 28 29 30 31 32 33 34 3.5 30 37 38 39 40 41 42 43 44 43 40 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.993:351 993;374 993396 993 ns 993440 993462 993484 993506 , 993528 993550 993572 9.99;i594 993016 993638 9930G0 993081 993703 993725 993746 993768 993789 9.993811 993832 993875 993897 993918 994003 .994024 994045 994066 994087 994108 994129 994150 994171 994191 994212 .994233 994254 994274 994295 994316 994336 994357 994377 994398 994418 .994438 994459 994479 994519 994-540 994560 991580 994600 994G20 .36 Tai PPl 10.753681 7.54421 7.55161 755903 756646 757390 758135 758882 759629 760378 701128 10.701880 702632 76414] 761897 765655 766414 707174 707935 10.769461 770227 770993 771761 772529 773300 774071 774844 775618 770393 10.777170 777948 778728 779508 780290 781074 781858 782044 783432 78^220 10.785011 785802 780595 787389 788185 789780 790580 791381 792183 10.792987 793793 791600 79.5408 790218 79T029 707841 7986.5,5 799171 800287 ; 12.32 1 12.34 12.36 12.38 12.40 12.42 12.44 12.46 12.48 12.50 12.52 12.54 12.56 12.58 12.60 12.62 12.65 12.67 12.69 12.71 12.73 12.75 12.77 12.79 12.81 12.84 12.86 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 13.26 13.28 13.31 13.33 13.35 13.38 13.40 13.42 13.45 13.47 13.49 13.52 13.54 13.57 13.59 13.61 PPl' ('otane. 5 4 3 2 1 >pi" m7 Sii 9.994G20 994040 994000 994080 994700 994720 994739 994759 994779 994798 994818 9.994&38 994857 994877 994890 994910 9949X5 994955 994974 994993 995013 9.995032 995051 993070 9950S9 995108 993127 995146 995165 993184 995203 9.995222 995241 995260 995278 995297 995316 995334 995X53 995372 995390 9.99.5109 995427 995446 995464 995482 995501 99.5519 995537 9955.55 99.5573 9.995591 993610 993028 995046 99.5004 995081 99.5717 99.57.35 99.57.5;^ PP ..33 .m .,33 .33 .3,3 .33 .33 ..33 .33 .33 .33 .33 .33 ..32 .,32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .32 .31 .31 .31 .31 .31 .31 .31 .31 .31 .31 .31 .31 .31 .,31 .31 .30 .30 .30 .30 .30 .,30 .30 .30 .30 .30 .30 .30 .30 10.800287 801100 801926 802747 803570 804394 80,5220 806047 800876 807706 808538 10.809,371 810206 811042 811880 812720 813561 814403 815248 810093 816941 10.817789 818640 819492 820345 821201 822058 822916 823776 824638 825501 10.826366 827233 828101 828971 829843 830716 831591 832468 833346 834226 10.835108 835992 836877 837764 839543 840435 841329 842225 843123 10.844022 844923 845826 846731 847637 848546 849456 850368 851282 852197 PPl' I Ootang. PPl" M 13.64 13. 13. 13.71 13.74 13.76 13.79 13.81 13.84 13. 13.89 13.91 13.93 13.90 13.99 14.02 14.04 14.07 14. 14.12 14.15 14.17 14.20 14. 14.25 14.28 14.31 14.33 14.36 14. 14.42 14.44 14.47 14.50 14.53 14.55 14.58 14.61 14.64 14.67 14.70 14.73 14.76 14.79 14.81 14.84 14.87 14.90 14.93 14.96 14.99 15.02 15.05 15.08 15.11 15.14 15.17 15.20 15.23 15.26 9« 412 82< SINES AND TANGENTS. 83° M. 1 2 3 4 5 Sine. PPl" Tang. PPl" M .995753 995771 995788 995823 995841 995876 995894 995928 .995946 995963 99S015 996032 996049 9960J6 9930S3 9.996117 996131 996151 996168 996185 998202 996219 996235 996252 996269 9.99628.5 996302 996318 996335 996351 996368 996384 996400 996417 996433 9.996449 996465 996482 996498 996514 996530 996546 996562 996578 996594 9.996010 996625 996641 996657 990673 996704 996720 9967;55 996751 .29 .29 .29 .29 .29 .29 •29 .29 .29 .29 .29 .29 .29 .29 .29 .29 .29 .29 .29 .28 .2S .2S .28 .28 .23 .28 .28 .28 .28 .28 .28 .28 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .27 .26 .26 .26 .26 .26 .26 .26 .26 .26 .23 10.8.52197 &53115 a54034 854956 855879 857731 858660 859591 860524 8614.58 10.862395 86333:3 864274 865216 866161 867107 868056 889006 889959 870913 10.871870 872828 873789 874751 875716 876683 877652 878623 879596 880.571 10.881548 882528 883509 884493 885479 88()467 887457 888149 889444 8J0141 10.891140 892441 893444 894150 895158 896468 897481 898496 899513 9005.32 10.901.554 902578 903605 901633 905664 PPl" 35*" 907734 908772 8108.56 15.29 15.32 15.35 15.39 15.42 15.45 15.48 15.51 15.55 15.58 1.5.61 15.64 15.67 1.5.71 15.74 15.77 1.5.81 1.5.84 15.87 91 15.94 5.97 16.01 16.04 18.07 10.11 1{;.15 16.18 16.22 16.25 16.29 16.32 16.36 16.39 16.43 16.46 16.50 16.54 16.58 16.61 16.65 16. 16.72 16.76 16.80 16.84 16.87 16.91 16.95 16.99 17.03 17.07 17.11 17.15 17.19 17.22 17.27 17.30 17.34 17.38 PPl' 9.996751 996766 996782 996797 996812 996828 996^43 996858 996874 996934 996^9 996979 997009 997024 997053 9.997068 997083 997112 997127 997141 997156 997170 997185 997199 9.997214 997228 997242 997257 997271 997285 997299 997313 997;^ 997341 9.997355 997369 997383 997397 997411 997425 997439 997452 997480 .997493 997507 997520 997534 997547 997561 997574 997588 997601 997614 .26 .26 .26 .26 .2,5 .25 .25 .25 .2.5 .25 .25 .25 .25 .25 .25 .25 .2.5 .25 .25 .25 .25 .24 .24 .24 .24 .24 .24 .24 .21 .24 .24 .24 .24 .24 .24 .24 .24 .24 .23 .23 .23 .23 .23 .2:3 .2:3 .23 .23 .23 .23 .23 .2:3 .23 .23 .22 .22 .22 .22 .22 .22 'Iju 10.910856 911902 912950 914000 915053 916109 917167 918227 919290 920356 921424 10.922495 92a568 924644 925722 926803 927887 928973 9,30062 9311.54 932248 10.933345 934444 935547 936652 937760 938870 PPl" 941100 942219 943341 10-944465 945593 &46723 ft47856 Cosine. |PPr 950131 951273 952418 953566 954716 10.9.55870 957027 958187 959349 9<,a515 961G84 962856 964031 965209 966391 10.967575 968763 971148 972345 973545 974749 975956 977166 978380 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 18.60 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 19.3:3 19.38 19.43 19.48 19.53 19.58 19.64 19. 19.74 19.79 19.85 19.90 19.95 20.00 20.06 20.11 20.17 20.23 Cotang. PPl" I M. 60 84" TABLE IV.— LOGARITHMLC 85° 1 2 3 4 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 9.997614 997628 997641 997^54 997667 997706 997719 997732 997745 .997758 997771 997784 997797 997809 997822 9978;^ 997847 997860 997872 .997885 997910 997922 99793.5 997947 997959 997972 997984 998020 998032 998056 998080 998092 998104 998116 .998128 9981 :J9 998151 998163 998174 998186 998197 998209 998220 998232 .998243 998255 998266 998277 998300 998311 998322 998344 Tan' .978380 979597 980817 982041 98:^268 984498 985732 10. 988210 989454 990702 991953 993208 994466 995728 996993 998262 999535 000812 002092 003376 004663 005955 007250 008549 009851 011158 012468 013783 015101 016423 017749 019079 020414 021752 023094 024440 025791 027145 028504 029867 031234 032606 033981 035361 036745 0:38134 039527 040925 042326 043733 045144 046559 047979 049403 0.50832 052266 053705 055148 056596 058048 Cosine. PPl" Cotaiigr. PPl" M. }'V 20.28 20. S3 20.40 20.45 20.51 20.56 20.62 20.68 20.74 20.80 20.8.5 20.91 20.97 21.03 21.09 21.15 21.21 21.27 21.. 34 21.40 21.46 21.52 21. .58 21.65 21.71 21.78 21.84 21.91 21.97 22.04 22.10 22.17 22.23 22.30 22.37 22.44 22.51 22.57 22.65 22.71 22.79 22.86 22.93 23.00 23.07 23.14 23.22 23.29 2:3.37 23.44 23.51 23.60 23.66 23.74 23.82 2:3.90 23.97 24.05 24.13 24.21 9.9!(8344 99835.5 998366 998377 998388 998399 998410 998421 998431 998442 998453 9.998464 998474 998485 998506 998516 998527 998537 998548 998558 ). 998568 998578 998619 998629 998649 998708 998718 998728 99873 99874' 99875 9.998766 998776 998785 998795 . 998804 99b813 99882:} 998832 998841 998851 9.998860 998878 998887 998905 998923 998932 rnug. PPl" M. I1.05804S 059506 000908 0624:35 OC:3907 065384 066866 068a53 069845 071:342 072M4 11.074351 075864 077381 078904 0804.32 081966 083505 085049 088154 11.089715 091281 094430 096013 097602 099197 10079' 102404 104016 11.10.5()34 107258 . 108888 110524 112167 113815 115470 1171,31 118798 120471 11.1221.51 123838 12.55:31 1272:30 128936 130(349 132368 134094 135827 137567 11.139314 141068 142829 144597 146372 1481.54 149943 151740 153545 155a56 PPl"! Cotaiig. 24.29 24.37 24.45 24.53 24.62 24.70 24.78 24.87 24.85 2.5.03 2,5.12 25.21 25.30 2.5.38 25.47 25.56 25.&5 25.74 25.83 25.92 26.01 26.10 26.20 26.29 26.38 26.48 26.58 26.67 26.77 26.87 26.97 27.07 27.17 27.27 27.37 27.47 27.58 27.68 27.79 27.89 28.00 28.11 28.21 28.32 28.43 28.54 28.66 28.77 28.88 29.00 29.11 29.23 29.:35 29.46 29.58 29.70 29.82 29.95 30.07 30.19 PPl' 414 S6o_ 3ir~ 1 2 3 4 5 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 ;ii .51 5.-) 56 60 SINES AND TANGENTS. 8T» ). 998941 998958 998976 998984 999002 999010 999027 .999036 999044 9990.5;^ 999061 999077 999094 999102 999110 9.999118 999126 999l;« 999142 999150 999158 999166 999174 999181 999189 9.999197 999205 999212 999220 999227 99f)2;i5 999242 999250 9992.)7 99926.5 9.999272 999279 999287 999294 999301 999308 999315 999322 999329 999336 9.999343 999;j.50 9993.57 999364 999371 999378 999384 999391 999398 999404 PFl 'I'aii^ .15 .15 .15 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 .14 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .11 .11 .11 .11 .11 .11 .11 11.1.5.53.56 1.57175 1.59002 160837 162679 164529 166387 1682.52 170126 172008 173897 11.17.5795 177702 179616 181.539 18;3471 18.5411 187359 189317 191283 1932,58 11.19.3242 1972,3.5 1992.37 201248 203269 205299 207338 209;«7 211446 213514 11.21.5592 217680 219778 221886 224005 226131 228273 2:30422 23258;^ 2;i47.51 11.2:369^5 239128 241:332 2ia547 215773 248011 2.)0260 252521 254793 2.57078 11.2.59:374 20168:3 264004 260337 268683 271041 273412 275796 278194 280604 30.32 :30.45 .,57 30.70 30.83 30.96 31.10 31.23 31.36 31.50 31.63 31.77 31.91 :32.05 32.19 32.33 32.48 32.62 .32.77 32.92 313.07 .22 3:3.37 33.52 33.68 3:3.83 33.99 34.15 ;34.31 34.47 34.64 34.80 :34.97 .3.5.14 3.3.31 :3.5.48 •3;5.ft3 :3.5.83 36.00 36.18 ;36.36 36.55 36.73 36.92 37.10 37.29 37.49 37.68 37.87 38.07 ;38.27 38.48 38.68 38.89 39. 39.30 39.52 39.74 ;39.95 40.17 PPl" Cotaiu M. 9.999404 999411 999424 999431 999437 999443 999450 9994.56 999463 999469 .999475 999481 999487 999493 999500 999512 999518 999524 999.529 .99953.5 9995.58 999564 999570 95)9.575 999581 999,58(i 9.999-592 999597 9'j94 691116 11.697,366 703708 710144 716677 72.3;»9 7:30044 736885 7438a5 758()7i PPI" (V)taim. PIM" M I'l' 60.62 61.13 61.65 62.18 62.72 63.26 63.82 64.:-}9 64.96 65.55 66.15 66.76 67.38 68.01 68.6.5 69.31 69.98 70.6(5 71. a5 72.06 72.79 73.52 74.28 75.05 75.83 76.63 77.45 78.29 79.14 80.02 80.91 81.82 82.76 83.71 84.70 85.70 86.72 87.77 88. &5 89.95 91.08 92.24 93.43 94.65 9.5.90 97.19 98.51 99.87 101.3 102.7 104.2 105.7 107.2 108.9 110.5 112.2 114.0 115.8 117.7 119.7 9.999934 9999:36 999938 999940 999942 999944 999946 999950 999952 9999,54 9.9999;56 999958 999961 999963 999964 999966 9999()8 9999(i9 999971 9.999972 999973 999975 999979 999981 999982 999988 99{}992 999993 9.999993 999994 999{)95 999995 99999(1 999996 999997 999997 999998 999998 9.999999 999999 999999 10.000000 000000 mmo 000000 OOWKX) 000000 Cosiiif 1>P1" Tii PPI 11.7,5^079 765:379 77280.5 780:359 788047 795874 803844 811964 8;^02:37 828672 837273 11.846048 8.55004 864149 873490 8a30:37 892797 902783 9i;j(K)3 923409 934194 11.94.5191 9.56473 968055 979956 992191 12.004781 017747 031111 044900 059142 12.07:3866 089106 101901 121292 138326 l,5()a56 174540 19.3845 214049 2:3,5239 12.257516 280997 1305821 3.32151 360180 390143 422328 457091 494880 536273 12.,5820:30 633183 091175 758122 837304 9:34214 13.0.59153 2:3.5244 5:36274 Infinite. ("otunt PPI" 121.7 123.8 125.9 128.1 1:30.4 1,32.8 135.3 1.37.9 140.6 143.3 146.2 149.3 152.4 155.7 1,59.1 162.7 166.4 170.3 174.4 178.7 18:3.3 188.0 193.0 198.3 20:3.9 209.8 216.1 222.7 229.8 237.3 245.4 2.54.0 263.2 273.2 283.9 295.5 308.0 321.7 336.7 3.53.2 371.2 ;391.3 413.7 4:38.8 467.1 499.4 .5:36.4 .579.4 629.8 689.9 762.6 8;52.5 966.5 1116 1320 1615 2082 29:35 5017 PPI" M 410 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 difierence 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 tlie 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 the 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 ^btract 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. 417 TABLE v.— AIDS TO FOR THE SINES OF SMALL ANGLES. Angles. Seconds. Diff, 0" 9' 15' 50" 20' 20" 23' 50" 27' 29' 50" 32' 30" 35' 37' 20" 39' 30" 41' 30" 43' 20" 45' 10" 47' 48' 40" 50' 20" 52' 53' 30" 55' 56' 30" 58' 59' 20" 1° 00' 40" 2' 3' 20" 4' 40" 5' 50" 7' 8' 10" 9' 20" 10' 30" 11' 40" 12' 50" 14' 15' 16' 10" 17' 10" 18' 10' 19' 20" 20' 20" 21' 20" 22 20" 23' 20" 24' 20" 2.3' 10" 26' 10" 27' 10" 28' 10" 29' 29' 60" 540 950 1220 H30 1620 1790 1950 2100 2240 2370 2490 2600 2710 2820 2920 3020 3120 3210 3300 3390 3480 3-^60 3640 3720 3800 39.50 4020 4090 4160 4230 4300 4370 4440 4.300 4570 4630 4690 4760 4820 4880 4940 5000 50(10 5110 5170 52.-50 5290 6340 AngU'S. Second s. Dili. 10 29' 50" 5390 50 51 52 53 54 SO' 50" 5450 31' 40" 5500 32' ,30" 5550 33' 30" 5C10 34' 20" 5G60 oo 35' 10" 5710 56 57 58 59 CO 36' 5760 36' 50" 5810 37' 40" 58(i0 38' 30" 5910 39' 30" 5970 61 62 63 64 65 40' 20" 60i:o 41' 10" 6070 41' 50" 6110 42' 40" 6160 43' 30" 6210 66 67 68 69 70 44' 10" 6250 45' 6300 45' 50" 63.50 46' 30" 6390 47' 20" 6440 71 72 73 74 75 48' 6480 48' ,50" 6530 49' 30" 6570 50' 20" 6620 51' 6660 76 77 78 79 80 51' 50" 6710 52' 30" 6750 63' 10" 6790 54' 6840 54' 40" 6880 81 82 83 84 85 55' 20" 6920 56' 10" 6970 56' 50" 7010 57' 30" 7050 58' 10" 7090 86 87 88 89 90 58' 50" 7130 59' ,30" 7170 2° 00' 10" 7210 50" 7250 1' 40" 7.300 91 92 93 94 95 2' 20" 7310 3' " 7380 3' S5" 7415 4' 10" 7450 4' 50" 7490 96 97 98 99 6' 30" 75^30 6' 10" 7570 6' 50" 7610 7' 30" 7650 Angles. Seconds. Diff, 2° 7' 30" 8' 10" 8' 45" 9' 20" 10' 10' 40" 11' 15" 11' 50" 12' 30" 13' 5" 13' 40" 14' 20" 15' 15' 35" 16' 10" 16' 46" 17' 20" 17' 56" 18' 30" 19' 5" 19' 40" 20' 15" 20' 50" 21' 25" 22' 22' 35" 23' 10" 23' 45" 24' 20" 24' 55" 25' 30" 26' 26' 35" 27' 5" 27' 40" 28' 10" 28' 45" 29' 15" 29' 50" 30' 20" SO' 55" 31' 25" 32' 32' 30" 33' 5" 33' 35" 34' 5" 34' 40" 35' 10" 35' 40" 36' 15" 7650 7690 7725 7760 7800 7840 7875 7910 7950 7985 8020 8060 8100 8135 8170 8206 8240 8275 8310 8345 8415 8450 8486 8520 8666 8590 8625 8660 8695 8730 8760 8795 8825 8925 8955 8990 9020 9055 9120 9150 91 &5 9216 9245 9280 9310 9340 9375 418 PRECISE CALCULATIONS. FOB TANGENTS AND COTANGENTS OF SMALL, ANGLES. Angk 0" 7' 10" 11' 10" 14' 10" 17' 19' 21' 23' 24' SC 26' 30' 27' 50' 29' 2^ 30' 40' 32' 33' 10' 34' 20' 35' 30' 36' 40' 37' 50' 38' 50' 39' 5C 40' 50' 41' 50' 42' 50' 43' 50' 44' 40' 45' 40' 46' 30' 47' 20' 48' 10' 49' 49' 50' 50' 40' 51' 30' 52' 20' 53' 53' 50' 54' 40' 55' 20' 56' 56' 50' 57' 30' 58' 10' 58' 50' 59' 30' 0'20' 1' 1'40' 2' 10' 2' 50' 3' 30' Seconds. Diff. 1 2 3 4 5 430 670 850 1020 1140 1260 6 7 8 9 10 1380 1490 1590 1670 1760 11 12 13 14 15 1840 1920 1990 2060 2130 2200 16 17 18 19 20 2270 2330 2390 2450 21 22 23 24 25 2510 2570 2630 2680 2740 2790 26 27 28 29 30 2840 2890 2940 2990 31 32 33 34 35 3040 3090 3140 3180 3230 3280 36 37 38 39 40 3320 a360 3410 3450 41 42 43 44 45 3490 3530 3.570 3620 3660 46 47 48 49 3700 3730 3770 3810 Angles. Seconds. Diff. 1° 3' 30" 4' 10" 4' 50" 5' 30" 6' 6' 40" 7' 20" 7' 50" 8' 30" 9' 9' 40" 10' 20" 10' 50" 11' 30" 12' 12' 30" 13' 10" 13' 40" 14' 10" 14' 50" 15' 20" 15' 50" 16' 20" 17' 17' 30" 18' 18' 30" 19' 19' 30" 20' 20' 30" 21' 21' 30" 22' 22' 30" 23' 23' 30" 24' 24' 30" 25' 25' 30" 26' 26' 30" 26' 50" 27' 20" 27' 50" 28' 20" 28' 40" 29' 10" 29' 40" 30' 10" 3810 3850 3890 3930 3960 4000 4040 4070 4110 4140 4180 4220 4250 4290 4320 4350 4390 4420 4450 4490 4520 4550 4580 4620 4650 4680 4710 4740 4770 4800 4830 4950 5010 5040 5070 5100 5130 5160 5190 5210 5240 5270 5300 5320 5350 5380 5410 67 87 Angles. 1° 30' 10' 30' 30' 31' 31' 30' 32' 32' 20' 32' 50' 33' 10' 33' 40' 34' 10' 34' 30' 35' 20' 35' 50' 36' 10' 36' 40' 37' 10' 37' 30' 38' 38' 20' 38' 50* 39' 10' 89^30' 40' 40' 20' 40' 5^ 41' IC 41' 40' 42' 42' 30' 42' 50' 43' 10' 43' 40' 44' 44' 30' 44' 50' 45' 20" 45' 40' 46' 46' 20' 40' 40' 47' 10' 47' 30' 48' 48' 20' 48' 40' 49' 49' 20' 49' 40' 50' 10' 50' 30' Seconds. I Diff. &410 5430 5460 5490 5520 5540 5570 5590 5620 56.50 5670 5700 5720 5750 5770 5800 5830 5850 5880 5900 5950 5970 6050 6070 6100 6120 6150 6170 6190 6220 6240 6270 6290 6320 6340 6360 6380 6400 6430 6450 6480 6500 6520 6540 6560 6580 6610 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 12:1 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 419 TABLE v.— AIDS TO PRECISE CALCULATIONS. FOR TANGENTS AND COTANGENTS OF SMALL ANGLES. Angle 1° 50' 30' 50' 50' 51' 10' 51' 30' 52' 52' 20' 52' 40' 53' 53' 20' 53' 50' 54' 10' 54' 30" 54' 60'' 55' 10' 55' 30'- 65' 50' 56' 10' 5G' 30' 56' 50' 57' 10' 57' 40' 58' 58' 20' 68' 40' 59' 59' 20' 59' 40' 2° 00' 00' 20' 40' 1' 1'20' 1'40' 2' 2' 20' 2' 40' 3' 3' 20' 3' 40' 4' 4' 20' 4' 40' 6' 5' 20' 5' 40' b' 6' 20' 6' 40' 7' 7'20' 7' 40' Seconds. Diff. 6630 6650 6670 6690 6720 6740 6760 6780 6800 6830 6850 6870 6890 6910 6930 6950 7010 7030 7060 7080 7100 7120 7140 7160 7180 7200 7220 7240 7260 7280 7300 7320 7340 7360 7380 7400 7420 7440 7460 7480 7500 7520 7540 7560 7580 7600 7620 7640 7660 I ir. 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 Angles. 2o 7' 40" 8' 8' 15' 8' 30' 8' 50' 9' 10' 9' 80' 9' 50' 10' 10' 10' 20' 10' 40' 10' 55' 11' 15' 11' 35' 11' 55' 12' 15' 12' 35' 12' 55' 13' 15' 13' 35' 13' 50' 14' 10' 14' 30' 14' 45 15' 5' 15' 20' 15' 40' 15' 56' 16' 15' 16' 30' 16' 50' 17' 5' 17' 25' 17' 40' 18' 18' 15' 18' 35' 18' 55' 19' 15' 19' 30' 19' 45' 20' 5' 20' 20' 20' 40' 20' 55' 21' 15' 21' 30' 21' 45' 22' 5' 22' 20' 22' 35' Seconds. Diff 7660 7680 7695 7710 7730 7750 7770 7790 7810 7820 7840 7855 7875 7895 7915 7935 7955 7975 7995 8015 8050 8070 8085 8105 8120 8140 8155 8175 8190 8210 8225 8245 8260 8280 8295 8315 aS35 8355 8370 8385 8405 8420 8440 8455 8475 8490 8j05 8525 8610 8555 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 237 238 239 240 241 242 243 244 245 246 247 248 249 Anglef 2° 22' 35" 22' 55" 23' 10" 23' 30" 23' 45" 24' 24' 20" 24' 35" 24' 55" 25' 10" 25' 25" 25' 45" 26' 26' 20" 26' 35" 26' 50" 27' 10" 27' 25" 27' 45" 28' 28' 15" 28' 35" 28' 50" 29' 10" 29' 25" 29' 40" SO' 15" 30' 30" 30' 50" 31' 5" 31' 20" 31' 35" 31' 55" 32' 10" 32' 25" 32' 40" 32' 55" 33' 15" 33' 30" 33' 45" 34' 34' 15" 34' 30" 34' 45" 35' 35' 20" 35' 35" 35' 50" 36' 5" 86' 20" 8555 8575 8590 8610 8625 8640 8675 8695 8710 8725 8745 8760 8780 8795 8810 8830 8845 8865 8880 8915 8965 9000 9015 9030 9050 9066 9080 9095 9115 9130 9145 9160 9176 9195 9210 9225 9240 9255 9270 9285 9300 9320 9335 9350 420 RETURN EDUCATION-PSYCHOLOGY LIBRARY TO— #^ 2600 Tolman Hall 642-4209 LOAN PERIOD ■ 1 MONTH ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 2- hour books must be renewed in person Return to desk from which borrowed DUE AS STAMPED BELOW UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DDIO, 5m, 3/80 BERKELEY, CA 94720 ®s U^ K^K^ y W^^k^ 9S4225 THE UNIVERSITY OF CALIFORNIA LIBRARY