*«; / ■-. n Vw LIBRARY OF THE University of California. GIFT OF .^......^rL^.......> This book owes mucli to criticisms, made by many of my f ellow-teacbers, upon my Treatise on Plane and Solid Geome- try. To my pupils also I am greatly indebted; their ques- tions and remarks have indicated many difficulties to be smoothed and errors to be avoided in an elementary work. It has been my aim — ^to state correctly the first principles; of the science; upon these premises to demonstrate rigor- ously and in good English the whole doctrine of Elementary Geometry; and to arrange the matter so as to proceed with easy gradation from the simple to the complex. Some teachers hold that the study of geometry is valuable only as a kind of intellectual gymnastics. Mr. Todhunter says that the admirers of Euclid value it "more for the process of reasoning involved than for the results obtained"; and Mr. De Morgan sums up the educational value of geometry in the habit of "tracing necessary consequences from given premises by elementary logical steps." With such views, much of the best use of the study is neglected. It is as im- portant to begin with truth as to argue logically. The cer- tain knowledge of the notions we begin with is as useful as the process of inference. In volumes of pretentious and false 184009 iv PREFACE, theories, almost the only logical errors are the assumptions made in a few sentences. It is a great misfortune to a youth to believe that the truth of premises is immaterial, or that the process of de- monstration is the paramount object. Injury to love of truth is more fatal than ignorance of logic. No science can grati- fy the desire for exact knowledge better than this, where the truth is unmixed with errors of observation. In England, the text-book on geometry in most common use has been Eobert Simson's version of the Elements of Euchd, published little more than a hundred years ago, but written in a style that was even then growing obsolete. A quaint language, not used by good writers on any other science, has been the custom in English works on geometry. Proba- bly I have not been able to avoid this entirely. A prime object of this study is to train the student in rigorous argu- ment, but the training is of more value when the argument is expressed in the common language. The undue importance allowed to syllogistic inference has shut the eyes of many geometers to the want of logical order in the arrangement of topics. EucHd's first proposition treats of a triangle. The order pursued by Legendre is somewhat more gradual. The example of these illustrious authors has induced many of their students to think that a gradual de- velopment of the subject is not consistent with rigorous de- monstration. Admitting the necessity of retaining this rigor, I have tried to show that it is aided and not hindered by an arrangement of the topics in the order of their increasing complexity, beginning with straight lines. In the earlier chapters I have usually given all the steps of demonstration, the general truth, the particular, the con- PREFACE. Y struction, the argument, and the conclusion, according to the ancient method. In the last two or three chapters more brevity is used. Graduation has been observed also in the exercises; hints are freely given in the beginning, but the student, as his knowledge increases, is supposed to have ac- quired skill by practice. The reasons that have governed my work, in some points about which geometers differ, are presented in the I^otes more fully than would be suitable in the limits of a preface. Eli T. Tappan. Gambier, Ohio, October, 1884. CONTENTS. PART FIRST.— ELEMENTARY. PAGE CHAPTER L— Preliminary . . . . . . .1 CHAPTER II.— The Subject Stated ..... 4 PART SECOND.— PLANE GEOMETRY. CHAPTER IIL— Straight Lines . . . . , .14 CHAPTER IV.— Circumferences . . . . . 38 Scholium on Geometrical Reasoning . . .57 Problems in Drawing .... 69 CHAPTER v.— Triangles . . . . . . . 6Y CHAPTER VI.— Quadrilaterals ..... 101 CHAPTER VII.— Polygons 123 CHAPTER VIII.— Circles 143 PART THIRD.— GEOMETRY OF SPACE. CHAPTER IX.— Straight Lines and Planes . . . .156 CHAPTER X.— PoLYEDRONS . . . . . . .185 CHAPTER XL— Solids of Revolution . . . . .212 NOTES . . 243 INDEX . . . 249 NOTICE In this work, reference to an article in tlie same chapter is made thus (5), that is, Article 5 of the chapter in which the reference is given. Reference to an article in a previous chapter, thus (Y, 6), that is. Article 6, Chapter Y. Eeference is made to a corollary by adding the number in small capitals, thus (5, ii) or (Y, 6, ii), that is. Corollary II of that article. Reference to a problem in drawing, or any other subdivision of an article, is made in the same way as to a corollary. When there is only one corollary in an article, it is referred to by the letter c. In the exercises, the added remark or reference [in brack- ets] is given as a hint to the student. ELEMENTS OF GEOMETRY. CHAPTER I. PRELIMINARY. Article 1.— The principles of Geometry are applied when- ever the size, shape, or position of anything is investigated. This science establishes the laws upon which all measurements are made. It is the basis of the sciences of Mechanics and Astronomy. Without Geometry, men could not build machines, survey the earth, or navigate the ocean. In addition to these uses of the science, Geometry is taught for the purpose of intellectual training. The student cultivates precision of language, by the use of precise terms ; his reason- ing power, in the various analyses and demonstrations ; his imagination, in conceiving and holding in his mind the com- binations of lines and surfaces ; and his inventive faculty, in making new solutions and demonstrations. The definitions and principles in this chapter are not peculiar to Geometry, but are much used in the study, and are placed here for convenience of reference. liOgical Terms. 3, Propositions in geometry are either theoretical or practi- cal. Theoretical propositions declare that a certain property belongs or does not belong to a certain object ; practical propositions declare that something can be. Some propositions are so simple and evident that they can not be made more so by any course of reasoning. They are therefore called indemonstrable. They are also called self- 2 ELEMENTS OF GEOMETRY. [Chap. I. evident, because every person that apprehends the meaning of such a proposition necessarily admits its truth. A Postulate is a self-evident practical proposition. A Problem is a practical proposition that can be proved. An Axiom is a self-evident theoretical proposition. A Theorem is a theoretical proposition that can be de- monstrated. A Corollary is a proposition that follows from previous principles without further reasoning, or the demonstration of which is brief and simple. Converse propositions are two propositions, such that the subject of each is the predicate of the other. When propositions are expressed hypothetically, then con- verse propositions are such that the hypothesis of each is the conclusion of the other. Every good definition is a proposition whose converse is true ; but the converse of a true statement may be false. Sometimes the hypothesis of a theorem is complex, i. e., con- sists of several distinct hypotheses ; in this case every theorem formed by interchanging the conclusion and one of the hypo- theses is a converse of the original theorem. The student should analyze every proposition, separating the subject or hypothesis from the predicate or conclusion. A Scholium is an introductory or an explanatory remark. Axioms that apply to all mathematics are called general, to distinguish them from those which relate to geometrical no- tions. The statement of general axioms in works on Geometry is generally defective, and it is unnecessary. Proportion. 3, The following properties of proportions are given here without demonstration. The student is referred to the algebra for a fuller treatment of this subject. If four quantities are in proportion taken in their direct order, then, I. They are in proportion when taken alternately ; II. They are in proportion when taken inversely ; III. They are in proportion when taken by composition ; IV. They are in proportion when taken by division ; Art. 3.] PRELIMINARY. 3 Y. They are in proportion when taken by composition and division. YI. If there is a series of equal ratios, then the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. YII. The products of the corresponding terms of two proportions are in proportion. YIII. The quotients of the corresponding terms of two proportions are in proportion. IX. If two proportions have the same antecedents, the consequents are in proportion in their direct order. X. If two proportions have the same extremes, the means of one proportion may be substituted for the extremes of the other. CHAPTER II. THE SUBJECT STATED. Article 1. — 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 properties may be considered distinct from all others. If the book is removed, the space which it has occupied remains, that is, the space exists independently, and this space has extent and form. 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 occupies space. The limit or boundary that separates a solid from the sur- rounding space is a surface. Every surface has extent and form, but a surface has no thickness or depth. Two geometrical solids may occupy the same space, entirely or partially. The space which has been occupied by a book may be occupied by a block of wood. Their surfaces may meet or cut each other. The boundaries of a surface are lines. The intersection of two surfaces, being the limit of the parts into which each divides the other, is a line. A line has extent and form, but neither breadth nor thickness. The ends of a line are points. The intersections of lines are also points. A point has neither extent nor form. 3. A Magnitude is that which has only extent and form. Geometry is the science of magnitude. A Solid IS a magnitude having length, breadth, and thickness. A Surface is a magnitude having length and breadth, with- out thickness. Akt. 2.] THE SUBJECT STATED. 6 A Line is a magnitude having length, without breadth or thickness. A Point has only position, without extent. A line may be measured in only one way, that is, a line has only one dimension. A surface has two, and a solid has three dimensions. Every magnitude belongs to one of these three classes. The extent of a line is called its length ; of a surface, its area ; and of a solid, its volume. The word magnitude has been used by some writers in the sense of quantity, " that of which greater or less can be predi- cated when two of the same kind are compared together," but generally the term is limited to lines, surfaces, and solids. The Postulates. 3« Postulate of Form. — A magnitude may he of any form. Postulate of Extent. — A magnitude may he of any ex- tent. The form may be the most simple, as of a straight line, or it may be as complex as the most intricate piece of machinery that can be imagined. Lines, surfaces, and solids, may be contained within the limits of the smallest spot that represents a point, or they may have such extent as to reach across the universe. The as- tronomer knows that his lines reach to the stars, and that his planes extend beyond the sun. The space attributed to an atom of matter is divisible into any number of smaller spaces. The postulates do not assert that every magnitude can be represented to the senses by a drawing or otherwise. It is not asserted that every magnitude can be represented even in im- agination ; no person can so represent the magnitudes of as- tronomy or of physics, although convinced of their reality. The postulates only assert that whatever magnitude can be definitely apprehended does exist ; it is a subject of geometrical thought. The form of a magnitude consists in the relative position of its parts. However the extent may vary, the form is unchanged if all the points keep the same relative position. 6 ELEMENTS OF GEOMETRY. [Chap. H. Similar Magnitudes are those that have the same form. Homologous points are similarly situated points in similar magnitudes. The term homologous is also applied to similarly- situated lines, or surfaces, of similar magnitudes. Equivalent Magnitudes are those that have the same extent. Superposition. 4, One magnitude may be mentally applied to another in order to compare their form or their extent. This is called the method of superposition — one magnitude being placed upon the other. Equal Magnitudes are such that one can be applied to the other, and the two coincide. Since they coincide, every part of one has its corresponding equal part in the other, and the parts are arranged the same way 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 magnitudes coincide throughout, which shows them to be equal. Hence, equal magnitudes have the same extent and the same form ; and conversely, if magnitudes are both similar and equivalent, they are equal. Two magnitudes composed of parts respectively equal, but differently arranged, are equivalent, for the sums of equal ex- tents are equal ; but such magnitudes may be unequal, that is, they may not coincide. Figures. 5, In order that a magnitude may be a subject of stated reasoning, the term that designates it must be definite, its mean- ing must be precise. A Geometrical Figure is any line, surface, solid, or combination of these magnitudes that can be described in exact terms. Figures are represented by diagrams, and by models. In common language, such representations are called figures ; a small spot is called a point, and a long mark a line, but they are not geometrical points and lines. They have not only extent and form, but also color, weight, and other properties. Art. 5.] THE SUBJECT STATED. 7 The subject of a proposition is said to be given. The addi- tions to the figure made for the purpose of demonstration or solution constitute the construction. In the diagrams, points are designated by capital letters. Thus, the points A and B at the extremities of the line. Figures are usually designated by naming some of their points, as the line AB, and the quadrilateral CDEF^ or simply DF. A C Sometimes, when a figure is designated by a single letter, the small letters are used. Thus, the line m, or the circle n. Liines. 6. A Straight Line is one that has the same direction through its whole extent. The path of a point moving in one direction, turning neither up nor down, right nor left, is a straight line. From one end to the other, a straight line has only one direc- tion ; but from the other end to the first it has exactly the opposite di- jy^ b rection. A straight line has only two directions. The direction from ^ to J5 is called the direction AB^ and the direction from B to A is called the direction BA. A Curve is a line that has a continuous change in direction. The Curvature of a line is the amount of change in direc- tion. A line may curve so as to return to the same point within a small space, or the change may be so slight that it can hardly be distinguished from a straight line. 8 ELEMENTS OF GEOMETRY, [Chap. II. Lines may have a uniform curvature, as the circmnference of a circle, defined in Article 15 ; or the curvature may vary, as in a spiral, an ellipse, or a reversed curve. A line may be composed of parts that are straight, or of curves, or of both curved and straight parts. A line composed of straight parts is called a broken line. Straight Liines. Axioms. 7. Axiom of Direction, — In one direction from a point, there can he only one straight line. Axiom of Distance. — The straight line is the shortest that can join two points. Direction is a term that is too simple to admit of definition. It is essentially distinct from any notion of extent. The direc- tion of a straight line is determined by any two points of the line, however near or far apart they may be. 8. As a corollary of the postulates, we have this Problem. — There may he a straight line from any point, in any direction, and of any extent. Therefore, there may be a straight line from any point to any other, and indefinitely beyond either. 9. From the Axiom of Direction we have these Corollaries. — ^I. From one point to another there can be only one straight line. Art. 9.] THE SUBJECT STATED. 9 II. If two straight lines have two points common, or if they have the same direction from a common point, they must coin- cide ; they are one line. III. The position of a straight line is determined by two points, or by one point and one direction. IV. If any figure revolves about a straight line, the line it- self remains fixed in position. This property is peculiar to the straight line. If the curve BG revolves upon the points JB and C, the straight line B G does ^ ^--.^^^ not move while the curve moves B ^- --^ c around it. An Axis is a line about which a figure revolves. Distance implies a limited extent, the shortest path from one point to another. The Axiom of Distance states that property of a straight line which was used by Archimedes as the definition. He has been followed by Legendre, one of the most distinguished of modern geometers. The defect of this definition is that it makes straightness a matter of extent, while it is essentially a matter of form. Euclid defined a straight line as " one that lies the same as to the points in it." A fine thread being stretched till it assumes that position which is the shortest path between its ends is a good representa- tion of a straight line. Surfaces. 10. Surfaces are classified, as lines are, according to uni- formity or change of direction. At one point of a line, there are only two directions along the line, but at one point of a sur- face there are an indefinite number of directions along the sur- face. A Plane is a surface that never varies in direction. A Curved Siirface is one in which there is a change of direction at every point. The surface of a small body of still water is perceptibly straight in every direction. A piece of paper may be bent to 10 ELEMENTS OF GEOMETRY. [Chap. II. illustrate a curved surface which, from any point, is straight in one direction and curved in others. A globe serves to illustrate a surface that is not straight in any direction. Planes. 11. Theorem. — If two planes coincide to any extent^ they must coincide throughout. This follows from the definition of a plane. If two planes could have a common portion and then diverge, some direction of one of them would vary. 12. Theorem. — A straight line that has two points in a plane lies wholly in the plane. This follows from the definitions of straight line and plane. Uniformity of direction characterizes both the plane surface and the straight line. 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 coincide. It is to be understood that geometrical lines and surfaces are not limited in extent, unless it is expressly stated or implied in the context. In the above theorems, if either of the magnitudes is limited, they coincide so far as they both extend. Euclid defined a plane surface as " one that lies the same as to the straight lines in it." The idea is that of invariable direc- tion. Another ancient geometer defined a plane as " that sur- face to all the parts whereof a straight line may be accom- modated." This is in substance the property stated in the above theorem. It is the definition used by many modern geometers. 13. Problem. — Through any three points there may be a plane. Through any two of the three points there may be a straight line (8). A plane may pass through this line, and if the third point is in the line it is also in the plane (12). If the third point is not in this line, the plane may turn on the line as an axis (9, iv) Art. 13.] THE SUBJECT STATED. 11 until it passes througli the third point. Then all three points are in the plane. Plane Figures. 14. When two straight lines have one common point they lie in one plane (13), and from this point the lines have different directions (7). An Angle is the difference of two directions. The angle is greater or less according as this difference is greater or less. Let the line AB be fixed, and the line A G revolve in a plane about the point A, taking every direction from A in the plane. The ^ angle or difference in direction of the -^ two lines increases 2l8 AO revolves. The Arms of an angle are the lines that form it. These lines have been called sides. Suppose the line AC revolves about the point A, in the same plane with the line BD. In the course of one revolution, it takes ■ every possible direction in the plane. ^ One of these is the direction BD. ' When A C has the same direction as BI>, then the direction GA, being the opposite of A C, is the same as the direction DB, which is the opposite of BD. The Vertex of an angle is the point of intersection of the arms. Parallel Ijines are straight lines that have the same direc- tions. Angular quantity consists in the difference of directions. Parallelism consists in the identity of directions. The definition of similarity can now be stated in a better form than the one given in Article 3. SimilsLT Figures are such that every possible angle in one has its corresponding equal angle in the other. 15. A Plane Figure is one whose points all lie in one plane ; for example, a straight line, or an angle, or two parallel 13 ELEMENTS OF GEOMETRY. [Chap. H. lines. None of the plane figures just mentioned can inclose a portion of the plane, for two straight lines can have only one common point (9). 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. A Triangle is a polygon of three sides. A quadrilateral has four sides ; a pentagon, five ; a hexagon, six ; an octa- gon, eight ; a decagon, ten ; a dodecagon, twelve ; and a pentedecagon, fifteen. The line AB may revolve in a plane about the end Aj which is fixed. The point B then describes a curve which returns upon itself. Every point of this curve is at the same distance from the point A, this dis- \ ^b tance being equal to the length of AB, A Circle is a portion of a plane bounded by a curve that is everywhere equally distant from a point within. The curve is the circumference, and the point is the center. Polygons and circles are by their definitions plane figures. Plane Geometry is that branch of the science which treats of plane figures. In this work plane figures are discussed in the following order : 1. The straight line and combinations of straight lines (Chap. Ill) ; 2. The circumference, and straight lines in com- bination with it (Chap. IV) ; 3. Inclosed rectilinear figures, that is, the triangle (Chap. Y), the quadrilateral (Chap. VI), and other polygons (Chap. VII) ; and 4. The circle (Chap. VIII). Figures in Space. 16. Geometry of Space treats of figures whose points are not all in one plane. It is frequently called Solid Geometry, but strictly a solid is an enclosed figure. The Figures in Space which are treated of in Elementary Geometry are : 1. Unenclosed figures that consist of straight Art. 16.] THE SUBJECT STATED. 13 lines and planes (Chap. IX) ; 2. Enclosed figures that are bounded by planes, and are called polyedrons (Chap. X) ; and, 3. Three inclosed figures bounded by curved surfaces, the cone, the cylinder, and the sphere (Chap. XI). 17. Scholium. — This chapter contains the first principles of Geometry and the definitions of the most important terms and figures. Four fundamental truths are the basis of the science — two postulates and two axioms. Every geometrical conception, however simple or complex, is composed of two kinds of elements — directions, and lengths or distances. The directions determine its form, and the dis- tances its extent. 18. Exercises. — 1. What geometrical principle is used to ascer- tain if a mark is straight, when one applies a straight edge to it and observes whether they coincide ? 2. What principle is used to ascertain if a surface is plane, when one applies a straight edge to it in many positions, observing whether it touches throughout? 3. In a diagram two letters suffice to mark a straight line, but it may require three to designate a curve. Why? 4. Which is the greater angle, a or 5, and why? 5. What is the greatest number of points in which two straight lines may cut each other ? In which tliree may cut each other ? Four ? 6. What principle is applied when a stretched cord is used to mark a straight line ? PLANE GEOMETRY. CHAPTER III. STRAIGHT LINES. Article 1. — Straight lines may be given in a plane in any position, and witli any length (II, 8), and either separately or combined. A line may be given in extent only : for example, " one inch," or, " a line equal to twice AB^ A line may be given in position only ; as, " a line through A and j5," or, " a line parallel to AB through 6V' A line may be given in extent and position ; as, " the line AB^'' the ends being designated. I*TO'blem.. — Straight lines may he added, subtracted, multi- plied, or divided. Straight lines may be of any length (II, 8). Therefore, there may be a line equal to any sum or to any difference of other lines, and there may be a line equal to any multiple or to any equal part of any other line, or having any ratio to it. A line may be thought of as the trace of a point in motion. As the point moves, the growing line has every length from zero to the entire distance passed over. It has an infinite num- ber of various lengths which no drawing could exactly represent, and its existence at every one of those lengths is as certain as that of a geometrical line can be. This problem does not assert that a line can be drawn or otherwise represented equal to the sum of two other lines, or that it can, by dividers or other instruments, be divided into equal parts, etc. This problem merely asserts the possible exist- ence of such sums, parts, etc. Problems in drawing may require Art. 1.] STRAIGHT LINES. 15 an application of geometrical principles, but such work is not pure geometry. Theorem. — Straight lines are similar figures. The only possible difference of directions in a straight line is the same as in any other straight line. For each of them has only two directions, which are exactly the opposite of each other. Straight lines have, therefore, the same form. See the definition of similar figures (II, 14). If one straight line is applied to another, they coincide so far as they both extend ; they differ only in extent. Corollary I. — The points in two lines which divide them in the same ratio are ho- mologous points. Thus, j^ c B if the lines AB and EI) are divided at the ^ Z ^ points C and F, so that AC '.AB = EF'.EB, then C and F are homologous points in these similar figures, and A G and EF are homologous parts. A point in a straight line is said to divide it internally, or simply to divide it ; and a point in the line produced is said to divide it externally. In both cases, the distances of the point from the extremities of the line are called segments of the line. Thus, the line AB is divided internally at the point F, and ext era ally at the point S. When divided at P ^ s_^ S, AS and BS are the A B segments. Corollary II. — A straight line is equal to the sum or to the difference of its segments, according as it is divided inter- nally or externally. A Secant is a line that cuts or passes through any other line or plane figure. The lengths of any two lines of definite extent, being quan- tities of the same kind, have a certain ratio. When two lines have a common measure, their ratio can be exactly expressed by two whole numbers, that is, by the num- bers that tell how many times the lines contain the common 16 PLANE OEOMETR Y. [Chap. III. measure. Conversely, if the lengths can be expressed by two whole numbers, the lines are commensurable, that is, they have a common measure. Frequently two lengths are incom- mensurable, and the ratio can not be expressed by ordinary numerals. Such ratios are expressed by reference to the quantities compared, as in the Corollary I above. Radicals are examples of numbers incommensurable with unity. S, A line is said to be divided in extreme and mean ratio when one segment of the line is a mean pro- ^ b c portional between the whole line and the other segment. Thus, \i AG \ AB = AB : BC, the line AG h dl vided at B in extreme and mean ratio. The student of algebra may prove that the ratio of the greater segment to the less is ^(|/5 + 1). It will be shown in the sequel how to mark the point of division of a line in this ratio (Y, 44, x). Theorem.. — If a U?ie is divided in extreme and mean ratio, the segments have no common measure. Suppose the line divided at B, so that A G : AB = AB : BG. If these segments have a common measure, it is also a n E b c ^ a common measure of their difference ; that is, if BD is taken equal to BG, any line contained an in- tegral number of times in AB and in BG, or its equal BI), must be contained an integral number of times in AB, for the difference of two whole numbers is a whole number. In the same way, if BJE is taken equal to AB, the common measure sought must be a measure of the remainder BJE, And so on, the common measure of the segments AB and BC must be a measure of every remainder as long as there is a remain- der found by such subtractions. Since BB equals B G, the ratio AB : BD is the same as A G : AB. Therefore, there was necessarily a remainder when BB was taken from AB, and AB : BB = BB : AB, That Art. 2.] STRAIGHT LINES. 17 is, AB is divided in extreme and mean ratio. For the same reason, when DE (equal to AD) is taken from DB there must be a remainder, and the ratio DE\EB is the same as the first. The part taken away has at every step the same ratio to the re- mainder. Therefore, there cannot cease to be a remainder ; and, as there is no last remainder, the segments have no common measure. A line is said to be divided harmonically when it is divided both internally and externally in the same ra- . 'PS tio. For example, if the ■ ^ — ^ ' line AB is divided at P and at /S, so that AP\AS=BP'.BB, then AB is divided harmonically. This agrees with the numerical definition of harmonic ratio, for the several distances from A to P^ B^ and S^ are such that " the first is to the third as the difference between the first and the second is to the difference between the second and the third." Broken Liiues. 3. A curve or a broken line is said to be concave to the side on which there can be a straight line joining two of its points, and convex to the other side. Part of a curve or of a broken line may be concave to one side and part concave to the other side. Theorem. — A broken linCy no part of which is concave toicard another line that unites its extreme points, is shorter than that line. It is to be proved that ^ the line ABCD is shorter ^/~y \^ than the line AEGD, to- //b ■ ____^ \ ward which no part of the /^ c \Q^ first line is concave. D Produce AB and ^(7 till they meet the outer line in F and JZ 2 18 PLANE GEOMETRY. [Chap. m. Then „ E, — i ,c AB+BF(7 and j5i>i<: Vertical angles are the opposite angles formed by two in- tersecting lines. A Right angle is formed when one line meets another, making the adjacent angles equal. A Perpendicular to a . line is another line making a right angle with it. An Oblique line is one that is neither perpendicular nor parallel to another line to which it is referred. An angle that is neither a right angle nor a multiple of a right angle is called oblique. An Acute angle is one that is less than a right angle. An Obtuse angle is one that is greater than a right angle, and less than a straight angle. Complementary angles are two whose sum is equal to a right angle. Each is the complement of the other. Supplementary angles are two whose sum is equal to a straight angle. Each is the supplement of the other. Art. '7.] STRAIGHT LINES. 21 7 . Theorem, — All straight angles are equal. For any straight angle may be placed on another, and the two must coincide, for the arms of each constitute one straight line (II, 9, ii). Corollaries. — I. All right angles are equal, for a right angle is half of a straight angle. II. The sum of all the successive angles formed in a plane, on one side of a straight line, is equal to a straight angle, or to two right angles ; for the straight line may coincide with the arms of a straight angle. III. Conversely, when the sum of several successive angles is equal to two right angles, the extreme arms form one straight line. For the sum being one straight angle, the extreme arms may coincide with the arms of a straight angle (II, 4). ly. The sum of all the suc- cessive angles formed in a plane around a point is equal to two straight angles, or to four right angles. For instance, the sum of two conjugate angles. V. Complements of the same or of equal angles are equal. VI. Supplements of the same or of equal angles are equal. VII. The supplement of an obtuse angle is acute, and con- versely. VIII. The greater an angle the less is its supplement. IX. Vertical angles are equal. For both AFC and DFB are sup- plements of the same angle AFD, 22 PLANE GEOMETRY. [Chap. III. In geometrical investigations, the right angle is the standard or unit of angular quantity. The right angle is divided into ninety equal parts, called degrees, marked thus °. The six- tieth part of a degree is a minute, marked thus '; and the Gixtieth part of a minute is a second, marked thus ". 8. Exercises. — 1. Of two conjugate angles, one is equal to twice the other ; how many degrees in the less ? 2. Prove that the hisectors of two adjacent supplementary angles are perpendicular to each other. 3. Find the angle between the bisectors of adjacent complementary angles. 4. If ^ is any angle, prove that 90° + A and 90° — A are supple- mentary. 5. State the converse of Corollary lY, Article 7. 6. If the arms of an angle and the bisector of it are produced beyond the vertex, the bisector also bisects the vertical angle. [7, ix.] Perpendicular and Oblique Lines. . 9. Theorem. — There can he only one line through a given point perpendicular to a given straight line. For, since all right angles are equal, all lines lying 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. When the point is in the given line, this theorem must be limited to one plane. Art. 10.] STRAIGHT LINES. 23 10. Theorem, — If a perpendicular and several oblique lines extend from a common point to a given straight line : 1. The perpendicular is shorter than any oblique line; 2. Two oblique lines that meet the given line at equal dis- tances from the perpendicular are equal/ 3. An oblique line that meets the given line at a greater dis- tance than another from the perpendicular is longer than the other. AD being perpeDdicular and AG oblique to BE, it is to be proved that AD is shorter than AC. Let the figure revolve upon BE as an axis (II, 12, c), until the point A falls upon the plane of the original figure at the point F, so that AD coincides with FD and A G with FG. Since the angle i/F Therefore, than AGF. FDG coin- cides with ADG, it also must be a right angle. ADF is a straight line (7, iii), and is shorter AD, the half of ADF, must also be shorter than AG, the half of AGF. Secondly, let AD be the per- pendicular, and A G and AE the oblique lines, making GD equal to DE. Let that portion of the figure on the left of AD turn upon AD. Since the angles ADB and ADF are equal, DB takes the direction DF', and, since i) (7 and DE are equal, the point G falls on E. Therefore, A G and AE coin- cide (II, 9, ii), and are equal. Lastly, if DF is longer than DG, it is to be proved that AF IS longer than A G, JJ ^ERSITY OF 24 PLANE GEOMETRY. [Chap. III. ^H On the line DF take a part DJE equal to D C, and join AE. Then let the figure revolve upon B Gy the point A falling upon IT, and the lines AD, AE and AF upon ^2>, ^Je' and HF Now, AEH is shorter than ^7^-£r (3) ; therefore, AE, the half of AEH, is shorter than ^i^ the half of AEH, But AC \^ equal to ^^. Hence, AF is longer than ^ (7, or ^j^, or any line from A meeting the given line at a less distance from D than DF. CorollSLries. — I. Conversely, two equal oblique lines extend- ing from a common point to a given line meet it at equal dis- tances from the perpendicular. II. If two such oblique lines are unequal, the foot of the shorter is nearer the perpendicular. III. The distance from a point to a straight line is the length of the perpendicular, for distance means shortest path. IV. Equal oblique lines make equal angles with the perpendicular ; for CAD, when applied, coincides with DAE. V. Equal oblique lines make equal angles with the given straight line ; that is, A CD and AED are equal. VI. A point may be at the same distance from two points of a straight line, one on each side of the perpendicular ; but it cannot be at the same distance from more than two points of one straight line. VII. Any point of a line that is perpendicular to another at its midpoint is equally distant from the two ends of the second line. 11, Theorem. — If a line is perpendicular to another at its midpoint, every point out of the perpendicular is nearer to that end of the line that is on the same side of the perpen- dicular. If BF\b perpendicular to ^ C at its midpoint B, then it is to Art. 11.] STRAIGHT LINES. 25 be proved that D, a point not in BF, is nearer to C, on the same side of the perpendicular as D, than it is to A. Join DA and D C ; also join C with G, the point where 1>A cuts the perpendicular. Then, B But Da^GC=I)G-\-GA = DA, DC and J5C. From the point A, where GF cuts the bisector, make AH perpendicular to CD, and join GH. Then GA + AH But GF, GE< GH< GA + AH Therefore GF is less than GF It may be that the bisector is not cut by either of the per- pendiculars from the point to the arms of the angle. The de- monstration of that case is left to the student. Corollaries. — I. If a line passes through two points, each of which is equally distant from the arms of an angle, the line bi- sects the angle (II, 9, iii). 11. A line that bisects an angle is the locus of the points that are equally distant from the arms. The locus of the points jf equally distant from two lines that cut each other is the two lines that bi- sect the angles made. Thus, FG and HF are the locus of the points equally distant from A C and DF, 15. Application. — Perpendicular lines are constantly used in architecture, carpentry, ma- chinery, etc. The instrument called a square consists of two pieces of wood, iron, or steel, having straight edges; the edges of one piece being at right angles to those of the other, lines or surfaces are required. It is used in any work where perpendicular 28 PLANE GEOMETRY. [Chap. III. In order to test the square, draw an angle with it on any plane sur- face, as BA G. Extend BA in tlie same straight line to D. Then turn the square so that the edge's by which BAG was drawn may be applied to the arms of the angle DAG. If the coincidence is exact, the square is correct as to these j^ 16. Exercises. — 1. Can two lines have the same projection on a third ? Wliat is the projection of a line that is perpendicular to the line of projection ? 2. What is the geometrical principle involved in the method of testing a square described in the preceding Article ? Parallels. 17. Parallel lines are straight lines that have the same directions. CoroUeLiy. — Lines that are parallel to the same line are par- allel to each other. 18. Theorem. — Through a given point there can he only one line parallel to a given line. For, in one direction from a point there can be only one straight line. Corollary. — Parallel lines can never meet. 19. Theorem. — Two parallel lines lie in one plane. Let AG and B-D be parallel lines. Now, there may be a plane ^ ^ through the points A, IB, and C (II, ' 13). The line AG lies wholly in ^^ jy this plane (II, 12). There may be ~ in this plane a line through 3 parallel to A G (II, 14) ; but there can be through J3 only one line parallel to A G. Therefore, BD is that line in the plane of A, B, and G. Parallel lines were defined by Euclid as " straight lines which. Art. 19.] STRAIGHT LINES. 29 being in the same plane, and being produced indefinitely to both sides, meet on neither side." It has just been demonstrated that these properties belong to two straight lines that have the same directions. The objections to the definition of Euclid are that it is double, and one of the attributes is negative. The defini- tion based on direction was introduced by an American geome- ter, James Hayward. 30, When two straight lines in the same plane are cut by a third, the angles are named as follows : Corresponding angles are any two having their vertices at different points, both being on the same side of the secant, and on the same side of the two lines cut ; for example, h and h. Alternate angles are any two having their vertices at different points and being on op- posite sides of the secant and on opposite sides of the two lines cut ; for example, f and 7c. Interior angles are those between the two lines, as /, g, A, ^-^^ ^"\^ \ and k. The others are exte- rior. Corollary. — The corresponding and the alternate of any given angle are vertical to each other, and therefore equal. For instance, h is the corresponding of A, and g is the alternate of h. 21. Theorem. — When the arms of an angle are parallel to those of another^ and have re- spectively the same directions from their vertices, the angles are equal. If the directions iTi^ and LH are the same and the directions -ffZ> and LB are the same, the angles FKD and HLB are equal, for each is the difference of the same directions. 30 PLANE GEOMETRY, [Chap. III. 33. Theorem. — Whoi two parallel lines are cut hy a secant^ each of the eight angles is equal to its corresponding angle. If the parallels AB and CD are cut by the secant EF^ the same reasoning as in the last demonstration proves that the angles FKJD and KIB are equal. The same of any two correspond- ing angles. Corollaries. — I. If two angles have their arms respectively par- allel and the directions of one angle opposite to those of the other, the angles are equal. Thus the angles OKI and BLN are equal, for one is vertically oppo- site to an angle that is equal to the other. II. When two parallel lines are cut by a secant, each of the eight angles is equal to its alternate. III. If two angles have their arms respectively parallel, and one pair of parallel arms have opposite directions from their vertices, and the other pair have the same directions from their vertices, the angles are supplementary. Thus the angles CKF and AIj G are supplementary. lY. Two interior angles on the same side of a secant are supplements of each other. V. When a line is perpendicular to one of two parallels, it is perpendicular to the other, and all the angles are right. 23. Theorem.— When two lines in 07ie plane are not parallel, and are cut hy a secant, the cor- responding angles are unequal. If KN is not parallel to AB, and if they are cut by FF at G and 11^ then through G there may be a line CB parallel to AB (II, 8). Since CD is parallel to AB and KJSF is not, the lines CD AST. 23.] STRAIGHT LINES. 31 and jOT make an angle with each other at G. Therefore, the angles which they make with UF are not equal. For instance, EGN is less than EGD ; but EGD is equal to its corresponding angle EHB, Therefore EGN is less than EHB, and so it may be proved that any corresponding angles made by AB and KJ^ with EF are unequal. 24, Theorem. — When two straight lines in the same plane are cut l)y a third, making the corresponding angles equal, the two lines so cut are parallel. For, ii GO and HA were not parallel, the corresponding angles would be unequal (23). Corollaries. — I. If the alter- nate angles are equal, the lines are parallel. II. The lines are parallel when the interior angles on the same side of the secant are supplementary. Angles with Perpendicular Arms. 25. Theorem. — Two angles that have their arms respective- ly perpendicular are equal or supplementary. If AB is perpendicular to I) G, and BC is perpendicular to EF, then it is to be proved that the angle ABC is equal to one and supplementary to the other of the angles formed by BG and EF. Make BI parallel to GB, -^x and BIT parallel to EF. The right angles ABI and CBH are equal (22, v). Sub- tracting the angle HBA from both, the remainders HBI and AB C are equal. But HBI is equal to FGB (21), which is the supplement of EGD, There- 33 PLANE GEOMETRY, [Chap. III. fore, the angle ABC is equal to one and is supplementary to the other of the angles formed by the lines Z> 6^ and ^i^ ^, Corollary. — When the vertex of each angle is between the arms of the other, or between the exten- sions of those arms produced be- yond the vertex, the angles are supplementary. Also, if the an- gles have their vertices at the same point, and both the arms of one are between the arms of the other, the angles are sup- plementary. When otherwise placed, the angles are equal. B M D Distances between Parallels. 26. Theorem.. — Two parallel lines are everywhere equally distant. A line perpendicular to one of the parallels at any point of it is also perpendicular to the other, as L3I. That part of the parallels on — one side of LM may be made to revolve on L3I as an axis. As the angles at L are equal, LA falls upon LB, and, as the an- gles at M are equal, 31 G falls upon 3IB. Since the axis L3f may be placed at any point, it follows that any part of the parallels may be made to coincide with any other part. Therefore, the distance between the lines is every- where the same. CoroUebry. — The parts of parallel lines included between perpendiculars to them are equal. For the perpendiculars are parallel. Secants and Parallels. 27- Theorem. — When several parallel lines are cut hy a secant, if one distance between two parallels is equal to another distance between two parallels, then the segments of the secant are equal. Art. 21.] STRAIGHT LINES. 33 n V G \o K \rs ^ c H If the distance between the parallels B and D is the same as between the parallels D and G, it is to be proved that the seg- ments JEI and 10 are equal. That part of the figure be- ^ tween BC and BF may be placed upon and coincide with the part between BF and GH. Since two parallels are every- where equally distant, they may be so placed that the point F falls upon I. Then, since the \ angles BEI and BIO are equal -^ (22), the line ^Z takes the direc- tion 10, and since BF falls on GB^, the point I falls on 0. Therefore, EI and 10 coincide and are equal. In like manner it is proved that any two of the segments of the line A Y are equal, if the distances between the parallels are equal. Corollaries. — I. Conversely, when several parallels are cut by a secant, if the segments of the secant are equal, the distances between consecutive parallels are equal. II. When several parallels intercept equal parts of a secant, then any other secant is divided into equal segments by the same parallels. 38, Theorem. — When three parallel lines are cut by two secants, the secants are divided proportionally. If AB, BE, and CF are parallel, it is to be proved that AB:BC=zBE'.EF. When AB and B C have a ratio that can be expressed by two whole numbers, they may be divided into equal parts corresponding in number to the terms of the ratio. Thus, m and n being integers, if AB :BC = m'.n, AB may be divided into m equal parts, and BG into n, and all the m -{■ n parts are equal. Then lines 34 PLANE GEOMETRY. [Chap. III. througli the points of division parallel to AD divide DF into equal parts (27, ii). The number of parts in DJE is the same as in AB^ and the number in EF is the same as in B G. Hence, AB:BC = DF:FF. When the ratio AB : B C can not be expressed by whole numbers, these segments have no common measure. Any meas- ure of AB, applied to B (7, leaves a part unmeasured, which is less than the measure. Say the part measured is BG. A second measure of AB may be taken less than the remainder G C. Applying this to B G, the remainder is less than the measure, and therefore less than G G. Using a measure regularly smaller than the re- mainder, the part oi BG measured by a measure of AB is a constantly in- creasing line and may become as nearly equal to BG as we choose. For a measure of AB may be taken smaller than any line that can be assigned (Postulate of Extent). Let BJT represent this increasing line, let a line parallel to AB pass through JT, and let Y' be the point where this par- allel is cut by BF. As shown above, since AB and B^ have a common measure, AB : BJl = BE\ FY. These ratios are variable, but always equal. As the lines BX and FY can approximate the limits BG and FF, the ratio AB : BJl can approximate AB : B G, and the ratio BF : FY can ap- proximate BF : FF, within less than any assignable differ- ence. Now the constant ratios AB : B G and BF : FF, which limit the variable ratios, must be equal. For if there were any difference, then one of these two variables could exceed the lesser constant ratio, while the other variable could not. That is, equals could be unequal, which is absurd. Therefore AB:BG = BF',FF. Corollary.— By alternation, AB : BF =BG: FF. Hence, when any number of parallels are cut by two secants, every Art. 28.] STRAIGHT LINES. 35 segment of one secant has tlie same ratio to the corresponding segment of the other. A Variable is a magnitude, or other quantity, that varies according to any rule made for the purpose of the investigation. A Constant is a magnitude, or other quantity, that remains un- changed. The Ijimit of a variable is a con- stant that the variable may approxi- mate indefinitely, but can never quite reach. Thus the line ^ C is the limit of BX, The law govern- ing the variation of BX is that it is a part of J3 G, commensur- able with AB^ and that the common measure used at every step is less than the last remainder, BC — BJC. A variable may have zero for its limit. For example, the remainder XC may become less than any assignable distance. Lines not Parallel meet. 39. Theorem.— If two straight lines that lie in one plane are not parallel, they meet when produced. If AB and CD lying in the same plane are not par- allel, it is to be proved that "^^^-^ when produced they must ^ — -^^TTr::;^^. _^ ™eet. ^ Vi^^L^. „...^ Through H and Z, two K \ l"-^^^^^^ ^ points of AB, make UF \ ^ and JSTL parallel to CD. — ^ Make IIG, joining the two outer parallels, and there- fore cutting the inner parallel. Then, as just demonstrated, UK is to KG as HI is to the segment of AB that lies between the parallels NL and CD. Therefore, at a definite distance from ij the lines AB and CD must meet and cut each other. 36 PLANE GEOMETRY. [Chap. III. Corollary. — ^If two straight lines are in the same plane and can never meet, they are parallel. Scholium. — The following four propositions about two straight lines that lie in one plane have a relation to each other that deserves attention : 1. If the two lines are parallel, they never meet ; 2. If the lines can not meet, they are parallel ; 3. If the lines can meet, they are not parallel ; and 4. If the lines are not parallel, they can meet. The first is a corollary from the definition of parallels and the Axiom of Direction. The second is the converse of the first, but is not a consequence of it and requires a separate demonstration. The third is a consequence of the first, and these two are called contrapositives. Each is the contrapositive of the other. The second and fourth are con- trapositives. Contrapositive propositions are such that the hy- pothesis of each consists in the denial of the conclusion of the other. Either of two contrapositives is a corollary of the other. Articles 23 and 24 ,of this chapter are con- trapositives. 30. Applications. — The instrument called the T-square consists of two straight rulers at right angles to each other, as in the figure. It is used to draw ^ , parallel lines. Lay the cross-piece of the in- strument along a straight line perpendicular to the direction of the intended parallels. The other piece, called the blade, may be moved, keeping the cross-piece coincident with the perpendicular, and lines parallel to each other may be drawn along the blade. The draughting instrument called a trian- gle is a flat, triangular piece of wood or other material, right angled at one corner. It is used for drawing parallel lines in the same way as the T-square ; but, instead of a straight line for direction, it is better to use a ruler held firmly to the paper. The uniform distance of two parallels is a common principle in manu- factures and the mechanic arts. It is constantly employed in the con- Art. 30.] STRAIGHT LINES. 37 struction of houses, furniture, and macbinerj. One of the simplest tools made on this principle is the joiner's gauge, used to draw a line on a board parallel to its edge. 31. Scholium. — Demonstration by the method of superposi- tion has been used several times in this chapter. There is a dif- ferent motion required in different cases, to place one figure on another. Every plane figure may be regarded from either side. Thus it has two faces. The upward face or obverse, and the down- ward face or reverse. There are two methods of superposition of plane figures : the first, called direct, when the reverse of one figure is applied to the obverse of the other, and the second, called inverse, when the obverse faces are applied to each other. In the former case, of which there is an example in Article 27, one figure may be supposed to slide along the plane till the coincidence takes place. In the latter case, of which there are examples in Article 10, one figure or part of a figure must be turned upon an axis in order that its obverse may be applied to the obverse of the other figure or part of the figure. 32. Miscellaneous Exercises. — 1. What is the principle in- volved in the use of the T-square ? 2. Find the locus of those points that are at a given distance from a straight line given in position. [Represent the given line by a drawing ; mark many points, all at the same given distance from the line ; then de- scribe the location of all such points.] 3. If a straight line joining two parallel lines is bisected, another line through the point of bisection, and joining the two parallels, is also bi- sected at that point. [Make a third parallel through the point of bisec- tion ; apply Corollary II of Article 27.] 4. What is the greatest number of points in which seven straight lines can cut each other, three of them being parallel ? 5. If two lines have a common measure, they have a common multi- ple ; and conversely. CHAPTER ly. CIR C UMFER ENCES, Article 1. — It follows, from the definition (II, 15), that a circumference of a circle is the locus of all the points of a plane that are at the same distance from one point, A Radius is a straight line from the center to the circum- ference. A Diameter is a straight line through the center with both ends in the circumference. Corollaries. — I. Radii of the same circle are equal. II. A diameter is equal to two radii. III. Every point of the plane is outside of, on, or inside of the circumference, according as its distance from the center is greater than, equal to, or less than the length of the radius. IV. Circles having equal radii are equal. Concentric circles are those having the same center. 2. Theorem. — Three points not in the same straight line de- termine the position and extent of a circumference. To demonstrate this, it must be shown that one circumfer- ence, and only one, may be made through three such points, jg A, B, and G. Join AB and jrj ,.-""" \ BC. At D and E, the mid- A„^-"""\ \,f points of those lines, let per- A y' \ ^ pendiculars be erected m the \ / -^ plane passing through the three /'\p points. Since AB and B have dif- ferent directions, BG is not perpendicular to BC. Hence Art. 2.] CIRCUMFERENCES. 39 D G and JEH are not parallel, and they must meet if produced (III, 29) Since every point oi DG is equidistant from A and S (III, 10, Yii), and every point of JS H is equidistant from £ and (7, their common point L is equidistant from A, B, and G. There- fore, with this point as a center, a circumference may be de- scribed through A^ B^ and G, There can be no other circum- ference through these three points, for there is no other point besides L equally distant from all three (III, 11). Corollary, — Two circumferences can cut each other in two points only. 3. Theorem. — A circumference is curved throughout. For a straight line can not have more than two points equally distant from a given point (III, 10, vi). Corollary. — A straight line may cut a circumference in two points only. An Arc is a portion of a curve. A Chord is a straight line joining the ends of an arc. 4. Theorem. — A diameter is lon- ger than any other chord of the same circumference. For any other chord is shorter than the sum of two radii (Axiom of Dis- tance). Symmetry of the Circumference. 5. A Center of Symmetry is a point with reference to which other points are symmetrical. Two points are symmet- rical with reference to a center of symmetry when it is the mid- point of the straight line that joins them. For example, the ends of a diameter are symmetrical with reference to the center of the circle. An Axis of Symmetry is a straight line with reference to which points are symmetrical. Two points are symmetrical with 40 PLANE GEOMETRY. [Chap. IV. reference to an axis of symmetry, when the line that joins them is bisected perpendicularly by the axis. There is also a plane of symmetry, but its consideration is deferred. Two figures are symmetrical with reference to a center or to an axis of symmetry, when every point in each has its symmet- rical point in the other. A symmetrical figure is one that can be divided into two parts that are symmetrical with reference to an axis of symmetry. Any straight line is a symmetrical figure, the axis being the perpendicular that bisects it. The halves of the line are also symmetrical with reference to the midpoint of the line as a center of symmetry. Corollary, — Two figures, or parts of a figure, symmetrical with reference to an axis, are equal. Every case of demonstra- tion of equality by rotation on an axis is an example of this (III, 10, 13, and 26). 6. Theorem. — Every diameter bisects the circumference and the circle. If the part on one side of the diameter is turned upon that line as an axis, the two parts coincide, for otherwise some points of the circumference would be unequally distant from the center. Corollaries. — I. Every circle is a symmetrical figure. II. Any diameter of a circle is an axis of symmetry. III. Every point of a circumference has its symmetrical point with reference to the center, as a center of symmetry. 7. Problem. — Arcs of equal radii may he added, subtracted, multiplied, or divided. For an arc, having a given radius, may be produced to any extent, or it may be diminished at will. The sum of several arcs may be greater than a circumfer- ence. Scholium. — 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. Art. 1.] CIRCUMFERENCES. 41 The circumference is the only line that can move along itself, around a center. For any line that can do this must have all its points equally distant from the center ; that is, it must be a cir- cumference. 8. Applications. — The axles of wheels, shafts, and other solid bodies that are required to rotate within a hollow mold or casing of their own form, must be circular. If they were of any other form, they could not turn without carrying the mold or casing with them. Wheels, intended to maintain a carriage at the same height above the road on which they roll, must be circular, with the axle at the center. Arcs and Chords. 9. Every chord divides the circumference into two parts. Each of such arcs is said to be conjugate to the other. A Major Arc is one greater than a semicircumference. A Minor Arc is one less than a semicircumference. In speaking of the arc of a chord, when the two are unequal, the minor arc is intended unless otherwise expressed. A chord is said to subtend its arc. Any arc or other line extending from a point in one arm of an angle to a point in the other arm is said to subtend the angle. Thus, any side of a tri- angle subtends the opposite angle, and the angle made by two radii is subtended by an arc. 10. Theorem. — If two arcs are equal, their chords are equal. For equal arcs may coincide ; then the straight lines joining their ends must coincide. 11. Theorem. — When a radius n bisects an arc, it is per^yendicular to y""^ the chord of the arc and bisects it. / Let CD bisect the arc AB. The / points D and C are each equally dis- tant from A and JB (10). There- \ fore, CD is perpendicular to AB at \ its midpoint, E (III, 11, c). \^^ 3 42 PLANE GEOMETRY. [Chap. IV. Corollaries. — I. Since two conditions determine the posi- tion of a straight line, if it has any two of the four con- ditions mentioned in the theorem, it has the other two. These conditions are : 1. The line passes through the center of the circle, that is, it is a radius. 2. It passes through the midpoint of the chord. 3. It passes through the midpoint of the arc. 4. It is perpendicular to the chord. II. The angles made by a chord with the radii at its ends are equal (III, 10, v). 12. Theorem. — Iii the same circle, or in equal circles^ if two minor arcs are unequal, the greater arc has the greater chord. The arc AMB being greater than CND, both being minor arcs, it is to be proved that AB is lon- ger than CD. Take AME, equal to CND. Make the chords AE and BE, and the radius 01, perpendicular toJ5^. Since the arc AMI is less than a semicircumference, the points A and E are on the same side of the radius that is perpendicular to BE. Therefore, AB is longer than AE, or its equal CD (HI, 11). Corollaries. — I. When both arcs are major arcs, the greater arc has the shorter chord. II. Conversely, if two chords are unequal, the greater chord has a greater minor arc and a less major arc. IIL When the chords are equal, the arcs are equal. Abt. 13.] CIRCUMFERENCES. 43 Distance from the Center. 13. Theorem. — The chords of equal arcs are equally dis- tant from the center. For the arcs, being equal, may coincide, also their chords (10). Then the perpendiculars, which measure the distance from the center, must coincide (2). 14. Theorem, — Of two unequal chords, in the same circle or in equal circles^ the less is the farther from the center. Since the greater chord, AB, has a greater arc than the other, GJDy take an arc, ^JF^ equal to CD, so that the arcs AB and EF have the same midpoint G. Then the radius HG is perpen- dicular to both the chords AB and EF (11), and ^i^is the farther from H, because its distance is measured by the whole of SK, and the dis- tance oi AB is measured by a part of HK. But FF and CD are equally distant from the center (13). Corollary. — Conversely, chords at equal distances from the center are equal ; and of two chords at different distances from the center, the farther is the less. 15. Exercises, — 1. What is the locus of the midpoints of all the equal chords in a given circle? [Make a diagram of the given figure, designate many points which satisfy the condition, describe their location, and prove that the points in this locus satisfy the condition and that no other points do so (13 and 1, iii).] 2. Find the locus of the centers of the circles having the same length of radius, whose circumferences pass through a given point. [1, iii.] 3. Find the locus of the centers of the circles whose circumferences pass through two given points. [Ill, 11, o.] 4. If any point, not the center, is taken in a diameter of a circle, of all the chords that pass through that point, that is the least which is at right angles to the diameter. [Make two chords through the point, one perpendicular and the other oblique to the diameter ; and consider their respective distances from the center.] 44 PLANE GEOMETRY. [Chap. IV. Curvature. 16. Theorem. — A straight line that is perpendicular to a radius at its extremity touches the circumference in only one point. Let AD be perpendicular to the radius J30 at its extremity J3. Then it is to be proved that A J) touches the circumference at no other point than B. If the center C is joined by a straight line with any point of AD, the perpendicular J3C is shorter than any such oblique line (III, 10). There- fore (1, III), every point of the line AD, except -S, is outside of the cir- cumference. 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. 17. Theorem. — A straight line that is oblique to a radius at its extremity cuts the circumference in two points. Let AD be oblique to the radius CB at its extremity B, From the center (7, let CJE fall perpendicularly on AD. On ED, take UF equal to FB, The distance from G to any point of the line AD between B and F is less than the length of the radius GB, and, to any point of the line beyond B and F, it is greater than the length of GB (III, 10). There- fore, that part of the line between B and F is within, and the parts be- yond B and F are without the cir- cumference. That is, the line cuts the circumference in two points. Art. 17.] CIRCU3IFERENCES. \ 47 Corollaries. — I. A tangent to a circumferenc\ % dius that extends to the point of contact are perj each other. II. At one point of a circumference, there can tangent (III, 9). III. The two directions that the curve has at the point of contact are the same as those of the tangent, for any other straight line through the point of contact cuts the curve. Whenever a curve is an arm of an angle, the direction of that arm is that of the curve at the vertex. A Normal is a line perpendicular to a curve. In the circle every radius is a normal. Sometimes a tangent to a curve is defined as a straight line having the same direction as the curve at anoint that is common to the two. 18, Theorem. — The curvature of an arc of a circle is eqital to that angle at the center which the arc subtends. The curvature of an arc is the amount of change in direc- tion (II, 6). Since the direction of the arc AJ^ at the extremity A is the same ^ CD as that of the tangent AD, and the direc- tion at the other extremity is the same as that of the tangent (7-S, the curvature of the arc is the angle D CJB ; but this an- gle is equal to AEB formed by the radii (III, 25). Scholium. — The amount of curvature being intended, an arc is an angular quantity. Thus an arc is said to be the complement of an angle, when the sum of the angle and the curvature of the arc is one right angle, or ninety degrees. As the subtending arc has the same angular quantity as the angle at the center, it is sometimes said that "the arc meas- ures the angle," but this is not a proper use of the word 19. Application, — Tangent lines are frequently used in the arts. A common example is when a strap is carried round a part of the cir- cumference of a wheel, and extends to a distance. 44 PLANE GEOMETRY. Angrles at the Center. [Chap, IV. 30. Theorem. — 171 the same circle^ or in equal circles^ tico angles at the center have the same ratio as the arcs that subtend them. This theorem presents three cases : 1st. If the arcs are equal. The equal arcs, being placed one upon the other, coincide. Then BG coincides with AO, and DC with UO. Thus the angles coincide and are equal, that is, they have the same ratio as the arcs. 2d. If the arcs have the ratio of two whole numbers. Let the arcs be divided into equal parts corresponding in number to the terms of the ratio ; these small arcs are all equal. If, for example BB : AE = 13 : 5, the thirteenth part of BB is equal to the fifth part of AE, Art. 20.] CIRCUMFERENCES. 47 Let radii be made to all the points of division. The small angles at the center thus formed are all equal, because their in- tercepted arcs are equal. But B CD is the sum of thirteen, and A OE of five of these equal angles. Therefore, angle BCD : angle J. 0^= 13 : 5 ; that is, the angles have the same ratio as the arcs. 3d. The arcs may have a ratio that can not be expressed by- two whole numbers, that is, the arcs may not have any common measure. It is to be proved that the angles have still the same ratio as the arcs, that arc BD : arc AE= angle B CD : angle A OE. If this proportion is not true, then, the first, second, and third terms being unchanged, the fourth term is either too large or too small. If it were too large, then some smaller angle, as A 07" would verify the proportion, and arc BD : arc AE= angle BCD : angle A 01. Let the arc BD be divided into equal parts, each of them 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 fall between I and E^ say at the point XT. Join OTI. Now, by construction, the arcs BD and A U have the ratio of two whole numbers. Therefore, arc BD : arc AU= angle B CD : angle A Oil. 48 PLANE OEOMETRY, [Chap. IY. These two proportions have the same antecedents respective- ly. It follows that their consequents are in proportion ; that IS, arc AE: arc AU= angle A 01 \ angle AOIZ But this is impossible, for the first antecedent is greater than its consequent, while the second antecedent is less than its con- sequent. Therefore, the supposition that 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 A OE is the true fourth term of the proportion, that is, the arc BD is to the arc AE as the angle BCD is to the angle A OE, Corollaries. — I. Any angle at the center has the same ratio to the sum of four right angles that the intercepted arc has to the whole circumference. II. If two diameters are perpendicular to each other, they divide the circumference into four equal parts ; and conversely, III. The curvature of an arc of a circle increases in the same ratio as its linear extent. That is, the curvature of a circumference of a circle is uniform throughout. A Quadrant is the fourth part of a circumference. It is divided into de- grees, minutes, and seconds, the same as a right angle. Art. 21.] CIRCUMFERENCES. 49 31. Exercises. — 1. "What is the locus of the midpoints of all the chords that are parallel to a given tangent ? 2. If two circles are concentric, a line that is a chord of the outer and a tangent of the inner one is bisected at the point of contact. [Make a radius through the point of contact.] 3. If two concentric circles are cut bj the same secant, the segments of the secant between the circumferences are equal. [Make a radius per- pendicular to the secant.] 4. If the length of a circumference is one meter, what is the length of the arc that subtends an angle at the center, of 7° 12'? 5. If the arc subtending a central angle of 25" is one inch in length, how long is the whole circumference ? 6. If a circumference is 3| times as long as its diameter, is an arc of 58° 48' longer or shorter than the radius ? Angles and Parallels in a Circumference. 22. An Inscribed Angle is one formed by chords, with its vertex on the circumference. An inscribed angle is said to stand upon the arc that subtends it ; the re- mainder of the circumference is said to contain the angle ; and the an- gle is in the arc that contains it. Thus the angle AEI stands upon the arc A 01, and is in the arc AEI, ■which contains it. Theorem. — An inscribed angle is equal to the curvature of half the arc it stands upon. The demonstration presents two cases. 1st. When one arm of the angle is a diameter, as JBA. It is to be proved that the angle JB is equal to the curvature of half the arc AEG. Make the radius DE parallel to BC, and join CD. The angles B and C are equal (11, ii) ; B is equal to its corresponding angle ABE (III, 22) ; and G is equal to its alternate angle GBE (III, 22, 50 PLANE GEOMETRY. [Chap. IV. ii). Therefore, the arcs AE and JEG are equal (20), and the angle B is equal to the curvature of the arc AE, which is half of the subtending arc AEC. 2d. When neither arm of the an- gle is a diameter. Make the diam- eter BD. The angle ABB is equal to the curvature of half the arc AB, as just proved, and the angle BBC is equal to the curvature of half the arc BG. There- fore, taking the sum or difference according as the center of the circle is between the arms of the angle or not, the angle AB G is equal to the curvature of half the arc A G. Corollaries, — I. When an inscribed angle and an angle at the center have the same subtending arc, the inscribed angle is half of the angle at the center. II. Several inscribed angles, in the same arc or equal arcs, are equal ; and conversely, if inscribed angles are equal and in the same circle or in equal circles, then their subtending arcs are equal. III. An inscribed angle is obtuse, right, or acute, according as the arc that subtends it is greater than, equal to, or less than a semicircumference. Art. 22.] CIRCUMFERENCES. 51 lY. The angles inscribed in conjugate arcs are supplemen- tary, for their sum is equal to the curvature of half the circum- ference. 33. Theorem, — The angle formed by a tangent and a chord at the point of contact is equal to the curvature of half of the subtending arc. Making the diameter J^D from the point of contact, DEC is a right angle, and therefore equal to the curvature of half the semi- circumference DIOEy but the in- scribed angle DEI is equal to the curvature of half the arc DL Sub- tracting this angle and arc from the right angle and semicircum- f erence, the remaining angle lEG is equal to the curvature of half the arc lOE. By addition, the angle lEA is equal to the curvatur#of half the subtending arc IDE. 24. Theorem. — Two parallel lines intercept equal arcs of a circle. 4 n For, joining B Cy the alternate angles AB G and B CD are equal. Therefore, the subtending arcs are equal. 52 PLANE GEOMETRY. [Chap. IY. 25. Theorem. — Every angle lohose vertex is within the circumference is equal to the curvature of half the sum of the arcs intercepted between its arms and between its arms produced. Thus, the angle Z/AE is equal to the curvature of half the sum of the arcs UJE and 10. To be demonstrated by the student, using the previous theorems (22 and 24). 26. Theorem. — Every angle whose vertex is outside of a cir- cumference, and whose arms are either tangent or secant, is equal to the curvature of half the difference of the subtending arcs. Thus, the angle A CF is equal to the curvature of half the difference of the arcs AF and AB ; the angle FOG, to that of half the difference of the arcs FGr and BI', and the angle ACE, to that of half the difference of the arcs AFGE and ABIE. This, also, may be demon- strated by the student. Corollaries. — I. Accord- ing as an angle subtended by a chord has its vertex outside, on, or inside the arc, it is less than, equal to, or greater than the angle contained by the arc. II. Conversely, according as an angle that is subtended by a chord is less than, equal to, or greater than the angle contained by the arc of that chord, its vertex is outside, on, or within the arc. III. The locus of the vertices of all angles that are equal to a given angle and are subtended by a given line is an arc of a circle. Art. 26.] CIRCUMFERENCES. 53 IV. If two angles are supplementary and have their vertices on opposite sides of a common subtending line, a circumfer- ence may pass through the vertices and the ends of the line (22, IV). Scholium. — All the cases of arcs between two lines that cut or touch a circumference are included in this general rule : The angle formed by the lines is equal to the curvature of half of the arc or sum of the arcs concave to the vertex, less half that arc, if any, which is convex to the vertex. The case of arcs between parallels is included in this rule, for there is no angle, no difference of direction of the two lines, and there is no difference between the intercepted arcs. 2i*7 • Applications. — Instruments for measuring angles, founded upon the principle that arcs are proportional to angles, consist of a part or an entire circle of metal, on which are engraved its divisions into degrees, etc. Many instruments used by surveyors, navigators, and astronomers, are constructed upon this principle. An instrument called a protractor is used in drawing angles, or meas- uring angles in a drawing. It consists of a semicircle, the arc of which is divided into degrees and parts of a degree. A method of surveying a railway ^ \ curve depends upon the principles just \ established. ^r'''-^-"*"""^H/\ Suppose that AB and CD are straight /y'^""'""^^^^-^<\ parts of the track arid are to be connected '^/" "V' by a curve, the points B and B being so / \ selected that the angles ABB and BBC /A (A are equal. Two transits (instruments for measuring angles) are used, one at B pointed toward B, the other at B pointed toward E in the con- tinuation of CB. If the first transit is turned toward E and the second toward B, both the same number of degrees, the point of in- tersection of their lines of sight is on the curve. Any number of such points may be ascertained without moving the instruments from B and B. 54 PLANE GEOMETRY. [Chap. IV. 28. Exercises. — 1. The opposite lines joining the ends of two diameters in a circle are parallel. [Ill, 7, ix ; IV, 22, i ; III, 24, i.] 2. If two equal circumferences cut each other at the points A and J5, and any line is made through B to cut the curves at G and at 2), then the chord AG \^ equal to the chord AD. [Join AB.] 3. If any two circumferences cut each other at the points A and J?, and a tangent is made to each curve at A and extended to the other curve at G and at i>, then the angles GBA and DBA are equal. [Produce DA and GA beyond Ai] 4. If from one point there ex- tend two lines tangent to a circum- ference, the angle contained by the tangents is double the angle contained by the line joining the points of con- tact and the radius extending to one of them. [There are several ways to demonstrate this. The construction for one way is : Make two diameters from the points of contact. For another way : From one of the points of contact make a chord parallel to the other tangent, and from the other point of contact make a diameter.] 5. Find the locus of the midpoints of all the chords in a circle which extend from the same point on the circumference. [11, i, and 26, III.] Positions of two Circumferences. 29. The various relative positions of two circles depend upon the distance between their centers compared with the siim or the difference of their radii. This is shown as follows : Since every diameter is an axis of symmetry of a circle, the line joining the centers of two circles is an axis of symmetry of the figure. This line is called the central line. A perpendicular to the central line, at a point where one of the curves cuts it, must be tangent to that curve (16). Art. 29.] CIRCUMFERENCES. 55 If both curves cut the central line at the same point, the per- pendicular at that point is a common tangent, and the circum- ferences are tangent to each other. Corollaries. — I. When two circumferences are exterior to each other, the distance between their centers is greater than the sum of their radii. \ h II. "When one of two circumferences is within the other, the distance between the centers is less than the difference of the radii. III. When they touch each other exteriorly the distance be- tween the centers is equal to the sum of the radii. lY. When they touch each other interiorly, the distance be tween the centers is equal to the difference of the radii. Y. When they cut each other, the distance between the centers is less than the sum, but greater than the difference of the radii. YI. The above five cases are all that are possible. Therefore the converse of each proposition is true. 56 PLANE GEOMETRY. [Chap, IV. VII. The common chord of two intersecting circumferences is perpendicular to the central line and is bisected by it (III, 11, c). VIII. Two concentric circumferences are equidistant at all points. 30. Exercises. — 1. If the radii of two circles are 67 and 78 milli- meters, and they are placed with their centers 140 millimeters apart, do the curves cut or touch each other ? If the centers are 145 millimeters apart? If 150? 2. How is it when the centers are 10 millimeters apart? "When 11? When 12? 3. What is the locus of the centers of those circles whose circumfer- ences touch a given line at a given point? 4. Of the centers of those which touch a given arc at a given point ? 5. When two circumferences have no common point, the least dis- tance between the curves is measured along the central line. 6. On any two circumferences, the two points which are at the great- est distance apart are in the central line. 7. In each of the five cases of Article 29, how many straight lines can be tangent to both circumferences? [The number is different for each case.] 8. If two circumferences are tangent to each other, and two secants are made through the point of contact, the chords which Join the ends of these secants are parallel. [At the point of contact, make a straight line tangent to the curves, and compare angles.] 9. Can two chords of the same circle, that are not both diameters, bisect each other? [Ill, 9.] 10. A circle of one inch radius being given, find the locus of the centers of the circles of three inches radius, all of which are tangent to the given circle. 31. Scholia, — The study of the circumference has served to develop the notion of symmetry ; also to show the close con- nection between angles and arcs of circles. This chapter completes the study of plane figures that consist in lines without reference to inclosed surface. Elementary Geometry treats only of figures every line of which is either straight or of uniform curvature. Higher Geom- etry treats of curves of varying curvature. Art. 31.] CIRCUMFERENCES. 57 Modes of Geometrical Reasoning, I. A Direct Demonstration proceeds from established premises by a regular deduction. An Indirect Demonstration begins with the conclusion. It proceeds by these steps : 1st. Suppose that the conclusion to be demonstrated is not true. This supposition is called the false hypothesis. 2d. Show, by reasoning upon the false hypothesis, that it involves a contradiction, or leads to an impossible conclusion. This contradiction or impossibility is called the absurd conclu- sion, and, hence, this method is called reductio ad absnr- dum. 3d. Since the supposition that the conclusion is false leads to an absurdity, the conclusion must be true. For example, take Article 11 of Chapter II. The false hypo- thesis is, that two planes that coincide to some extent may diverge. It would follow from this that some of the directions of at least one of them may vary. As this contradicts the defi- nition of a plane it is absurd. This method of argument is as common in other matters as in mathematics. Some modes of reasoning, such as the method of super- position (II, 4, and III, 31), are peculiar to the science of Magnitude. II. The Method of Exhaustions was used by Euclid in the demonstration of a theorem that involved a ratio between magnitudes that have no common measure. For an example of this method, see the third case of Article 20. It involves a double application of the reductio ad ahsurdum. In modern times, various attempts have been made to substi- tute some demonstration equally logical and less tedious. This has not been done without a substantial adherence to the mode of thought pursued by Euclid. III. The IVCethod of Limits is one of the substitutes for the Method of Exhaustions. It is used in the demonstration of the theorem in Article 28, Chapter III. The use of this method has generally been attended with fallacies. The most common 58 PLANE GEOMETRY. [Chap. IV. error is the neglect to prove that the constant quantity is the limit of the variable. It has also been assumed, without proof, that any rule of measurement which applies to a variable magni- tude holds true of its limit. Dr. Whewell, of Trinity College, Cambridge, claimed this as axiomatic, but whatever is true in the statement can be demonstrated with as much rigor as is found in the ancient method of exhaustions. Theorem. — "If two variables are equal at every step of their variation and if each has a limit, the limits are equals The following demonstration is substantially the same as that of Duhamel, who first stated this theorem. If the limits were unequal, that is, if they differed by any quantity, then one of the variables might pass over the limit of the other variable, in order to reach within less than this quan- tity of difference from its own limit. Then one of the two varia- bles would be greater and the other less than the limit passed ; which is absurd, as they remain equal by hypothesis. Therefore the limits can not be unequal. By the Euclidean method of exhaustions, the magnitude ex- hausted was a limit. An essential part of the demonstration consisted in showing that this magnitude could be exhausted within less than any assignable quantity. lY. The Method of Indivisibles consists in regarding magnitudes as composed of infinitely small elements, called Infinitesimals or Indivisibles. " A line is said to consist of points, a surface of parallel lines, and a solid of parallel sur- faces." Two equal lines consist of an equal number of points, for the two lines may coincide so that every point in each has its corre- sponding point in the other. It follows that if one line is twice as long as another, there are twice as many points in the first as in the second ; and universally, the numbers of the points in two lines have the same ratio as the lengths of the lines. The num- bers are infinite, but their ratio is finite. Apply this to the discussion of angles at the center and their intercepted arcs. Radii are supposed to extend from every point of the two arcs. These radii make equal infinitesimal an- gles at the center, one such angle for every point of the arc. It Art. 81.] CIRCUMFERENCES. 59 follows that the whole angles have the same ratio as the whole arcs, as in the second case in the demonstration in Article 20. These methods are an economy, not merely of words. They have saved much intellectual labor, and contributed largely to the progress of mathematical science. 32. Miscellaneous Exercises, — 1. 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 witiiin the circumfer- ence are equal. [Ill, 13.] 2. What is the locus of the centers of those circles that have a radius of the same given length, and which are tangent to the same given circle? [Generalize Exercise 10, Article 30.] 3. If two circumferences are such that the radius of one is the diam- eter of the other, any straight line extending from their point of contact to the outer circumference is bisected by the inner one. [Join the center of the larger circle and the point where the straight line cuts the smaller circumference.] 4. If two circumferences cut each other, and from either point of in- tersection a diameter is made in each, the extremities of these diameters and the other point of intersection are in the same straight line. [Make the common chord.] 6. In a given circle find the locus of the midpoints of the chords that pass through a given point within the curve. 6. If, from the ends of a diameter, perpendiculars fall on any straight line that cuts the circumference, the segments of the lino intercepted be- tween those perpendiculars and the curve are equal. Problems in Drawing. 33. The solution of Problems in Drawing is a test of a student's knowledge of geometrical principles and an exercise of his skill in their application. Except the pencil or crayon, the only instruments used are the ruler and compasses. The ruler has a straight edge. 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. The ruler is used to 60 PLANE GEOMETRY. [Chap. IV. draw straight lines. The compasses are used to draw circum- ferences, or arcs of circles. That this much should be taken for granted was expressed by Euclid in these three postulates : Postulates of Euclid. — 1. A straight line can he drawn from any point to any point. 2. A given straight line can he produced any length, in the same direction. 3. A circumference can he descrihed with any center and with any radius. Since the straight line and the circumference are the only- lines treated of in elementary geometry, these Euclidian Postu- lates contain all that need be granted for the solution of elemen- tary problems in drawing. The rule forbidding the use of any instruments except the ruler and compasses is, in effect, a restric- tion to the use of these elementary lines. By means of the compasses a part of a line may be taken equal to a given line. Problems in geometry are distinct from problems in drawing. The former are pure mathematical principles ; the latter are ap- plications of principles to handiwork. The former are state- ments that certain magnitudes can exist. Their truth is made manifest by an explanation which shows that the definition or description of the figure is not incompatible with itself. Prob- lems in drawing are tasks that are to be done under certain con- ditions. In the Elements of Euclid, which, for many ages, was the only text-book on Elementary Geometry, the problems in draw- ing occupy the place of problems in geometry. At present nearly all mathematicians put them aside as not forming a neces- sary part of the theory of the science. The Postulates of Euclid express no complete scientific truth. They are only a partial statement of elementary possibilities. Therefore it is an erroneous theory that makes them the basis of any demonstration of principle. The student is advised to make a drawing of every problem. First draw the parts given, then the construction requisite for solu- tion. Endeavor to make the drawing as exact as possible. Let the Art. 33.] CIRCUMFERENCES. 61 lines be fine and even, as they better represent the abstract lines of geometry. Accuracy in drawing, like precision in language, is an aid to correct thought. Problems. — I. To find, the least common multiple of two straigJit lines. Take for example the lines ^ g ^ and c. From a point, A^ draw an in- definite straight line, AE. Apply each of the given lines to it a number A ' r-^ ' ^ ' i-i ' ^ of times in succession. If the ends coincide for the first time at E^ then AE \^ the least common multiple of the two lines. If two incommensurable lines are given, as those in Article 2, Chap- ter III, theoreticallj the coincidence is never reached, but apparently a coincidence can be obtained, and such a result gives apparently the least common multiple. This exercise and the following may be made tests of accuracy in drawing. II. — T'o find the greatest common measure of two straight lines. Subtract the smaller line from the greater as many times as it can be taken. Subtract the remainder, if any, from the smaller line, in the same way. Then use the remainder as a subtrahend, and so on, until after some subtraction there is no remainder. The last subtrahend is the greatest common measure. Compare this with the rule for numbers, or algebraic quantities. Also compare the demonstration in Article 2, Chapter III. III. — To find the ratio of two straight lines. Count how many times each is contained in the least common multi- ple. These numbers express the ratio inversely. Or, having found the greatest common measure of the lines, reverse the steps and count how many times the common measure is contained in each. Both these methods are 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 num- bers thus found express the ratio nearly. Whenever two lines have any geometrical dependence upon each other, the ratio is found by calculation with an accuracy no measurement by the hand can reach. 62 PLANE GEOMETRY. [Chap. IV. lY. — To lisect a given straight line. With A and JB as centers, and V2) with the same radius, which must be • greater than the half of AB, describe arcs which intersect at D. In the same way describe two arcs intersect- ing at some other point, as E. Join j DE. I The line DE has two points each j/ equally distant from the ends of AB, ^^^ and therefore bisects it. Y. — To erect a perpendicular on a given line. To erect at C a perpendicular to AB^ take CB and GA equal. De- scribe, as in the last problem, two arcs intersecting at i), and draw DC. It is perpendicular. The demonstration of this and of some that follow is left to the student. YI. — To let fall a perpendicular from a given point on a given straight line. With the given point as a center, and a radius long enough, describe j i an arc cutting the given line BG in j the points D and E. With D and E i as centers, and with a radius greater j Ey_c than the half of DE^ describe arcs jj n^j9 cutting each other in F. The straight ""~-~ line joining A and F is perpendicular to DE. ^ YIL— To describe a circumference through three given points. The solution is evident from Article 2. YIII. — To find the center of a circle when all or a part of the circum- ference is given. Take any three points of the arc and proceed as in the last problem. IX. — To draw 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. X. — To Used a given arc. Draw the chord of the arc and erect a perpendicular at its mid- point. Corollary.— To bisect an angle, first draw an arc with the vertex as center. Art. 33.] CIRCUMFERENCES, " The most simple case of the division of an arc, after its bisectiol, .o its trisection, or its division into three equal parts. This problem accord- ingly exercised, at an early epoch in the progress of geometrical science, the ingenuity of mathematicians, and has become memorable in the his- tory of geometrical discovery, for having baffled the skill of the most illustrious geometers. "Its object was to determine meaos of dividing any given arc into three equal parts, without any other instruments than the rule and com- passes 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." — Lardner'% Treatise. It must not be inferred that there is any impossibility in the trisection of an angle. Greek geometers solved the problem with other instru- ments, by which they drew curves that have not a uniform curvature. XI. — To draw an angle equal to a given angle. Let it be required to draw a line making, with the line BG^ an angle at B equal to the angle' A. With ^ as a center, and any as- sumed radius, AD^ draw the arc DE cutting the arms of the angle A. With jB as a center, and the same radius as before, draw an arc, FG. With i^ as a center, and a radius equal to the chord BE^ draw an arc cutting FG at the point G. Join BG. Then GBF is the required angle. For the arcs DE and FG have equal radii and equal chords, and therefore are equal. Hence, they subtend equal angles. Corollary, — An arc equal to a given arc may be drawn in the same way. XII. — To draw an angle equal to the sum of two given angles. Let A and B be the given angles. First, make the angle DOE equal to A^ and then at C, on the line CE^ draw the angle EGF equal to B. The an- gle FGD is equal to the sum of A and 5. 64 PLANE GEOMETRY. [Chap. IV. XIII. — To erect a perpendicular to a given line at its extreme pointy without producing ilie line. A right angle may be made separately, and then, at the end of the given line, an angle equal to the right angle. This is the method employed by mechanics and draughtsmen to construct right angles and perpendiculars by the use of the square. XIV. — To draw a line tJirough a given point parallel to a given line. Draw a line from the given point to any point of the given line. Then through the given point draw a line making the alternate angles equal. XV. — Through a given point out of a circumference^ to draw a tan- gent to the circumference. Let A be the given point, and G the center of the given circle. Join AG. Bisect AG at the point £. "With 5 as a center and £G as a radius, describe a cir- cumference. It cuts the given circumference in two points, JD and E. Draw straight lines from A through I) and E. AD and AE are tangents to the given circumference. Join GD and GE. The an- gle GBA, inscribed in a semi- circumference, is therefore a right angle. AD, perpendicular to the radius GD, is tangent to the cir- cumference. XVI. — Upon a given chord to describe an arc that contains a given angle. Let AB be the chord, and G the angle. Make the angle DAB equal to G. At A erect a perpendicular to AD, and erect a perpendicular to ^^ at its midpoint. Produce these till they meet at F (III, 29). With i^ as a center, and FA as a radius, describe a circumference. BGHA is equal to the given angle G. For AD, being perpendicular to the radius FA, is a tangent. Any Any angle inscribed in the arc OF THt MVERS Akt. 83.] r.^^^^^^^,^x^h^IRCUMFERENCES. 65 angle contained in the arc AEOB is therefore equal to BAD, which was made equal to G (22 and 23). 34. Exercises in Drawing. — Many geometers have ad- vised the use of what is called the analytic method for discov- ering the solutions of exercises in drawing. Its steps are : 1. Suppose the problem solved and the figure completed. 2. Find, with or without the aid of auxiliary lines, the geometrical relations of the parts of the completed figure ; and, 3. Make a solution from these principles. However, the truth and the proof can generally he discov- ered as readily hy reasoning from what is given as hy heginning with the figure completed. When the student does not succeed hy one method, he may try the other. No method can give success without rigorous thought and a thorough knowledge of principles. The term analytic is also applied to a mode of discovering solutions of problems by the use of algebraic formulas. A Determinate Problem is one that admits of a definite number of solutions ; as the problem, to draw a circumference through two given points, with a given radius. A problem is indeterminate when it admits of an indefinite number of solu- tions. Every locus is an answer to an indeterminate problem. For instance, to find a point at a given distance from a given line is an indeterminate problem. There are an infinite number of such points. Their locus is the answer. When two conditions are named in a problem, each may determine a locus, and the point sought is the intersection of the loci. Such a problem is determinate. For instance, Problem VII in the preceding Arti- cle. In such cases, there may be more than one point of intersec- tion of the loci, and there may be none. In the former case, the problem has more than one solution, in the latter, it is impossi- ble. One problem may present several cases. The complete discussion requires a statement of all the possible cases and the conditions of each. 66 PLANE GEOMETRY. [Chap. IV. Exercises. — 1. To draw an angle equal to the difference of two given angles. 2. To draw an angle of 45*. 8. From a given point, to draw the shortest line possible to a given straight line. 4. To find a point in a given straight line, at a given distance from another given straight line. [Find the locus.] 5. To find a point in a given straight line at equal distances from two other straight lines. [Ill, 14, ii.] 6. With a given length of radius to draw a circumference through two given points. [Third Postulate of Euclid.] 7. From two given points, to draw two equal straight lines ending at the same point of a given line. [Find the point in the given line.] 8. From two points on the same side of a straight line, to draw straight lines that meet in the first and make equal angles with it. [Ill, 7, IX, and III, 10, iv.] 9. From a given point out of a straight line, to draw a second line making a required angle with the first. [Ill, 22.] 10. To draw a line through a point such that the perpendiculars upon this line, from two other points, are equal. [Join the given points.] 11. To draw a circumference with a given radius, and a. Through a given point and tangent to a given line ; or 5. Touching two given lines; or c. Touching two given circumferences. [In every case the center is the intersection of loci.] 12. To draw a circumference touching a given line at a given point (Article 30, Exercise 3), and a. Having a given radius ; or 5. Passing through a second given point. 13. To draw a circumference through two given points, with the cen- ter in a given line. CHAPTER y. TRIANGLES, Article 1. — Next in order is the consideration of plane figures that inclose a surface ; and, first, of those whose bound- aries are straight lines. Less than three straight lines can not inclose a surface, for two can have only one common point. Therefore, the triangle is the simplest polygon. From a consideration of its properties those of all other polygons are derived. An Acute-angled triangle has all its angles acute. A Right-angled triangle has one of the angles right. An Obtuse-angled triangle has one of the angles obtuse. The Hypotenuse of a right-angled triangle is the side that subtends the right angle. An Equilateral triangle has three sides equal. An Isosceles triangle has only two sides equal. A Scalene triangle has no two sides equal. The angles at the ends of one side of a triangle are said to be adjacent to that side. The angle formed by the other two sides is opposite. Thus, a side subtends its opposite angle. The Altitude of a triangle is the perpendicular distance be- tween one side and the vertex of the opposite angle. This side 68 PLANE GEOMETRY. [Chap. V. is called the base, and the opposite vertex is called the vertex of the triangle. Any side of a triangle may be taken as the base. Conse- quently the altitude may be any one of three distances. When two sides of a triangle have been mentioned, as in the case of the isosceles triangle, the remaining side is often called the base. In the same manner, after one side has been spoken of, as the base or the hypotenuse, the other two are sometimes called the sides. "When one of the angles at the base is obtuse, the perpen- dicular from the vertex falls out- side of the triangle. Corollary, — The altitude of a triangle is equal to the dis- tance between the base and a line thrpugh the vertex parallel to the base. A Medial of a triangle is a line from the vertex to the mid- point of the base. Hence there may be three medials in a tri- angle. luscribed and Circumscribed. 2. When a circumference passes through the vertices 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 circum- ference, the circle is inscribed and the polygon circum- scribed. A circle that touches one of the three sides of a tri- angle and the other two sides produced is called an escribed circle. Problem, — About every triangle there may he a circumscribed circle. For a circumference may pass through any three points not in the same straight line (IV, 2). Art. 2.] TRIANGLES. 69 Corollary. — The three lines that bisect perpendicularly the three sides of a triangle meet in one point, the center of the circumscribed circle (lY, 11, i). 3, Problem. — In every triangle there may he an inscribed circle. In the triangle AB G, let lines bisecting the angles A and B be produced until they meet. The point i>, where the two bisecting lines meet, is equally distant from the sides AB and B (7, since it is a point of the line which bisects the angle B (III, 13). For a similar reason, the point B is equally distant from the sides AB and A G. is, it is equally distant from the three sides of the triangle. Therefore, a circle having I) as its center, with a radius equal to the distance from B to either side, is the inscribed circle. Corollary. — The three lines that bisect the several angles of a triangle meet at one point, the center of the inscribed cir- cle. That Sum of the Angles, 4, Theorem. — The sum of the angles of a triangle is equal to two right angles. Let the line BU pass through the vertex of one angle, B, parallel to the opposite side, A G. Then the angle A is equal to its alternate angle BBA (III, 22, ii). For the same reason, the an- gle G is equal to the angle EBG. Hence, the three angles of the triangle are equal to the three consecutive angles at the point _B, whose sum is two right angles (III, 7, ii). Therefore, the sum of the three angles of the triangle is two right angles. 70 PLANE GEOMETRY, [Chap. V. Corollaries. — I. Every angle of a triangle is the supplement of the sum of the other two. II. If one side of a triangle is produced, the exterior angle is equal to the sum of the two interior angles not adjacent to it, and is greater than either one of them. Thus -S Cl> is equal to the sum of A and B. III. In every triangle, at least two of the angles are acute. lY. If two angles of a tri- angle are equal, they are both acute. V. In a right-angled trian- gle, the two acute angles are complementary. VI. If two angles of a triangle are respectively equal to two angles of another, then the third angles are also equal. Sides. 5. Theorem. — Each side of a triangle is less than the sum of the other two, and greater than their difference. The first part of this theorem is an immediate consequence of the Axiom of Distance ; that is, AC, the side AB to the side BF, and AC to BE, 74 PLANE GEOMETRY. [Chap. V. Apply the side AG to its equal DE. Since the angles A and I) are equal, AB takes the direction DF, and, these lines being equal, B falls upon F, Therefore, B G and FF coincide, and the triangles coincide throughout. One Side and Two Angles. 13, Theorem. — Two triangles are equal when one side and two angles of one are respectively equal to the corresponding ele- ments of the other. Suppose the side AG equal to BF, and any two of the angles A, B, and G equal respectively to the corresponding angles of the other triangle. Then the third angles must be equal also (4, yi). Apply the side AG to its equal BF, so that the vertices of the equal angles come together, A upon B, and G upon F, and so that both triangles fall upon one side of the common line. Then, since the angles A and B are equal, AB takes the direction BF, and the point B falls somewhere in the line BF. Since the angles G and F are equal, GB takes the direction FF, and B is also in the line FF. Therefore, B falls upon F, the only point common to the lines BF and EF Hence, the sides of the one triangle coincide with those of the other, and the two triangles are equal. Art. 12.] TRIANGLES. 75 Corollary. — Two right-angled triangles are equal, when an acute angle and any side of one are equal to the corresponding elements of the other. 13. Scholium. — Two equal triangles may be so placed as to be symmetrical figures, either about an axis or about a center of symmetry. In the diagram of Article 10 the triangles are symmetrical with reference to an axis of symmetry. in Article 12 the triangles are symmetrical with reference to a point at the intersection of the lines AD and £M In Article 10 the superposition is effected by turning one tri- angle around the axis of symmetry ; in Article 11, by sliding one triangla along the plane without rotation of either kind ; and in Article 12, by turning one triangle in the plane and around the center of symmetry. In some positions of figures, all three of these motions are requisite for superposition. Compare III, 31. Two Sides and an Opposite Angle. 14. Theorem. — Two triangles are equal when one of them has two sides, and the angle opposite to the side which is not less than the other given side respectively equal to the corresponding elements of the other triangle. Let the sides AE and EI^ EI being equal to or greater than AE, and the angle A, be respectively equal to the sides B G, CD, and the angle B. Let the side AE be placed on its equal BG. Since the angles A and B are equal, AI takes the direction BD^ and the 76 PLANE GEOMETRY. [Chap. V. point I falls on the line JBI>. Since EI and CD are equal, the point I is in the circumference whose center is at (7, 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 DC ib equal to or less than CD. Therefore, the point I falls on Z>, AI and DD are equal, and the triangles are equal (10). Corollary. — Two triangles are equal when they have an obtuse or a right angle and any two sides in one respectively equal to the corresponding elements in the other (7, c). Exceptions to the General Hule. - r~>tLk — \ 15, Three elements are always necessary, and they are usually enough to determine the triangle. There are two excep- tions to this rule. 1. When the three angles are given. Two unequal triangles may have their angles respectively equal. ! 2. When two unequal sides and the angle opposite to the less are given. With the sides AD and B C and the angle A, there are two triangles, AB C and ABD, Art. 16.] TRIANGLES. 77 Unequal Triangles. 16. Theorem, — When two triangles have two sides of one respectively equal to two sides of the other, and the included angles unequal, the third side in that triangle that has the greater angle is greater than in the other. Let B CD and AEI be two triangles, having B C equal to AE, BD equal to AI, and the angle A less than B, It is to be proved that CD is greater than EI. Apply the triangle AEI to BCD, making AE co- incide with its equal BC, Since the angle A is less than B, the side ^Z falls between B C and BD, in the position B G, and EI has the posi- tion CG, Join GD, and join B with K, the midpoint of GD, Now, BG is equal to BD, Therefore, BE is perpendicular to GD (III, 11, c). Since the points C and G are on the same side of BE, CD is greater than CG (III, 11), or its equal EI, 17. Theorem. — Conversely, if two triangles have two sides of one equal to two sides of the other, and the third sides un- equal, then the angle subtended by the greater side is greater than the angle subtended by the less side. For, if it were less, the opposite side would be less, and, if it were equal, the opposite sides would be equal ; both of which conclusions are contrary to the hypothesis. I 18. 'Exercises. — 1. The lines that bisect the angles at the base of an isosceles triangle, and extend to the other sides of the triangle, are equal. 2. If, from the midpoint of the base of any triangle, lines are made parallel to the other sides, and if a line is made joining the points where these parallels reach the sides of the triangle, the triangle is divided into four equal parts. 78 PLANE GEOMETRY. [Chap. V. 3. The two taDgents to a circle from one point — that is, the lengths from the common point to the points of contact — are equal. 4. If two triangles have two sides and an opposite angle in one equal respectively to corresponding elements in the other triangle, if the angles opposite to the other two equal sides are not equal, they are supplement- ary. [Consider the second diagram of Article 15.] 5. If, from any point within a circle, except the center, lines are made to various points on the circumference, not more than two such lines can be equal. Similarity of Triangles. 19, By the definition of similarity (II, 3 and 14), two figures are similar when every angle that can be formed by join- ing points of one has its corresponding equal angle in the other. It has been usual to define similar triangles as those whose angles are respectively equal, and whose homologous sides have the same ratio ; but it is demonstrated in all works on geometry that two triangles that agree in either one of these respects must agree also in the other. It is also true, and will be demon- strated in the sequel, that when any two figures have every pos- sible angle in one, formed by diagonals or other lines, equal to its corresponding angle in the other, such figures have all their homologous lines proportional. This shows that the ancient definition is redundant. .^ f3^A CL - Angles Equal. 30. Theorem.. — Two triangles are similar when two angles of one are respectively equal to two angles of the other. It follows immediately that the third angles are equal (4, vi), that is, all the angles formed by the sides of the triangles are respectively equal. It is to be shown that any other angle in one of the trian- gles has its homologous equal angle in the other. Let the angles A, B, and C be respectively equal to the angles JD, E^ and F, and suppose, first, that the angle to be con- sidered has one arm passing through a vertex of the triangle, as Art. 20.] TRIANGLES, 79 IG. From the point F, homologous to G, make FZ, making the angle LFF equal to IGB, Subtracting these from the given equal angles F and (7, the remainders DFL and A GI are equal. Since the angles A and D are equal, DLF must be equal to AIG (4, yi), and FLE must be equal to GIB. Suppose now a line that does not pass through one of the vertices, as RI. First connect IG^ and make FL as before, and XJf with the angle JliXii^ equal to RIG. It is shown, by reason- ing as above, that every angle made at R or at Zhas its homolo- gous equal angle at M ox at L. Therefore, the relative directions of all their points are the same in both triangles ; that is, they have the same form, they are similar triangles. Corollaries. — I. Two similar triangles may be divided into the same number of triangles respectively similar and similarly arranged. II. If two sides of a triangle are cut by a line parallel to the third side, the triangle cut off is similar to the original triangle (III, 22). This is true when the sides are divided externally as well as when the division is internal ; and the external division may be \7 on the sides produced either beyond the base or beyond the vertex. 80 PLANE GEOMETRY. [Chap. V. III. Two right-angled triangles are similar when an acute angle in one is equal to one in the other. 31. Theorem. — Two triangles are similar, lohen the sides of one are parallel to those of the other ; or, when the sides of one are perpendicular to those of the other. The angles formed by lines that are parallel are either equal or supplementary ; and the same is true of angles whose arms are perpendicular. It is to be shown that the angles can not be supplementary in two triangles. If even two angles of one triangle could be respectively sup- plementary 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 impossible (4). Hence, when two triangles have their sides respectively parallel or perpendicular, two of the angles of one triangle must be equal to two of the other. Therefore, the triangles are similar. Sides Proportional. 3 2. Theorem. — When two triangles are similar, every side of one has the same ratio to the homologous side of the other. r B Suppose the angles A^ E, and I respectively equal to B, C, Art. 22.] TRIANGLES. 81 and D, Then AE and B C are homologous sides, also EI and GB. Take CF equal to EA^ and (76^ equal to EI^ and join EG. The angle Ci^6r is equal to ^ (H)* 3'^. FA:GB = AI:BJ). Since GF is equal to FAj FG is equal to AT; and the tri- angles GFG and AFI are equal (10). Therefore, the triangles AFI and B GD are similar. 36. Theorem., — If from two vertices of a triangle perpen- diculars fall on the opposite sides, a line joiniyig the feet of these perpendiculars cuts off a triangle which is similar to the first one. It is to be proved that the tri- angle BED is similar to BA G. The triangles BDA and BEG, being right-angled at D and E, and having the common angle B, are similar (20). Therefore, BA'.BG=BD\BE, and the triangles BED and BGA are similar (24). It may be necessary to pro- duce one or both of the sides. The sides of the triangle ABG are inversely proportional to their segments EB and DB, Centers of Similarity. 27. Theorem. — If lines are made from the several vertices of a triangle to any point of the plane, and these lines are divided in the same ratio, the points of division are the vertices of a triangle similar to the first. 84 PLANE GEOMETRY, [Chap. V. From Af B, and C, make lines to G. Take any point on GA, as Z>, and take the points E and F, so that GD'.GA= GE'. GB = GF-. GO-, then it is to be proved that BEF is similar to AB C. In the triangle GAB, BE is parallel to AB (23, c). Like- wise, EF and BF are respectively parallel to BG and AG. Therefore, the triangle BEF is similar to AB C, The division may be external. Then GHI is similar to ABG (24) ; and B[I is parallel to AB (III, 24, i). In the same way the other sides are respectively parallel, and the tri- angle HIK is similar to AB C, 28. Theorem. — Conversely, when tioo triangles have their sides respectively parallel, the three lines made through homolo- gous vertices meet at one point. If the triangles ABG and BEF have their sides respective- ly parallel, it is to be proved that the lines AB, BE, and CF must meet at one point. Produce these lines till they meet. Suppose AB and CF meet at G, and AB and BE meet at N. Since the triangles AGC and B GF are similar (20, ii), AC'.BF=AGxBG. Likewise, AB : BE = AJST : BJST. Art. 28.] TRIANGLES. 85 But by hypothesis, AG:DF=AB'.DK Therefore, AG.DG^ AJST : BJST, That is, the line AD is divided in the same ratio at N and Gy which is absurd, unless iV and G are one point. /. The demonstration is the same for the triangle HIK, A Center of Similarity is a point similarly situated with reference to two similar figures, whose homologous lines are -^ parallel. It is an internal or external center of similarity accord- ing as it divides internally or externally the lines joining homol- ogous points. The center of similarity may be at any point of the plane. It may be at one of the vertices, as when a triangle is cut by a line parallel to the base ; or it may be on one side. Either an internal or an external center of similarity may be within both polygons, or it may be within neither. Let the student illus- trate each of these cases. Corolla,ry. — If two triangles have their homologous sides parallel, a line joining homologous points is divided at the cen- ter of similarity in the linear ratio of the two figures. 86 PLANE GEOMETRY. [Chap. V. Right-angled Triangles. 29, Every triangle may be divided into two right-angled triangles, by a perpendicular from the vertex to the base. The investigation of the properties of right-angled triangles leads thus to many of the properties of triangles in general. Theorem. — In a right-angled triangle, if a perpendicular falls from the vertex of the right angle upon the hypotenuse, then, 1. I^ach of the triangles thus formed is similar to the origi- nal triangle ; 2. Either side of the original triangle is a mean propor- tional between the hypotenuse and the adjacent segment of the hypotenuse; and, 3. The perpendicular is a mean proportional heticeen the two segments of the hypotenuse. The right-angled triangles AEO and AEI have the acute angle A common. Therefore, these two triangles are similar (20, III). That the triangles EOI and *MIA are similar is proved by the same reasoning. Since the triangles are simi- lar, the homologous sides are proportional, and AI'.AE^AE\AO\ that is, the side AE is a mean proportional between the whole hypotenuse and the segment A which is adjacent to that side. In like manner, EI is a mean proportional between AI and OL Lastly, the triangles AEO and EIO are similar (20), and therefore, A0\ 0E=^ 0E\ 01', that is, the perpendicular is a mean proportional between the two segments of the hypotenuse. Art. 29.] TRIANGLES. 87 Corollary. — A perpendicu- lar from any point of a circum- ference upon a diameter is a mean proportional between the two segments of the diameter (lY, 22, III). 1/ \> 30. Theorem. — The second power of the length of the hy- potenuse is equal to the sum of the second powers of the lengths of the other two sides of a right-angled triangle. Let h be the hypotenuse, a the perpendicular let fall upon it, h and c the other sides, and d and e the corresponding seg- ments of the hypotenuse made by the perpendicular. That is, these letters represent the ratios of these lines to some unit of length. By the second conclusion of the last theorem, h'.h '.:J) : d, and hic.-. c.e. Hence, hd=^J)^y and he=.c^. Adding these two, h (d -\- e) =i b^ •\- c^ . But c? + e = A ; therefore, A2 = ^»2 + c^. 31. Theorem. — In any triangle^ if a perpendicular falls from the vertex upon the hase^ the sum of the segments of the 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 PLANE GEOMETRY. [Chap. V. Then, two right-angled triangles are formed, in one of which a2 + ^2 = d^ ; and in the other, Subtracting, Factoring, Whence, a2 + ^2— e2 (* + «) (* — 6) i-\-e:d-{-c {d-^c) (d — c). d — c : i — e. The base is equal to the first or to the last term of this pro- portion, according as the perpen- dicular divides it internally or externally. Corollary. — The segments of the base may be expressed in terms of the sides of the triangle d^ — c^ 2 =F e = — , . Therefore, For i ±e = h, and I = 53 + C?2 26 , and e h^-\-c' d^ 2b Segments of Chords. 32. Theorem. — If two chords of a circle cut each other y the segments of one are the extremes^ and the segments of the other the means of a proportion. If from a point without a circle^ two secants extend to the farther side, then the ichole of one secaiit and its exterior part are the extremes, and the whole of the other secatit and its exte- rior part are the means of a proportion. i Art. 82.] TRIANGLES. 89 The second proposition is usually stated as a separate theo- rem. It is that case of the first when the chords divide each other externally. Join AB and BG. The triangles AED and CEB have the angles B and B equal (IV, 22), and the angle E vertical in one case and common in the other. Therefore, the triangles are similar (20), and EA:EG = EB:EB, 33. Theorem. — If from the same point there are a tangent and a secant, the tangent is a mean proportional between the secant and its exterior part. This may be demonstrated by the student. For construc- tion, make chords from the point of tangency to the points where the secant cuts the curve. Medials of Triangles. 34. Theorem. — Two medials of a triangle cut each other in the same ratio, the smaller segment being half of the larger. It is to be proved that the medials BE and CB, of the trian- gle ABC, are divided at (9, so that BO is half oi OC and EO is half of OB. Join BE. Since B and E are the midpoints of AB and AG, /j the triangle ABE is similar /I to AB G (23), and BE is half ^ / L oi BG (22). The triangles /C''l OEB and OB G are mutually /■'''''' \ / equiangular (III, 22, ii) and /' \l similar. Therefore, B0\ OG = EO: 0B = BE'.BG=1 : 2. 35. Theorem. — The three medials of a triangle cut each other in one point. For one is divided by both the others in the same ratio, and therefore at the same point. 5 90 PLANE GEOMETRY. [Chap. V. JVf 36. Theorem. — The medial to the base of a triangle is less than half the sum of the other two sides. Join the midpoint of the base, M^ to X, the mid- point of the side £ C. The triangle MLC is similar to AB (7, and LM is half of AB. Therefore, the sum of BL and LM'vs, half of the two sides, but BM is shorter than BLM. CoroUai^. — The sum of the medials is less than the perim- eter of a triangle. 37. Theorem. — The sum of the medials is greater than three fourths of the perimeter of a triangle. The sum of the three lines from any point in a triangle to the vertices is greater than half of the perimeter. For A0-{- OB >AB; OB^ OG > BG; and 0G+ AG "^ AG, By addition, 2 (AO-i- 0B-\- OG)>AB + BG+AG. Since each of the lines from is two thirds of a medial, two thirds the sum of the medials is greater than half the perimeter. Therefore, the sum of the medials is greater than three fourths of the perimeter. 38. Theorem. — The perimeter and three fourths of the perimeter are the limits of the sum of the medials of a tri- angle. It only remains to be proved that the sum of the medials may be within less than any assignable difference of these limits. The student may demonstrate the superior limit by a triangle having one very small side, and the inferior limit by a triangle with one very small medial. Aet. 39.] TRIANGLES, 91 39. Theorem. — One medial of a triangle is less than the sum and greater than the difference of the other two. In any triangle ABC^ having the medials AL, JBM, and CJ^y the line NL may be made, and it may be produced to P, making LP equal to iVX, and the lines BP and MP may be made. As AM is half of A (7, it is equal to iVZ, and therefore to LP, to which it is also parallel, making the figure AMPL a parallelogram. Therefore, MP is equal to AL. As the trian- gles LLP and CZA^have two sides and the in- cluded angle of one equal to the corresponding ele- ments of the other, the third side, BP, is equal to NC. Thus, the triangle BMP has for its sides the medials of the given triangle AB G. Since this construc- tion is possible for any given triangle, every medial must be less than the sum and greater than the difference of the other two. 40. Applications. — Principles demonstrated in this chapter are used to measure heights and distances, even the distances of the stars. One side and the angles of a triangle determine the other sides. By measuring a base and the angles at each end made by lines extending to a distant object, the distance is ascertained. Trigonometry teaches how- to calculate these distances. Such problems may be solved approxi- mately by drawing a triangle similar to the one measured, measur- ing its sides, and applying the principle that homologous lines are proportional. Similarly, the height of any object may be found when its angular elevation and its distance are known. "Without any drawing, or consider- ation of the angles, the height of a house or tree may be found by meas- uring its shadow, and at the same time the length of shadow of a yard- stick held vertically. 92 PLANE GEOMETRY. [Chap. V. 41. Exercises. — 1. Given tlie sides of a triangle, 17 and 18, and the base, 23 ; to find the segments of the base made by a perpendicular from the vertex. 2. The parts of two parallel lines, intercepted by several straight lines that meet at one point, are proportional. 3. Find the locus of the mid- points of all lines that extend from a given point to a given line. 4. If tangents to two inter- secting circles are made from any point on the common chord produced, such tangents are equal. [33.] 5. The common chords of every pair of three intersecting circles meet in one point. [32.J 43. Scholium. — The cases of equality of triangles are : 1. Three sides equal ; 2. Two sides and included angle ; 3. One side and two angles ; and 4. Two sides and an oppo- site angle. The cases of similarity are : 1. Three sides proportional ; 2. Two sides proportional and included angles equal ; 3. Two angles equal (including the cases of parallel and perpendicular sides). A fourth case might be stated, when two sides are pro- portional, and the angles opposite the greater of the given sides are equal. Thus, each case in the theory of equality has its cor- responding case in the theory of similarity. In every case of equality of triangles, at least one line in one must be equal to the corresponding line in the other. In no case of similarity of triangles is any equal dimension given. Any case of similarity becomes a case of equality, if we add to the hypothesis that some two homologous lines are equal. That is, when triangles are similar, and when the linear ratio is unity, they are equal. In similarity, equality of corresponding angles is assumed in the definition, equality of homologous ratios is proved as the universal consequence. Art. 43.] TRIANGLES. 93 43, Miscellaneous Exercises. — 1. If the diameter of a circle is one of tbe equal sides of an isosceles triangle, the circumference bisects the base of the triangle. [IV, 26, ii.] 2. If two circles have for their diameters two sides of a triangle, the circumferences cut the third side at the same point. It may be necessary to produce the third side. 3. If three circles have the vertices of a triangle as centers, and the circumferences pass through the point of contact of the sides with the inscribed circle, those three circles are tangent each to the others. 4. If an isosceles and an equilateral triangle are 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 is i of, or 2^ times its vertical angle. 6. The semi-perimeter of a triangle is greater than any one of the sides, and less than the sum of any two. [5.] 6. The angle at the base of an isos- celes triangle, being one fourth of the angle at the vertex, if a perpendicular is erected to the base at its extreme point, and this perpendicular meets the opposite side of the triangle produced, then the part produced, the remaining side, and the perpendicular form an equi- lateral triangle. 7. Of all triangles on the same base, and having their vertices in the same line parallel to the base, the isosceles has the greatest vertical angle. [Circumscribe a circle about the isosceles triangle.] 8. If, from a point without a circle, two tangents are made to the circle, and if a third tangent is made at any point of the circumference between the first two, then, at whatever point the last tangent is made, the perimeter of the triangle formed by these tangents is twice the length of one of the tangents first made. 9. The midpoint of the hypotenuse is equally distant from the three vertices of a right-angled triangle. 94 PLANE GEOMETRY. [Chap. V. 10. If a circle is inscribed in a right-angled triangle, the differ- ence between the hypotenuse and the sum of the two sides is equal to the diameter of the circle. [Make a radius to each point of tangency.] 11. In a right-angled triangle, if one of the acute angles is equal to twice the other, the hypotenuse is equal to twice the shortest Bide. [Join the vertex of the right angle to the midpoint of the hypotenuse.] 12. Two triangles are similar, when two sides of one are proportional 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 cor- responding angle in the other. [14.] 13. If perpendiculars fall from the three vertices on the opposite sides of a triangle, the triangle formed by joining the feet of these perpendiculars has its angles (either interior or exterior) bisected by the perpendiculars. [The bisection of an exterior angle occurs when the given triangle is obtuse-angled, as in the second diagram of Article 26.] 14. If an equilateral triangle is in- scribed in a circle, and from any point on the circumference lines extend to the three vertices, one of these is equal to the sum of the other two. [lY, 22, n; V, 22.] 15. Find the locus of the points such that the sum of the distances of each from the two arms of a given angle is equal to a given line. [The point on one arm of the angle at a distance from the other arm equal to the given line must be one point of the locus ; the corresponding point on the other arm must be a second point.] 16. Find the locu^ of the points such that the difference of the dis- tances of each from two arms of a given angle is equal to a given line. Discuss both of these upon the supposition that two indefinite lines are given instead of two arms of an angle. Akt. 44.] TRIANGLES, 95 Problems in Drawing. I 44:. Every case of equality of triangles has its corresponding problem in drawing. I. — To draw a triangle when the three sides are given. Let a, &, and e be the given lines. Draw the line IE eqnal to c. With 7 as a center, and with the line & as a radius, describe an arc, and with £J as a center and the line a as a ra- dius describe a second arc, so that the two may cut each other. Join Oj the point of intersection of these arcs, with / and with K JOE is the required triangle. In the same way draw a triangle equal to a given triangle. The demonstration of this and of some of the following problems is left to the student. II. — 7o draw a triangle^ two sides ^ and the included angle leing given. -^- Let a and & be the given lines, and E the angle. Draw FG eqnal to I. At C make an angle equal to E. Take DC equal to a, and join FD. Then FDG is a triangle having the given elements. III. — To draw a triangle when one side and two angles a/re given. If one of the angles is opposite the given side, find the supplement of the sum of the given angles; this is the other adjacent angle. Then, « let a be the given side, and D and E the adjacent angles. Draw EG equal to a. At B make an angle equal to i), and at G an angle equal to E. Produce the sides till they meet at the point F. FBG is a triangle having the given side and angles. 96 PLANE GEOMETRY. [Chap. V. IV. — To draw a triangle when two aides and an angle opposite to one of them are given. Construct an angle equal to the given angle. Lay off on one side of the angle the length of the given adjacent side. With the extremity of this side as a center, and with a radius equal to the side opposite the given angle, draw an arc. If this arc cuts or touches the other side of the angle, join the point of intersection or tangeucy with that point which was taken as a center. A triangle thus formed has the given elements. The student can better discuss this problem after drawing several triangles. Let the given angle vary from y^tj obtuse to very acute; and let the opposite side vary from being much larger to much smaller than the side adjacent to the given angle. y. — To find a fourth proportional to three given lines. Let a be the given extreme, and 5 and e the given means. On a line extending from the point I) take DG equal to a, and Bff equal to c; from G draw GF equal to 5; join DF; from ff draw J2X parallel to GF; and produce ffK and BF (if necessary) till they meet. HK is the required fourth proportional. For the triangles DGF and DRK are similar. Hence, DG:GF=DE:HK. That is, a : J = c : HK. yi. — To divide a line in given ratios. Let LD be the line to be divided into parts proportional to the lines a, 5, and c. I Akt. 44.] TRIANGLES. 97 a From L draw the line LE^ making LF equal to a, FG equal to &, and GE equal to c. Join DE^ and draw (?/ and FH parallel to DE. LR^ HI^ and ID are the parts required (III, 28). yil.—To divide a line hy a given nwrriber. This may be done by the last problem ; bat, when the line is small, the following method is preferable. To divide the line AB by ten, draw A G indefinitely, mak- ing of it ten equal parts. Join BC^ and from the several points of division of ^ C draw lines par- allel to AB^ and produce them to BG. The parallel nearest to G is one tenth of AB, the next is two tenths, and so on. This depends upon similarity of triangles. This is the method employed in the common scale used for draughting. VIII. — To divide a line externally in a given ratio. Let LD be the line to be di- vided in the ratio a : &. Draw LG equal to 5, the greater, and from G take FG, toward Z, equal to a. Join FD, and draw GE parallel to it. Then LD is divided at H, so that EDxEL^a-.l). IX. — Tojind a mean 'proportional to two given lines. Make a straight line equal to the sum of the two. Upon this as a diameter, describe a semi- circumference. Upon this diameter erect a per- pendicular at the point of meeting of the two given lines. Produce this to the circumference. The line last drawn is the required mean propor- tional. X. — To divide a line in extreme and mean ratio. Let AG \)Q the line. At G erect a perpendicular, 67, equal to half of AG. Join AL Take ID equal to CI, and AB equal to AD. The line AG \& divided at the point B in extreme and X,- mean ratio. That is, AG:AB=zAB:BG. as With / as a center and IG a radius, describe an arc 98 PLANE GEOMETRY. [Chap. V. DGE^ and produce AI till it meets this arc at E. Then, -4(7 is a tangent to this arc, and therefore (33), Hence, AE'.AG=AG'.AD. AG:AE—AG=AD'.AG—AD. But AG — DE, Therefore, AE — AG = AB = AB, and ^C AD = BG. Substituting these equals, AG'.AB = AB:BG. Corollary. — AD is divided externally in extreme and mean ratio, at the point JS', for AD: DE= DE'.AE. The numerical value of these ratios is shown as follows. If the line J.C is 1, GI is ^. Then AI= \/ {AGY + {Giy = y^Ii = i |/5; BG = AG—AB=^--\\r5^ and XI. — To draw an isosceles triangle having an angle at the l>ase twice as great as the angle at the vertex. Divide the line AB, in extreme and mean ratio, at D. Then draw the triangle AGD with the base equal to the smaller segment AD, and with the sides GD and GA each equal to the larger segment BD. y'' Join BG. Since AB\AG= /' AG: AD, the triangles ABG /' and A.GD are similar (24), and ^ j) j^ the angle B equals Z)C^. Since BD equals GD, the angles B and BGD are equal. Therefore, BGA is double the angle B, and the angle A is double the angle DGA. Art. 46.] TRIANGLES. 99 45. Exercises in Drawing. — The suggestions of Article 34, Chapter IV, should be remembered in connection with these exercises. 1. The base of an isosceles triangle is to one side as three to two. Find, by construction and measuring, whether the angle at the vertex is acute or obtuse. 2. Construct a right-angled triangle when a. An acute angle and one side are given ; h. An acute angle and the hypotenuse are given ; c. A side and the hypotenuse are given ; and when, d. The two sides are given. 3. To describe a circle touching three given straight lines. [In what loci must the center be ?] 4. To draw a hne DE parallel to the base, BG^ of a triangle, ABG^ making DE equal to the sum of BD and CE. [6.] A* //* -/ \ n^ 7 B^^^' o a C 6. To draw a line BE parallel to the base, BG, of a triangle, ABG^ making DE equal to the difference of BD and GE. 6. Through a given point to draw a line such that the parts of it between the given point and perpendiculars let fall on it from two other given points are equal. [Ill, 27.] 7. To describe a circumference through two given points and touch- ing a given straight line. [33.] 8. To draw an isosceles triangle when one side and one angle are given. [This includes several cases.] 9. To draw a triangle with a given base, altitude, and angle at the vertex. [Use loci. IV, 26, iii.] 10. To draw a triangle with a given base, altitude, and radius of cir- cumscribed circle. 11. In a given circle to inscribe a triangle similar to a given triangle. [What arc must subtend each angle of the required triangle ?] 100 PLANE GEOMETRY. [Chap. V. JD 12. To divide a right angle into five equal parts. [38, xi.] 13. Througli a given point be- tween two given lines, to draw a line such that the part intercepted by the given lines is bisected at the given point. [That is, given the point P and the lines A G and BD^ to draw AB 80 that AP = PB. 14. From a point without two lines, to draw a line such that the part intercepted between the lines is equal to the part between the given point and the nearest line. 15. To draw a triangle, given the base, one angle at the base, and the difference of the sides. [Draw the base AB and the angle at A as given. On the other arm of the angle A^ take AD equal to the given difference. It may be necessary to produce the arm lelow the base a distance equal to the given difference.] 16. To describe a circumference tangent to three given equal circum- ferences that are tangent to each other. CHAPTER VI. Q UADRILA TERALS. Article 1. — Quadrilaterals are classified according to the parallelism of sides and the equality of angles. A Trapezoid is a quadri- lateral that has two sides paral- lel. The parallel sides are called bases. No two of the angles are necessarily equal, but the angles at the ends of one base may be equal. Then the other two angles, being respectively supple- ments of these (III, 22, iv), are also equal. Such a figure is symmetrical, the axis of symmetry being the line joining the midpoints of the bases. A Parallelogram is a quadrilateral that has its oppo- site sides parallel. Corollary. — Two consecu- tive angles of a parallelogram are supplementary, and the oppo- site angles are equal (III, 7, vi). Hence, if one angle is right, they are all right. A Rectangle is a right-angled parallelogram. A Rhombus, or Lozenge, is a parallelogram that has all its sides equal. It will be shown (3) that this is a consequence when two adjacent sides are equal. 102 PLANE GEOMETRY. [Chap. VI. A Square is a quadrilateral having its sides equal and its angles right. It will appear that such a figure is a rectangle with equal sides. The Altitude of any quadrilateral having parallel sides is the distance between the parallels. Then either of these is called the base, and sometimes the two are called the bases. The altitude of a parallelogram may be either of two dis- tances. A Diagonal of a polygon is a straight line joining two vertices, except, of course, two consecutive vertices, which are joined by a side. 2. Problem, — There may he a quadrilateral inscribed in a circle. For any four points of a circumference may be joined by chords. Corollaries. — I. The opposite angles of an inscribed quadri- lateral are supplementary (IV, 22, iv). II. When a quadrilateral has its opposite angles supplement- ary, a circle may be circumscribed about it (IV, 26, rv). III. If the opposite angles of a quadrilateral are not supple- mentary, a circle can not be circumscribed. IV. The rectangle and the square are the only parallelo- grams that can be inscribed in a circle. Parallelograms. 3« Theorem.. — The opposite sides of a poLraUelogram are equal. For, joining -4C by a diagonal, the triangles formed have the side A C common ; the angles A GB and DA C equal, for they are alternate ; and A CD and BA C equal, for the same reason. Therefore (V, 12), the triangles are equal, the side AD is equal to B (7, and AB to CD, AST. 8.] QUADRILATERALS, 103 Corollaries. — I. When two systems of parallels cross each other, the parts of one system in- cluded between two lines of the other are equal. II. A diagonal divides a parallelogram into two equal tri- angles. 4, Theorem, — If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. Join A C, The triangles AJS G and CD A are equal, for the side AD is equal to J3 (7, and D O is equal to AJB, by hypothesis ; and they have the side A C com- / ^ mon. Therefore, the angles DA O ^ ^^ and BOA are equal. But these angles are alternate with reference to the lines AD and B G, and the secant A G. Hence, AD and j5 G are parallel, and, for a similar reason, AD and D G are par- allel ; that is, the figure is a parallelogram. 5, Theorem. — If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram. If AD and DG are equal and parallel, it is to be proved that AB is parallel to D G. Joining BD, the triangles formed are equal, since they have v -■-""' \ the side BD common, the side \ ----"T"" A AD equal to BG, and the angle ^ ^ ADB equal to its alternate DB G (V, 11). Hence, the angle ABD is equal to BDG. But these are alternate with reference to the lines AB and D G, and the secant BD. Therefore, AB and D G are parallel, and the figure is a par- allelogram. 104 PLANE GEOMETRY. [Chap. VI. 6, Theorem. — The diagonals of a parallelogram bisect each other, and conversely. For the triangles ABH and CDH have the sides AB and CD equal, the an- gles ABU and CDH equal, and the angles BAH and DGII equal. Therefore, the triangles are equal and AH is equal to CH, and BH to DH. The converse may be demonstrated by the student. Corollary. — The midpoint of the diagonals of a parallelo- gram is a center of symmetry for the figure (IV, 5). 7. Applications. — The rectangle is the most frequently used of all quadrilaterals. "Walls and floors of apartments, doors and windows, books, paper, and many other articles, have this form. Carpenters make an ingenious use of a geometrical principle in order to make door and window frames rectangular. Having made the frame, with its sides equal and its ends equal, they measure the two diagonals, and make the frame take such a shape that these also become equal. A rhombus inscribed in a rectangle is the basis of many or- naments used in architecture and other work. The symmetrical trapezoid is used in architecture, sometimes for or- nament, and sometimes as the form of the stones of an arch. A B An instrument called parallel rulers^ used in drawing parallel lines, consists of two rulers, connected by cross pieces with pins, on which the rulers may turn. The distances between the pins along the rulers, that Art. 7.] QUADRILATERALS. 105 is, AB and GD^ must be equal ; also, along the cross pieces, tliat is, A G and BD. If one ruler is held fast while the other is moved, lines drawn along the edge of the other ruler, at different positions, are parallel. 8. Exercises, — 1. The sum of two opposite sides of any quadri- lateral circumscribed about a circle is equal to the sum of the other two sides. [V, 18, Ex. 3.] 2. The two angles at the center of the circle which are subtended by opposite sides of the circumscribed quadrilateral are supplementary. 3. The diagonals of a rectangle are equal ; and conversely. 4. The diagonals of a rhombus bisect its angles; and are perpendicu- lar to each other ; and conversely. 5. If the four midpoints of the sides of any quadrilateral are joined by straight lines, those lines form a parallelogram. [Make the diagonals of the quadrilateral, and see V, 23.] 6. If four points are taken, one in each side of a square, at equal distances from the four vertices, the figure formed by joining these successive points is a square ; for example, the points A^ B, G, and B. 7. If the same thing is done in a paral- lelogram, the figure formed is a parallelo- gram. 8. The lines that bisect the angles of a parallelogram form a rectangle. 9. The diagonals of the rectangle in the last exercise are parallel to the sides of the parallelogram, and are midway between them. [Produce each bisector till it reaches both the other sides of the parallelogram, one of them being produced.] 10. When one diagonal of a quadrilateral divides the figure into equal triangles, is the figure necessarily a parallelogram ? 11. If two equal circumferences intersect at right angles, the common chord is equal to the distance between the centers. [lY, 17, i.] 12. If equal parallels are projected on the same line, the projections are equal. [Ill, 27.] 13. Parallels that have equal projections on the same line are equal. 14. What geometrical principles are applied in making a win- dow frame rectangular? In the construction and use of parallel rulers? 106 PLANE GEOMETRY. Measure of Area. [Chap. VI. 9, The standard measure of surfaces is a square. That is, the unit of area is a square, having for its side the unit of length. It is used because of its simplicity. It has the same length throughout its breadth, and the same breadth throughout its length ; and the length and breadth are equal. Area of Kectangles. Theorem. — The area of a rectangle is equal to the product of its base and altitude. That is, if the number of units of length in the base is multiplied by the number of units of length in the al- titude, the product is the number of units of area in the rectangle. Any straight line may be the unit of length. The square on it is the unit of area. There are two cases : 1st, when the base and the altitude of the rectangle have a common measure ; and 2d, when these lines have no common meas- ure. 1st. Divide the base and the altitude into segments, each equal to the common measure, and let it be the unit of length. Through every point of division of the base make lines parallel to the altitude, and through every point of division of the alti- tude make lines parallel to the base. The lines of one set of parallels are perpendicular to those of the other, and all the segments are equal to the unit of length. Thus the rectangle is divided into equal squares, each square equal to the unit of area. The number of these squares is the product of the number in one row by the number of rows, that is, the number of times the unit of length is in the base by the number of times it is in the altitude. f t -j h • [ 1 |- 1- III! 1 I I I Art. 9.] QUADRILATERALS. 107 Therefore the area of the rectangle is to the unit of area as the product of the base and altitude of the rectangle is to unity. This may be expressed Here h and a are the numerical ratios of the base and altitude of the rectangle to the unit of length. In this 1st case, h and a are integers. 2d. When the base and the altitude of the rectangle have no common* measure, if the same method is pursued as before, a part of the rectangle may be measured by using a square whose side is an aliquot part of the altitude. By using a square whose side is regularly smaller than the unmeasured part of the base of the rectangle, the part of the rectangle measured is an increasing variable rectangle, and its limit is the given rectangle. Also the base of this variable rectangle is a variable, whose limit is the given base. For an aliquot part of the altitude may be taken (Postulate of Extent) so small that the base of the given rectangle and its area are exhausted within less than any assign- able difference. At every step the variable area is to the unit of area as the product of the variable base and the altitude is to the square of unity, as demonstrated above. Substituting their limits for these always equal variable ratios, M '. V = b a : 1. Hence, ji=zba zr. Here b and a represent the ratios of the base and altitude to the unit of length. They may be any numbers either commen- surable or incommensurable with unity. Area of Parallelograms. 10. Theorem. — The area of a parallelogram is equal to the product of its base and altitude. At the ends of the base AB erect perpendiculars, and produce them till they meet the opposite side, in the points £J and JFl Now the right-angled triangles IT D 108 PLANE GEOMETRY. [Chap. VI. F C V AED and BFC are equal, haviDg the side BF equal to AE, since they are perpendiculars between parallels ; and the side BC equal to AD. If each of these equal triangles is subtracted from the entire figure, ABGE, the remainders ABFE and ABCD must be equivalent. But ABFE is a rectangle having the same base and altitude as the parallelogram AB CD. Hence, the area of the parallelogram is equal to the product of the base and altitude. Corollaries. — I. Any two parallelograms have their areas in the same ratio as the product of the base and altitude of one to the product of the base and altitude of the other. II. Parallelograms of equal altitude have the same ratio as their bases, and parallelograms of equal bases have the same ratio as their altitudes. III. Two parallelograms are equivalent when they have equal bases and equal altitudes ; also, when the dimensions of the one are the extremes, and the dimensions of the other are the means, of a proportion. The base and altitude are the dimensions of a parallelogram. A rectangle is said to be contained by its dimensions. IV. If two chords of a circle cut each other, the rectangle contained by the segments of one is equivalent to that contained by the segments of the other. Area of Triangles. 11. Theorem. — The area of a triangle is equal to half the product of its base and altitude. For any triangle is one half of a parallelogram having the same base and altitude. Corollaries. — I. The areas of two triangles have the ratio of the product of the base and altitude of one to the product of the same dimensions of the other. i Art. 11.] QUADRILATERALS. 109 IL Triangles of equal altitudes have the same ratio as their bases ; and triangles of equal bases have the same ratio as their altitudes. III. Two triangles are equivalent when they have equal bases and equal altitudes. IV. If a parallelogram and a triangle have equal bases and equal altitudes, the area of the parallelogram is double that of the triangle. 12. Theorem, — If from half the sum of the three sides of a triangle each side is subtracted^ and if these rem^ainders and the half sum are multiplied together y then the square root of the pro- duct is the area of the triangle. Let a, by and c be the sides of the triangle, h the 5'^^^: 7^""^"J^ altitude, and m and n the x n \ m — -^^ segments of the base, a ; ^ that is, these letters repre- sent the ratios of these lines to some unit of length. Then (V, SI, i) m = «-l+|i=l£!. Since 7i, m, and 5 make a right-angled triangle (Y, 30), Substituting for m its value from the preceding equation, and bearing in mind that the difference of two squares is equal to the product of the sum and difference of the roots, we have = V + 2^ ) V 2^ ) _ g" + 2 g5 + 5" — c' c'' — g" + 2 a5 — 5" ~ 2~« ^ 2 a = {(i + ^)'' — c^ c' — ja — by ~ 2a ^ 2a _ (a + h + c) (a + i — c) (a + c — h) (b + c — a) " 4 a' 110 PLANE GEOMETRY. [Chap. VI. Taking the square root and multiplying by ^ a, ■^ ah = -^ a Y a) 4.0" l)(l> + c — a) = V tV (a + & + c) (a + & — c) (a + c But i ah is the measure of the area of the triangle. The expression may be reduced by letting p repre- sent the perimeter. Then, a + & — c = p — 2c = 2(|-j9 — c), a + c — &=^ — 2 5 = 2(1-^ — &), I + c — a=p — 2a=2 {\p — a). Substituting these values, we have, area = V ^pikp — a) (ii? — &) {^p — c). 13. Theorem. — The areas of similar triangles are to each other as the squares of homologous lines. Let AEI and B CD be similar triangles, and 10 and DH homologous altitudes. Then, IO'.DH=AE\BC. But, \AE'.\BCz=AE'.BC. Multiplying, ^AExIO-.^BCxBIT^AE : BC\ That is. area JL^J: area J5aZ> = ^^ : BC, The second power of a ratio is sometimes called the dupli- cate of it. Thus, the superficial ratio of similar triangles is the duplicate of their linear ratio. Art. 14.] QUADRILATERALS. HI 14. Theorem, — If two triangles have an angle in one equal, to an angle in the other, their areas are to each other as the pro- ducts of the sides including the equal angles. Place the triangles AB C and ADF, so that the equal ^ angles may coincide at A, Join CD, Taking AB and AD as the bases of the triangles ABC and ADC, we have (11,11) ABC',ADC = AB'.AD. Likewise, AD C : ADF =AC: AF, Multiplying, and canceling the factor AD 6', ABC ,ADF= AB ,AC '.AD , AF, Area of Trapezoids. 15. Theorem. — The area of a trapezoid is equal to half the product of its altitude by the sum of its bases, ' The trapezoid may be divided by a diagonal into two triangles, each having for its base one base of the trape- zoid. The altitude of each of these triangles is equal to that of the trapezoid. The area of each triangle being half the product of the common altitude by its base, the area of their sum, or of the whole trapezoid, is half the product of the altitude by the sum of the bases. Corollary. — The area of ^ ^ a trapezoid is equal to the product of its altitude by a line midway between the bases and parallel to them ; for FF is half the sum of AB and CD, FF is the arithmetical mean of the K, 112 PLANE GEOMETRY. [Chap. VI. 16. Applications. — Enlightened nations attach great importance to exact standard measures. The standards generally used for the measure of surface are the squares described upon linear standards, such as a yard, a meter, or some other ; but the acre, the common unit for measuring land in this country, is an exception. The public lands sold by the United States are divided into square townships, each containiag thirty-six square miles, called sections, but in some of the townships there are only twenty-five square miles. If through one vertex of any polygon, however irregular, an in- definite straight line is drawn, and on this line a perpendicular is dropped from every other vertex, the whole is divided into trapezoids and trian- gles. By measuring these, the entire area is ascertained. This is the usual mode of surveying land. 17. Exercises. — 1. What is the area of a lot, in the shape of a right-angled triangle, the longest side being 100 yards, and one of the other sides 36 yards? 2. Two parallelograms having the same base and altitude are equiva- lent. To be demonstrated without using Articles 9, 10 or 11. [V, 12.] 3. The diagonals of a parallelogram divide it into four equivalent triangles. 4. Any straight line through the midpoint of the diagonals of a parallelogram divides it into equivalent parts. 5. What is the locus of the vertices of those triangles that have equal areas and a common base ? 6. The area of a triangle is equal to half the product of the perimeter by the radius of the inscribed circle. [Extend a line from the center to each vertex and to each point of contact.] 7. A triangle is divided into equivalent parts by a medial line. To be demonstrated without using any theorem in this chapter. [From the foot of the medial extend lines parallel to the sides.] 8. If is any point in a parallelogram ABCB, the triangles OAB and OCD are together equivalent to half of the parallelogram. [Art. 8.] 9. If perpendiculars fall from any point within an equilateral triangle to the three sides, their sum is equal to the altitude of the triangle. How should this be stated when the point is outside of til e triangle? 10. If two triangles have two sides of one respectively equal to two sides of the other and the included angles supplementary, the triangles are equivalent. [Ill, 7, in.] Art. 18.] QUADRILATERALS. 113 The Algebraic Method. 18« In the last chapter and in this, the length of a line has been represented several times by a single letter, the letter representing the number of times some unit of length is con- tained in the line, that is, the numerical value of the line. The measure of area is always a product of two such numbers. The area of a square is the second power of its length, etc. Any product of two numerical factors may be taken as representing a rectangular area, the factors being the two dimensions. When any homogeneous equation of the second degree is known to be universally true, one form of this truth is a principle of geometry. We owe this algebraic method of investigating geometric truth to the mathematicians of the seventeenth century. The method of reasoning directly upon the geometrical mag- nitudes was pursued by the ancients twenty centuries before ; and it is usually called, by way of distinction, the ancient method. 19, For example of the algebraic method, we know that whatever are the values of a and h, (a + 5)2 =^2 + 52 -\-2ah. This formula includes the following Theorem. — The square on the sum of two straight lines is equivalent to the sum of the squares on the two lines, increased by twice the rectangle contained by those lines. This may be demonstrated by the ancient method, as fol- lows : The sum of two lines CD and I>JE is CK Upon GE erect the K_ square JEK\ upon CD erect the square DII\ and produce the sides of this second square to iV and P. Then HK, being the difference between (7^ and CH, is equal to 6 /v JJ 114 PLANE GEOMETRY, [Chap. VL DE. PG and FK are also equal to I>Ey being parallels between parallels. Likewise, MG and GD are each equal to CD, Therefore, the square on C£J, the sum, is composed of the squares on CD and on DE, and of two rectangles, each contained by these two lines. ff Ti N- E 20. Theorem. — The square on the difference of two straight lines is equivalent to the sum of the squares on the two lines, diminished hy twice the rectangle contained hy ^ those lines. This is a consequence of the truth of the equation, {a — hy = a^—2ab-\-bK 0.-6 The student may demonstrate it also by the ancient method. 21. Theorem. — The rectangle contained hy the sum and the difference of two straight lines is equivalent to the difference of the squares on those lines. This is proved by the principle expressed in the equation, {a-\-h) (a — b) = a"-—bK It may also be demonstrated by aid of this diagram. 22. Theorem. — The square on the side that subtends an acute angle of a triangle is equivalent to the sum of the squares on the other two sides, diminished by twice the rectangle con- tained by one of those sides and the projection of the other on that side. Art. 22,] Q UADRILA TERAL8, 115 Let a, hf and c be the sides of the triangle, h the base, the angle opposite to a being acute, m and n respectively the pro- jections of a and of c on 5, and h the altitude. Then (Y, 30, and VI, 20), aS = A2 + m2 23. Theorem, — 7%e square on the side that subtends an obtuse angle of a triangle is equivalent to the mini of the squares on the other two sides, increased by twice the rectangle of one of those sides and the projection of the other on that side. Representing the lines as before, the angle opposite to a being obtuse, a3 = A2 4- m2 = A2 _|_ ^2 _|. 52 J^2nb = c2 + J2 _|_2n5. Corollary. — If the square on one side of a triangle is equivalent to the sum of the squares on the other two sides, then the opposite angle is a right angle. For it can be neither acute nor obtuse. 116 PLANE GEOMETRY, [Chap. VI. 34. Theorem. — Selecting one side of a triangle as the base, the rectangle contained by the other two sides is equivalent to the rectangle contained by the altitude of the triangle and the diam- eter of the circumscribed circle. Taking BC 2.9, the base of the triangle, AD is the altitude and AE the diameter of the circum- scribed circle. Join EC. The triangles ABB and AEG have the angles B and E equal (IV, 22, 11), also the angles A CE and ABB. Therefore, AC'.AB = AE'.AB. The product of the means being equal to that of the extremes, the proposition is proved (10, in). 25. Theorem. — The area of a triangle is measured by the product of the three sides multiplied together , divided by four times the radius of the circumscribed circle. Let b represent the base, a and c the sides, h the altitude, and r the radius. Then, as just proved. Therefore, 2rh = ac. h h _ab c 26. Theorem. — If a quadrilateral is inscribed in a circUy the sum of the two rectangles, each of which is contained by op- posite sides of the quadrilateral, is equivalent to the rectangle con- tained by its diagonals. Make the angle ABE equal to BBC. Then ABB is equal to EBC. Since BBA and BCE are equal angles, the triangles ABB and EB C are similar, and BG:CE=BB'.BA, Art. 26.] QUADRILATERALS. 117 Likewise, from the similar triangles BEA and B CD, CI):EA = BD:AB, From these proportions we have B C ' DA = BD * CE and CD • AB = BD ' EA, By addition, BC'DA -{-CD'AB = BD' CA. This is the Ptolemaic Theorem. It is found in the Almagest, written by Claudius Ptolemy in the second century. 27. Theorem. — A line bisecting the vertical angle of a tri- angle divides the base in the ratio of the other two sides. Taking AB and BC slb the bases of the two triangles ABD and CBD, they have equal altitudes (III, 13). There- fore (11, ii), area ABD : area CBD = AB : CB. Regarding AD and DC sls the bases, area ABD : area CBD = AD : CD. Therefore, AD : CD = AB : CB. Corollaries, — I. If the ex- terior angle at the vertex is bisected, the base is divided externally, in the same ratio. The words of the above demon- stration apply to this. II. Representing the sides by a and c and the base by b, the segments of the base when divided internally are externally, they are — a -{- G a b and - b and — c a a-\-G c b. When it is divided 5. 118 PLANE GEOMETRY. [Chap. VI. III. If the angle at the base and the adjacent exterior angle are both bisected, the base is divided harmonically (III, 2). IV. The locus of the points, the distances of each of which from two fixed points, A and G, are in a given ratio, is a cir- cumference cutting the line AC in the required ratio both internally and externally, and having for its diameter the dis- tance between the points of intersection. The Pythagorean Theorem. 28. Since numerical equations represent geometrical truths, the following theorem might be inferred from Article 30, Chap- ter V. This theorem, discovered by Pythagoras, is known as the Forty-seventh Proposition, that being its number in the First Book of Euclid's Elements. It has been demonstrated in a great variety of ways. The first demonstration following is from Euclid. Theorem.— The square on the hypotenuse of a right-angled triangle is equivalent to the sum of the squares on the sides that contain the right angle. Let AB C be a right-angled triangle, having the right angle JBA C. It is to be proved that the square on BC is equivalent to the squares on BA and A C. Through A, make AL parallel to BJD, and join AB and FC. Then, because each of the angles BAC and BAG is right, their sum is two right angles, and GA C is one straight line. For the same reason, BAH is one straight line. The angles FB G and BBA are equal, since each is the sum of a right angle and the angle ABG. The two triangles JF!Z? (7 and BBA are equal, for the side FB is equal to BAy and the side BG h equal to BB, and the included angles are equal, as just proved. Art. 28.] QUADRILATERALS. 119 Kow, the area of the parallelogram BL is double that of the triangle DBA, because they have the same base BB and the same altitude BL ; and the area of the square BG is double that of the triangle BB C^ because they have the same base BB^ and the same altitude jP6r. But doubles of equals are equal. Therefore, the parallelogram BB and the square B G are equiva- lent. In the same manner, by joining ABJ and BB, it is demon- strated that the parallelogram CB and the square CBT are equiva- lent. Therefore, the whole square BB, on the hypotenuse, is equivalent to the squares BG and Cir, on the other two sides of the right-angled triangle. /T\^ Another mode of demonstration / i , n. is by dividing the three squares into / ^ | N. parts, such that the several parts of /. | the large square ^re respectively l^v I / equal to the several parts of the two i , /\^ \ / ' others. The diagram shows the J .^bsZ. construction. 29. Exercises.— 1. What kind of a triangle is that whose sides are 3, 4, and 5 ? That whose sides are 14, 15, and 20 ? 14, 15, and 5 ? 12, 13, and 5 ? 2. What is the radius of the circle inscribed in the triangle whose sides are 8, 10, and 12 ? [Find the area of the triangle.] 3. The sum of the areas of the squares on the sides of a parallelogram is equal to the sum of the areas of the squares on its diagonals. [22 and 23.] 4. If m and n are any numbers, and a triangle has sides equal to m^ + n^, m^ — n^, and 2mn units respectively, the triangle is right-angled. 5. In a circle of 26 inches diameter, a chord is distant one foot from the center; how long is the chord? [lY, 11, i.] 6. If two equal chords of a circle cut each other, their segments are respectively equal. [Find the value of a segment of one in terms of a segment of the other.] 7. If two chords of a circle cut each other at right angles, the sum of the squares on the four segments is equivalent to the square on the diam- eter. 120 PLANE GEOMETRY. [Chap. VI. 8. If, from any point in a square, lines are made to the four vertices, also perpendiculars to the four sides, the sum of the areas of the squares on the first four lines is double the sura of the areas of the squares on the perpendiculars. [28.] 9. If a straight line is divided internally or externally, the sum of the areas of the squares on the segments is double the sum of the areas of the squares on half the line and on the line between the point of division and the midpoint of the given line. 10. Given the lengths of the bases and of the altitude of a trapezoid, to find the distance from either base to the point of meeting of the oblique sides produced. [V, 22, i.] 11. When the base of a triangle is given, also the difference of the squares of the sides, the locus of the vertex is a straight line perpendicu- lar to the base. [Y, 31.] 30. Scholium. — The study of triangles developed the doc- trine of similarity. The principles of mensuration of surfaces and the theory of equivalent figures are developed from the properties of quadrilaterals. The algebraic method, applied to investigations in the higher mathematics, has led to the greatest achievements in the science. The Pythagorean Theorem and the theory of similar trian- gles are the basis of Trigonometry. 31, Miscellaneous Exercises. — 1. If two triangles have a common base, but are on opposite sides of it, the line Joining their ver- tices is cut by the base in the ratio of the areas of the triangle. [11, II.] 2. In any parallelogram, the distance of one vertex from a straight line passing through the opposite vertex is equal to the sum or difference of the distances of the line from the other two vertices, according as the line is without or within the parallelogram. [3.] 3. From any point in the base of a triangle, lines are made parallel to the two sides: find the locus of the centers of symmetry of the parallelo- grams so formed. [6.] 4. If, from the sides of the triangle ABC^ AD^ BE, and CF are cut off, each equal to two thirds of the side from which it is cut, then the area of the triangle DEF is one third that of ABG. [14.] 5. Find the locus of the points such that the sum of the squares of Art. 31.] QUADRILATERALS. 121 the distances of eacli from two given points is equivalent to the square of the line joining the given points. [28.] 6. If a triangle is equilateral, the radii of the inscribed, circumscribed, and escribed circles are in the ratio of 1, 2, 3. [V, 43, Ex. 11.] Problems in Drawing. 33, The Squaring or Quadrature of a figure consists in drawing a square equivalent to it. I. — To draw a polygon equivalent to a given one, hut with a less num- ter of sides. Let ABODE be the given polygon. Join DA. Produce BA, and through E draw ^F parallel to DA. Join DF. The triangles DAF and DAE are equivalent, for they have the same base DA, and their vertices are in the line EF parallel to the base. To each of these equals add the figure ABGD, and we have the quad- rilateral FBGD equivalent to the polygon ABODE. In this manner, the number of sides may be diminished till a triangle is formed equivalent to the given poly- gon. In this diagram it is the triangle FDG. II. — To represent \/ab geometrically. ^ The square root of the product of two numbers is their mean propor- tional. The problem assumes that the letters represent lines. Therefore, find a mean proportional between two lines of the lengths a and & (V, 44, ix). Geometrical representation of algebraic formulas is used in the solution of problems in drawing. When a relation between the magnitudes given and the magnitudes required can be stated as an equation, the algebraic solution may be made first. All that remains is to represent the result geometrically. III. — To draw a rectangle with a given base, equivalent to a given par- allelogram. The altitude of the rectangle is to be found. If c is the given base, and & and a the dimensions of the given parallelogram, let x represent the required altitude. Then c : 5 = a : a; (V, 44, v). 122 PLANE GEOMETRY. [Chap. VL ly. — To draw a square equivalent to a given parallelogram. If & and a are the dimensions of the parallelogram, let x represent the side of the square. Then x^ = 5a. Therefore a; = Via. Y. — To draw a square equivalent to a given triangle. aj* = i &a. Therefore ^l):x = x:a. 33. Exercises in Drawing. — 1. To divide a given triangle into any number of equivalent triangles. 2. To divide a given trapezoid into any number of equivalent trape- zoids. 3. To divide a straight line into two segments such that the area of the square on one is four times that of the square on the otlier. 4. Represent this formula by a diagram: (a + 5 + c)^ = a^ + 52 + c^ + 2 a& + 2 ac + 2 &c. 5. To represent \^{a + l) {a — I) geometrically. 6. To draw a square equivalent to two given squares. To three. 7. To divide any two squares into parts that may be combined to form one square. 8. To draw a square with a given straight line as its diagonal. 9. To draw a triangle, having the base and one of the angles at the base given, and an area equal to that of a given square. 10. To draw a parallelogram equivalent to a given triangle, and having an angle equal to a given angle. 11. From a given isosceles triangle, to cut off a trapezoid having the same base as the triangle, and the remaining three sides equal to each other. 12. To divide a triangle into two equivalent parts, by a line drawn from a given point in one of the sides ; viz., to divide the triangle ABG into equiva- lent parts by a line from D. 13. To divide a triangle into three equivalent triangles by lines that meet at a point within the given tri- angle. 14. Can every triangle be divided into two equal parts? Into three? Into nine ? CHAPTER YII. POLYGONS. Article 1. — Polygons have been defined and classified according to the number of sides (II, 15). A Convex Polygon has all its diagonals interior. A Concave Polygon has at least one diagonal exterior. If the angles are reojarded as toward the figure, a concave polygon must have a reflex angle. A Regular Polygon is both equilateral and equiangular. The square and the equi- lateral triangle are regular polygons. Diagonals. 2. Theorem. — The number of diagonals from any vertex of a polygon is three less than the number of sides. For, from any vertex, a diagonal may extend to every other vertex except one on each side. Corollaries. — I. The diagonals from one vertex divide a polygon into as many triangles as the polygon has sides, less two. A polygon may be divided into a greater number of trian- gles, in various ways ; but not into a less number than here stated. II. The whole number of diagonals possible in a polygon of n sides, is i ti (w — 3). For, if we count the diagonals at all the 124 PLANE GEOMETRY. [Chap. VII. n vertices, we have n {n — 3), but this is counting each diagonal at both ends. Sum of the Angles. 3. Theorem. — The sum of all the angles of a polygon is equal to twice as many right angles as the polygon has sides, less two. For the polygon may be divided into as many triangles as it has sides, less two ; and the angles of these triangles coincide altogether with those of the polygon. The sum of the angles of each triangle is two right angles. Therefore, the sum of the angles of the polygon is equal to twice as many right angles as it has sides, less two. In applying this theorem to a concave figure, the value of the re-entrant angle must be taken on the side toward the poly- gon, and therefore as reflex. Let B, represent a right angle ; then the sum of the angles of a polygon of n sides is 2 (/i — S) i2 ; or, it may be written thus, {2n — 4) 72 ; that is {n — 2) straight angles. 4. Theorem:. — If each side of a convex polygon is produced, the sum of all the exterior angles is equal to four right angles. Let the sides be produced either all to the right or all to the left. Then, from any point in the plane, extend lines parallel to the sides thus produced, and in the same directions. The angles thus formed are equal in number to the exterior angles of the polygon, and are respectively equal to them (III, 21). But the sum of ^/\ those formed about the / ^v?- \ ' point is equal to four / j ^V\^. right angles. /' \ 1 Therefore, the sum V^^;- — — — J. « of the exterior angles \ of the polygon is equal to four right angles. i") Art. 4.] POLYGONS, 135 This is also true of concave polygons, if the angle formed by producing one side of the re-entrant angle is taken negatively. Thus, the sum of the exterior angles at -S, (7, JD, E, F, and G, less the angle j5j is four right angles. In going from B, around the polygon, in the order of the letters, every exterior angle is a divergence, or difference of direction, to the right ; but at H the divergence is to the left. Subtracting H from the sum of the others, the result is four right angles. Equal Polygons, 5. Theorem. — Two polygons are equal when they are com- posed of the same number of triangles respectively equal and similarly arranged. This is an immediate consequence of the definition of equal- ity (II, 4). Corollary. — Conversely, two equal polygons may be divided into the same number of triangles respectively equal and simi- larly arranged. 6. Theorem. — Two polygons are equal when all the sides and all the diagonals from o?ie vertex of one are respectively equal to the same lines in the other and are similarly arranged. For each triangle in one, having its sides respectively equal to those of the similarly situated triangle in the other polygon, is equal to it. Corollary. — 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. 7. Theorem. — Two polygons are equal when all the sides and angles of one are respectively equal to the same elements of the other and are similarly arranged. For each triangle in one is equal to its homologous tri- 126 PLANE GEOMETRY. [Chap. VII. angle in the other, since they have two sides and the included angle equal. It is enough for the hypothesis of this theorem, that all the angles except three are equal. Corollaries. — I. Two quadrilaterals are equal when the four sides and an angle of one are respectively equal to the four sides and the similarly situated angle of the other. II. Two parallelograms are equal when two adjacent sides and the included angle of one are respectively equal to those elements of the other. III. Two rectangles are equal when two adjacent sides of one are respectively equal to those elements of the other. IV. Two squares are equal when a side of one is equal to a side of the other. 8. Exercises. — 1. "What is the number of diagonals that can be in a pentagon ? 2. How many sides has that polygon the number of whose diagonals is seven times the number of sides? 3. "Wiiat is the sum of the angles of a hexagon? Of a dodeca- gon? 4. A convex polygon can not have more than three acute an- gles. 5. Join any point witbin a given polygon with every vertex of the polygon, and with the figure thus formed demonstrate the theorem, Article 3. 6. Demonstrate Article 4 by means of Article 3. Akt. 9.] POLYGONS. 127 Similar Polygons. 9. Theorem. — Similar polygons are composed of the same number of triangles, respectively similar and similarly arranged. Since the figures are similar, every angle in one has its cor- responding equal angle in the other. If homologous diagonals divide the polygons into triangles, every angle formed has its corresponding equal angle. Therefore, the triangles are respect- ively similar and are similarly arranged. Corollary. — Two similar polygons may be so placed that their homologous sides are parallel. Then there is a center of similarity as in the case of triangles (V, 27 and 28), which is external when the corresponding sides are arranged in the same order, and interaal when they are in the reverse order. 10. Theorem. — If two polygons are composed of the same number of triangles respectively similar and 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 any other angle formed by lines in one has its corresponding equal angle in the other poly- gon. Whatever line is made in one polygon, a homologous line may be made in the corresponding similar triangle or triangles of the other polygon. Since the triangles are similar, every angle made by one of these lines has its corresponding equal angle in the other polygon. 128 PLANE GEOMETRY. [Chap. VII. 11, Theorem. — Two polygons are similar when the angles formed by the sides are respectively equal, and there is the same ratio between each side of one and its homologous side of the other polygon. Let all the diagonals possible extend from a vertex A of one polygon, and the same from the homologous vertex B of the other polygon. The triangles AEI and B CD are similar, because they have two sides proportional, and the included angles equal. Therefore, EI : CD = AI : BD. But, by hypothesis, EI\CD = IO\DF. Then, AI.BjD = IO'. DF. Subtract the equal angles EIA and CBB 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, every triangle of the first polygon is similar to its corresponding triangle in the other. Therefore, the figures are similar. As in the case of equal polygons (7), 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. 1 2 . Theorem.. — In similar polygons the ratio of two hoinolo- gous lines is the same as of any other two homologous lines. Art. 12.] POLYGONS. 129 For, since the polygons are similar, tlie triangles which com- pose them are also similar, and AE\BC = EI', CD = AI: J3D = 10 : DF, etc. This common ratio is the linear ratio of the two figures. Corollary. — The perimeters of similar polygons are to each other as any two homologous lines. 13, Theorem. — The area of any polygon 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, FGH, and AEI, OUT, be divided into triangles by homologous diagonals. The triangles are re- spectively similar. 8 2 Therefore (YI, 13), area BCD : area AEI — BD I'AI = area BDE : area AIO = BE : 'AO = area BEG : area AOV = BG'.AU= area BGH: area A UY. Selecting from these equal ratios the triangles, area B CD : area AEI = area BDE : area AIO = area BEG : area AOZT = area B GIT : area A TIY. 130 PLANE GEOMETRY. [Chap. VII. Therefore (I, 3, vi), area BCDFGHB : area AEIOUTA 8 Z = area B CD : area A£JI ; or, as ^ (7 : AF ; or, as the areas of any other homologous parts ; or, as the squares of any other homologous lines. Corollary, — The superficial ratio of two similar polygons is the second power or duplicate of their linear ratio ; and con- versely, the linear ratio is the square root of the superficial ratio. 14, Exercises. — 1. Compose two polygons of the same number of triangles respectively similar,- but not similarly arranged. 2. What is the relation between the areas of the equilateral triangles described on the three sides of a right-angled triangle? 3. Two parallelograms are similar when they have an angle in one equal to an angle in the other, and these equal angles included between proportional sides. 4. Two irregular garden plats, of the same shape, contain, respect- ively, 18 and 32 square yards: required their linear ratio. 5. If, through any point in the diag- onal of a parallelogram, lines are made parallel to the sides, four parallelograms are formed ; the two through which the diagonal does not pass are equivalent. These two are called the complements of the parallelograms about the diagonal. Thus a and 5 are the complements. Regular Polygons. 15. Theorem. — Lines bisecting the several angles of a regu- lar polygon all meet in one point ; this point is equally distant from all the vertices, and is also equally distant from all the sides of the polygon. Every triangle formed by one side of the polygon and two con- ^ b secutive bisecting lines is isosceles (V, 6), and all the triangles so formed are equal (V, 12). Hence, BO is cut by the lines AO and (70 at the same point. Thus all Art. 15.] POLYGONS. 131 the bisecting lines meet at one point which is equally distant from the vertices. These equal isosceles triangles must have equal altitudes, that is, the point is equally distant from all the sides of the poly- gon. Corollaries. — I. Every regular polygon may have a circle circumscribed about it. II. Every regular polygon may have a circle inscribed in it. III. 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. IV. Every regular polygon may be divided into as many equal isosceles triangles as the polygon has sides. V. Conversely, a polygon composed of equal isosceles tri- angles having their vertices at a common point is regular. The Center of a Regular Polygon is the point equally distant from the vertices. The Radius is the distance from the center to a vertex. The Apothem is the distance from the center to a side. VI. If a regular polygon has an even number of sides, a line from one vertex through the center, being produced, passes through the opposite vertex, and is an axis of symmetry. VII. If a regular polygon has an odd number of sides, a line from one vertex through the center, being produced, bisects the opposite side at right angles, and is an axis of symmetry. VIII. Every straight line through the center of a regular polygon of an even number of sides cuts the perimeter at equal distances from the center ; and the center of such a polygon is a center of symmetry. 16, Theorem, — If the circumference of a circle is divided into equal arcs, the chords of those arcs are the sides of a regu- lar polygon. For the sides are all equal, being the chords of equal arcs ; and the angles are all equal, being inscribed in equal arcs. 132 PLANE GEOMETRY. [Chap. YII. lY. Theorem. — If a circumference is divided into equal arcs, and lines tangent at the several points of division are pro- duced 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 6^^, ^^, and^a The triangles GAF, ABB, 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 G equal, for each is formed by a tangent and chord which intercept equal arcs. Therefore, these triangles are all isosceles, and all equal ; and the angles F, D, and E are equal. Also, FB and BE, be- , ing doubles of equals, are equal. In the same manner, it is proved that all the angles of the polygon FBE are equal, and that all its sides are equal. That is, it is a regular polygon, circumscribed about the circle. Regular Polygons Similar. 18. Theorem., — Regular polygons of the same number of sides are similar. For, making all the radii in the polygons compared, each is seen to be composed of as many triangles as it has sides (15, iv). All these triangles are similar, since their angles are respectively equal ; and they are similarly arranged in the polygons. There- fore, the polygons are similar (10). Corollaries. — I. The areas of two regular polygons of the same number of sides are to each other as the squares of their homologous lines. II. In regular polygons of the same number of sides, any two certain lines have a constant ratio (12). Art. 18.] POLYGONS. 133 For instance, the ratio of the diagonal to the side of a square is always as the square root of 2 is to 1, since the *\ square on the diagonal is equiv- alent to the sum of the squares on two sides (13, c). Similarly, the ratios between the side and apothem, between the side and radius, and between the radius and apothem of a regular polygon of a certain num- ber of sides, are constant. III. If two similar regular polygons of an even number of sides have their sides respectively parallel when taken in the same order, they are also respectively parallel when taken in re- verse order. Therefore two such polygons so placed have two centers of similarity, one external and one internal (9, c) ; but, if the polygons are also placed concentrically, the two centers of similarity coincide at the center of the polygons. 19. Exercises.— \. First in right angles, and then in degrees, ex- press the value of an angle of each regular polygon, from three sides up to ten. 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. The exterior angle of a regular polygon being one third of a right angle, find the number of sides of the polygon. 4. A plane may be so covered by equilateral triangles as to leave no intervening surface. 5. A plane may be covered in the same way by equal squares. 134 PLANE GEOMETRY. [Chap. VII. 6. A plane may be covered in the same way by regular hexagons. 7. A plane can not be covered in the same way by equal regular polygons of any other number of sides. 8. Find the ratios between the side, the radius, and the apothem, of a square. [The apothem, radius, and half side of every regular polygon form a right-angled tri- angle. If we know any one of the ratios from the nature of the polygon, the Pythagorean Theorem enables us to find the others.] 9. Find the ratios between the side, the radius, and the apothem, of a regular hexagon. [What sort of a triangle is made by a side and two radii?] 10. The area of an inscribed regular hexagon is three fourths that of a regular hexagon circumscribed about the same circle. Maxima and Minima. 30. A Maximum magnitude is the greatest of its kind. A Miuimum magnitude is the least of its kind. For example, a diameter is the maximum chord in a given circle ; and a perpendicular is the minimum line from a given point to a given straight line. Isoperimetrical Figures are inclosed surfaces whose perimeters have the same extent. Theorem. — The shortest line that extends from one point to another, through some point of a given straight line, mahes equal angles with that line. Suppose first that the points are on opposite sides of the line. Then the shortest line joining them is a straight line cutting the given line and making equal angles. If the points are on the same side of the straight line, the shortest line from one to the other through a point of that line must be composed of two straight lines. 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, It is to be proved that AEG and BUB are equal angles. Art. 20.] POLYGONS. 135 Make ^J9^ perpendicular to CD, and produce it to F, making HF equal to AH. Every point of the line CD is equally distant from A and F. Therefore, every line joining B to F through some point of CD is equal to a line joining jB to ^ through the same point. Thus BGF is equal to B GA, and BFF is equal to BFA. Since BFA is the shortest line from B to A through any point of CD, BFF is the shortest line from B o F and it is therefore a straight line. Then the angles FFH and BFD, being vertical, are equal ; but FFjB: and AFBT are equal (III, 10, iv). Therefore AFBT and BFD are equal. 21. Theorem. — Of all equivalent triangles of a given hose, the one having the least perimeter is isosceles. The equivalent triangles having the same base, AE, have also the same altitude, and their vertices are in the same line parallel to the base. The shortest line that can be made from ^ to ^ through some point of DB constitutes the other two sides of the tri- angle of least perimeter. This shortest line is the one making equal angles with DB, that is, making A CD and FCB equal. The angle A CD is equal to its alternate A, and the angle FCB to its alternate F. Therefore, the angles at the base are equal, and the triangle is isosceles. 22. Theorem. — Of all equivalent polygons of the same number of sides, the one having the least perimeter is regu- lar. 136 PLANE GEOMETRY. [Chap. VII. Of several equivalent polygons, suppose AB and ^ C to be two adjacent sides of the one having the least perimeter. It is to be proved, first, that these sides are equal. Join AC. Now, if AB and BG were not equal, there could be constructed on the base ^ (7 an isosceles triangle equivalent to AB (7, whose sides would have less extent. Then, this new tri- angle, with the rest of the polygon, would be equivalent to the given polygon, and have a less perimeter, which is con- trary to the hypothesis. It follows that AB and B G must be equal. So of every two adjacent sides. Therefore, the polygon is equilateral. It remains to be proved that the polygon has all its angles equal. AB, BG, and GI) being consecutive sides, produce AB and GD till they meet at E. Now the triangle BGE is isosceles. For, if EG, for example, were longer than EB, we could then take EI equal to jES, .and EF equal to EG, and, joining EI, make the two triangles EB G and EIF equal (V, 11). Then, the new polygon, having AFID for part of its perimeter, would be equivalent and isoperimetrical to the given polygon having ABGD as part of its perimeter. But the given polygon has, by hypothesis, the least possible peri- meter, and, as just proved, its sides AB, BG, and GD are equal. If the new polygon has the same area and perimeter, its sides also must be equal ; that is, AF, FI, and ID. But this is ab- surd, for AF is less than AB, and ID is greater than GD. Therefore, the supposition that EG is greater than EB, which Art. 22.] POLYGONS, 137 supposition led to this conclusion, is false. Hence, EB and JEC are equal. Therefore, the angles EBQ and EGB are equal, and their supplements ABC and BCD are equal. Thus, it is shown that consecutive angles are equal. It being proved that the polygon has its sides equal and its angles equal, it is regular. 33. Theorem. — Of all isoperimetrical polygons of the same number of sides, that which is regular has the greatest area. Compare two isoperimetrical polygons of the same number of sides, one of the two being regular. Designate the regular poly- gon as By the other as I. There may be a third polygon similar to I and equivalent to B (II, 3). By the last theorem, the perimeter of this third polygon is greater than that of B, therefore greater than the perimeter of I. As the third polygon is similar to I and has a greater perimeter, it must have a greater area (13, c). Therefore its equivalent B has a greater area than I, 24, Theorem. — Of regular equivalent polygons, that which has the greatest number of sides has the least perimeter. It will be sufficient to demonstrate the principle, when one of the equivalent polygons has one side more than the other. In the polygon having the less number of sides, join the ver- tex C to any point, as H, of the side B G, Then, on CH con- struct an isosceles triangle, GKI£, equivalent to CBS, 7 138 PLANE GEOMETRY, [Chap. VII. Then JIK and KC are less than SB and B G ; therefore, the perimeter GHKCDF is less than the perimeter of its equivalent polygon GB CDF. But the perimeter of the regu- lar polygon AO \s> less than the perimeter of its equivalent irregular polygon of the same number of sides, GHKCDF, Therefore, it is less than the perimeter of GB CDF. 25. Theorem, — Of several regular isoperimetrical polygons the greatest is that which has the greatest number of sides. Designate two regular isoperimetrical polygons, one having m sides, as M: and one having more than m sides, as N', There may be a third polygon similar to M and equivalent to JST, By the last theorem, the perimeter of this third polygon is greater than of N", or its equal, the perimeter of 3f, As this third polygon is similar to M and has a greater perimeter, it must have a greater area (13, c). Therefore its equivalent iV has a greater area than M. 26. Application. — When a ray of light is reflected, the incident and reflected parts of the ray make equal angles with the surface. That is, light, when reflected, adheres to the law that requires it to take the shortest path. 27. Exercises. — 1. Three houses are built with walls of the same aggregate length ; the first in the shape of a square, the second of a rectangle, and the third of a regular octagon. Which has the greatest amount of room, and which the least? Art. 27.] POLYGONS, 139 2. Of all equivalent parallelograms having equal bases, what one has the minimum perimeter? 3. 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. 4. 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 ? S8. Scholium. — Certain principles which had been demon- strated as to equality and similarity of triangles and as to cen- ters of similarity have been generalized in this chapter. That is, they are proved to be applicable to all polygons. The knowl- edge of the regular polygon has been added, and some princi- ples of maxima and minima. 29. Miscellaneous Exercises. — 1. Show that the side, the radius, and the apothem, of a regular triangle have the ratios of 2V^3; 2; 1. 2. Show that the corresponding ratios in a regular octagon are as 2 ; V4+ 2 V2; (1 + /2). 3. If two diagonals of a regular pentagon cut each other, both are divided in extreme and mean ratio. [Circumscribe a circle about the polygon. Compare V, 44, xi.] 4. All the diagonals being formed in a regular pentagon, the figure inclosed by them is a regular pentagon. 6. Show that the side, the radius, and the apothem, of a regular pen- tagon have the ratios of i i^lO — 2 V"5 ; 1 ; i (1 + V^). 6. If a regular pentagon, hexagon, and decagon are inscribed in a circle, a triangle having its sides respectively equal to the sides of these three polygons is right-angled. [YI, 23, c] 7. If, from any point within a given regular polygon, perpendiculars fall on all the sides, the sura of these perpendiculars is a constant quan- tity. [Make a line from the point named to every vertex of the poly- gon.] 140 PLANE GEOMETRY. [Chap. VU. Problems in Drawing. 30. Problems. — I. To draw a polygon equal to a given polygon. II. — To draw a polygon when all its sides and all the diagonals from one vertex are given in their order. These depend on Y, 44, i. III. — To draw a polygon when the sides and angles are given in their order. It is enough for this problem if all the angles except three are given. Suppose first that the angles not given are consecu- tive, as at D, Bj and C. Draw the trian- gles «, c, i, and 0. Having DC, complete the polygon by drawing the triangle DBG from its three known sides. Suppose the angles not given are 2>, G, and F. Then draw the triangles a, e, and i, and separ- ately, the triangle u. Having the three sides of the triangle o, it may be drawn, and the polygon completed. The problem, to iDscribe a regular polygon in a circle by- means of straight lines and arcs of circles, can be solved in only a limited number of cases. The solution depends upon the divisiop of the circumference into equal arcs ; and this depends upon the division of the sum of four right angles into equal parts. ly. — To inscribe a square in a given circle. Draw two diameters perpendicular to each other. Join their extrem- ities by chords. These chords form an inscribed square. For the angles at the center are equal by construction. Therefore, the subtending arcs are equal, and the chords of those arcs are the sides of a regular polygon. V. — To inscribe a regular hexagon in a circle. Suppose the figure completed; then drawing two successive radii, we have ^^ ^ the triangle ABG. /f Y\ The angle G is 60° (15, iii). Hence n \\ the sum of the angles at A and B is 120°, // \l ^ V C t A and, as they are equal, each is 60°. There- K \ A fore, the triangle is equilateral, and one \\ \ /I side of a regular hexagon is equal to its \\ \l^ radius. ^- P^-ff Art. 30.] POLYGONS, 141 The solutiou of the problem is— apply the radius to the circumfer- ence six times as a chord. YI. — To inscribe a regular decagon in a circle. Divide the radius CA in extreme and mean ratio at the point B (V, 44, x), making BG the greater part. Draw the chord AD equal to BG. This chord is one side of the inscribed decagon. For the angle G is one fifth of a straight angle (Y, 44, xi). Hence, the arc AD is one tenth of the circumfer- ence. Corollary, — Many other regular poly- gons may be drawn by means of the above three. To inscribe an equilateral triangle, join the alternate vertices of the hexagon. To inscribe a regular pentagon, join the alternate vertices of the decagon. To inscribe a regular pentedecagon, subtract the arc subtended by the side of the decagon, 36°, from that subtended by the side of the hexa- gon, 60°. The remainder, 24°, is one fifteenth of the circumfer- ence. Having an inscribed regular polygon, to inscribe one of double the number of sides, bisect each arc subtended by a side of the given polygon, and draw the chords of all these half arcs. YH. — To circumscribe a regular polygon about a circle. Divide the circumference into the requisite number of equal arcs by the above methods for inscribing regular polygons. Through the points of division draw tangents. These, produced til] they meet, form the re- quired polygon (17). Scholium. — It has been demonstrated by Gauss that any regular polygon can be drawn, using only straight lines and arcs of circles, if the number of sides is prime and is included in some value of the expression 2" -j- 1, the exponent n being any whole number. This includes the following numbers : 3 ; 5 ; 17 ; 257 ; and 65537. No other numbers included in this for- mula are known to be prime. Any other such prime number must exceed the sixty-fourth power of 2, 143 PLANE GEOMETRY. [Chap. VII. 31. Exercises in Drawing. — 1. To draw a quadrilateral when the four sides and one angle are given. 2. To draw a parallelogram when two adjacent sides and an angle are given. 3. To draw a parallelogram, having the diagonals and one side given. [VI, 6.] 4. To bisect any quadrilateral by a line from a given vertex, that is, to divide it into two equivalent parts. 5. To draw the mimimum tangent from a given straight line to a given circumference. [Ill, 10.] 6. To draw a triangle similar to a given triangle, but with double the area. [13, o.] 7. To construct a regular octagon of a given side. 8. Given a regular inscribed polygon, to circumscribe a similar poly- gon whose sides are parallel to the former. 9. To inscribe a square in a given segment of a circle. CHAPTER VIII. CIRCLES. Article 1. — The properties of circumferences and of straight lines in connection with them have been developed in Chapter IV. A Segment of a circle is the part cut off by a secant or chord ; asAJBC.oT GDE. A Sector of a circle is the part between two radii and the arc intercepted by them ; as GRL The liimit of Polygons. 3. Theorem. — If a polygon inscribed in a circle is made to vary by bisecting every arc subtended by a side and Joining these midpoints to the ends of the sides so as to double the number of sides of the polygon at every step, the circle is the limit of this variable polygon. Also, if a circumscribed polygon is made to vary by bisect- ing every arc between the points of tangency and making tan- gents at these midpoints so as to double the number of sides at every step, the circle is the limit of this variable circumscrihed polygon. That is, the variable polygon can approximate the circle within less than any assignable difference, but can never become equal to or pass it. 144 PLANE GEOMETRY. [Chap. Via Suppose the chord AB a Bide of an inscribed polygon, and CD and CJE sides of a circumscribed polygon, tangent at A and B. Make the tangent FG par- allel to AB, and join AH and BH, By a similar construction at every arc subtended by a side of the inscribed polygon, we have a second inscribed polygon, having for two of its sides the chords AH and HB^ and a new circumscribed polygon having for part of its perimeter the tangents AF, FG, and GB, The difference between the first inscribed and the first circumscribed polygon is the sum of the triangle ABC and the other similarly situated triangles. The difference between the second inscribed and the second cir- cumscribed polygon is the sum of the triangles AFH, HB G, and the others similarly situated. The triangle AHB, having the same altitude as the trape- zoid AFGB and its base the greater base of the trapezoid, has more than half the area of the trapezoid. Hence, the area of the triangles AFH and HGB is less than half the area of the trapezoid, and much less than half of the triangle ABC. As the same reasoning applies to the construction upon every side, the second inscribed and the second circumscribed polygon approximate each other by more than half the previous differ- ence of extent. Repeating this process, the inscribed and circumscribed poly- gons are variable figures which approximate each other in such a way that less than half of the difference of their areas remains at*every step. It can in this manner become less than any designated extent of surface. But at every step the circle con- tains the inscribed polygon and is therefore greater than it ; and is contained by the circumscribed polygon and is therefore less than it. Art. 2.] CIRCLES, 145 Neither perimeter can ever pass the curve, nor coincide with it, for, however short the sides may be, they are straight lines. The circle is therefore the limit of both the variable poly- gons. The variable polygon may be regular or not, and the curve may be that of the circle or any other. Corollaries. — I. The circumference is the limit of the perimeters of the variable inscribed and circumscribed poly- gons. 11. The superficial ratio of similar plane figures, whether curvilinear or rectilinear, is the square of their linear ratio. 3. Theorem. — A curve is shorter than any line thai Joins its ends, and toioard which it is convex. For a broken line having its vertices in JBD G may be made to jP approximate BDG within less than any assignable difference. But such broken line is less than BFC (III, 3). Therefore, BDG " must be less than BFG. Corollary. — The circumference of a circle is shorter than the perimeter of a circumscribed polygon. That the circumference is longer than the perimeter of any inscribed polygon, follows immediately from the Axiom of Dis- tance. 4. Theorem. — A circle has a less perimeter than any equiva- lent polygon. For, of equivalent polygons, that has the least perimeter which is regular (VII, 22), and has the greatest number of sides (VII, 24). Corollary. — A circle has a greater area than any isoperi- metrical figure. 146 PLANE GEOMETRY. [Chap. VIII. Similarity of Circular Figures. 5. Theorem. — Circles are similar figures. Whatever lines are made in one circle, homologous lines making equal angles may be made in another (IV, 20 to 26). Corollaries. — I. Two arcs are similar when they subtend equal angles at the center. II. Sectors are similar when their arcs are similar. Segments likewise. III. Similar arcs have the same amount of curvature ; and conversely. 6« Theorem, — The ratio of the circumference to the diam- eter is a constant quantity. For the perimeters of similar polygons have the same ratio to homologous lines. Then, the circumferences, which are the limits of the perimeters, must have the same ratio to their re- spective diameters. This constant ratio is designated by the Greek letter tt, the initial of perimeter. Then, if the radius is H, the circumference is 27ri2. Corollaries. — ^I. Any homologous lines in similar circular figures (that is, circles or parts of circles) are proportional. II. The curvatures of two equivalent arcs of different radii are inversely as the lengths of their radii. III. Of two arcs having a common chord, that having less curvature has less extent. lY. If two minor arcs have a common chord, the shorter arc is the one that has the longer radius ; and conversely. V. If two major arcs have a common chord, the shorter arc has the shorter radius ; and conversely. 7. Theorem.. — Any two circles in one plane have an inter- nal and an external center of similarity , viz., those points in the line joining the centers, where that line is divided in the ratio of the radii. Let B and C be the centers of two circles, and F and G Art. 7.] CIRCLES. 14T the points where the central line is divided internally and exter- nally in the ratio of the two radii. Let BH and CE be any two parallel radii lying in the same direction from B and O. Then aB:GC = BH'.CE, Let BH and CD be any two parallel radii, in opposite di- rections from B and C. Then FB\FG-BH\ CD, Therefore, the points 6r, E^ and H are in one straight line ; also, the points Z>, F, and 11. Compare VII, 18, iii. Rectification of the Circumference. 8. The RectifLcation of a curve consists in finding a straight line of the same extent. It has been demonstrated that this cannot be done for the circumference of a circle by the methods allowed in problems in drawing ; but the length of the curve may be calculated to any required degree of approxima- tion. The number tt is less than 4 and greater than 3. For, if the diameter is 1, the perimeter of the circumscribed square is 4, but this is greater than the circumference ; and the perimeter of 148 PLANE GEOMETRY. [Chap. VIU. the inscribed regular hexagon is 3, but this is less than the cir- cumference. To calculate this number more accurately, use the following Theorem. — When one of two isoperimetrical polygons has half as many sides as the other, 1. The apothem of the second is half the sum of the radius and apothem of the first ; and 2. The radius of the second is a inean proportional between the radius of the first and the apothem of the second. Let ^0 be the apothem and BO the radius of the first polygon. Then, making the right-angled triangle BA O, AB is half the side of the polygon, O is the center, and BFG, described with OB as a radius, is an arc of the circum- scribed circle. Produce AO to the circumference at (7, join BC, and make 01) perpen- dicular to B Cy and I)£J perpendicular to^a The line OB being half the line CB (IV, 11, i), JEJB is half of AB (Y, 20, ii). Therefore, JEJB is half of the side of the second polygon, which has double the number of sides and same perimeter as the first. Since the angle ECD is half of the angle A OB (lY, 22, i), DG\^ the radius and EC the apothem of the second polygon. Now, EG is half of A G, that is, half of the sum of the apothem and radius of the first polygon. In the right-angled triangle 0J9C (Y, 29) CO'.GD=GB: GE. That is, the radius of the second polygon is a mean proportional between the radius of the first polygon and the apothem of the second. Representing the apothem of the first polygon by a, the radius by r, and the apothem of the polygon of double the num- ber of sides by x, and its radius by y, x-=. \{a-\-r), and y = V ^^' Art. 9.] CIRCLES. 149 9, Problem, — To find the approximate value of the ratio of the circumference to the diameter of a circle. In any regular polygon, the apothem, the radius, and half the side form a right-angled triangle. Hence, r^ = a^ + (|.s)3... a= ^^s — i^s z= ^ \/ 4.r^ —s^. Suppose a regular hexagon, whose perimeter is unity. Then its side is ■}- or .167 nearly, and its radius is the same. The apothem h ^ V f^ — "sV = 1*2- '^ ^> ^^ -1^^ +• Then, by the formula, x ^ \ {a -\- r), find the apothem of the regular dodecagon, whose perimeter is unity. It is .156 nearly. The radius of the same, by the formula, y = V a;r, is .161 nearly. Proceeding in this way, find the apothem and radius of the iso- perimetrical polygon of 24 sides, and so on. Number op Sides. Apothem. Radius. 6 .144 + .167 — 12 .156 — .161 — 24 .158 + .160 — 48 .159 — .159 + Since the perimeter in the above calculation is always unity, these figures indicate the ratios of the apothems and radii of the polygons to their respective perimeters. These are the same as the ratios of the radius of a circle to the perimeters of the cir- cumscribed and inscribed polygons. Since the circumference is less than the former and greater than the latter, the ratio of the radius to the circumference must be between the values of any pair of the above ratios. It is shown, at the first step, that it must be greater than .144 and less than .167. As the calculation proceeds, the ratios approximate. As far as they agree, the ratio of the radius to the circumference of a circle is ascertained. Hence we know that .159 is this ratio within one thousandth. By calculating to a greater number of places, and continuing the table, the ratio may be calculated to a greater degree of accuracy. The ratio of the diameter to the circumference must be twice .159 = .318. Therefore, TT = 1 -f- .318 = 3.14 +. 150 PLANE GEOMETRY, [Chap. VIII. 10. Exercises. — 1. 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 territory ? 2. If two circles touch each other, any line through the point of con- tact cuts off similar segments. 3. What is the locus of the midpoints of the lines extending from a given point to a given circumference ? 4. If two circles touch each other, any two straight lines extending through the point of contact are cut proportionally by the circumfer- ences. 5. If two circles are tangent to each other externally and to a straight line on the same side of them, that part of the common tangent which is between the points of contact is a mean proportional between the two diameters. [At the point of contact of the two circles, erect a perpen- dicular to the central line ; from the point where this meets the tangent, make a line to the center of each circle.] 6. Two wheels, whose diameters are twelve and eighteen inches, are connected by a belt, so that the rotation of one causes that of the other. The smaller makes twenty-four rotations in a minute ; what is the velocity of the larger wheel ? 7. Two wheels, whose diameters are twelve and eighteen inches, are fixed on the same axle, so that they turn together. A point on the rim of the smaller moves at the rate of six feet per second, what is the ve- locity of a point on the rim of the larger wheel ? 8. If the radius of a car-wheel is thirteen inches, how many rotations does it make in traveling one mile? 9. If the equatorial diameter of the earth is 7926 miles, what is the length of one degree of longitude on the equator? Quadrature of tlie Circle. 11. The Quadrature or Squaring of a Circle consists in finding an equivalent rectilinear figure. Theorem. — The area of a circle is equal to half the product of its circumference hy its radius. First consider the area of any polygon, circumscribed about a circle, as ABDEF. From the center of the circle, let straight lines extend to the vertices of the polygon, also to the points of tangency. The lines extending to the points of tangency are radii, and Art. 11.] CIRCLES. 151 therefore perpendicular to the sides of the polygon, which are tangents of the circle. The polygon is divided by the lines ex- tending to the vertices into as many triangles as it has sides, ACB, BCD, etc. Regarding the sides of the polygon, AB^ BBj etc., as the bases of these several triangles, they have equal altitudes. Now, the area of each triangle is equal to half the product of its base by the common altitude. But the area of the polygon is the sum of the areas of the triangles, and the perimeter of the polygon is the sum of their bases. It follows that the area of the polygon is equal to half the product of the perimeter by the radius. As this is true of all circumscribed polygons, it is true of the circle, which is the limit of a variable circumscribed poly- gon. Corollaries. — I. The area of a regular polygon is measured by half the product of its perimeter by its apothem. II. Let r represent the radius. Then the circumference is 2 TT r, and half the product of this by r is tt r^ ; that is, the area is equal to the square of the radius multiplied by the ratio of the circumference to the diameter. III. The area of a circle is to the square described on the radius as the circumference is to the diameter. This shows that the problems of rectification and of quadrature are in effect the same. IV. The areas of two circles are to each other as the squares of their radii ; or, as the squares of their diameters. 152 PLANE GEOMETRY. [Chap. VIII. V. The area of a sector is measured by half the product of its arc by its radius. For the sector is to the circle as its arc is to the circumference. 13. Exercises. — 1. What is the length of the radius when the arc of 60° is 10 feet? 2. What is the value, in degrees, of the angle at the center, whose arc has the same length as the radius ? 3. What is the area of the segment whose arc is 60** and radius 1 foot ? [Subtract the triangle from the sector.] 4. One tenth of a circular field, of one acre, is in a path extending round the whole ; required the width of the path. 5. A semicircle described on the hypotenuse as a diameter is equiva- lent to the sum of the two semicircles described on the sides of a right- angled triangle. 13, Scholium, — It was shown by Archimedes that the value of TT is less than 3|, and greater than S^J-. The number 3^ is in common use for mechanical purposes. About the year 1600, Metius found the approximation f ff, which is true to six places of decimals. By the calculus, shorter methods have been discovered for calculating the approximate value of tt. The calculation has been extended to several hundred decimals. The first thirty- nine are 3.141 592 653 589 793 238 462 643 383 279 502 884 197. 14. Miscellaneous Exercises,— Tha following may be used while reviewing the Plane Geometry. 1. Take some principle of general apphcation, and state all its conse- quences that are found in the chapter under review, arranging as the first class those immediately deduced from the given principle, then those derived from these, and so on. 2. Reversing the above, take some theorem in the latter part of a chapter, state all the principles upon which its proof immediately de- pends, then all upon which these depend, and so on, back to the ele- ments of the science. Art. 14.] CIRCLES. 153 8. If two straight lines are not parallel, the difference between the alternate angles formed by any secant is constant. 4. If two opposite sides of a parallelogram are bisected, straight lines from the points of bisection to the opposite vertices trisect the diag- onal. 5. In any triangle ABG^ if BE and OF are perpendiculars to any line through A^ and if D is the midpoint of BG^ then DE is equal to DF. 6. If, from the vertex of the right angle of a triangle, there extend two lines, one bisecting the hypotenuse, and the other perpendicular to it, the angle of these two lines is equal to the difference between the two acute angles of the triangle. 7. In the base of a triangle, find the point from which lines extending to the sides, and parallel to them, are equal. 8. Can two unequal triangles have a side and two angles in one equal to a side and two angles in the other ? 9. Of all triangles on the same base, and having the same vertical angle, the isosceles has the greatest area. 10. The lines that bisect the angles formed by producing the sides of a quadrilateral that is inscribed in a circle are perpendicular to each other. [IV, 26.] 11. Two quadrilaterals are equivalent when their diagonals are re- spectively equal and form equal angles. [Compare VI, 31, Ex. 1.] 12. Lines joining the midpoints of the opposite sides of any quadri- lateral bisect each other. 13. In the triangle ABC, the side AB = 13, BG = 15, the altitude = 12 ; required the base AG. 14. The sides of a triangle have the ratio of 65, 70, and 75 ; its area is 21 square inches ; required the length of each side. 15. The area of a triangle that has one angle 30° is one fourth the product of the two sides containing that angle. [Compare V, 43, Ex. 11.] 16. An inscribed equilateral triangle has one fourth the area of a similar circumscribed triangle. 17. A chord is 8 inches, and the altitude of its segment 3 inches; re- quired the area of the circle. [V, 29, o.] 18. What is the area of the sector whose arc is 50° and whose radius is 10 inches? 19. The altitudes of a triangle, that is, the perpendiculars let fall from the several vertices on the opposite sides, meet in one point. [Through each vertex of the triangle make a line parallel to the opposite side ; and see V, 2, c] 154 PLANE GEOMETRY. [Chap. VIII. 20. If the medials of one triangle are respectively eqnal to the sides of a second, and the medials of the second are respectively equal to the sides of a third, the third is similar to the first. What is the linear ratio ? 21. The sum of twice the square on the medial of a triangle and twice the square on half the base is equivalent to the sum of the squares on the other two sides. [VI, 22 and 23.] 22. Three times the sum of the squares on the sides of a triangle is equivalent to four times the sum of the squares on the medials. 23. If the oblique sides of a trapezoid are produced till they meet, the point of meeting, the point of intersection of the two diagonals of the trapezoid, and the midpoints of the two bases are all in one straight line. [Compare V, 34 and 35.] 24. What is the area of the segment whose arc is 36° and chord 6 inches ? [This chord is one side of an inscribed decagon.] 25. The sum of the squares on the sides of any quadrilateral is equivalent to the sum of the squares on the diagonals, increased by four times the square on the line joining the midpoints of the diagonals. [This de- pends on Ex. 22.] 26. If, from any point in a circum- ference, perpendiculars fall on the sides of an inscribed triangle, the three points of intersection are in the same straight line. [From /), the perpendiculars DE^ DF^ DG^ fall on the sides of the triangle ABG.^ 15. Exercises in Drawing. — 1. Given aline divided internally in extreme and mean ratio, to produce it to the point where it is divided externally in the same ratio, without using any other lines. 2. From two points, one on each side of a given straight Kne, to draw lines making an angle that is bisected by the given line. 3. To describe a circumference through a given point, and touching two given straight lines. 4. To describe four equal circumferences, each touching two of the others exteriorly, and all touching a given circumference interiorly. 5. To draw lines having the ratios V 2 : 1, -/ 3 : 1, V 5 : 1, etc. [V, 38, IX.] 6. To draw a right angle by means of VI, 23, o. Art. 15.] CIRCLES, 155 7. To divide a circle into two or more equivalent parts by concentric circumferences. 8. To divide a given straight line in the ratio of the areas of two given squares. [Compare Euclidean demonstration of VI, 28.] 9. To construct a right-angled triangle when the area and hypote- nuse are given. 10. Given one angle, a side opposite to it, and the difference of the other two sides, to construct the triangle. [Let CD be the given differ- ence, CB the side, and ^ the angle.] c-^ 11. Given one angle, a side opposite to it, and the sum of the other two sides, to construct the triangle. [Let AB\)Q the side, ^(7 the sum, and ADB the angle.] 12. To construct a triangle when the three medials are given. 13. To construct a triangle when the three altitudes are given. [Two sides of a triangle are inversely pro- portional to the corresponding alti- tudes.] 14. To construct a triangle, when the altitude, the medial, and the line bisecting the vertical angle are given. [Let CD be the altitude, CE the bi- sector of the vertical angle, and CF the medial.] GEOMETRY OF SPACE. CHAPTER IX. STRAIGHT LINES AND PLANES. Article 1. — ^A knowledge of the properties of plane figures is the basis for the study of figures that do not lie in a plane. The student who understands the former will find little diffi- culty in mastering the latter. Every plane figure could be illustrated by the diagram on the surface of the paper. That is not always possible in the sequel. "Whenever the student finds it difficult to see the figure, he should make use of sticks, wires, or threads, to represent lines, and of sheets of paper to represent surfaces. These, how- ever, are only helps to realize the perfect figure which exists in the imagination. liines in Space. 3. The definitions of angle and parallel are as applicable in space as to plane figures. Also, those principles concerning straight lines which are immediately derived from the Postulates and Axioms of Geometry. Through any point in space there may be a straight line in any given direction, and through any other point there may be one and only one straight line having the same directions as the former. Two such lines, having no difference in direction, are parallel and lie in one plane. Any other line that is parallel to one of these two is also Art. 2,1 STRAIGHT LINES AND PLANES, 157 parallel to the other. The three may, or may not, lie in one plane. There may be any number of parallel lines in space, no three of them being in one plane. Two straight lines in space may have different directions, and two such lines may lie in one plane, or they may not. If they lie in one plane they meet when produced, and the angle is evident. They may not lie in one plane, as for example the eastern boundary of the floor and the northern boundary of the ceiling. Two such lines can never be produced to meet ; yet they are not parallel. The amount of angle of two straight lines that do not lie in one plane may be made evident by a third line parallel to one of the two and through any point of the other. There is, in the third line, the same divergence from the directions of the par- allel not met as from those of the parallel that is met. CoroUary. — If two lines are respectively parallel to two other lines in space, either the angle of the first two is equal to the angle of the second two, or they are supplementary. The angles are equal or supplementary as in the case of angles in one plane (III, 21 and 22). One arm of an angle remaining fixed as an axis, the other may rotate about it, keeping all the while the same angle with it. Every position of the rotating arm is in fact a distinct line. Thus there may be in space any number of straight lines making the same angle with a given line, and meeting it at the same point, but the axis is not in the same plane as any two of the other lines, unless the two are on directly opposite sides of the axis. Planes and Lines. 3. Theorem. — Three points that are not in one straight line fix the position of a plane. That is, there can be one and only one plane through three such points. It has been explained (II, 13) how there can be a plane through any three points. If there can be two planes through three points, as A, J?, and C, then each of the lines ABy BG, and AC, lies wholly in 158 GEOMETRY OF SPACE, [Chap. IX. both of the supposed planes (II, 12) ; that is, the two planes coincide in all the points of these three lines, so far as they ex- tend. Then they must coincide in the surface of the triangle AB (7, for through any point of this surface a straight line may extend from the perimeter on one side and be produced to the peri- meter beyond, and such line must lie in both planes. But two planes that coincide to any extent of surface must coincide throughout, and are in fact one plane (II, 11). Corollary. — The position of a plane is determined by any plane figure, except a straight line. If two points are one above and one below a plane, a line joining them must pass through the plane and have a point in it, and only one ; for a straight line having two points in a plane lies wholly in it. A straight line and a plane may, therefore, have a single point in common. When a line and a plane have only one common point, the line is said to pierce the plane, and the plane to 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. 4. Theorem. — The intersection of two planes is a straight line. One plane may have points on opposite sides of another. For there can be a plane through any three points. Two such planes must have more than one point in common. For, in the first plane, there may be a circle with the center on one side and part of the circumference on the other side of the second plane. Several radii to this arc must pierce the second plane at several points, all of which are common to both planes. Other plane figures, such as a triangle or other polygon, may be used to demonstrate that the two planes have several common points. All these points lie in one straight line, for otherwise the two planes would be one (3). Art. 5.] STRAIGHT LIFE8 AND PLANES, 159 ^h?-v Perpendicular Lines. 5o Theorem. — A straight line that is perpendicular to each of two straight li?ies at their poitit of intersection^ is perpen- dicular to every straight line that 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 MN, the point A being above, and I below that plane. If the line AB is perpendicu- lar to BG and to BD, it is to be proved that it is also perpen- dicular to every other line lying in the plane MN", and passing through the point B ; as, for example, BE. Produce AB, making BI equal to BA, and let any line, as FH, cut the lines BC, BE, and BD, in F, G, and JI. Then join AF, A G, AH, and IF, IG, and IH. Now, since B G and BB are perpendicular to AI at its mid- point, the triangles AFH and IFH have ^i^ equal to IF (III, 10, vii), AH equal to IH, and FH common. Therefore, the angle AHF is equal to IHF. Then the triangles AHG and IHG are equal (V, 11), and the lines AG and IG are equal. Therefore, the line BE, having two points each equally distant from A and I, is perpendicular to the line AI (III, 11, c). 6. Theorem. — Conversely, if several straight lines are per- pendicular to a given line at the same point, these several lines lie in one plane. If BA is perpendicular to BC, to BB, and to BE, it is to be proved that these three all lie in one plane. BD, for instance, must be in the plane GBE. For the inter- section of the plane of ABB with the plane of GBE is a straight 160 GEOMETRY OF SPACE. [Chap. IX line, which is perpendicular to AJB at the point B (5). Therefore, it coincides with BB (III, 9). Thus, any line, per- pendicular to AB at the point J5, is in the plane of (7, B, D, and E. This theorem is the ground of the fol- lowing definition. A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line that passes through its foot in that plane, and the plane is said to be perpendicular to the line. Corollaries. — I. If one of two perpendicular lines revolves about the other, the revolving line describes a plane, which is perpendicular to the axis. II. Through one point of a straight line there can be only one plane perpendicular to that line. III. If a plane cuts a straight line perpendicularly at the mid- point of the line, then every point of the plane is equally dis- tant from the two ends of the line. IV. If a line extends from the foot of the perpendicular above the plane, it makes an acute angle with the perpendicular ; and conversely. V. The locus of those points in space that are equally distant from two given points is the plane perpendicular to and bisect- ing the straight line joining those points (III, 12). 7. Theorem. — Through one point there can he only one line perpendicular to a plane. If there could be two, these two lines through one point would determine the position of another plane, which would in- tersect the given plane. Then the line of intersection would lie in the plane of the supposed perpendiculars,^nd be perpendicu- lar to both of them. Thus there would be in one plane two perpendiculars to the same line through one point, which is absurd. Therefore, the two supposed perpendiculars to a plane through one point can not exist. Art. Y.] STRAIGHT LINES AND PLANES. 161 Corollaries. — I. If one of two parallel lines is perpendicular to a plane, the other is. For, to every line in the plane through the foot of the first given parallel, there may be in the plane a parallel line through the foot of the second given parallel. The second given parallel is perpendicular to these lines (2, c). II. Conversely, lines perpendicular to one plane are par- allel. III. A line perpendicular to a plane is perpendicular to every straight line in the plane (2). Oblique Liiues and Planes. 8. Theorem. — If, from a point without a plane, a perpen- dicular line and oblique lines extend to the plane, two oblique lines that meet the plane at equal distances from the foot of the perpendicular are equal. Let AB be perpendicular, and -4 C and AD oblique to the plane MN, and the dis- tances i?(7 and BD equal. Then the triangles AB O and ABB are equal (V, 11), and AG i^ equal to AB. Corollaries. — I. A per- pendicular 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. II. If the distances from the perpendicular are unequal the oblique lines are unequal ; and the greater the distance the greater the line. The Axis of a Circle is the straight line perpendicular to the plane of the circle at its center. HI. All points of the circumference of a circle are equidis- tant from any given point of its axis. IV. The locus of those points that are equally distant from three given points is the axis of the circle whose circumference passes through those three points. 8 162 GEOMETRY OF SPACE. [Chap. IX. •-S A^^' 9. If, from all points of a line, perpendiculars fall on a plane, the line thus described on the plane is the projection of the given line on the given plane. Theorem. — The projection of a straight line on a plane is a straight line. Let AB be the given line, and M the given plane. From the point B let the perpendicu- lar BI) fall upon the plane. Every perpendicular to M let fall from a point of AB must be parallel to BI), and must therefore lie in the plane AI), and meet the plane M in some point of the intersection of the two planes. Hence the intersection CD is the projection of the straight line AB on the plane M. There is one exception. When the line is perpendicular to the plane, its projection is a point. Corollary. — ^A straight line and its 'projection on a plane both lie in one plane. M 10. Theorem. — The angle made hy a straight line and its projection on a plane is smaller than the angle it makes with any other line in the plane. Let ^C be the given line, and BG its projection on the plane. It is to be proved that the angle ACB is less ^ than the angle made hj AG with any other line in the plane, as GB. Make AB perpendicular to the plane. Then the point B is in the projection of AG. Take GB equal to GB, and join AB. The triangles A GB and AGB have two sides of one respectively equal to two sides of the other, and the third side K • li"v\ H^nX |\ X \ I?' ^ ^^ f* \ . d'^ Art. 10.] STRAIGHT LINES AND PLANES. 163 AI) longer than the third side AB. Therefore, the angle A CD is greater than the angle A CB. The angle a line makes with its projection on a plane is called the angle of the line and plane. 11. Applications. — Three points, however placed, must be in the same plane. It is on tliis principle that stability is more readily obtained by three supports than by a greater number. A three-legged stool must be steady. It is frequently important in machinery that a body shall have what is called a parallel motion ; that is, that all its parts shall move in parallel lines. , The piston of a steam-engine and the rod which it drives receive such a motion. A ready way of constructing a line perpendicular to a plane is by the use of two squares (III, 15). Place the corner of each at the foot of the desired perpendicular, one edge of each square resting on the plane sur- face. Bring their perpendicular edges together. The position of these coincident edges must be that of a perpendicular to the plane, for it is perpendicular to two lines in the plane. When a circle rotates on its axis, the figure undergoes no change of position, each point of the circumference taking successively the position deserted by another point. In the turning lathe, the axis on which the material is made to rotate is the axis of the circles formed by the cutting tool, as it 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. 13, Exercises. — 1. Designate two lines that are everywhere equally distant, but which are not parallel. 2. Designate four points that do not all lie in one plane. 3. Demonstrate the proposition of Article 5 without using lines below the plane MN. [Let the given lines in the plane extend beyond B^ and construct triangles there.] 4. Two equal oblique lines from the same point make equal angles with the plane ; also, the angles they make with the perpendicular are equal. 5. The angle that a line makes with its projection produced is larger than the angle made with any other line in the plane. 164 GEOMETRY OF SPACE. [Chap. IX. Diedrals. 13, A Diedral is the figure formed by two planes which meet. It is also called a diedral angle. The planes are its faces and the intersection is its edge. In naming a diedral, four points are designated, one in each face, and two on the edge ; the letters designating the points on the edge are placed between the other two, as the diedral BAEC or the diedral (7^^2>. When there would be no ambiguity, a diedral may be designated by two letters, or by one. The quantity of a diedral is the difference of direction of the two faces from the edge, as indicated by a line in each face per- pendicular to the edge. Problem. — A diedral may have any angular quantity. The plane BE may be fixed in position, also the line AE and the line FO perpendicular to AE. A plane may rotate from the position BEj around AE as an axis. The perpendicular describes the plane EGO (6, i). The difference of direction of the planes BE and CE from the common line AE is indicated by the angle EO G. Thus the diedral angle BAEC may increase from zero without limit. Corollaries. — I. A diedral may be equal to any sum or to any difference of angles, or it may have any ratio to another angle. II. Diedral angles are acute, obtuse, supplementary, etc., as linear angles are. III. All right diedral angles are equal. Art. 13.] STRAIGHT LINES AND PLANES. 165 IV. When two planes cut each other, the opposite or vertical diedral angles are equal. Bight Diedrals. 14. Theorem. — If a line is perpendicular to a plane^ then any plane in which this line lies is perpendicular to the other plane. If AB in the plane P§ is perpendicular to the plane M, then AB is perpendicular to every line in M which passes through the point B ; thus, to B Q, the intersection of the two planes, and to B O, which is made perpendicular to the intersec- tion BQ. Then, ABC is a right angle. But the diedral angle AB Q C is the same as the linear angle ABC. ' '^ Corollaries. — I. If one plane is perpendicular to another, a straight line perpendicular to one of them at some point of their intersection must lie wholly in the other plane ; for there may be in the second plane such a per- pendicular to the first at any point of their intersection, and there can not be two perpendiculars to the same plane at one point. II. If a plane is perpendicular to another plane, a line in one of them perpendicular to their intersection must be perpendicu- lar to the other plane ; for it is perpendicular to two lines in that other plane. III. If two planes are perpendicular to a third, the intersec- tion of the first two is a line perpendicular to the third plane. Oblique Diedrals. 15. Theorem. — If the arms of an angle whose vertex is on the edge of a diedral are respectively perpendicular to the faces of the diedral, and if each arm is on the same side of the plane to which it is perpendicular as the other face is, then the angle and diedral are supplementary. 166 GEOMETRY OF SPACE. [Chap. IX. The plane of the angle GDH is perpendicular to both faces of the diedral (14), and to the edge AB (14, iii). Therefore the intersections FD and DE of this plane with the faces of the diedral are perpendicular to AB, and the angle FDE is the same as the diedral. Now, as the angles FDE and GHD lie in one plane, their sum is equal to the sum of the right angles FDH and GDE. Corollary. — If the arms of an angle are respectively perpen- dicular to the faces of a diedral, the angle and diedral are either equal or supplementary ; for the arms are respectively parallel to those mentioned in the theorem (2, c). Compare III, 25. 16. Theorem. — Every point of a plane that bisects a die- dral is equally distant from its two faces. Let the plane EC bisect the diedral BB CE. It is to be proved that every point of this plane, as A, for example, is equally distant from the planes B C and EC. Pass a plane through A perpendicular to B C. The intersec- tions with the given planes, OH, OA, and 01, are all perpen- dicular to B C. Then the line AO bisects the angle HOI, and the point A is equally dis- tant from the lines OH and 01 (III, 13). But the distance of A from these lines is meas- ured by the perpendiculars AH and AI, which measure its dis- tances from the two faces B C and EC. Therefore, any point of the bisecting plane is equally distant from the two faces of the given diedral. Corollary. — The locus of the points equally distant from two Art. 16.] STRAIGHT LINES AND PLANES. 167 given planes consists of two planes bisecting the adjacent die- drals formed by the given planes. 17. Applications. — The theory of diedrals is as important in the study of magnitudes bounded by planes as the theory of angles in the study of polygons. In the science of crystallography, crystals of many species may be classified by measuring their diedrals. The plane of the surface of a liquid at rest is called horizontal^ or the plane of the horizon. The direction of a plumb-line is qsX\qA vertical. A vertical plane or line is perpendicular to the horizon, the positions of both being governed by the same causes. Horizontal and vertical planes are in frequent use. Floors, walls, ceilings, etc., are examples. The methods of using the builder's level and plummet are among the simplest applications of geometrical principles. Civil engineers, astronomers, and navigators refer to the horizon, or to a vertical plane, in their observations. 18. Exercises. — 1. If a line is perpendicular to a plane, and if from its foot a perpendicular falls on some other line which lies in the plane, then this last line is perpendicular to the plane of the other two. 2. What is the locus of those points in space, each of which is equally distant from two given straight lines that lie in the same plane ? 3. What is the locus of those points in space, each of which is equally distant from three given straight lines that lie in the same plane ? 4. "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? Parallel Lines and Planes. 19. Parallelism consists in the identity of the directions of lines, or of a line and a plane, or of planes. A Parallel Line and Plane are such that the line is par- allel to a line in the plane ; that is, the plane has directions which are the same that the line has. Corollaries. — I. When a line and a plane are parallel, there may be through any point of the plane a line parallel to the given line. 168 GEOMETRY OF SPACE. [Chap. IX. II. A line that is parallel to a plane is parallel to its projec- tion on that plane. III. A line parallel to a plane is everywhere equally distant from it. Parallel Planes. 30. Parallel Planes are such that every straight line in one has a parallel line in the other ; that is, the planes have the same directions. If the parallel lines AB and CD re- volve 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 as that of the other, and in the course of a revolution they take all the directions of the two planes. Corollaries. — I. Two planes parallel to a third are parallel to each other. II. Two planes perpendicular to the same straight line are parallel to each other. III. A straight line perpendicular to one of two parallel planes is perpendicular to the other. ly. 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. V. Parallel planes can not meet ; for, if they had a common point, there could be two straight lines in one direction from that point. 21. Theorem. — The intersections of two parallel planes hy a third plane are parallel lines. Since the planes 3f and N" are par- allel, the intersections AB and CD can never meet. Since AB and CD lie in one plane, P, and can never meet, they are parallel (III, 29, c). .f] V __. i\ / ~UI y N / Art. 22.] STRAIGHT LINES AND PLANES. 169 33. Theorem, — The parts of parallel lines intercepted be- tween parallel planes are equal. The lines AB and CD, being parallel, lie in one plane. Then AG and J3D, the intersections of this plane with the parallel planes M and P, are parallel lines. Therefore, AB is equal to CD (VI, 3). M Ar C ^ B n 23. Theorem. — Two parallel planes are everywhere equally distant. For the distance is measured by perpendiculars (8, i). Such perpendicular lines are parallel (7, ii), and therefore equal. ^B 24. Theorem. — If the arms of an angle are both parallel to a given plane, the plane of that angle is parallel to the given plane. Let BA C be the given angle and 31 the given plane. Make AB perpendicular to iHf, and make BE and BF in that plane respectively parallel to AB and A C (19, i). Then BA, being perpendicular to BE and to BF, is perpendicular to AB and io AG (III, 22, v), and to the plane of BA G ; and the two planes are parallel (20, ii). 25. Theorem.. — If two straight lines that cut each other are respectively parallel to two other straight lines that cut each other, then the plane of the first two is parallel to the plane of the second two. 170 GEOMETRY OF SPACE. [Chap. IX. Let AB be parallel to EF, and CD parallel to GIT, Both AB and CB must be parallel to the plane N', since they are parallel to lines in that plane. Therefore the plane 31 is parallel to iV^ 26. Theorem, — Straight lines cut hy three parallel planes are divided proportionally. The line AB is cut at ^, E, and B, and the line GB is cut at (7, F, and B, by the parallel planes Jf, iV, and P. Joining AB, it pierces the plane iV at (r. Then the plane ABB makes the inter- sections FG and BB with the planes iVand P, and the plane ABC makes the intersections A C and GF with the planes JIf and J^. Since FG is parallel to BB (21), AF'.FB=^AG\GB. For a like reason, ^6?: GB= CF'.FB. Therefore, AF xFB=CF'. FB. Corollary. — The segments are proportional to the distances between the planes, that is, AF is to FB as the distance M — 2^ is to the distance K — P. 27. Applications. — The general problem of perspective in draw- lug consists in representing upon a plane surface the apparent form of objects in sight. This plane, the plane of the picture, is supposed to be Art. 27.] STRAIGHT LINES AND PLANES. 171 between the eye and the objects to be drawn. Then each object is to bo represented upon the plane, at the point where it is pierced by the visual ray from the object to the eye. All the visual rays from one straight object, such as the top of a wall, or one corner of a house, lie in one plane. Their intersection with the plane of the picture must be a straight line. Therefore, all straight ob- jects, whatever their position, are drawn as straight lines. If parallel straight objects are also parallel to the plane of the picture, they are drawn parallel, for the lines drawn must be parallel to the ob- jects, and therefore to each other. Two parallel objects, which are not parallel to the plane of the pict- ure, are not parallel in the perspective. The lines meet, if produced, at that point where the plane of the picture is pierced by a line from the eye parallel to the objects. 38. Exercises. — 1. A straight line makes equal angles with two parallel planes. 2. The projections of two parallel lines on a plane are parallel. 3. Two parallel lines make equal angles with a given plane. 4. When two planes are both perpendicular to a third, and their in- tersections with the third plane are parallel lines, the two planes are par- allel to each other. [Make a line in the third plane perpendicular to the intersections.] 5. Given any two straight lines in space; either one plane may pass through both, or two parallel planes may pass through them, that is, one through each point. 6. Demonstrate the last sentence of Article 27. [If two faces of a diedral have a Kne in one parallel to a line in the other, both these lines are parallel to the edge of the diedral.] 172 GEOMETRY OF SPACE. [Chap. IX. Triedrals. 29. When three planes cut each other, three cases are possible. 1st. The intersections may coincide. Then six die- drals are formed, having for their common edge the in- tersection of the planes. 2d. The three intersec- tions may be parallel lines, intersection of the other two. Then each plane is parallel to the 3d. The three intersections may meet at one point. Then the space about the point is divided by the three planes into eight parts. Notice that two intersecting planes make four diedrals, and a third plane divides each diedral into two parts. Each of these parts is called a triedral. A fourth case is impossible. For, since any two of the in- tersections lie in one plane, either they are parallel, or they meet. If two intersections meet, the point of meeting is common to the three planes, and therefore common to all the intersections. Art. 29.] STRAIGHT LINES AND PLANES. 173 Hence, either the three intersections coincide, or they are par- allel, or they have only one common point. But these are the three cases just considered. A Triedral is the figure formed by three planes meeting at one point. The point where the planes and intersections all meet is called the vertex of the triedral. The intersections are its edges, and the angles formed by the edges are its faces. The planes of the angles are sometimes called faces. Al- ways when the term face designates a quantity, it means the angles. The corners of a room, or of a chest, are illustrations of trie- drals with rectangular faces. The point of a triangular file, or of a small-sword, has the form of a triedral with acute faces. The triedral has many things analogous to the plane trian- gle. It has been called a solid triangle ; and more frequently, but with less propriety, a solid angle. The three planes make three diedrals. The diedrals and the faces are the six elements of a triedral. They are all angular quantities. A triedral that has one rectangular diedral is called a rect- angular triedral. If it has two, it is birectangular ; if it has three, it is trirectangular. A triedral that has two of its faces equal is called isos- celes ; if all three are equal, it is equilateral. A triedral may be named by a single letter at the vertex, but, when two triedrals have the same vertex, add three letters, one on each edge, thus : S — ABC, Supplementary Triedrals. 30. Theorem. — If, at the vertex of a triedral, there is a perpendicular to the plane of each face, and on the same side of the plane as the other parts of the triedral, the three perpendicu- lars are the edges of a second triedral ; Each edge of the first triedral is perpendicular to the plane of (^ fctce of the second, and on the same side of it as the other parts of the figure ; 174 GEOMETRY OF SPACE. [Chap. IX. ^Each diedral of the first is a supplement of the opposite face of the second triedral ; and Each face of the first is a supplement of the opposite diedral of the second. Let S—ABC be the first trie- dral, SF perpendicular to the plane C \ A SB and on the same side of it as > SO J jSI) perpendicular to the plane CjSB and on the same side of it as ^^^:^^rr~-— _^ /SAy and jSF perpendicular to the ^^^ / ^-^.^ plane ASC and on the same side of -^ / ^ it as SB. Then S—BEF is the '"^ second triedral. Since SF is perpendicular to the plane ASB^ it is perpendic- ular to the line 8A. Since SE is perpendicular to the plane CSA, it is perpendicular to SA. Then SA, being perpendic- ular to the two lines SF and SE, is perpendicular to the plane FSE. Since SB and SA are on the same side of the plane CSB the angle A SB is acute (6, iv) ; and SB and SA are on the same side of the plane FSE. In the same manner, it is proved that SB is perpen- dicular to the plane FSB and on the same side of it as SE'y also that SG i^ perpendicular to BSE and on the same side of it as SF. Since SF and SE are perpendicular to SAy the diedral whose edge is SA is the supplement of the face ESF (15). Similarly, the diedral SB is the supplement of the face BSF and the diedral SC \^ the supplement of the face BSE. Also, the faces A SB, ASC, and BSC, are respectively supplements of the diedrals SF, SE, and SB. Supplementary Triedrals are two triedrals, in which the faces and diedrals of the one are respectively the supplements of the diedrals and faces of the other. Two triedrals can have this reciprocal relation without having a common vertex. This posi- tion is used here for purposes of demonstration only. Art. 31.] STRAIGHT LINES AND PLANES. 175 Faces of a Triedral. 31. Theorem. — Each face of a triedral is less than the sum of the other two. The theorem is demonstrated, when it is shown that the greatest face is less than the sum of the other two. Let ASB be the greatest of the three faces of the triedral S. From the face ASB take the part ASD, equal to the face ASC. Join the edges SA and SB by any straight line AB, Take SC equal to SB, and join ^ (7 and ^ C. Since the triangles ASB and ASC are equal (V, 11), AB is equal to A C. But AB \i less than the sum oi AG and BG, and from these, subtracting the equals AB and AG, we have BB less than BG. Hence, the triangles BSB and BSG have two sides of the one equal to two sides of the other, and the third side BB less than the third side B G. Therefore, the included angle BSB is less than the angle BSG. Adding to these the equal angles ASB and ASG,wq have the face ASB less than the sum of the faces ASG and BSG. 33. Theorem.— The sum of the faces of a triedral is less than four right angles. Through any three points, one in each edge of the triedral, let the plane AB G pass, making the intersections ABy B G, and A G, with the faces. There is formed a triedral at each of the points A, B, and G. Then the face BA G is less than the sum of BAS and GAS, the face ABG is less than the sum of ABS and GBS, and the face B GA is less than the sum of 176 GEOMETRY OF SPACE. [Chap. IX. A CS and B CS. Adding, we find that the sum of the angles of the triangle ABC, which is two right angles, is less than the ^ sum of the six angles at the A\ bases of the triangles whose ver- / \ \ tices are at /SL / \ \ The sum of all the angles of ^ / \ \^ these three triangles is six right / W' \ angles. Therefore, since the sum of those at the bases is more than two right angles, the sum of those at the vertex S is less than four right angles. Take any three points on the paper or blackboard for A, B, and C. Take S at some distance from the surface, so that the angles formed at S> are quite acute. Then let & approach the surface of the triangle AB C. The angles at S increase until the point S reaches the surface of the triangle, when the sum becomes four right angles, and the triedral becomes a plane. Sum of tlie Diedrals. 33. Theorem. — In every triedral the sum of the diedrals is greater than two right angles, and less than six. Consider the supplementary triedral, with the given one. Now, the sum of the diedrals of the given triedral, and of the faces of its supplementary triedral, is 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, and is greater than zero. Subtracting this sum from the former, the remainder, being the sum of the diedrals of the given triedral, is greater than two and less than six right angles. Symmetrical Triedrals. 34, Symmetrical Triedrals are two triedrals whose ele- ments are respectively equal, but arranged in reverse order. If the edges of a triedral are produced beyond the vertex, they form the edges of a new triedral. The faces of these two triedrals are respectively equal, for the angles are vertical. Art. 34.] STRAIGHT LINES AND PLANES, 177 Thus, the angles ASO and ESD are equal ; also, the angles JBSO and FSE are equal, and the angles A8B and BSF. The diedrals whose edges are FS and BS are also equal, since, being formed by the same planes, FFSBG and DFSBA, they are vertically opposite diedrals. The same is true of the diedrals whose edges are BS and SA, and of the diedrals whose edges are FS and SO. The point ^iS' is a center of symmetry, and the triedrals are symmetrical with reference to this center (IV, 5). Symmetry with reference to a point, line or plane is symmetry of posi- tion, but triedrals such as defined in the first sentence of this article need not have a common vertex. Their symmetry is called the symmetry of form. This distinction was not needed in plane geometry, for symmetrical plane figures are equal. Two points are symmetrical with reference to a plane Ol symmetry when the line that joins them is bisected perpen- dicularly by the plane. Two figures, or two parts of one figure, are symmetrical with reference to a plane of symmetry, when every point in one has its symmetrical point in the other. The diagram in Article 5 of this chapter illustrates a figure whose two parts are symmetrical with reference to the plane of symmetry MIST, Every point in the lines AB, AH, AF, has its symmetrical point in the lines IB, IH, IF. Models of symmetrical triedrals may be made as follows. Draw four lines from a common point A, making three angles, BAC, GAB, and BAE, which are unequal, neither one larger than the sum of the other two, and the sum of the three less than four right angles. Cut out the remaining angle BAE. Fold the 178 GEOMETRY OF SPACE, [Chap. IX. paper at the lines AG and AD toward the obverse face, and bring the edges AE and AJ3 together. Make a second model with angles equal to these, cut, and fold toward the reverse face. These two models may be placed vertex to vertex, with the equal angles vertically opposite, as in the diagram. Also they may be placed with a face of one against the equal face of the other. Then these faces illustrate a plane of symmetry. This, however, depends on a principle to be demonstrated (35). A triedral having three unequal faces is not equal to its sym- metrical, for they can not be made to coincide. The elements are arranged in reverse order. This is evident, if an attempt is made to insert one of the models in the other. Symmetry is also illustrated by two gloves, which are composed of equal parts arranged in reverse order. Equality of Triedrals. 35. Equality may be shown by the possible coincidence of the respective faces, for then the edges must coincide ; or, by the possible coincidence of the edges, for then the faces must coincide. Theorem, — When two triedrals have th^ir faces respectively equaly their diedrals are respectively equal. Suppose the faces ASB and DTE equal, also ASC and DTF, also DSC and ETF. On the several edges take equal lengths, SA, SB, SC, TD, TE, and TF\ join AB, AC, BC, DE, DF, and EF The isosceles triangles ASB and DTE are equal (V, 11) ; so are Art. 35.] STRAIGHT LINES AND PLANES. 179 ASG and DTF\ also JBSG and JETF. Hence the triangles AB C and DFF are equal. From any point on AS make MN" and MP, both perpendic- ular to AS, and in the planes ASB and ^/S'(7 respectively. Since SAB is an angle at the base of an isosceles triangle, it is acute, and MN" must meet AB. For a like reason, MP must meet A C. Join iVP. Take 7> (x equal to AM and repeat in this triedral the same construction. The triangles AMN' and J) GK are equal (V, 12, c), and AN is equal to DK. For a like reason, AP is equal to DH. The triangles ^iVP and DKH are equal (V, 11), and NP is equal to KH. Then the triangles MNP and GKH are equal (V, 10), and the angles NMP and KGH are equal, that is, the diedrals >S14. and TB are equal. In the same manner, it is shown that the diedrals SB and TF are equal ; also the diedrals SG and TF 36. Theorem. — Two triedrals that have their diedrals re- spectively equal have their faces respectively equal. Let G and g represent the given triedrals, and S and s their respective supplementary triedrals. The faces of S are respec- tively equal to those of s, for they are the supplements of die- drals that are equal by hypothesis. Therefore the diedrals of S are respectively equal to that of s (35). But the faces of G and of g are respectively the supplements of the diedrals of aS' and of s. Therefore these faces are respectively equal. 37. Theorem. — Two triedrals that have two faces and the included diedral of one respectively equal to the corresponding elements of the other have the remaining elements respectively equal. There are two cases to be considered. 1st. Suppose the angles AEO and BG G equal, and the angles AEI and B GD equal, also the included diedrals whose edges are AE and B G. Let the arrangement be the 180 GEOMETRY OF SPACE. [Chap. IX. same in both, so that, if we go around one triedral in the order O, Ay I, O, and around the other in the order G, JB, J9, G, in both cases the triedral is on the right. Place the face BCD di- rectly upon its equal, AEL Since the diedrals are equal, and are on the same side of the plane AEI, the planes BOG and AEO coincide. Since the faces BCG and AEO are equal, the lines CG and £10 coincide. Thus, the angles DCG and lEO coincide, and the two triedrals coincide throughout. 2d. Let the angles AEO and BOG he equal, and the angles AEI and B CB, also the included diedrals, whose edges are AE and B C. But let the arrangement be the reverse ; that is, if we go around one triedral in the order 0, Ay ly Oy and around the other in the order (r, i>, By Gy in one case the triedral is to the right, and in the other to the left. Then it may be proved that the two triedrals are symmetrical. One of the triedrals can be made to coincide with the sym- metrical of the other ; for, if the edges BG, GCy and B G are produced beyond (7, the triedral G — EBEhas two faces and the included diedral respect- ively equal to those parts of the triedral E — A Oly and arranged in the same order ; that is, the reverse of the triedral G—BGB. Hence, as just shown, the triedrals G — FHK and E^AOI are equal. Therefore, E — A 01 and G—BGB are symmetrical triedrals. In both cases, all the ele- ments of one triedral are respectively equal to those of the other. \Kr jt-^ / Art. 38.] STRAIGHT LINES AND PLANES, 181 38# Theorem. — Two triedrals that have one face and the two adjacent diedrals of one respectively equal to the correspond- ing elements of the other, have the remaining elements respective- ly equal. If the arrangement of the corresponding elements is the same in both, the equality of the triedrals may be demonstrated by superposition. If the arrangement in one is the reverse of the other, the equality of the remaining, elements may be demon- strated by the symmetrical triedral, as in the preceding article. This theorem may also be demonstrated by the principle of supplementary triedrals (30). Let G and g represent the given triedrals, and S and s their respective supplementary triedrals. Since G and g have one face and two adjacent diedrals of one respectively equal to the corresponding elements of the other, S and s must have two faces and the included diedral of one re- spectively equal to the corresponding elements of the other. Therefore (37), S and s have their remaining elements, one face and the adjacent diedrals, respectively equal. Then the remain- ing elements of G and g, being the supplements of these, must be respectively equal. Corollaries. — I. In all cases where two triedrals have all their elements respectively equal, if the arrangement is the same the triedrals are equal, and if the arrangement is the reverse the triedrals are symmetrical. II. An isosceles triedral and its symmetrical are equal (37). III. In an isosceles triedral, the diedjals opposite the equal faces are equal. IV. If two diedrals of a triedral are equal, the faces opposite these diedrals are equal. 39. Exercises. — 1. Of two unequal diedrals of a triedral the face opposite the greater is greater than the face opposite the less ; and con- versely. 2. What is the locus of those points in space, each of which is equally distant from three given planes ? 182 GEOMETRY OF SPACE. [Chap. IX. Polyedrals. 40. A Polyedral is a figure formed by several planes that meet at one point. The vertex, edges, and faces are defined, as those of a triedral. A triedral is a polyedral formed by three planes. If the edges of a polyedral are produced beyond the vertex, they form the edges of a second polyedral, symmetrical to the former. Several properties of triedrals are common to other polye- drals. Problem. — Any polyedral of more than three faces may he divided into triedrals. For a plane may pass through any two edges that are not consecutive. Thus, a polyedral of four faces may be divided into two triedrals ; one of five faces, into three, and so on. This is like the division of a polygon into triangles. The plane passing through two edges not consecutive is called a diagonal plane. A polyedral is called convex, when every diagonal plane lies within the figure ; otherwise it is called concave. When any figure is cut by a plane, the figure that is defined on the plane by the surfaces of the figure so cut is called a plane section. Corollaries, — I. If the plane of one face of a convex polye- dral is produced, it does not cut the polyedral. II. A plane may pass through the vertex of a convex polye- dral without cutting any face. Art. 40.] STRAIGHT LINES AND PLANES. 183 III. A plane may cut all the edges of a convex polyedral. The section is a convex polygon. 41. Theorem. — The sum of the faces of a convex polyedral is less than four right angles. This is demonstrated in the same manner as the correspond- ing theorem on triedrals (32). 42. Theorem. — Tn any convex polyedral^ the sum of the diedrals is greater than the sum of the angles of a polygon haviyig the same number of sides that the polyedral has faces. Let the given polyedral be divided by diagonal planes into triedrals. Then this theorem is demonstrated like the analogous proposition on polygons. 43. Scholium.. — The theory of equality of triedrals has some analogy to the theory of similarity, and some to the theory of equality of triangles. In the discussion of triedrals there is no consideration of the ratios of lengths as in similarity, nor of the equality of magnitudes as in the equality of triangles. Every element of a triedral is an angle. The analogy of trie- drals with triangles will be more apparent in the theoiy of spherical triangles, which is based upon the theory of trie- drals. At first it appears impossible to adapt problems in draw- ing to the Geometry of Space ; for a drawing is made on a plane surface, while the figures investigated are not plane figures. This, 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 Geome- try. Descriptive Geometry is that branch of mathematics which teaches how to represent and determine, by means of drawings on a plane surface, the absolute or relative position of points or 184 GEOMETRY OF SPACE. [Chap. IX. magnitudes in space. It is beyond the design of the present work to do more than allude to this interesting and very useful science. 44. Exercises. — 1. If a straight line is perpendicular to a plane, every plane parallel to the given line is perpendicular to the given plane. 2. If every diedral of a triedral is bisected, the three planes have one common intersection. 3. If two straight lines are not in the same plane, there is one straight line and only one that is perpendicular to both of them. 4. In the second case of Exercise 5, Article 28, a line that is per- pendicular to both the given lines is also perpendicular to the two planes. 5. When two parallel planes are cut by a third plane, the corre- sponding diedrals are equal. Is the converse true ? 6. When two triedrals have two faces of the one respectively equal to two faces of the other, and the included diedrals unequal, the third faces are unequal, and that face is greater which is opposite the greater die- dral. Is the converse true ? T. If two convex polyedrals have the same number of faces, the sum of the diedrals of one is within four right angles of the sum of the die- drals of the other. CHAPTER X. POLYEDRONS. Article 1. — Figures that inclose a portion of space are classified according to their surfaces. This chapter treats of some that have plane surfaces ; the following, of some that have curved surfaces. A Polyedron is a solid, or portion of space, bounded by plane surfaces. Each of the surfaces is a face, their intersec- tions are edges, and the points of meeting of the edges are ver- tices of the polyedron. Corollary. — The edges of a polyedron are straight lines (IX, 4), and the faces are polygons. A Diagonal of a polyedron is a straight line joining two vertices that are not in the same face. A Diagonal Plane is a plane passing through three ver- tices that are not in the same face. Tetraedrons. 2. A Tetraedron is a polyedron having four faces. Three planes can not inclose a solid (IX, 29), but if a plane is passed through three points, one on each edge of a triedral, the triangular plane section and the three triangles cut off in- close a polyedron, which has four faces, four vertices and six edges. Problem. — A7iy four points, not all in one plane, may he the vertices of a tetraedron. Straight lines joining these, two and two, form the six edges, and these bound the four faces. 9 186 * GEOMETRY OF SPACE [Chap. X. The Altitude of a tetraedron is the perpendicular distance from one face to the opposite vertex. This face is called the base, and the vertex is called the vertex of the tetraedron. Any face of a tetraedron may be taken as the base ; then the other faces are called the sides. Consequently the altitude may be either of four distances. When a diedral edge of the base is obtuse, the perpendicular falls outside of the figure, on the base produced. Corollary. — ^The altitude of a tetraedron is equal to the dis- tance from the base to a plane through the vertex parallel to the base (IX, 23). 3. Theorem. — There is one point equally distant from the four vertices of a tetraedron. Every point that is equally distant from the three points B, C, and jP] lies in the axis of the circle whose circumference passes through those three points (IX, 8, iv). Every point equally distant from the two points G and D lies in the plane that bisects the edge CD perpendicularly (IX, 6, V). Since the line CD is not parallel to the plane B CF, the plane perpendicular to CD and the line perpendicular to B CF are not parallel, and the plane must cut the line at some point. The point common to these two loci is the only point equally distant from the four vertices. The point is not necessarily within the tetraedron. Corollaries. — I. The axes of the four circles that circum- scribe the faces of a tetraedron meet at one point. II. The six planes that bisect perpendicularly the edges of a tetraedron meet at the same point. 4. Theorem. — There is one point xoithin every tetraedron equally distant from the four faces. Art. 4.] P0LYEDR0N8. 187 Let OB be the strai^t line formed by the intersection of two planes, one of which bisects the diedral A 0, and the other the diedral EO. Every point of the first ^ bisecting plane is equally ^-^ which can be done, for the sides of these triangles are given. Then, cut out the whole figure from the paper and fold it at the lines AB^ B (7, and CA. Since BF is equal to BI), CF to GE, and AB to AE, the points F, I), and E may be united to form a vertex. To construct models of symmetrical tetraedrons make the drawings equal, but fold up in one and down in the other. A regular pyramid : Draw a regular polygon of any number of sides. Upon these sides, as bases, draw equal isosceles trian- gles, with altitudes greater than the apothem of the base. Cut out and fold. Right prisms present no difficulty. Other prisms and pyramids and frustra may be constructed by trial with tolerable accuracy. The five regular polyedrons : For the regular tetraedron, draw an equilateral triangle, join the midpoints of the three sides, cut, and fold. For the cube, draw six equal squares, as in the diagram. For the octaedron, draw eight equal regular triangles. Art. n.] P0LYEDR0N8. 197 For the dodecaedron, draw twelve equal regular penta- gons. For the icosaedron, draw twenty equal regular triangles. 18, Exercises. — 1. Are two tetraedrons equal when five edges of one are respectively equal to five edges of the other, the edges being similarly arranged ? 2. Are two regular hexagonal pyramids equal, if they have equal alti- tudes and a side cf one hase equal to a side of the other ? 3. If two triangular prisms are composed of tetraedrons respectively equal, must the prisms be equal ? 4. If a right prism is symmetrical to another, they are equal. 5. Two regular pyramids, symmetrical to each other, are equal; and conversely. Similarity of Polyedrons. 1 9. Since similarity consists in having the same form, so that every difference of direction in one of two similar figures has its corresponding equal difference of direction in the other, it follows that, when two polyedrons are similar, their homolo- gous faces are similar polygons, their homologous edges are of equal diedral angles, and their homologous vertices are of equal polyedrals. If the similar faces are not arranged similarly, but in reverse order, the polyedrons are symmetrically similar. Theorem. — When two polyedrons are similar, any edge or other line in one is to the homologous line in the second as any other line in the first is to its homologous line in the second. If the proportion to be proved is between sides of homolo- 398 GEOMETRY OF SPACE. [Chap. X. gous triangles, it follows at once from the similarity of the tri- angles. Suppose the edges taken in one of the polyedrons are not sides of one face ; as, AE and 10, Then it is to be proved that AE:BG = IO:DF. AE '.BC = IE'.CD,2i% just proved, and IO'.DF=zIE:CD. Therefore, AE\BC = IO\ DF, Again, compare the homologous altitudes AK and BH. Join KO and HF. Then the planes KA O and HBF are re- spectively perpendicular to the bases EIO and CDF, and the angles KOA and JIFB are homologous and equal. Then the right-angled triangles KOA and HFB are similar, and AKiBR=AO:BF. Thus any homologous lines can be united to homologous edges by similar triangles. Corollary, — When two polyedrons are symmetrically similar homologous lines are proportional. 20. Theorem. — Two tetraedrons are similar when there is the same ratio of every edge of one to the corresponding edge of the other, and they are similarly arranged. Since the corresponding faces are similar triangles, every angle on the surface of one has its corresponding equal angle on the surface of the other. If a line is made through the figure, it is shown, by the aid of auxiliary lines, as in the corresponding proposition of similar tri- Art. 20.] POLYEDRONS. 199 angles, that every angle in one figure has its homologous equal angle in the other. Corollaries. — The corresponding elements being similarly- arranged, two tetraedrons are similar : I. When three faces of one are respectively similar to those of the other ; II. When two triedral vertices of one are respectively equal to two vertices of the other. III. If a plane cuts a tetraedron parallel to one face, the part cut off is similar to the whole. 21. Theorem. — Two similar polyedrons are composed of tetraedrons respectively similar and similarly arranged. For, however the one is divided into tetraedrons, the con- struction of homologous lines and planes divides the other in the same manner. Then the similarity of the corresponding tetrae- drons follows from the proportionality of the lines. Corollaries. — I. Conversely, two polyedrons are similar when composed of tetraedrons respectively similar and similarly arranged. II. When two polyedrons are similar there is the same ratio of any line of one to the homologous line of the other. III. When a pyramid is cut by a plane parallel to the base, the pyramid cut off is similar to the whole. 23, Theorem. — If lines are tnade from the vertices of a polyedron to any point in space, and if these lines are divided either internally or externally in the same ratio, the points of division are the vertices of a polyedron, which is either similar or symmetrically similar to the first. Conversely, if two similar or symmetrically similar polye- drons have their homologous edges parallel, lines made through homologous points all meet in one point. The demonstration of these propositions is similar to that of the corresponding propositions in Plane Geometry (V, 27 and 28). The detail of the argument is left to the student. The definition of internal and of external centers of simi- larity is as in Plane Geometry. 200 GEOMETRY OF SPACE. [Chap. X. Scholium. — The contrary arrangement of the parts in plane figures arises from viewing the figures from an opposite direc- tion. The arrangement of the parts viewed from the reverse face is contrary to that which they have when viewed from the obverse face of the figure. Change in the arrangement of the parts of a solid figure can not be effected by any change of the position of the observer ; the contrary arrangement makes a different figure. Surfaces of Polyedrons. 23. The area of the surface of a polyedron is the sum of the areas of the faces. When several faces are equal, the process is shortened by multiplication. Theorem. — The area of the lateral surface of a regular pyramid is equal to half the product of the perirneter of the base by the slant height. The area of every side is equal to half the product of its base by its altitude. But the altitude of every side is the slant height of the pyramid, and the sum of the bases of the sides is the perimeter of the base of the pyramid. Therefore, the area of the lateral surface of the pyramid, which is the sum of all the sides, is equal to half the product of the perimeter of the base by the slant height. Corollary. — The area of the lateral surface of a regular pyra- mid is equal to the product of the slant height by the perimeter of a section, midway between the vertex and the base. For the perimeter of the midway section is one half the perimeter of the base. 34. Theorem, — The area of the lateral surface of the frus- tum of a regular pyramid is equal to half the product of the sum of the perimeters of the bases by the slant height. The area of every trapezoidal side is equal to half the prod- uct of the sum of its parallel bases by its altitude, which is the slant height of the frustum. Therefore, the area of the lateral surface, which is the sum of these equal trapezoids, is equal to Abt. 24.] P0LYEDR0N8. 201 the product of half the sum of the perimeters of the bases multi- plied by the slant height. Corollary, — ^The area of the lateral surface of a frustum of a regular pyramid is equal to the product of the slant height by the perimeter of a section midway between the bases. 25. Theorem. — The area of the lateral surface of a prism is equal to the product of one of the lateral edges by the perim- eter of a section^ made hy a plane perpendicular to those edges. Since the plane UN' is perpendicular to the edges of the prism, every side of the polygon HJST is perpendicular to the edges which it con- nects. Every side of the prism is a par- allelogram with one lateral edge of the prism for its base and one side of the poly- gon IIN" for its altitude ; and the area of the parallelogram is the product of these two factors. Now, the lateral edges of the prism are equal. Therefore, the sum of the areas of these parallelograms, that is, the lateral surface of the prism, is equal to the product of one edge multi- plied by the perimeter of the polygon. 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. 26. Theorem.— The areas of homologous faces of similar tetraedrons are to each other as the squares of homologous lines. This is a corollary of the theorem that the areas of simi- lar triangles are to each other as the squares of homologous lines. Corollary. — The areas of homologous surfaces of similar polyedrons are to each other as the squares of homologous lines. 202 GEOMETRY OF SPACK [Chap. X. 27. 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 have the same ratio as the areas of the bases. The frustra have equal altitudes. Subtracting these from the given common altitudes, the remainders are equal, that is, the tetraedrons A GHK and BLNP have equal altitudes. The tetraedrons AEIO and A GHK are similar. There- fore, 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. 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'. iiVP. By alternation, EIO : CDF= GHK'.LNP. Corollaries. — ^I. "When the bases are equivalent the sections are equivalent. II. When the bases are equal the sections are equal. For they are similar and equivalent. 28. Exercises. — 1. Find the area of the surface of a regular octa- gonal pyramid whose slant height is 5 inches, and a side of whose base is 2 inches. 2. The area of the entire surface of a regular tetraedron, the edge being 1 inch, is V^ square inches. Art. 28.] POLTEBRONS. 203 3. 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 sec- tions are proportional to the areas of the bases. 4. A right prism has less surface than any other prism of equal base and equal altitude ; and a regular prism has less surface than any other right prism of equivalent base and equal altitude, and of same number of sides. 5. A regular pyramid and a regular prism have equal hexagonal bases, and altitudes equal to three times the radius of the base ; required the ratio of the areas of their lateral surfaces. 6. In what case of the first proposition of Article 22 are the figures similar, and in what case are they symmetrically similar ? 7. Demonstrate the Corollary of Article 25, without using the Theorem ? Measure of Volume. 39. The cube, on account of its simple form and the unity of its three dimensions, has the same place among solids that the square has among plane figures. It is the unit of volume. That is, whatever straight line is taken as the unit of length, the cube whose edge is of that length is the unit of volume, as the square whose side is of that length is the unit of area. Volume of Parallelopipeds. Theorem. — The volume of a rectangular parallelopiped is equal to the product of its length, breadth, and altitude. That is, the number of cubical units contained in a rectangu- lar parallelopiped is equal to the product of the numbers of linear units in the length, the breadth, and the altitude. If the altitude AE, the length EI, and the breadth 10, have a common measure, let each be di- vided by it ; and let planes, par- allel to the faces of the prism, pass through all the points of division, B, (7, D, etc. \ 0\ ^^^^V I 3 H 204 GEOMETRY OF SPACE. [Chap. X. All tlie 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, multiplied by the num- ber in each row ; that is, the product 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. It follows that if two rectangular parallelopipeds are such that there is a common measure of all their dimensions, then one volume is to the other as the product of the dimen- sions of the first is to the product of the dimensions of the second. When the length, breadth, and altitude of the parallelepiped have no common measure, let any line be assumed as the unit of length, and the cube having this edge as the unit of volume. Applying this to the parallelopiped, a part only is measured, but, if a cube whose edge is an aliquot part of the unit of length is used, a greater part of the parallelopiped may be measured ; and, by using a unit regularly smaller, the part measured is a variable parallelopiped whose limit is the one given. Also the product of the three dimensions of this variable is a variable whose limit is the product of the dimensions of the given parallelopiped. At every step the volume of the variable is to the unit of volume as the product of the dimensions of the variable to the product of the dimensions of the unit. As these ratios are always equal, their limits are equal, that is, the volume of the parallelopiped is to the unit of volume as the product of the three dimensions of the parallelopiped is to the cube of unity. Therefore a rectangular parallelopiped is measured by the product of its length, breadth, and altitude. 30, Theorem. — The volume of any parallelopiped is equal to the product of its length, breadth, and altitude. As this has been demonstrated for the rectangular parallel- i Art. 30.] POLYEDRONS. 205 opiped, it is sufficient to show that any parallelopiped is equivalent to a rectangular one having the same linear dimen- sions. Let the lower bases of the two prisms be on the same plane. Then their upper bases are in one plane. Through every point of the altitude AE pass a plane parallel to the base AI, and let it cut both prisms. Now every section in either prism is equal to the base ; but the bases, having the same dimensions, are equivalent. Therefore every section in one solid has its corresponding equivalent section in the other. Therefore the sum of the sections in one is equivalent to the sum of the sections in the other ; but the volume of each prism consists of this sum of an infinite number of plane figures. Besides the above demonstration by the method of in- divisibles (IV, 31), the theorem may be demonstrated by the superposition and coincidence of equal figures, as fol- lows : Let AF be any oblique parallelopiped. It is equivalent to the parallelopiped AL, which has a rectangular base, AH; since the prism LHEO is equal to the prism DGAI. But the parallelopipeds AF and AL have the same length, breadth, and altitude. By similar reasoning, the prism AL is shown to be equivalent to a prism of the same base and altitude, with two 206 GEOMETRY OF SPACE. [Chap. X. of its opposite sides rectangles. This third prism is then shown to be equivalent to a fourth, which is quite rectangular, and has the same dimensions as the others. Corollaries. — I. The volume of any parallelopiped is equal to the product of its base by its altitude. II. The volumes of any two parallelopiped s are to each other as the products of their three dimensions. Volume of Prisms. 31. Theorem. — The volume of a triangular prism is equal to the product of its base by its altitude. The base of a right triangular prism may be considered 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 is one (16, c). Therefore, the given prism is measured by half the product of its altitude by the base of the parallelopiped ; that is, by the product of its own base and altitude. It remains to be proved that an oblique triangular prism is equivalent to a right triangular prism of the same base and alti- tude. This may be demonstrated by the method of indivisibles used in the preceding Article. Corollary. — The volume of any prism is equal to the product of its base by its altitude. For any prism is composed of trian- gular prisms, having the common altitude of the given prism, the sum of their bases forming the given base. Volume of Tetraedrons. 33. Theorem. — Two tetraedrons of equivalent bases and of the same altitude are equivalent. Suppose the bases of the two tetraedrons to be in the same plane. Through every point of the common altitude pass a plane parallel to the base and produce it through both solids. Every section in one tetraedron is equivalent to the section Art. 32.] P0LTEDR0N8, 207 made by the same plane in the other (27, i). Therefore the sum of all the sections in one is equivalent to the sum of all the sec- tions in the other, but the volume of each tetraedron consists of this sum of an infinite number of plane triangles. 33. 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 tetraedron a triangular prism may be erected, having the same altitude, and one edge coincident with an edge of the tetraedron. This prism may be divided into three tetraedrons, the given one and two others, which, taken two and two, have equal bases and equal altitudes. These three tetraedrons are equivalent ; therefore the volume of the given tetraedron is one third of the volume of the prism ; that is, one third of the product of its base by its altitude. Volume of Pyramids. Corollaries. — I. The volume of any pyramid is equal to one third of the product of its base by its altitude. For any pyra- mid 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. II. Symmetrical prisms are equivalent. The same is true of symmetrical pyramids. III. The volume of a frustum of a pyramid is found by sub- 208 GEOMETRY OF SPACE. [Chap. X. tracting the volume of the pyramid cut off from the volume of the whole. When the two bases and the altitude of a frustum are given, a rule for calculating the volume is made as follows. Represent the lower base by 5^, the upper base by a^, and the altitude by A. Then, if x represents the altitude of the part cut off. Therefore, x = h t and h-\-x = h t . The volume ' — a ' o — a J a b^ a^ of the frustum is \b^ h t \ a^ h v =: \ h , = J A (J2 _|_ j^ _|_ ^2)^ That is, the volume of the frustum is equal to the sum of the volumes of three pyramids having the altitude of the frustum, and bases respectively equal to its lower base, its upper base, and a mean proportional between them. Ratio of Similar Volumes. 34. Theorem. — The volumes of similar polyedrons are pro- portional to the cubes of homologous lines. First, suppose the figures to be tetraedrons. Let AH and BGhe. the altitudes. Then (26), EIO : CDF= EI^ : CF^ = AH^ : BG^. By the proportionality of homologous lines, 4 AH: iBG = EI'. CF= AH.BG, Aet. 34.] P0L7EDR0NS. Multiplying (T, 3, vii), AJEIO \BCFD = EI^ : 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 homol- ogous lines. By arranging these in a continued proportion, as in Article 13, Chapter YII, it is shown that the volume of one polyedron is to the volume of the other as the cube of any line of the first is to the cube of the homologous line of the second. 35. Exercises. — 1. If two tetraedrons have a triedral vertex in each equal, their volumes are in the ratio of the products of the edges which contain the equal vertices. 2. The plane that bisects a diedral angle of a tetraedron divides the opposite edge in the ratio of the areas of the adjacent faces. 3. The volume of a prism is equal to the product of one of its lateral edges by the area of a section perpendicular to that edge. 4. What is the ratio between the edges of two cubes, one of which has twice the volume of the other ? The duplication of the cube was one of the celebrated problems of ancient times. The oracle 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 solution ; that is, the edge of one cube being given, to draw a line equal to the edge of the other. Whilst using no instruments but the rule and compasses, they failed. The student will find no difficulty in making an arithmeti- cal solution, within any desired degree of approximation. 5. How may a pyramid be cut by a plane parallel to the base, so as to make the area or the volume of the part cut off have a given ratio to the area or the volume of the whole pyramid? 6. The area of the lower base of a frustum of a pyramid is five square feet, of the upper base one and four-fifths square feet, and the altitude is two feet; required the volume. 36. Scholium. — Among solid as well as plane figures, simi- larity involves equality of homologous angles and equality of 10 ''210 GEOMETRY OF SPACE. [Chap. X. the ratios of homologous lines — ^the former is the essential char- acter as expressed in the definition, and the latter is a necessary consequence. Equality of some corresponding lines is never a condition of similarity, but it is always a condition of equality, both in plane and in solid figures. On the other hand, symmetry shows a great contrast be- tween figures on a plane and figures in space. Plane symmetri- cal figures are equal, because, by turning one, they may coin- cide ; but symmetrical solids may not coincide. Symmetry in- volves equality of corresponding angles, and of corresponding straight lines, and of corresponding plane surfaces ; also it in- volves equivalence both as to area and volume. When a polyedron is symmetrical to a second and similar to a third, the second and third are called symmetrically similar. The proportion of homologous lines of similar polyedrons is equally true of those that are symmetrically similar. There are three ratios between similar figures : viz., the linear between homologous lengths, the superficial between homologous areas, and the cubic between homologous volumes — the second and third being respectively the second and third powers of the first. The use of the term cube to designate the third power arose in this way. In the measure of an area there are two linear dimensions ; and in the measure of a volume three linear, or one linear and one superficial. 37. Miscellaneous Exercises. — 1. The opposite vertices of a parallel opiped are symmetrical triedrals. 2. The diagonals of a rectangular parallelopiped are equal. 3. The square of the diagonal of a rectangular parallelopiped is equiv- alent to the sum of the squares of its length, breadth, and altitude. 4. The diagonals of a parallelopiped bisect each other; the lines that join the midpoints of the opposite edges bisect each other; the lines that join the centers of the opposite faces bisect each other; and the point of intersection is the same for all these lines, and is a center of symmetry. Art. Z1.] POLYEDROKS. 211 5. A cube is the largest parallel opiped of the same extent of surface (VII, 23). 6. The centroid of a triangle is the point of intersection of the rae- dials. The four lines that join the vertices of a tetraedron to the cen- troids of the opposite faces intersect each other in one point. In what ratio do the lines just described in the tetraedron divide each other? [Let a triangular diagram represent a plane section of a tetraedron made through one edge and the midpoint of the opposite edge.] 7. Two different tetraedrons, and only two, may be formed with the same four triangular faces ; and these two tetraedrons are symmetrical. [Use paper models.] 8. In any polyedron the sum of the number of vertices and the num- ber of faces exceeds by two the number of edges. 9. In what way is each of the five regular polyedrons symmetrical ? CHAPTER XL SOLIDS OF REVOLUTION, Article 1. — Solids of Revolution are generated by the revolution of a plane figure about an axis. A Cone is a solid described by the revolution of a right- angled triangle about one of its sides as an axis. The other side describes a plane surface, a circle, having for its radius the line by which it is described. The hy- potenuse describes a curved surface. The plane surface of a cone is called its base. The opposite extremity of the axis is the vertex. The altitude is the distance from the vertex to the base, and the slant height is the distance from the vertex to the circumference of the base. A Cylinder is a solid described by the revolution of a rectangle about one of its sides as an axis. The sides per- pendicular to the axis describe circles ; the opposite side describes a curved sur- face. The plane surfaces of a cylinder are called its bases, and the perpendicular distance between them is its altitude. These figures are a cone and a cylin- der of revolution ; other cones and cylin- ders are not usually discussed in Elementary Geometry. Art. 1.] SOLIDS OF REVOLUTION. 213 A Sphere is a solid described by the revolution of a semi- circle about its diameter as an axis. The center, radius, and diame- ter of the sphere are the same as those of the generating circle. The spherical surface is described by the circumference. Corollaries, — I. Radii of the same sphere are equal. The same is true of the diameters. II. Spheres having equal radii are equal. III. A plane passing through the center of a sphere divides it into equal parts. The halves of a sphere are called hemi- spheres. IV. A conical surface is the locus of those points in space the distance of each of which from a straight line given in posi- tion (the axis) has a constant ratio to the distance from a given point of that line (the vertex). This constant ratio is that of the radius of the base to the slant height of the cone. y. A cylindrical surface is the locus of those points that are at a uniform distance from a straight line given in position (the axis). This distance is equal to the radius of the base. VI. A spherical surface is the locus of those points that are at a uniform distance from a given point (the center). This dis- tance is the radius of the sphere. As the curved surfaces of the cone and cylinder are gener- ated by the motion of a straight line, both of these surfaces are straight in one direction. A straight line from the vertex of a cone to the circumfer- ence of the base lies wholly in the surface. So a straight line perpendicular to the base of a cylinder at its circumference lies wholly in the surface. For, in each case, the position had been occupied by the generating line. One surface is tangent to another when it meets, but being produced does not cut it. The place of contact of a plane with a conical or cylindrical surface is a straight line ; since, from 214 GEOMETRY OF SPACE. [Chap. XI. any point of one of those surfaces, it is straight in one direc- tion. 3, Theorem.— r A plane that is perpendicular to a radius of a sphere at its extremity is tangent to the sphere. If straight lines extend from the center of the sphere to any- other point of the plane, they are oblique and longer than the radius, which is perpendicular (IX, 8, i). Therefore, every point of the plane except one is beyond the surface of the sphere, and the plane is tan- gent. Corollary. — A spherical sur- face is curved in every direction. Every possible section of it is a curve. 3. Scholium. — Models of cones, of frusta, and of 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. The curvature of the sphere in every direction renders it im- possible to construct an exact model with plane paper. But the student is advised to procure or make a globe, upon which he can draw diagrams, representing spherical figures better than it is possible to do on a plane surface. Secant Planes. 4. Every point of the line that describes the curved surface of a cone, or of a cylinder, moves in a plane parallel to the base (IX, 20). Therefore, if a cone or a cylinder is cut by a plane parallel to the base, the section is a circle. A Frustum of a cone is that part of the cone between the base and a plane parallel to the base. "When a conical surface is cut by a plane, the form of the Art. 4.] SOLIDS OF REVOLUTION. 215 line of intersection varies according to the relative position of the plane and the axis of the cone. It may be a circumference of a circle, an ellipse, a parabola, a hyperbola, or it may be two straight lines. These Conic Sections are not considered in Elementary Geometry. Their properties are usually investigated by the application of algebra. If a c}^inder is cut by a plane obliquely, the intersection of the plane and the curved surface is an ellipse. Since every point in the arc that describes a spherical sur- face moves in a plane perpendicular to the axis, every plane sec- tion perpendicular to the axis is a circle. Since every point in the spherical surface is at the same distance from the center, any part of the surface may coincide with any other part so far as they both extend, and any diameter may be considered as the axis by which the figure was described. Corollaries. — I. Every plane section of a sphere is a circle, for every plane is perpendicular to some axis. 11. Three points on a spherical surface determine a circum- ference which lies in that surface. Similar Solids of Revolution, 5. Theorem. — Solids of revolution, described hy the revolu- tion of similar plane figures about homologous lines are simi- lar. The generating figures being so placed that they have a common axis, they have a center of similarity on this axis. Then, whatever line is made in one, there is a parallel homologous line in the other, making equal angles. Corollaries. — I. Cones are similar when the ratio of the radius of the base to the slant height is the same. II. If a cone is cut by a plane parallel to the base, the cone cut off is similar to the whole. 216 GEOMETRY OF SPACE. [Chap. XI. III. Cylinders are similar when the ratio of the diameter to the altitude is the same. IV. All spheres are similar, for all semi- circles are similar. V. In whatever position they are placed, two spheres have an internal and an external center of similarity. *i^ Cones. 6. 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 in- scribed in the cone. Theorem. — A cone is the limit of the variable pyramid that is produced when the number of sides of an inscribed or of a cir- cumscribed pyramid is doubled continually, bisecting the arcs at the base at every step of the variation. The base of the cone is a circle, which is the limit of the variable polygons that are the bases of the inscribed and cir- cumscribed pyramids. At every step of the variation, the sides of the circumscribed polygon are tangent to the circle ; hence every side of the pyramid is tangent to the curved surface, and the variable pyramid is always circumscribed. The vertices of Art. 6.] SOLIDS OF REVOLUTION. 217 the polygon inscribed in the base are always in the circumfer- ence, the lateral edges of this variable pyramid must at every step lie in the curved surface, and this variable pyramid is al- ways inscribed in the cone. It has been demonstrated (YIII, 2) that the circumscribed and inscribed polygons can approach each other within less than any assignable difference of surface. It follows that the varia- ble circumscribed and inscribed pyramids can approach each other indefinitely. As the cone is always between them, it must be the limit of both. Corollaries. — I. The area of the curved surface of a cone is equal to half the product of the slant height by the circumfer- ence of the base. Also, it is equal to the product of the slant height by the circumference of a section midway between the vertex and the base. It may be expressed by 'nRH. II. The volume of a cone is equal to one third of the prod- uct of the base by the altitude. It may be expressed by III. The area of the curved surface of the frustum of a cone is equal to half the product of its slant height by the sum of the circumferences of its bases. Also, it is equal to the product of its slant height by the circumference of a section midway be- tween the bases. 218 GEOMETRY OF SPACE. [Chap. XL Cylinders. 7. A cylinder is inscribed in a prism or circumscribed about it in the same way that a cone is inscribed or circumscribed about a pyramid. ;7^ Theorem. — A cylinder is the limit of the variable prism that is produced when the number of sides of an inscribed or circum- scribed prism is doubled continually, bisecting the arcs at the base at each step of the variation. The demonstration is similar to the last. Corollaries, — I. The area of the curved surface of a cylinder is equal to the product of the altitude by the circumference of the base. It may be expressed by 2r:ItA. II. The volume of a cylinder is equal to the product of the base by the altitude. It may be expressed by ttR^A. Spheres. 8. A sphere is said to be inscribed in a polyedron when the faces are tangent to the curved surface ; and the polyedron is circumscribed about the sphere. A sphere is circum- scribed about a polyedron when the vertices all lie in the curved surface ; and the polyedron is inscribed in the sphere. Thus the center of the sphere is equally distant from all the faces of the circumscribed polyedron. It is also equally distant from all the vertices of the inscribed polyedron. Let ASF be a right-angled triangle and BFD a semicircle, the hypotenuse AF being a secant, and the vertex F in the cir- cumference. From -EJ the point where AF cuts the arc, let a perpendicular EG fall upon AD. Art. 8.] SOLIDS OF REVOLUTION. 219 Suppose this figure to revolve about AD. The trian- gle AFH describes a cone, the trapezoid EGHF describes the frustum of a cone, and the semicircle describes a sphere. The points E and F describe the cir- cumferences of the bases of the frustum ; and these circumferences lie in the surface of the sphere. A frustum of a cone is said to be in- scribed in a sphere when the circumfer- ences of its bases lie in the surface of the sphere. A Great Circle of a sphere is a section made by a plane through the center. A Small Circle of a sphere is a section made by a plane not through the center. Corollaries. — The following are derived from the definitions and properties of the sphere, regular polyedrons (X, 11, c), and other figures. I. All great circles of a sphere are equal, having the same radius as the sphere. II. A great circle of a sphere is larger than a small circle. For a plane through the diameter of a small circle and the cen- ter of the sphere is the plane of a great circle, which has the diameter of the small circle for a chord. III. Of two unequal small circles of a sphere, the greater one is nearer the cen- ter. Equal small circles are equally distant from the center. lY. Two great circles bisect each other, and their intersec- tion is a diameter of the sphere. V. The axis of every circle passes through the center of the sphere. The Poles of a circle are the points where its axis pierces the spherical surface. VI. Circles whose planes are parallel to each other have the same axis and the same poles. 220 GEOMETRY OF SPACE, [Chap. XI. VIL Through any three points on the surface a circumfer- ence of a small circle may pass, provided the center of the sphere is not in the plane of the points given. VIII. If two points on the surface are diametrically opposite, any number of arcs of great circles may join them, all of which are semi-circumferences. In any other position two points on the surface determine one great circle. IX. A sphere may be circumscribed about any polyedron that has a point equally distant from all its vertices. X. A spherical surface may pass through any four points that are not all in one plane. XI. A sphere may be inscribed in any polyedron that has a point equally distant from all its faces. XII. Every face of an inscribed polyedron may have a circle circumscribed about it ; for the section of the sphere made by the plane of the face is such a circle. XIII. The points of contact of two adjacent faces of a cir- cumscribed polyedron are equally distant from the edge that is between those faces. XI Y. The points of contact of the faces that constitute one polyedral vertex of a circumscribed polyedron are equally dis- tant from that vertex ; for the point of coutact, the polyedral vertex, and the center of the sphere, are the vertices of a right- angled triangle, and all such triangles for one polyedral vertex are equal. XV. The points of contact of all those faces of a circum- scribed polyedron that constitute one polyedral vertex lie in one plane. XVI. The points just mentioned may be the vertices of one face of an inscribed polyedron. XVII. The remaining faces of an inscribed polyedron being made in the same way, in two such polyedrons, one circum- scribed and one inscribed in a sphere, every vertex of one cor- responds to a face of the other. XVIII. The centers of the faces of a cube being the vertices of a regular octaedron, a sphere that is inscribed in the cube is circumscribed about the octaedron. XIX. The centers of the faces of a regular octaedron being Art. 8] SOLIDS OF REVOLUTION, 221 the vertices of a cube, a sphere that is inscribed in the octaedron is circumscribed about the cube. 9. Exercises. — 1. A section of a cone, by a plane passing through the vertex, is an isosceles triangle. 2. A section of a cylinder made by a plane perpendicular to the base is a rectangle. 3. If a plane is tangent to a sphere at a point on the circumference of a section made by a second plane, the intersection of these planes is tan- gent to that circumference. 4. Of all the small circles through two given points on the surface of a sphere, the smallest is that which has for its diameter the straight line joining the given points. 5. Can every regular pyramid have a sphere circumscribed? In- scribed ? 6. Can a sphere be circumscribed about every regular prism? In- scribed ? 7. Show upon what definitions or principles the corollaries in the last Article depend. Spherical Arcs and Angles. 10. There are some remarkable analogies and some striking contrasts between lines made on the surface of a sphere and those made on a plane. Distance is usually measured along a straight line, but any path on a spherical surface must be a curve (2, c). Theorem, — Of all the arcs of circles from one point to another on the surface of a sphere, the shortest is the minor arc of the great circle, and the longest is the major arc of the great circle. For, when several arcs have a common chord, the minor arc of the circle having the greatest radius is the shortest, and the major arc of the same circle is the longest (VIII, 6, iv). CoroUsbries, — I. Distance on a spherical surface is estimated along the minor arc of a great circle. If two points are diamet- rically opposite, none but the arc of a great circle can join them, and the distance is a semi-circumference (8, viii). 11. The pole of a circle is at the same distance from every 222 GEOMETRY OF SPACE. [Chap. XI. point of the circumference, whether the distances are measured by chords or on the surface. The Polar Radius of a spherical circle is the distance along the surface from the pole to the curve. III. If two circles are equal, their polar radii are equal. IV. The polar radius of a great circle is a quadrant. V. If a point on the surface is at the distance of a quadrant from two points of an arc of a great circle which are not diamet- rically opposite to each other, that point is a pole of the circle (IX, 5). VI. Of two points on the surface, one of which is on the cir- cumference of a great circle, the one not on the circumference is nearer to the axis of the circle. In the following Articles distance is reckoned on the surface, unless otherwise stated. 11. Application. — The equator is a great circle of the earth, which is nearly a sphere. The parallels of latitude are small circles, having the same poles as the equator. The meridians are great circles perpendicular to the equator. The theorem of Article 10 is applied in navigation. A vessel crossing the ocean by the shortest route from one port to another, in the same latitude, does not sail along the parallel of latitude, for that is an arc of a small circle. 13. Two straight lines, having a common point, can not meet again, to whatever distance they are produced ; and, the longer they are, the farther apart are the ends. On a spherical surface, two lines as nearly straight as possible, that is, two arcs of great circles, proceeding from one point, meet again if pro- duced to the opposite point of the sphere, and return eventually to the point of starting. A Spherical Angle is the difference in the directions of two arcs of great circles at their point of meeting. Since the direction of an arc at a given point is the direction of a straight line tangent to the arc at that point, a spherical angle is the same as the angle formed by lines tangent to the given arcs at their point of meeting. Thus, the spherical angle Art. 12.] SOLIDS OF REVOLUTION. 223 DB G is the same as the angle HBK^ the lines HB and BK being respectively tangent to the arcs BD and B G. Corollaries. — I. A spherical angle is the same as the diedral angle formed by the planes of the two arcs. For the intersection BF of the planes of the arcs is a diameter, the tangents IIB and KB are perpendicular to it, and their angle is the same as the die- dral. II. A spherical angle is equal to the angle made by the same arcs on the opposite side of the sphere. Thus, the angles B and F are equal. III. A spherical angle is equal in angular quantity to the arc of a circle included between the arms of the angle, the pole of the arc being at the vertex. For such an arc, as J) 6r, has its center on the line BF, and its curvature is equal to the diedral BF lY. If two arcs of great circles are perpendicular to each other, each passes through the poles of the other (IX, 14, i). V. If two arcs of great circles are perpendicular to a third, they meet at the poles of the third. 13. Theorem. — From a point on the surface of a sphere there are two perpendiculars to the circumference of a great cir- cle ; one of which is the shortest, and the other the longest, arc that can extend from .the given point to the given circumference. Let BFJD be the great circle, the center of the sphere, and A the point on the surface. Make the plane BACO perpendicular to the circle BFJD. Then the arcs A C and AB 224 GEOMETRY OF SPACE. [Chap. XL are arcs of a great circle and both are perpendicular to the curve JBFD. Make AH perpendicular to the plane DCB, Then H is on the diameter JBG\ the nearest point to H on the given circum- ference is C, and the farthest from JS is B, Therefore the chord AG i^ shorter and AB \^ longer than a chord from A to any other point of the curve BFD (IX, 8, ii). Therefore, of all the arcs of great circles from A to points of the given circum- ference, the shortest \^ AC and the longest is AB. Corollaries. — I. From the pole of a great circle any number of perpendiculars extend to the circumference. II. The two perpendiculars from one point to a circumfer- ence of a great circle together make a semi-circumference through the pole. III. Distance between a point on the surface and an arc is reckoned by the shorter perpendicular. IV. Two points on the surface at the same distance from the plane of a circle are also equally distant from the circumference. For the arcs on the surface subtend equal angles at the center of the sphere (V, 14, c). V. If a point on the surface is at the same distance from the planes of two great circles, it is also equidistant from the circum- ferences. YI. If two points on the sur- face are at unequal distance from the plane of a circle, the point farther from the plane is also far- ther from the circumference. VII. Every point of the arc bisecting a spherical angle is equi- distant from the arms of the an- Art. 13.] SOLIDS OF REVOLUTION, 225 gle. A plane bisecting the diedral -Si^cuts the spherical surface in an arc that bisects the angle B. VIII. The arms of a spherical angle are farthest apart at a quadrant's distance from the vertex ; for that point of the arc is farthest from the edge of the diedral. An arc of a great circle that is perpendicular to the diameter BF bisects the arcs B GF and BDF at right angles, and its length is the maximum dis- tance between the arms of the angle B, Spherical Surfaces. 14. A Luue is the part of the surface of a sphere between two halves of great circles. That part of the sphere 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. A Zone is a part of the surface of a sphere between two parallel planes. That part of the sphere itself is called a seg- ment. 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. A spherical Sector is the part of a sphere described by the revolution of a circular sector about a diameter of the circle. It may have ^^Jl two or three curved surfaces. If AB is the axis, and the generat- ing sector is AFC, the sector has one spherical and one conical surface ; but, if, with the same axis, the generating sector is FCG, the sector has one spher- ical and two conical surfaces. A Spherical Polygon is part of the surface of a sphere included between three or more arcs of great circles. Let (7, the center of a sphere, be also the vertex of a polye- 226 GEOMETRY OF SPACE. [Chap. XI. dral. Then the planes of the faces cut the surface of the sphere in arcs of great circles, which form a polygon as BDFGH. Conversely, if a ^ -^ spherical polygon has the planes of its / -^ \ several sides produced, they form a poly- / f\ /\f \ edral whose vertex is at the center of I l.-""''P/ j the sphere. \ ^^y J Each angle of the polygon is the ^'^^^^.^-^ same as the corresponding diedral of the polyedral, and each side of the polygon is the same in curvature or angular quantity as the correspond- ing face of the polyedral. CoroUa,ries, — I. The limit of the sum of the sides of a con- vex polygon is the same as of the faces of a convex polyedral (IX, 41) ; and the limit of the sum of the angles is the same as of the diedrals of a polyedral (IX, 42). The Spherical Excess of a polygon is the excess of the sum of its angles over the sum of the angles of a plane polygon of the same number of sides. If the polygon is concave, the re- flex angle is counted, as in plane figures. II. The spherical excess of a convex polygon is less than four right angles. A Spherical Pyramid is part of a sphere included between a spherical polygon and its corresponding polyedral. The poly- edral is its base. Spherical Triangles. 15. Three great circles divide the surface of a sphere into eight triangles, as three planes divide the space about a point into eight parts (IX, 29). Thus every spherical triangle has seven others, whose elements (sides and angles) are either equal or supplementary to those of the given triangle. The one opposite the given triangle, as GKH to FDB^ has elements respectively equal to the elements of the given trian- gle, but arranged in reverse order, the corresponding triedrals being symmetrical. As defined by some geometers, the sides of a spherical trian- Art. 15.] SOLIDS OF REVOLUTION-. 227 gle may exceed a semi-circumference. For example, the figure bounded by the arcs JBKHD, DF, and FB is called a spheri- cal triangle. The properties of such figures are readily deduced from those of triangles whose perimeters are less than 180°. The sides and angles of a spherical triangle are equal in angular quantity to the faces and diedrals of the correspond- ing triedral at the center of the sphere. Hence a spherical tri- ^ angle may be isosceles, or it may be equilateral, or rectangular, etc. Also two triangles may be symmetrical, or they may be supplementary. The planes of three great circles, each of which is perpendic- ular to the other two, make eight equal trirectangular triedrals at the center of the sphere. The eight corresponding triangles have their angles all right angles and their sides all quad- rants. A Quadrant al triangle is a spherical triangle whose sides are all quadrants. It is also tri- rectangular. Its area is one eighth that of the sphere. Its spherical excess is one right an- gle. Spherical triangles differ from triedrals in this respect — a triangle has a certain area, and its perimeter has a certain length, a triangle is a magnitude. The elements of a triedral are angular quantities. Two triangles can not be equal, unless upon the same sphere or upon equal spheres. If two triangles on unequal spheres have their elements re- spectively equal in angular quantity, and arranged in the same 228 GEOMETRY OF SPACE, [Chap. XI. order, the triangles are similar. Placing the spheres concentric- ally, the demonstration is evident. Corollaries, — I. The sum of the angles of a spherical trian- gle is greater than two and less than six right angles (IX, 38). II. An isosceles spherical triangle is equal to its symmetrical. It has equal angles opposite the equal sides, and conversely (IX, 38, c). The radius being the same, when two triangles have the fol- lowing elements respectively equal, the remaining elements are respectively equal : III. The three sides (IX, 35), or IV. The three angles (IX, 86), or V. Two sides and the included angle (IX, 87), or VI. One side and the adjacent angles (IX, 38). VII. In every case of equal elements, when the arrangement is the same the triangles are equal ; when reversed, they are symmetrical. VIII. If one arc of a great circle bisects another perpendicu- larly, every point of the first is equidistant from the ends of the second arc. IX. From one point on the surface four oblique arcs make equal angles with the same cir- cumference. If the arc AG bisects DF at right angles, the angles ADC, AFC, AFC, and AGO are equal. X. Four unequal right-an- gled triangles may have three elements respectively equal. For instance, the triangles ADG,AGF,AGF,aTidAGG. There are several similar cases of unequal triangles that have three elements respectively equal ; some of which are the same as analogous cases in plane triangles. XL Bisectors of the angles of a triangle meet at a common point, which is equally distant from the sides. If perpendicu- Art. 15.] SOLIDS OF REVOLUTION. 229 lars are made from this point to the several sides and a small circle is made through the feet of these perpendiculars (8, vii), this circle is inscribed in the triangle. Compare Article 3, Chap- ter V. XII. The three arcs that bisect perpendicularly the sides of a triangle meet in one point, which is equally distant from the vertices, and is the pole of a small circle circumscribing the tri- angle. XIII. If two triangles are equal or symmetrical, the circum- scribed circles are equal. 16. Theorem. — In a spherical triangle^ the greater side is opposite tJie greater angle, and conversely. If, in the spherical triangle AB C, the angle B is greater than the angle (7, it is to be proved that the side AC i^, greater than AB. Making the angle CBD equal to (7, we have DB Cy an isosceles triangle. Now, AC = AD-{-I>C = AI)^2)B>AB. Conversely, the greater angle is opposite the greater side. 1*7. Theorem. — Iii a right-angled spherical triangle, one of the other angles and its opposite side are either both equal to ninety degrees, or both less, or both greater. Suppose A the right angle, and the center of the sphere. Then, 1st, if the side AB is a quadrant, B is the pole of the arc A C, for the radius BO is perpendicular to the plane of A C, Therefore the arc BC is perpendicular to A C, that is, the angle opposite the quadrant AB is right. 2d. If the side AJEJ is less than a quadrant, £J being on the arc AB and ^ B JE'jP being the hypotenuse, then join I^B. / Now BFA is a right angle, but EFA is / a part of BFA. / 3d. If E is beyond B, that is, if the / side is greater than the quadrant AB, ^ \I^ by joining the vertex of the opposite 230 GEOMETRY OF SPACE. [Chap. XL angle to B it is shown that that angle is greater than a right angle, the whole being greater than a part. Two angular quantities are said to be of the same species when both are less than, or both equal to, or both greater than ninety degrees. Polar Triangles. 18. If, at the vertex of a triedral, a perpendicular is erected to every face, these lines form the edges of a supplementary triedral. If the vertex is at the center of a sphere, there are two spherical triangles corresponding to these triedrals, and they have the same relations as two supplementary triedrals. Since each edge of one triedral is perpendicular to the oppo- site face of the other, the vertex of each angle of one triangle is the pole of the opposite side of the other. Such triangles are called polar triangles, and sometimes supplementary. Theorem. — If, with the vertices of a spherical triangle as poles, three arcs of great circles are made, a second triangle is formed whose vertices are poles of the sides of the first. Let ABC he the given triangle, and EF, BF, and BF arcs of great circles whose poles are respectively y^^,^^ A, B, and G. / ^V\ Since A is the pole of the arc / / \ \ FF, the distance from J. to i^ is / / \ \ one quadrant, and, since B is the / / jj^.''' \ pole of the arc BF, the distance /;(:::""" i'^'^^J..^-^^ from B to F IB one quadrant. Therefore F is the pole of the arc AB. In the same way it is proved that B is the pole of the arc B C, and that F is the pole of the arc A C. If BFF were the given triangle, the polar triangle formed from it would be ABC\ each is the polar triangle of the other. 19. Theorem. — Of two polar triangles, every angle of either triangle and the opposite side of the other are supple- mentary. Art. 19.] SOLIDS OF REVOLUTION, 231 In the two triangles AB C and DEF^ the pole of AB is 2>, the pole oi AC IB F, etc. The sum of FB and GH is equal to the sum of the two quadrants FH and GB, but the arc GB[ p is equal in angular quantity to [\ B the anffle A. Therefore the / \^\ angle A and the arc i^2> are /| \ X supplementary. By producing X I \ \(7 the side BF both ways to meet jr/ \ m^-^^ \ BA at ^ and B C produced at ^^-^^^^.e.rj c^"'\ ^^ -^iV iVJ it may be shown that B and ^' JJ BF are supplementary. Simi- larly F is the supplement of A C, etc. These two theorems are corollaries of the theorem on sup- plementary triedrals. Compare the two modes of demonstra- tion. The name " polar " refers to the position of the triangles as described in the first theorem. Two supplementary triangles do nob necessarily have any relative position. Another triangle equal to BFF might be on any part of the sphere, and such a triangle and the triangle AB C would have the reciprocal rela- tion stated in this second theorem. The student may derive assistance from diagrams on a globe. Draw the polar triangle of a birectangular triangle, of a trirect- angular triangle, of a very small triangle, etc., etc. 30. Exercises. — 1. A plane tangent to a sphere at the pole of a circle is parallel to the plane of that circle. 2. In a right-angled spherical triangle, if one of the oblique angles is acute, it is greater in angular quantity than its opposite side ; if obtuse, it is less. But, when two angles of a triangle are right, the third angle and its opposite side are equal in angular quantity. [Consult the diagram of Article 15, Corollary IX.] 3. If arcs of any two circles cut each other, their angle is equal to the diedral formed by two planes, each of which passes through a line tan- gent to one of the curves at their intersection, and through the center of the sphere. 4. If arcs on a sphere are tangent to the same straight line, they are tangent to each other. GEOMETRY OF SPACE. [Chap. XL Spherical Areas. 21. Theorem. — The area of the curved surface of an in- scribed 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 perpendicular let fall from the center of the sphere upon the slant height of the frustum. Let AEFJ) be the semicircle that de- scribes the given sphere, and EBHF the trapezoid that describes the frustum. Make EK perpendicular to FH. Then EK is the altitude of the frustum. Let IC be the perpendicular from the center of the sphere on EF. Then the circum- ference of a circle of this radius is 2rcCI. It is to be proved that the area of the curved surface of the frustum is equal to the product of 2r:CIxEK. The chord EF is bisected at the point I. Make GI perpendicular to AD. The point I in its revolu- tion describes the circumference of the section midway between the two bases of the frustum. GI is the radius of this circum- ference, which is therefore 2tt GI. The area of the curved sur- face of the frustum is equal to the product of the slant height by this circumference ; that is, ^ttGIxEF. The triangles EFK and IG (7, having their sides respective- ly perpendicular to each other, are similar. Therefore, EF'.EK = CI:GL Therefore, GI X EF =CIX EK, and 2t:GIxEF= 2irCIxEK. As the first member of this equation is equal to the area of the curved surface of the frustum, the second is equal to the same area. Corollary. — If the vertex of the cone were at the point A, the cone itself would be inscribed in the sphere. The curved surface of an inscribed cone is equal to the product of its alti- tude by the circumference of a circle whose radius is a perpen- Art. 21.] SOLIDS OF REVOLUTION. 233 dicular let fall from the center of the sphere upon the slant height. 23# Theorem, — The area of the surface of a sphere is equal to the product of the diameter by the circumference 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 revolves with it about the common diameter AB. Then, the surface described by the side AD is equal to 2?: GI by AH. The surface described by DE is equal to 27r GI by HK^ for the perpendicular let fall upon DE is equal to GI\ 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. Add- ing these equal quantities, the sums are equal. That is, the en- tire surface described by the revolution of the regular polygon about its diameter is equal to the product of the circumference whose radius is Glhj the diameter AB. Let the polygon vary, doubling the number of sides at every step. The radius of the sphere is the limit of the variable apo- them GI\ therefore the limit of the circumference having this variable radius is the circumference of a great circle. The limit of the variable perimeter is the semi-circumference ; therefore the limit of the surface described by the perimeter is the surface of the sphere. Therefore the area of the surface of a sphere is equal to the product of the diameter by the cii'cumference of a great circle. Corollaries. — I. 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 diame- ter. The spherical area may be expressed by 4:TtII^, II. The area of the quadrantal triangle is one half that of a great circle. It may be expressed by ^ttB^, III. The area of a zone is equal to the product of its altitude by the circumference of a great circle. For the area of the 11 234 GEOMETRY OF SPACE, [Chap. XI. zone described by the arc AD is equal to the product of AH by the circumference whose radius is the limit of the variable apothem. TV, The area of the surface of a sphere is equal to the area of the curved surface of a circumscribing cylinder. If a semicircle is inscribed in a rect- angle and the figure is revolved about the diameter as an axis, a sphere and a cylinder are generated, and the sphere is inscribed in the cylinder ; that is, the curved surface and both bases of the cylinder are tangent to the sphere. The altitude of the cylinder, the diam- eter of its base, and the diameter of the sphere are equal. V. The area of a lune is to the area of the whole spherical surface as the angle of the lune is to four right angles. For the extent of the lune and of its surface vary, as the angle D GE^ or its equal DAE. Let n be the number of right angles in the angle of the lune, then the area is - of the area of the sphere, and it is ^ 2n times the area of the quadrantal triangle. It may be ex- pressed by TTfili^. Areas of Triangles. !33. Theorem. — Two symmetrical spherical triangles are equivalent. Let AEI and B CD be two symmetrical triangles, the angle A being equal to B, E to C, and I to D. Then the side AE is equal to BC, AI to BD, and EI to CD, but the triangles are not superposable. It is to be proved that they have equal areas. Let a circle be described about each triangle, the poles being O and F. These two circles are equal (15, xiii). Art. 23.] SOLIDS OF REVOLUTION. 235 By arcs of great circles join OA, OE, and 01 \ and, in the same way, join FB, FC, and FB. The triangles A 01 and BFB are isosceles, and mutually equi- lateral ; for A 0, 10, BF, and BF are equal arcs (10, ii). Hence, these triangles are equal (15, iii). For a similar reason, the triangles I OF and GFB are equal ; also, the triangles A OF and BFC. Therefore the areas of the trian- gles AFI and BOB, being the differences of equals, are equal. The pole of the small circle may be inside of the given trian- gle, in which case each of the original triangles is the sum of three isosceles triangles. 34, Theorem. — If the quadrantal triangle is taJcen as the unit of spherical area, and the right angle as the unit of angu- lar quantity, the area of any spherical triangle is indicated by its spherical excess. That is, if there are a right angles in the spherical excess, the area is a times that of the quad- rantal triangle. Let AB G be any spherical tri- angle. Produce the sides around the sphere and consider the great circle BGFB as the plane of ref- erence. Let m, n, and p represent the number of right angles in the sev- eral angles A, B, and G. Then m, n, and p may each represent any number less than 2. The spherical excess is {in-\-n-\-p — 2) right angles, and it is to be proved that the area of the triangle is {m -\- n -\- p — 2) times the quadrantal triangle. The area of the lune AB GA is 2;?^ quadrantal triangles. The triangle BOG, which is a part of this lune, is equivalent to 236 GEOMETRY OF SPACE. [Chap. XL its opposite and symmetrical triangle DAF. Therefore the area of the two triangles ABC d^ndi DAF= 2m quadrantal triangles. The area of the lune BCFAB = 2n quadrantal triangles. The area of the lune GBDA C ~2p quadrantal triangles. In the sum of these three lunes, the triangle AB C is included three times, and the rest of the spherical surface, above the plane of reference, once. The sum is the area of the hemisphere, plus twice the area of the triangle AB C, but the area of the hemi- sphere is equal to that of four quadrantal triangles. Then, the areas of 4 quad. tri. -j- 2 triangles AB C = 2{m-{-n-{-p) quad. tri. Transposing the first term and dividing by 2, area AB C = {m-\-n-\-p — 2) quad. tri. Corollaries. — I. If the square of the radius is the unit of area, then the area of a spherical triangle i% ^ {m -\- n -{- p — 2) II. The area of any spherical polygon is indicated by its spherical excess. For the spherical excess of the polygon is the sum of the spherical excesses of the ti'iangles that compose it ; and its area is the sum of their areas. Triedrals and other polyedrals are measured by their spher- ical excess. The sum of the diedral angles over the sum of the angles of a plane polygon of the same number of sides indicates what portion of space is included between the faces. Ordinarily it is called the spherical area for a sphere having unity for its radius. This is applied in measurements of anything radiating in all directions from a point, such as light, heat, etc. The portion in- tercepted by any body, as compared with the whole amount radiated, is accurately measured by means of the spherical ex- cess ; or it may be called the spherical area for radius unity. Art. 25.] SOLIDS OF REVOLUTION. 237 Volume of the Sphere. 35. Theorem. — The volume of a sphere is equal to one third of the product of the surface by the radius. Suppose a polyedron circumscribed about the sphere. A plane may pass through each edge of the polyedron, and extend to the center of the sphere. These planes 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 to the point of tangency is perpendicular to the tangent plane. The volume of each pyramid is one third of its base by its alti- tude. Therefore, the volume of the polyedron is one third the sum of the bases by the common altitude, or radius. A sphere is the limit of a variable circumscribed polyedron which varies according to a certain law. This may be demon- strated by a reasoning like that in the analogous case of the cir- cle, showing that at every step the space between the sphere and the variable polyedron is diminished by more than a given frac- tional part of the remainder. The following demonstration by the method of infinites is more elegant and equally rigorous. Make the square CADB and the quadrant BA, having its center at (7, and join CD. If this figure revolves upon B Cy the square gener- ates a cylinder, the quarter circle BAG generates a hemisphere, and the triangle BJDG generates a cone. This cylinder, segment, and cone have equal bases, that is, the circle whose radius is CA or BJDy and they have the same altitude, BG. Through any point of BG, say E, make a plane parallel to the base. Three circular sections are made, a section of the sphere having the radius EH, a section of the cylinder 238 GEOMETRY OF SPACE. [Chap. XI. having the radius EF, and a section of the cone having the radius EG. Join GS. Then, since CEH is a right angle, EH^= CS^—EG^. But GE:= GA = EF, and EG = EG. Substituting and multiplying by tt, ttEE:^ = ttEF^ — ttEG^, that is, the section of the sphere is equivalent to the difference of the sections of the cylin- der and of the cone. The equation between these sections is true when E is at any point of the axis £G. Suppose a plane par- allel to the base through every point of £G. The hemisphere is composed of the sum of this infinite num- ber of parallel spherical sec- tions, that is, the sum of the terms represented by ttEIT^. The cylinder is composed of the infinite number of terms represented by ttEF^, and the cone of those represented by ttEG^. Every term in the first of these three infinite series has its correspond- ing term in the second series, and in the third. Therefore the same equality exists between the sums as between three corre- sponding terms ; and the volume of the hemisphere is equal to the difference of the volumes of the cylinder and the cone. Since the volume of the cylinder is equal to the product of the base by the altitude, and the volume of the cone to one third of that product, the volume of the hemisphere is equal to two thirds of the product of a great circle by the radius, and the vol- ume of a sphere is equal to four thirds of the product of a great circle by the radius, ^nH^. The area of the surface being four times that of a great circle, the volume is one third of the prod- uct of the surface by the radius. Art. 25.] SOLIDS OF REVOLUTION. 239 Corollaries. — I. The volume may be expressed by ^rrD^, or by iirUK II. The volume of a spherical pyramid, or of a spherical wedge, or of a spherical sector, is equal to one third of the prod- uct of the area of its spherical surface by the radius. The volume of a spherical segment of one base is equivalent to the sum or to the difference of the volume of a cone and that of a sector. For the sector ABCI> is composed of the segment ABC and the cone A CD. 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 AEFG is equal to the segment ABC less EBF. 2G» 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. Let D and d represent the diameters of any two spheres. Then the areas are 7r7>2 and 'nd'^^ whose ratio is J>2 . <^2^ The volumes are ^-rxD^ and ^ttc^^, whose ratio is D^ : d^. Since solids with curved surfaces are the limits of variable polyedrons, any similar figures have for their superficial ratio the square of the linear ratio, and for their solid ratio the cube of the linear. 2*7. Scholium. — Our subject has developed from the sim- plest linear figures to the doctrine of the cone and the sphere. The topics are nearly the same as were discussed two thousand years ago by Euclid and Archimedes. Euclid arranged the matter more for convenience of demonstration than with refer- ence to a classification of magnitudes (II, 15 and 16). His first theorems were about triangles. From these the properties of 240 GEOMETRY OF SPACE. [Chap. XI. other inclosed figures and of angles and parallel lines were de- duced. Until within a century, geometers have pursued sub- stantially the same order. Vincent was (I believe) the first to introduce, about 1830, the arrangement according to a logical classification of magnitudes. This logical order lets the student see that there are fields of geometrical research on both sides of the path he pursues. The science of geometry is not merely a specimen of rigorous demon- stration, although some famous teachers have seen nothing else in it. During the present century the theory of proportional lines has been very much developed. Centers of similarity, the ratios arising from lines meeting at one point and cut by other lines called transversals, and many theorems more or less related to these notions, constitute almost a distinct science, which has been called Modern Geometry. The properties of the Conic Sections have been developed in ancient times by the Euclidean method, and more exhaustively in modern times by algebraic methods. The application of al- gebra to the investigation of magnitudes constitutes the science of Analytic Geometry. The calculation of angles and of distances belongs to the science of Trigonometry. Mensuration, or the art of measuring, is an application of Geometry and Trigonometry. No real progress can be made in these studies without fre- quent use of the elementary principles of Geometry. 38« Exercises. — 1. Find the area of the earth's surface, suppos- ing it to bo a sphere with a diameter of 7,912 miles. 2. Find what part of the surface is between the equator and the par- allel of 30° north latitude. 3. Find what part of the surface is between two meridians which are ten degrees apart. 4. Find the area of a triangle described on a globe of 13 inches diam- eter, the angles being 100°, 45°, and 53°. 5. Discuss the possible relative positions of two spheres. 6. If a spherical triangle has one side a quadrant, another side and its opposite angle are of the same species. Abt. 28.] SOLIDS OF REVOLUTION, 241 7. The surface of a sphere can bo completely covered with either 4, or 8, or 20 equilateral spherical triangles. 8. The volume of a cone is equal to the product of its whole sur- face by one third the radius of the inscribed sphere. 9. If, about a sphere, a cylinder is circumscribed, also a cone whose slant height is equal to the diameter of its base, the area and the volume of the sphere are two thirds of the area and the volume of the cylinder ; and the area and the volume of the cylinder are two thirds of the area and the volume of the cone. ee y- NOTES. NOTE A. General Axioms, Chapter I, Article 2. The attempts of logicians and geometers to state the first principles of inference, or laws of thought, have not resulted in concurrence of opinion. In the Elements of Euclid there are nine ' ' common notions " as he called them. Legendre retained two of these: 1. Two quantities equal to a third are equal to each other ; and 2. The whole is greater than its part. Six of Euclid's axioms have been summed up in these two : 1. If the same operation is performed on equal quantities the results are equal, and 2. If the same operation is performed on unequal quantities the results are unequal. "With these Euclid stated an axiom, which is geometrical, and which Legendre has also retained, asserting the equiva- lence of congruent magnitudes. Logicians have differed as to whether this is merely a definition. The above apply only to relations of extent ; which vary by addition, subtraction, multiplication, or division. An axiom that covers the whole truth should include changes in form, or in any other attribute of things. For example : If two things that agree in a common attribute are changed in the same way and to the same amount, the results must agree as to that attribute ju^t as the two things agreed. Any statement of the axioms or primary laws of reasoning is omitted from the text, as they pertain rather to metaphysics than to geometry, and no statement yet made seems suf- cient. 244 NOTES, NOTE B. Postulates and Problems, Chapter I, Article 2 ; Chapter II, Articles 3, 8, 13, etc. It was taken for granted by Euclid that a straight line may have any position, and that its length may be increased to any extent, also that a circumference of a circle may have any position and any extent (IV, 34). It does not appear that he regarded these as self-evident truths, though in modern times they have received that character. They are only a j)ar- tial statement of the truth, for a line may be of any extent, small or great ; so may any magnitude; and a magnitude may be of any form. All this truth is as indemonstrable and as universal as the part demanded by Euclid. He carefully refrained from stating anything more than he thought was necessary to explain the construction of the figm-es con- tained in his Elements. Legendre assumed tacitly the possibility of every figure that can be precisely defined. He stated no postulates, but made any construction that was not self-contradictory. This method removes all occasion for problems in the theory of pure geometry. Accordingly, in Legendre's work, the problems are all problems in drawing. These are relegated to a separate place, as an application of geometric principles to an art. Clearly, Euclid did not regard his problems in that way. He pro- ceeded upon the rule that the possibility of every figure must be demon- strated. All that was deemed necessary to show this of the figures with- in the scope of his work was stated in his postulates concerning the straight line and the circle. He therefore limited his demands to these. The rule and compasses are used to draw straight lines and circles, and (long after Euclid's day) the problems came to be associated with these instruments, and hence to be regarded as problems in drawing. But the postulates of Euclid demand less than can be done with these instru- ments in some respects, and more in others. We may infer that his pos- tulates and the problems which depend on them were intended as a part of pure geometry. They stated what is possible with geometrical lines, which are not visible to the material eye, nor can be made with rule and compasses. There are two reasons for stating the Postulates of Form and of Ex- tent : 1. Every premise should be expressed. This is of no less importance if the premise is assumed as a first principle, for the student ought to see and examine the grounds of the argument. 2. The statement should be as broad as the nature of the truth; for the student may infer that the NOTES. 245 statement made is the whole truth. This has in fact occasioned general error as to the character of certain problems. That a square can be equivalent to a circle has been, by the half -learned, regarded as impos- sible, because the drawing is beyond the power of rule and compasses. Professor De Morgan, of University College, London, among his fine criticisms on Euclid, suggests the following as a postulate used in de- monstration by superposition : "Any figure maybe removed from place to place without alteration of form, and a plane figure may be turned round on the plane." Demonstration by superposition does assume this. It is, however, a corollary of the postulates in the text. If a figure can be of any form or extent, then a figure may be the result of combining in any way two that are given. The proposed postulate that **a magnitude may have any position " / is also a corollary. Position is relative ; the position of anything is its / direction and distance from another thing to which it is referred. / Therefore the notion of a position involves two things in one geometricali figure. The two are parts of one magnitude or combination, and their \ relative position is one element of the form of that total. Tliis so-called "postulate of position" is therefore included in the Postulate of Form. Form and extent are the only properties of magnitude. Therefore these two postulates assert the possible existence of whatever can be thought of magnitudes. NOTE C. "Direction" in Definition, Chapter II, Articles 6, 7, 10, and 14. Direction is as certain and definite a term as distance. Both words are used, without ambiguity, by every mathematician. No one attempts a definition of either. Two directions may be identical, and two distances may be equal. If the distances vary equally and in the same way, the re- sults are equal. If the directions vary equally and in the same way, the results are identical. Objectors to the use of direction in defining angle and parallel admit the propriety of saying the direction of a straight line is uniform — that the straight line AB may be produced "in the same direction" to G. Now, if, beyond B, there can be one line and only one in the direction AB^ it is equally true that from i>, a point to one side, there can be a straight line having the same direction AB. Not only is this possible, but the idea is more simple and precise than any other definition of par- allels. The possibility is an immediate inference from the Postulate of Form. 246 NOTES, In a finite straight line there are two elements, one of extent and one of form. For purposes of definition, Archimedes used the element of ex- tent: *^A straight line is that which is the shortest from one point to an- other." In this he was followed by Legendre and other eminent modern geometers. Euclid used the element of form : " A straight line is one that lies evenly as to its points." This is better, for the essential character of a geometrical figure is its form. But this Euclidean definition was ob- scure and barren ; it was not used in the Elements. The Greeks made the same word serve for "straight " and for "direction." The second postu- late has evdeiav iTr* evdtiag kK^dlleiv. Euclid's definition of a "plane angle" was "the leaning, K)daiQ^ to each other of two lines that meet and are in one plane and not in the same direction." "When the lines are straight, the angle is called recti- linear." Here is a near approach to the simplest idea of angular quan- tity. The ancient geometer saw that two lines must be "not in the same direction " from the point of meeting, but he did not see that two separate lines may be in the same direction. Hence liis definition of par- allels (III, 19) fails to show the true relation between angles and paral- lels. This is done only when they are seen to be species of one genus. Early in the eighteenth century, the German philosopher, Wolff, pro- posed to define parallels as straight lines that are everywhere equidistant, in which he has been followed by several eminent authors. The use of distance is as objectionable in the definition of parallels as in that of straight lines. It belongs with that definition of angles which makes them contain a portion of the indefinite or infinite plane. De Morgan, the most rigorous logician of the English mathematicians of the present century, said ' ' permanency of direction and straightness are equivalent notions " (Penny CyclopoBdia, Direction). In another ar- ticle, he said, "also the notion of differing directions is suggested by two lines which make an angle"; and he added, "we may readily see that the relation of situation which, adopting Euclid's term, may be called parallelism, is really that which would be also conveyed by the words sameness of direction." This was in 1838. He does not seem to have known that straightness and parallelism had been defined in this way, nine years previously, by an American author, Hayward. The English "Association for the Improvement of Geometrical Teaching " declares that an angle is incapable of definition, but says that "the angle is greater as the quantity of turning is greater " when one arm turns about the vertex. The idea of rotation is good for the explanation of variable angles, which become greater than a straight angle. The turning or rotation is a change of direction, a variation of angular quan- tity. But an angle may be a constant. The variation of quantity and NOTES, 247 the rotation of an arm are not essential characteristics. When divested of these accidental attributes, an angle is merely the difference of two directions. When this difference is zero, that is when the directions are identical, the result is either parallelism or coincidence. NOTE D. Axioms of Direction, Chapter II, Article 7. A statement of the principles that have been assumed by geometers to be primary truths concerning angles and parallels may assist the stu- dent to judge correctly what are the fundamental principles of direction. Euclid said, in addition to the three postulates in the text (IV, 33), ** Let it be granted *'4. That all right angles are equal ; *'5. That if a straight line meeting two straight lines makes the in- terior angles on the same side less than two right angles, the two straight lines being produced infinitely meet each other on that side on which the angles are less than two right angles ; and *' 6. That two straight lines do not enclose a space." Legendre expressed no postulates and only one geometrical axiom: "From one point to another only one straight line can be drawn." This is generally followed by geometers on the Continent. In England editors of Euclid have proposed various substitutes for the fifth demand. Simson's Axiom. — *'A straight line can not first come nearer to another straight line, and then go further from it, before it cuts it ; and, in like manner, a straight line can not go further from another straight line, and then come nearer to it ; nor can a straight line keep the same distance from another straight line, and then come nearer to it, or go further from it; for a straight line keeps always the same direction." Playf air's Axiom. — ''Two straight lines which intersect one another can not be both parallel to the same straight line." This author makes the following definition, which takes the place of Euclid's sixth postu- late : " If two lines are such that they can not coincide in any two points, without coinciding altogether, each of them is called a straight line." Among American authors, Hayward has two axioms — the one given by Legendre, and the axiom of distance. Peirce gives the axiom of distance and this: ''The direction of any point of a straight line from any preceding point is the same as the direction of the line itself. " 248 NOTSS. Loomis states two axioms, those of Playfair and of Legendre, and he follows Archimedes in defining a straight line. NOTE E. Some Teems variously used. Equal and Equivalent, Chapter II, Articles 3 and 4. — ^English au- thors, following Euclid or his translators, call figures that have the same extent "equal." Legendre used the word "equivalent," and this has been generally followed in America, and on the Continent of Europe. In order to express what we call "equal," the Association for the Im- provement of Geometrical Teaching says "identically equal." The terms "congruent," and "equal in all respects," have been used with the same meaning. Figure, Chapter II, Article 5. — ^Euclid's definition of axvi^o-^ figure, included lines, surfaces, and solids of limited extent, whatever is con- tained by one or more boundaries or ends. Legendre, whose first edition of the Elements was published in 1794, and Playfair, whose first edition of Euclid was published about the same time, both applied the word figure only to limited plane surfaces, and to inclosed portions of space. This restricted use of the word was generally adopted till within the last twenty years. In 1864, seeing that a term was needed to designate "any magnitude or combination of magnitudes," I made this the definition of figure. It has been adopted by several authors. Polygon, Chapter II, Article 15. — Euclid did not include triangles or quadrilaterals in this term. Legendre applied it to all plane, rectilinear, inclosed figures. There should be one word to designate this class of ob- jects. Trapezoid, Chapter VI, Article 1. — In American geometries and dic- tionaries, a quadrilateral with two sides parallel is called a trapezoid, but in England this figure is generally called a trapezium. INDEX. [This Index includes the terms defined in the text, and a few other words.] Acute angle, iii, 6. Acute-angled triangle, v, 1. Adjacent angles, iii, 6. Adjacent angles in a triangle, v, 1. Algebraic method, vi, 18. Alternate angles, iii, 20. Altitude of a triangle, v, 1. of a quadrilateral, vi, 1. of a tetraedron, x, 2. of a pyramid, x, 5. of a prism, x, 6. of a cone, xi, 1. of a cylinder, xi, 1. of a spherical zone and segment, xi, 14. Ancient method, vi, 18. Angle, ii, 14. in an arc, iv, 22. of a line and plane, ix, 10. diedral, ix, 13. spherical, xi, 12. Apothem, vii, 16. Arc, iv, 3. containing angle, iv, 22. Area, ii, 2. Arms of an angle, ii, 14. Axiom, i, 2. of Direction, ii, Y. of Distance, ii, 7. Axis, ii, 9, of Symmetry, iv, 5. of a Circle, ix, 8. Axles of wheels, iv, 8. Base of a triangle, v, 1. of a quadrilateral, vi, 1. of a tetraedron, x, 2. of a pyramid, x, 5. of a prism, x, 6. of a cone, xi, 1. of a cylinder, xi, 1. of a spherical segment, xi, 14. of a spherical pyramid, xi, 14. Birectangular triedral, ix, 29. Broken line, ii, 6. Center of a circle, ii, 15. of symmetry, iv, 5. of similarity, v, 28. of a regular polygon, vii, 15. of a regular polyedron, x, 11. of a sphere, xi, 1. Central line, iv, 29. Centroid of a triangle, x, 37. Chord, iv, 3. Circle, ii, 15. great, xi, 8. small, xi, 8. Circumference, ii, 15. Circumscribed circle and polygon, v, cone and pyramid, xi, 6. cylinder and prism, xi, 7. sphere and polyedron, xi, 8. Commensurable lines, iii, 1. Compasses, iv, 33. Complementary, iii, 6. 250 INDEX. Complements of angles, iii, 6. of parallelograms, vii, 14. Concave line, iii, 3. polygon, vii, 1. polyedral, ix, 40. Concentric circles, iv, 1. Cone, xi, 1. Conic sections, xi, 4. Conjugate angles, iii, 6. arcs, iv, 9. Constant, iii, 28. Construction, ii, 5. Contact, point of, iv, 16. Contained, angle by arc, iv, 22. rectangle by lines, vi, 10. Contrapositive, iii, 29. Converse propositions, i, 2. Convex line, iii, 3. polygon, vii, 1. polyedral, ix, 40. Corollary, i, 2. Corresponding angles, iii, 20. Cube, X, 6. Curvature, ii, 6. Curve, ii, 6. Curved surface, ii, 10. Cut, line by plane, ix, 3. Cylinder, xi, 1. Edge of a diedral, ix, 13. of a triedral, ix, 29. of a polyedron, x, 1. Elements of a triangle, v, 9. of a triedral, ix, 29. Equal and equality, ii, 4. Equilateral triangle, v, 1. triedral, ix, 29. Equivalent, ii, 3. Escribed circle, v, 2. Excess, spherical, xi, 14. Exhaustions, method of, iv, 31. Extent, Postulate of, ii, 3. Exterior angles, iii, 20. External center of similarity, v, 28. Externally, line divided, iii, 1. Extreme and mean ratio, iii, 2. Face of a diedral, ix, 13. of a triedral, ix, 29. of a polyedron, x, 1. Figure, geometrical, ii, 5. plane, ii, 15. Foot of line in a plane, ix, 3. Form, Postulate of, ii, 3. symmetry of, ix, 34. Frustum of a pyramid, x, 5. of a cone, xi, 4. Decagon, ii, 15. Determinate problem, iv, 34. Diagonal of a polygon, vi, 1. of a polyedron, x, 1. plane, ix, 40 ; and x, 1. Diameter, iv, 1 ; and xi, 1. Diedral, ix, 13. Direct demonstration, iv, 31. superposition, iii, 31. Direction, Axiom of, ii, Y. Distance, Axiom of, ii, 7. Division of lines, iii, 1. Dodecaedron, regular, x, 11. Dodecagon, ii, 15. Duplicate ratio, vi, 13. Gauge, iii, 30. Geometry, ii, 2. Given, ii, 5 ; and iii, 1. Great circle, xi, 8. Harmonic division, iii, 2. Height, slant of a pyramid, x, 5. of a frustum, x, 5. of a cone, xi, 1. Hemisphere, xi, 1. Hexaedron, regular, x, 11. Hexagon, ii, 15. Homologous, ii, 3. Horizontal, ix, 17. Hypotenuse, v, 1. INDEX. 251 Icosaedron, regular, x, 11. In, angle in arc, iv, 22. Incommensurable lines, iii, 1 and 2. Indeterminate problem, iv, 34. Indirect demonstration, iv, 31. Indivisibles, method of, iv, 31. Infinitesimals, method of, iv, 31. Inscribed angle, iv, 22. circle and polygon, v, 2. cone and pyramid, xi, 6. cylinder and prism, xi, T. sphere and polyedron, xi, 8. frustum of cone, xi, 8. Interior angles, iii, 20. Internal center of similarity, v, 28. Internally, line divided, iii, 1. Inverse superposition, iii, 31. Isoperimetrical, vii, 20. Isosceles triangle, v, 1. triedral, ix, 29. Lateral edges of a pyramid, x, 5. of a prism, x, 6. Length, ii, 2. Light, its path, vii, 26. Limit, iii, 28. Limits, method of, iv, 31. Line, ii, 2. of projection, iii, 12. Linear ratio, v, 22. Locus, iii, 12. Lozenge, vi, 1. Lune, xi, 14. Magnitude, ii, 2. Major arc, iv, 9. Maximum, vii, 20. Mean, and extreme, ratio, iii, 2. Medial, v, 1. Method of superposition, ii, 4 ; iii, 31. of exhaustions, iv, 31. of limits, iv, 31. of indivisibles, iv, 31. of infinitesimals, iv, 31. Method, for exercises in drawing, iv, 34. algebraic, vi, 18. ancient, vi, 18. Minimum, vii, 20. Minor arc, iv, 9. Modes of reasoning, iv, 31. Normal, iv, 17. Oblique lines and angles, iii, 6. Obtuse angle, iii, 6. Obtuse-angled triangle, v, 1. Obverse, iii, 31. Octaedron, regular, x, 11. Octagon, ii, 15. Opposite angles in a triangle, v, 1. Orthogonal projection, iii, 12. Parallel lines, ii, 14. (Euclid's definition, iii, 19). rulers, vi, 7. line and plane, ix, 19. planes, ix, 20. Parallelism, ix, 19. Parallelogram, vi, 1. Parallelopiped, x, 6. Pass, plane through line, ix, 3. Pentagon, ii, 15. Pentedecagon, ii, 15. Perimeter, ii, 16. Perpendicular lines, iii, 6. line and plane, ix, 6. Perspective, ix, 27. Pierce, line, plane, ix, 3. Plane, ii, 10. (Euclid's definition, ii, 12). figures, ii, 15. Geometry, ii, 15. of symmetry, ix, 34. section, ix, 40. Point, ii, 2. of contact, iv, 16. Polar radius, xi, 10. triangles, xi, 18. 252 INDEX. Poles, xi, 8. Polyedral, ix, 40. Polycdron, x, 1. Polygon, ii, 15. spherical, xi, 14. Position, symmetry of, ix, 34. Postulate, i, 2. of Form and of Extent, ii, 3. of Euclid, iv, 33. Practical propositions, i, 2. Prism, X, 6. Problem, i, 2. in drawing, iv, 33. Projection, iii, 12. of line on plane, ix, 9. Protractor, iv, 2*7. Ptolemaic Theorem, vi, 26. Pyramid, x, 6. spherical, xi, 14. Pythagorean Theorem, vi, 28. Quadrant, iv, 20. Quadrantal triangle, xi, 15. Quadrature, vi, 32. of the circle, viii, 1 1. Quadrilateral, ii, 15. Eadius of a circle, iv, 1. of a regular polygon, vii, 15. of a sphere, xi, 1. polar, xi, 10. Railway curve, iv, 27. Ratio, extreme and mean, iii, 2. Rectangle, vi, 1. Rectangular triedral, ix, 29. parallclopiped, x, 6. Rectification of a curve, viii, 8. Reductio ad absurdum^ iv, 31. Reflex angle, iii, 6. Regular polygon, vii, 1. pyramid, x, 5. prism, X, 6. polyedron, x, 11. Reverse face, iii, 31. Revolution, solid of, xi, 1. Rhombus, vi, 1. Right angle, iii, 6. prism, X, 6. solid, X, 6. Right-angled triangle, v, 1. Ruler, iv, 83 ; and vi, 7. Scalene triangle, v, 1. Scholium, i, 2. Secant, iii, 1. Section, plane, ix, 40. Sector of a circle, viii, 1. of a sphere, xi, 14. Segment of a line, iii, 1. of a circle, viii, 1. of a sphere, xi, 14. Side of a polygon, ii, 15. of a tetraedron, x, 2. of a pyramid, x, 5. of a prism, x, 6. Similar, ii, 3 ; and ii, 14. ancient definition, v, 19. Similarity, center of, v, 28. Slant height of a pyramid, x, 5, of a frustum, x, 5. of a cone, xi, 1. Small circle, xi, 9. Solid, ii, 2. of revolution, xi, 1. Space, Geometry of, ii, 16. Species, angles of same, xi, 17. Sphere, xi, 1. Spherical angle, xi, 12. excess, xi, 14. Square, vi, 1. an instrument, iii, 15 and 30. Squaring, vi, 32. of the circle, viii, 11. Stand, angle upon arc, iv, 22. Straight line, ii, 6. (definition of Euclid, ii, 9). (definition of Archimedes, ii, 9.) angle, iii, 6. INDEX. 253 Subtend, iv, 9. Superposition, ii, 4 ; iii, 31 ; and v, 13. Supplement of angle, iii, 6. Supplementary angles, iii, 6. triedrals, ix, 29. spherical triangles, xi, 18. Surface, ii, 2. Surveying land, vi, 16. Symmetrical figure, iv, 5. points, iv, 5. triedrals, ix, 34. Symmetrically similar, x, 19. Symmetry, center and axis of, iv, 5. of form and of position, ix, 34. plane of, ix, 34. Tangent lines, iv, 16. surface, xi, 1. Tetraedron, x, 2. Theorem, i, 2. Theoretical proposition, i, 2. Trapezoid, vi, 1. Triangle, ii, 15. an instrument, iii, 30. Triangle, spherical, xi, 15. quadrantal, xi, 15. Triedral, ix, 30. Trirectangular triedral, ix, 29. T-square, iii, 30. Truncated pyramid, x, 5. Upper base of frustum, x, 5. Variable, iii, 28. Vertex of an angle, ii, 14. of a triangle, v, 1. of a triedral, ix, 29. of a polyedron, x, 1. of a tetraedron, x, 2. of a pyramid, x, 5. of a cone, xi, 1. Vertical angles, iii, 6. line, plumb, ix, 1*7. Volume, ii, 2. Wedge, ix, 14. Zone, xi, 14. THE END. MATHEMATICS. Gillespie's Land-Surveying. Comprising the Theory- developed from Five Elementary Principles ; and the Practice with the Chain alone, the Compass, the Transit, the Theodolite, the Plane Table, etc. Illustrated by 400 Engravings and a Magnetic Chart. By W. M. Gillespie, LL. D., Civil Engineer, Professor of Civil Engi- neering in Union College. 1 vol., 8vo. 608 pages. A double object has been kept in view in the preparation of the volume, viz., to make an introductory treatise easy to be mastered by the young scholar or the practical man of little previous acquirement, the only prerequisites being arithme- tic and a little geometry ; and, at the same time, to make the instruction of such a character as to lay a foundation broad enough and deep enough for the most com- plete superstructure which the professional student may subsequently wish to raise upon it. Gillespie's Higher Surveying*. Edited by Cady Staley, a. M., C. E. Comprising Direct Leveling, Indirect or Trig- onometric Leveling, Barometric Leveling, Topography, Mining, Sur- veying, the Sextant, and other Reflecting Instruments, Hydrograph- ical Surveying, and Spherical Surveying or Geodesy. 1 vol., Svo, 173 pages. Elements of Plane and Spherical Trigonom- etry, with Applications. By Eugene L. Rich- AED8, B. A., Assistant Professor of Mathematics in Yale College. 12mo. 295 pages. The author has aimed to make the subject of Trigonometry plain to begin- ners, and much space, therefore, is devoted to elementary definitions and their ap- plications. A free use of diagrams is made to convey to the student a clear idea of relations of magnitudes, and all difficult points are fully explained and illustrated. Williamson's Integral Calculus, containing Appli- cations to Plane Curves and Surfaces, with numerous Examples. 12mo. 375 pages. Williamson's Differential Calculus, containing the Theory of Plane Curves, with numerous Examples. 12mo. 416 pages. Perkins's Elements of Algebra. i2mo. 244 pages. Inventional Geometry. Science Primer Series. 18mo. The Universal Metric System. By Alfred Colin, C. E. 12mo. D. APPLETON & CO., Publishers, NEW YORK, BOSTON, CHICAGO, SAN FRANCISCO. ASTRONOMY AND GEOLOGY. Lockyer's Elements of Astronomy. Accom- panied with numerous Illustrations, a Colored Kepresentation of the Solar, Stellar, and Nebular Spectra, and Arago's Celestial Charts of the Northern and Southern Hemisphere. American edition, revised, enlarged, and specially adapted to the wants of American schools. 12mo. 312 pages. The author's aim throughout the book has been to give a connected view of the whole subject rather than to discuss any particular parts of it, and to supply facts and ideas founded thereon, to serve as a basis for subsequent study. The arrange- ment adopted is new. The Sun's true place in the Cosmos is shown, and the real movements of the heavenly bodies are carefully distinguished from their apparent movements, which greatly aids in imparting a correct idea of the celestial sphere. The fine STAR-MAPS OF ARAGO, showing the boundaries of the constella- tions and the principal stars they contain, are appended to the volume. Science Primer of Astronomy. i8mo. Elements of Astronomy. By Robert S. Ball, Pro- fessor of Astronomy in the University of Dublin, and Eoyal Astron- omer of Ireland. 12mo. 459 pages. Le Conte's Elements of Geolog^y. A Text-Book for Colleges and for the General Reader. Revised and enlarged edition. 8vo. 633 pages. This work is now the standard text-book in most of the leading colleges and higher-grade schools of the country. The author has just made a thorough re- vision of the work, so as to embrace the results of all the latest researches in geological science. Nicholson's Text-Book of Geology. For Schools and Academies. 12mo. 277 pages. This book presents the leading principles and facts of Geological Science in as brief a compass as is compatible with the utmost clearness and accuracy. Lyell's Principles of Geology; or, The Modern Changes of the Earth and its Inhabitants considered as illustrative of Geology. Illustrated with Maps, Plates, and Woodcuts. 2 vols., royal 8vo. Sir Charles Lyell was one of the greatest geologists of our age. In this work are embodied such results of his observation and research as bear on the modem changes in the earth's structure and the organic and inorganic kingdoms of Nature. Science Primer of Geology. i8mo. D. APPLETON & CO., Publishers, NEW YORK, BOSTON, CHICAGO, SAN FRANCISCO. / ^ Z'^- THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. C SEP 6 193'' MAR 31 1948 ' * LD21-100w-7,'33 184009