,,marks' ,,first ,,lessons ,,in ,,geometry4 ,,in ,,two ,,parts4 ,,objectively ,,presented1 ,,and ,,designed ,,for ,,the ,,use ,,of ,,primary ,,classes ,,in ,,grammar ,,schools1 ,,academies1 ,,etc4 ,,by ,,bernhard ,,marks1 ,,principal ,,of ,,lincoln ,,school1 ,,san ,,francisco4 ,,new ,,york3 ,,published ,,by ,,ivison1 ,,phinney1 ,,blakeman1 ,,co4 ,,philadelphia3 ,j4 ,b4 ,,lippincott ,,co4 ,,chicago3 ,s4,c4 ,,griggs ,,co4 #ahfi4 ,entered1 according to ,act of ,congress1 in the year #ahfh1 by ,,bernhard ,,marks1 ,in the ,clerk's ,office of the ,district ,court of the ,united ,states for the ,district of ,california4 ,geo4 ,c4 ,rand ,avery1 ,electrotypers and ,printers1 #c ,cornhill1 ,boston4 ---------------------------------------- -------------------------------- ,,preface4 ,how it ever came to pass that ,arithmetic should be taught to the extent attained in the grammar schools of the civilized world1 while ,geometry is almost wholly excluded from them1 is a problem for which the author of this little book has often sought a solution1 but with only this result2 viz41 that ,arithmetic1 being considered an elementary branch1 is included in all systems of elementary instruction2 but ,geometry1 being regarded as a higher branch1 is reserved for systems of advanced education1 and is1 on that account1 reached by but very few of the many who need it4 ,the error here is fundamental4 ,instead of teaching the elements of all branches1 we teach elementary branches much too exhaustively4 ,the elements of ,geometry are much easier to learn1 and are of more value when learned1 than advanced ,arithmetic2 and1 if a boy is to leave school with merely a grammar-school education1 he would be better prepared for the active duties of life with a little ,arithmetic and some ,geometry1 than with more ,arithmetic and no ,geometry4 ,thousands of boys are allowed to leave school at the age of fourteen or sixteen years1 and are sent into the carpenter-shop1 the machine-shop1 the mill-wright's1 or the surveyor's office1 stuffed to repletion with ,interest and ,discount1 but so utterly ignorant of the merest elements of ,geometry1 that they could not find the centre of a circle already described1 if their lives depended upon it4 ,unthinking persons frequently assert that young children are incapable of reasoning1 and that the truths of ,geometry are too abstract in their nature to be apprehended by them4 ,to these objections1 it may be answered1 that any ordinary child1 five years of age1 can deduce the conclusion of a syllogism if it understands the terms contained in the propositions2 and that nothing can be more palpable to the mind of a child than forms1 magnitudes1 and directions4 ,there are many teachers who imagine that the perceptive faculties of children should be cultivated exclusively in early youth1 and that the reason should be addressed only at a later period4 ,it is certainly true that perception should receive a larger share of attention than the other faculties during the first school years2 but it is equally certain that no faculty can be safely disregarded1 even for a time4 ,the root does not attain maturity before the stem appears2 neither does the stem attain its growth before its branches come forth to give birth in turn to leaves2 but root1 stem1 and leaves are found simultaneously in the youngest plant4 ,that the reason may be profitably addressed through the medium of ,geometry at as early an age as seven years is asserted by no less an authority than ,president ,hill of ,harvard ,college1 who says1 in the preface to his admirable little ,geometry1 that a child seven years old may be taught ,geometry more easily than one of fifteen4 ,the author holds that this science should be taught in all primary and grammar schools1 for the same reasons that apply to all other branches4 ,one of these reasons will be stated here1 because it is not sufficiently recognized even by teachers4 ,it is this3-- ,the prime object of school instruction is to place in the hands of the pupil the means of continuing his studies without aid after he leaves school4 ,the man who is not a student of some part of ,god's works cannot be said to live a rational life4 ,it is the proper business of the school to do for each branch of science exactly what is done for reading4 ,children are taught to read1 not for the sake of what is contained in their readers1 but that they may be able to read all through life1 and thereby fulfil one of the requirements of civilized society4 ,so1 enough of each branch of science should be taught to enable the pupil to pursue it after leaving school4 ,if this view is correct1 it is wrong to allow a pupil to reach the age of fourteen years without knowing even the alphabet of ,geometry4 ,he should be taught at least how to read it4 ,it certainly does seem probable1 that if the youth who now leave school with so much ,arithmetic1 and no ,geometry1 were taught the first rudiments of the science1 thousands of them would be led to the study of the higher mathematics in their mature years1 by reason of those attractions of ,geometry which ,arithmetic does not possess4 ,,to ,,the ,,professional ,,reader4 ,this little book is constructed for the purpose of instructing large classes1 and with reference to being used also by teachers who have themselves no knowledge of ,geometry4 ,the first statement will account for the many1 and perhaps seemingly needless1 repetitions2 and the second1 for the suggestive style of some of the questions in the lessons which develop the matter contained in the review-lessons4 ,attention is respectfully directed to the following points3-- ,first the particular1 then the general4 ,see page #be4 ,why is m n g an acute angle8 ,what is an acute angle8 ,here the attention is directed first to this particular angle2 then this is taken as an example of its kind1 and the idea generalized by describing the class4 ,see also page #bi4 ,why are the lines e f and g h said to be parallel8 ,when are lines said to be parallel8 ,many of the questions are intended to test the vividness of the pupil's conception4 ,see page #bi4 ,also page #gh4 ,if the circumference were divided into #cfj equal parts1 would each arc be large or small8 ,many of the questions are intended to test the attention of the pupil4 ,the thing is not to be recognized by the definition2 but the definition is to be a description of the thing1 a description of the conception brought to the mind of the pupil by means of the name4 ,,contents4 ,,part ,i4 ,,lines #i ,,points #i ,,crooked ,,lines #aj ,,curved ,,lines #aa ,,straight ,,lines #aa ,,other ,,lines #aa ,,positions ,,of ,,lines #ad ,,angles #ag ,,relations ,,of ,,angles #bj ,,adjacent ,,angles #bj ,,vertical ,,angles #ba ,,kinds ,,of ,,angles #bc ,,right ,,angles #bc ,,acute ,,angles #bd ,,obtuse ,,angles #bd ,,relations ,,of ,,lines #bg ,,perpendicular ,,lines #bg ,,parallel ,,lines #bh ,,oblique ,,lines #bh ,,interior ,,angles #cj ,,exterior ,,angles #ca ,,opposite ,,angles #cb ,,alternate ,,angles #cc ,,problems ,,relating ,,to ,,angles #ch ,,polygons #dj ,,triangles #dd ,,isosceles ,,triangles #dh ,,problems ,,relating ,,to ,,triangles #ec ,,quadrilaterals #ee ,,parallelograms #ei ,,comparison ,,and ,,contrast ,,of ,,figures #fb ,,measurement ,,of ,,surfaces #ff ,,problems ,,relating ,,to ,,surfaces #ga ,,the ,,circle ,,and ,,its ,,lines #gc ,,arcs ,,and ,,degrees #gh ,,parts ,,of ,,the ,,circle #hb ,,part ,,ii4 ,,axioms ,,and ,,theorems4 ,,axioms4 ,,illustrated #he ,,theorems4 ,,illustrated #hh ,,first ,,lessons ,,in ,,geometry4 ,,part ,,first4 ,,lesson ,,first4 ,,lines4 ,,note ,,to ,,the ,,teacher4--,in all the development-lessons1 the pupils are to be occupied with the diagrams1 and not with the printed matter4 ,see ,note ,a1 ,appendix4 ,refer to ,,diagram #a1 and show that ,what are here drawn are intended to represent length only4 ,they have a little width1 that they may be seen4 ,they are called lines4 ,a line is that which has length only4 ,,points ,show that ,position is denoted by a point4 ,it occupies no space4 ,it has some size1 that it may be seen4 ,the ends of a line are points4 ,a line may be regarded as a succession of points4 ,the intersection of two lines is a point4 ,a point is named by placing a letter near it4 ,illustration3 ,diagram #a4 ,a point may be represented by a dot4 ,the point is in the center of the dot4 ,a point is that which denotes position only4 ,a line is named by naming the points at its ends4 ,read all the lines in ,diagram #a4 ,,crooked ,,lines4 ,see ,note ,b1 ,appendix4 ,does the line m n change direction at the point #a8 ,at what other points does it change direction8 ,it is called a crooked line4 ,a crooked line is one that changes direction at some of its points4 ,,curved ,,lines4 ,the line o p changes direction at every point4 ,it is called a curved line4 ,a curved line is one that changes direction at every point4 ,,straight ,,lines4 ,does the line i j change direction at any point8 ,it is called a straight line4 ,a straight line is one that does not change direction at any point4 ,,other ,,lines4 ,the line q r winds about a line4 ,it is called a spiral line4 ,the line w x winds about a point4 ,it also is called a spiral line4 ,a spiral line is one that winds about a line or point4 ,the line #g #ha looks like waves4 ,footnote #a3 ,to be read seven1 eight1 not seventy-eight4 ,it is called a wave line4 99 99 99 99 99 ,what kind of a line is a b8 ,why8 ,what is a straight line8 ,what kind of a line is #aa #af8 ,why8 ,what is a crooked line8 ,what kind of a line is o p8 ,why8 ,what is a curved line8 ,what kind of a line is s t8 ,why8 ,what kind of a line is #i #aj8 ,why8 ,what is a spiral line8 ,what kind of a line is w x8 ,why8 ,,lesson ,,second4 ,,review4 ,read all the straight lines4 7,,diagram #b47 ,why is m n a straight line8 ,define a straight line4 ,read all the crooked lines4 ,why is #g #h a crooked line8 ,define a crooked line4 ,read all the curved lines4 ,why is #e #f a curved line8 ,what is a curved line8 ,read all the wave lines4 ,read all the spiral lines4 ,why is #c #d a spiral line8 ,why is u v a spiral line8 ,what is a spiral line8 ,illustration3 ,diagram #b4 ,illustration3 ,diagram #c4 ,,lesson ,,third4 ,,positions ,,of ,,lines4 ,let the pupils hold their books so that they will be straight up and down like the wall4 ,,vertical ,,lines4 ,the straight line a b points to the center of the earth4 7,,diagram #c47 ,it is called a vertical line4 ,name all the vertical lines4 ,a vertical line is a straight line that points to the center of the earth4 ,,horizontal ,,lines4 ,the straight line o p points to the horizon4 ,it is called a horizontal line4 ,read all the horizontal lines4 ,a horizontal line is a straight line that points to the horizon4 ,,oblique ,,lines4 ,the line s t points neither to the center of the earth nor to the horizon4 ,it is called an oblique line4 ,read all the oblique lines4 ,an oblique line is a straight line that points neither to the horizon nor to the center of the earth4 ,,note4--,after going through with the lessons on angles1 the pupils may be told that oblique lines are so called because they form oblique angles with the horizon4 ,,lesson ,,fourth4 ,,review4 ,read all the vertical lines4 7,,diagram #d47 ,why is q r a vertical line8 ,what is a vertical line8 ,read all the horizontal lines4 ,why is #e #f a horizontal line8 ,define a horizontal line4 ,read all the oblique lines4 ,why is s t an oblique line4 ,what is an oblique line8 ,,note4--,lines that point in the same direction do not approach the same point4 ,illustration3 ,diagram #d4 ,illustration3 ,diagram #e4 ,,lesson ,,fifth4 ,,angles4 ,do the lines a b and c d 7,,diagram #e47 point in the same direction8 7,see note1 page #ae47 ,then they form an angle with each other4 ,what other line forms an angle with a b8 ,which of the two lines c d1 e f1 has the greater difference of direction from the line a b8 ,then which one forms the greater angle with a b8 ,what line forms a still greater angle with the line a b8 ,an angle is the difference of direction of two straight lines4 ,if the lines a b1 e f1 were made longer1 would their direction be changed8 ,then would there be any greater or less difference of direction8 ,then would the angles formed by them be any greater or less8 ,then does the size of an angle depend upon the length of the lines that form it8 ,if the lines a b1 e f1 were shortened1 would the angle formed by them be any smaller8 ,if two lines form an angle with each other1 and meet1 the point of meeting is called the vertex4 ,what is the vertex of the angle formed by the lines k j1 i j8--i j1 i l8 ,an angle is named by three letters1 that which denotes the vertex being in the middle4 ,thus1 the angle formed by k j1 i j1 is read k j i1 or i j k4 ,read the four angles formed by the lines m n and o p4 ,the eight formed by r s1 t u1 and v w4 ,,lesson ,,sixth4 ,,review4 ,read all the lines that form angles with the line a b4 7,,diagram #f47 ,which of them forms the greatest angle with it8 ,illustration3 ,diagram #f4 ,which the least8 ,of the two lines c d1 g h1 which forms the greater angle with e f8 ,read all the angles whose vertices are at o on i j4 ,which angle is the greater1 l o m1 or m o j8--i o k1 or i o l8--l o j1 or m o j8 ,read all the angles formed by the lines v w and x y4 ,read all the angles above the line n p4 ,below the line n p4 ,above the line q r4 ,at the right of the line #e u4 ,at the left4 ,at the right of the line s t4 ,at the left of the line s t4 ,which angle is the greater1 n #a #c1 or n #b #d8 ,if the lines x y and v w were lengthened or produced1 would the angles v z x1 y z w be any greater8 ,if they were shortened1 would the angles be any less8 ,what is an angle8 ,does the size of an angle depend upon the length of the lines which form it8 ,illustration3 ,diagram #g4 ,,lesson ,,seventh4 ,,relations ,,of ,,angles4 ,,adjacent ,,angles4 ,are the angles a e c1 c e b 7,,diagram #g471 on the same side of any line8 ,what line8 ,by what other straight line are they both formed8 ,then1 because they are both on the same side of the same straight line a b1 and are both formed by the second straight line c d1 they are called 8adjacent angles40 ,the angles c e b1 b e d are both on the same side of what straight line8 ,they are both formed by what second straight line8 ,then what kind of angles are they8 ,why are they called adjacent angles8 ,read the adjacent angles below the line a b4 ,below the line c d4 ,how many pairs of adjacent angles can be formed by two straight lines8 ,read all the adjacent angles formed by the lines l m and n p4 ,,vertical ,,angles4 ,are the angles a e c1 b e d formed by the same straight lines8 ,are they adjacent angles8 ,they are called 8vertical angles40 ,vertical angles are angles formed by the same straight lines1 but not adjacent to each other4 ,read the other pair of vertical angles formed by the lines a b1 c d4 ,read all the vertical angles formed by the lines f g1 i h4 ,by l m1 n p4 ,why are the angles l o n1 n o m adjacent angles8 ,why are the angles l o n1 p o m vertical angles8 ,illustration3 ,diagram #h4 ,,lesson ,,eighth4 ,,review4 ,read the pairs of adjacent angles above the line a b4 7,,diagram #h47 ,why are they adjacent8 ,what are adjacent angles8 ,read the adjacent angles below the line a b4 ,on the right of the line c d4 ,on the left4 ,how many pairs of adjacent angles are formed by the intersection of two lines4 ,read the pairs of adjacent angles formed by the lines f g and i h4 ,read all the adjacent angles formed by the lines l m1 n p4 ,read all the pairs of vertical angles formed by the lines a b1 c d4 ,why are c e b and a e d called vertical angles8 ,what are vertical angles8 ,read all the pairs of vertical angles formed by the lines h i1 f g4 ,how many pairs of vertical angles are formed by the intersection of two lines8 ,read all the pairs of vertical angles formed by the lines l m1 n p4 ,,lesson ,,ninth4 ,,kinds ,,of ,,angles4 ,,right ,,angles4 ,what do we call the angles a o c1 c o b8 7,,diagram #i47 ,are they equal to each other8 ,then they are called right angles4 ,a right angle is one of two adjacent angles that are equal to each other4 ,are the adjacent angles c o b1 b o d equal to each other8 ,then what are they called8 ,read the right angles below the line a b4 ,on the left of c d4 ,read three right angles whose vertices are at p4 ,illustration3 ,diagram #i4 ,,acute ,,angles4 ,is the angle m p q greater or less than the right angle m p r8 ,then it is called an acute angle4 ,an acute angle is one which is less than a right angle4 ,read four acute angles whose vertices are at p4 ,acute means sharp4 ,why is r p s an acute angle8 ,what is an acute angle8 ,,obtuse ,,angles4 ,is the angle m p s greater or less than the right angle m p r8 ,then it is called an obtuse angle4 ,an obtuse angle is one which is greater than a right angle4 ,what other obtuse angle has its vertex at p8 ,obtuse means blunt4 ,read three obtuse angles whose vertices are at x4 ,acute and obtuse angles are also called oblique angles4 ,,lesson ,,tenth4 ,,review4 ,read all the right angles formed by the lines a b and c d4 7,,diagram #aj47 ,why are the adjacent angles c e b1 b e d1 right angles8 ,what is a right angle8 ,read four right angles whose vertices are at n4 ,which is the greater1 the right angle p q r1 or the right angle t s u8 ,can one right angle be greater than another8 ,read six acute angles whose vertices are at n4 ,why is m n g an acute angle8 ,what is an acute angle8 ,which is greater1 the acute angle m n g1 or the acute angle l n m8 ,may one acute angle be greater than another8 ,what three acute angles are equal to one right angle8 ,illustration3 ,diagram #aj4 ,which of the two acute angles v f w1 y x z is the greater8 ,read four obtuse angles whose vertices are at n4 ,why is f n m an obtuse angle8 ,what is an obtuse angle8 ,what does obtuse mean8 ,acute8 ,by what other name are both called8 ,which is greater1 the large acute angle #a #d #b1 or the small obtuse angle #a #d #c8 ,how much greater than the right angle is the obtuse angle f n l8 ,how much less than a right angle is f n i8 ,illustration3 ,diagram #aa4 ,,lesson ,,eleventh4 ,,relations ,,of ,,lines4 ,,perpendicular ,,lines4 ,what kind of angles do the lines a b and c d make with each other8 7,,diagram #aa47 ,then they are perpendicular to each other4 ,what line is perpendicular to x y8 ,why is it perpendicular to it8 ,what line is perpendicular to z #a8 ,when is a line said to be perpendicular to another8 ,can a line standing alone be properly called a perpendicular line8 ,what two lines are perpendicular to the lines r s8 ,is the line g h perpendicular to the line i j8 ,why8 ,what other line is perpendicular to the line i j8 ,read three lines that are perpendicular to the line a b4 ,,parallel ,,lines4 ,do the lines k l1 m n1 differ in direction8 ,then do they form any angle with each other8 ,they are said to be parallel to each other4 ,read four other lines that are parallel with k l4 ,what line is parallel with #b #aj8 ,why8 ,lines are parallel with each other when they do not differ in direction4 ,,oblique ,,lines4 ,what kind of angles do the lines u t and #h #i form with each other8 ,then they are said to be oblique to each other4 ,lines are oblique to each other when they form oblique angles4 ,see ,note ,c1 ,appendix4 ,illustration3 ,diagram #ab4 ,,lesson ,,twelfth4 ,,review4 ,read five lines that are perpendicular to the line a b4 7,,diagram #ab47 ,five that are perpendicular to c d4 ,two that are perpendicular to u v1 and meet it4 ,three that do not meet it4 ,why are o p and m n perpendicular to each other8 ,when are lines said to be perpendicular to each other8 ,read four lines that are parallel with e f4 ,why are the lines e f and g h said to be parallel to each other8 ,when are lines said to be parallel to each other8 ,read four lines that are parallel to #e #f4 ,four that are parallel to o p4 ,is any line parallel to u v8 ,can a single line be properly called perpendicular8 ,parallel8 ,if two lines are perpendicular to each other1 what angle do they form8 ,if parallel1 what angle8 ,if oblique8 ,illustration3 ,diagram #ac4 ,,lesson ,,thirteenth4 ,,relations ,,of ,,angles4 ,,interior ,,angles4 ,is the angle a m n between the parallels1 or outside of them8 7,,diagram #ac47 ,it is called an interior angle4 ,read three other interior angles between the same parallels4 ,why is b m n an interior angle8 ,an interior angle is one that lies between parallel lines4 ,read the interior angles between the parallel lines g h and k l4 ,why is o p l an interior angle8 ,what is an interior angle8 ,,exterior ,,angles4 ,is the angle a m e between the parallels1 or outside of them8 ,it is called an exterior angle4 ,read three other exterior angles formed by the lines a b1 c d1 and e f4 ,why is the angle c n f an exterior angle8 ,an exterior angle is one that lies outside of the parallels4 ,,lesson ,,fourteenth4 ,,review4 ,read all the interior angles formed by the lines a b1 c d1 and e f4 ,why is m n d an interior angle8 ,what is an interior angle8 ,read all the exterior angles formed by the same lines4 ,why is d n f an exterior angle8 ,what is an exterior angle8 ,read all interior angles formed by the lines g h1 k l1 and i j4 ,all the remaining interior angles in the diagram4 ,all the exterior angles4 ,illustration3 ,diagram #ad4 ,,lesson ,,fifteenth4 ,,relations ,,of ,,angles4 ,,opposite ,,angles4 ,are the angles e m b1 b m n1 on the same side of the intersecting line e f8 ,are they adjacent8 ,are e m b1 m n d1 on the same side of the intersecting line e f8 ,are they adjacent8 ,then they are called opposite angles4 ,opposite angles lie on the same side of the intersecting line1 but are not adjacent4 ,are the angles e m b1 f n d1 on the same side of the intersecting line8 ,are they adjacent8 ,then are they opposite8 ,are they interior or exterior angles8 ,then they are 8opposite exterior angles40 ,why are they exterior8 ,why are they opposite8 ,are the angles b m n1 m n d1 opposite angles8 ,are they interior or exterior angles8 ,then they are 8opposite interior angles40 ,why are they opposite8 ,why interior8 ,read the opposite exterior angles on the left of the line e f4 ,read the opposite interior angles on the same side4 ,are the opposite angles e m a1 m n c1 both exterior or interior8 ,then they are opposite exterior and interior angles4 ,read two pairs of opposite exterior and interior angles on the right of e f4 ,on the left4 ,,alternate ,,angles4 ,do the angles b m n1 m n c1 lie on the same side of the intersecting line e f8 ,are they adjacent to each other8 ,are they vertical angles8 ,then they are alternate angles4 ,alternate angles lie on different sides of the intersecting line1 and are neither adjacent nor vertical4 ,are the alternate angles b m n1 m n c1 exterior or interior8 ,then they are called 8interior alternate angles40 ,read another pair of interior alternate angles between a b and c d4 ,are the angles e m b1 c n f1 alternate angles8 ,why8 ,are they exterior or interior8 ,then what may they be called8 ,read another pair of exterior alternate angles4 ,why are e m a1 d n f1 alternate angles8 ,why exterior alternate8 ,,lesson ,,sixteenth4 ,,review4 ,read the exterior opposite angles on the right of the line e f4 7,,diagram #ad47 ,on the left4 ,on the right of r s4 ,on the left4 ,why are e m a1 c n f1 exterior angles8 ,why are they opposite angles8 ,what are opposite angles8 ,read the interior opposite angles on the right of the intersecting line e f4 ,on the left of it4 ,on the right of r s4 ,on the left4 ,read the interior alternate angles formed by the lines a b1 c d1 and e f4 ,which pair are acute angles8 ,which pair are obtuse angles8 ,why are b m n1 m n c1 interior angles8 ,why alternate8 ,what are alternate angles8 ,read the exterior alternate angles of the same lines4 ,read the acute interior alternate angles of the parallels t u1 v w4 ,the obtuse4 ,the acute exterior alternate angles4 ,obtuse4 ,read the pair of opposite exterior angles on the right of the line e f4 ,on the left4 ,on the right of r s4 ,on the left4 ,illustration3 ,diagram #ae4 ,,lesson ,,seventeenth4 ,,review4 ,read thirteen or more angles whose vertices are at c4 7,,diagram #ae47 ,read four obtuse angles4 ,read two right angles4 ,what three acute angles equal one right angle8 ,which is greater1 the right angle #d1 or the right angle #e8 ,the obtuse angle #f1 or the acute angle #g8 ,read twelve pairs of adjacent angles formed by the lines w x1 c4 ,read six pairs of vertical angles formed by the same lines4 ,read all the interior angles formed by the lines i j1 k l1 and m n4 ,read all the exterior angles formed by the same lines4 ,two pairs of opposite exterior angles4 ,two pairs of opposite interior angles4 ,four pairs of opposite exterior and interior angles4 ,two pairs of alternate interior angles4 ,two pairs of alternate exterior angles4 ,why are the angles i o m1 m o j1 called adjacent8 ,what are adjacent angles8 ,what kind of an angle is i o m8 ,why8 ,what is an acute angle8 ,what kind of an angle is m o j8 ,why8 ,what is an obtuse angle8 ,why are a c f1 f c b1 right angles8 ,what is a right angle8 ,why are m o i1 j o p1 vertical angles8 ,what are vertical angles8 ,why is m o i an exterior angle8 ,what is an exterior angle8 ,why is j o p an interior angle8 ,what is an interior angle8 ,why are m o i1 o p k1 opposite angles8 ,what are opposite angles8 ,why are j o p1 o p k1 alternate angles8 ,what are alternate angles8 ,,lesson ,,eighteenth4 ,,problems4 ,draw an obtuse angle which shall be only a little larger than a right angle4 ,draw one which shall be much greater than a right angle4 ,draw an acute angle which shall be only a little less than a right angle4 ,draw one which shall be much less than a right angle4 ,draw an obtuse angle with lines about one inch long4 ,draw an acute angle with sides three inches long4 ,which is greater1 the obtuse angle1 or the acute angle8 ,draw a right angle with lines an inch long4 ,draw one with lines five inches long4 ,which is the greater1 first or the second8 ,illustration3 ,diagram #af4 ,,lesson ,,nineteenth4 ,,polygons4 ,name any thing besides your desk that has a flat surface4 ,a flat surface is called a plane4 ,how many sides has the plane ,fig4 ,a8 7,,diagram #af47 ,it is called a triangle4 8,tri0 means 8three40 ,what other triangles do you see4 ,triangles are sometimes called trigons4 ,a triangle is a plane figure having three sides4 ,how many sides has the plane figure marked ,b8 ,how many angles8 ,it is called a quadrangle1 or quadrilateral4 8,quad0 denotes 8four40 ,what other quadrangles do you see8 ,why is ,fig4 ,b a quadrangle8 ,a quadrangle is a plane figure having four sides4 ,how many sides has the ,fig4 ,c8 ,it is called a pentagon4 ,what other pentagon do you see8 ,why is ,fig4 ,c a pentagon8 ,a pentagon is a plane figure having five sides4 ,in like manner1-- ,a hexagon is a plane figure having six sides4 ,a heptagon is a plane figure having seven sides4 ,an octagon has eight sides4 ,a nonagon has nine sides4 ,a decagon has ten sides4 ,all these figures are called polygons4 8,poly0 means 8many40 ,what do you call a polygon of three sides8 ,of four sides8 ,of six sides8 c4 ,if the length of each side of triangle ,a is one inch1 how long are the three sides together8 ,the sum of the sides of a polygon is its perimeter4 ,which of the triangles has unequal sides8 ,which has equal sides8 ,the latter is called a regular polygon4 ,which pentagon has one side longer than any one of its other sides8 ,which has its sides all equal to each other8 ,are its angles also equal8 ,it is therefore a regular polygon1 or regular pentagon4 ,name a hexagon that is not regular4 ,name a regular hexagon4 ,a regular octagon4 ,a regular heptagon4 ,a polygon is a plane figure bounded by straight lines4 ,,lesson ,,twentieth4 ,,review4 ,name all the triangles4 7,,diagram #af47 ,why is ,fig4 ,a a triangle8 ,what is a triangle8 ,what other name is sometimes given to triangles8 ,name all the quadrilaterals4 ,why is ,fig4 ,b a quadrilateral8 ,what is a quadrilateral1 or quadrangle8 ,name all the pentagons1 hexagons1 heptagons1 octagons1 and nonagons4 ,why is ,c a pentagon8 ,what is a pentagon8 ,a hexagon8 ,a heptagon8 c4 ,how many polygons in the diagram8 ,what is a polygon8 ,if each side of ,fig4 ,b is one inch1 how many inches are there in its perimeter8 ,when is a polygon regular8 ,name all the regular polygons in diagram #af4 ,name all the irregular polygons4 ,illustration3 ,diagram #ag4 ,,lesson ,,twenty-,,first4 ,,triangles4 ,,acute-,,angled ,,triangles4 ,in the triangle #a1 what kind of an angle is b a c8 a c b8 c b a8 7,,diagram #ag47 ,then it is called an acute-angled triangle4 ,an acute-angled triangle is one whose angles are all acute4 ,read three other acute-angled triangles4 ,,obtuse-,,angled ,,triangles4 ,in the triangle #d1 what kind of an angle is l k m8 ,then it is called an obtuse-angled triangle4 ,an obtuse-angled triangle is one that has one obtuse angle4 ,name two others4 ,,right-,,angled ,,triangles4 ,in the triangle #c1 what kind of an angle is g i j8 ,then it is called a right-angled triangle4 ,a right-angled triangle is one that has one right angle4 ,name three other right-angled triangles4 ,upon which side does the triangle #c seem to stand8 ,then i j is called the base of the triangle4 ,what letter marks the vertex of the angle opposite the base8 ,then the point g is said to be the vertex of the triangle4 ,if1 in the triangle #g1 we consider t v the base1 what point is the vertex8 ,if v be considered the vertex1 which side will be the base8 ,in the triangle #c1 what side is opposite the right angle8 ,then g j is called the hypothenuse of the triangle4 ,the hypothenuse of a triangle is the side opposite the right angle4 ,read the hypothenuse of each of the triangles #e1 #f1 and #aa4 ,either side about the right angle may be considered the base4 ,then the other side will be the perpendicular4 ,in the triangle #c1 if i j is the base1 which side is the perpendicular8 ,if g i be considered the base1 which side is the perpendicular8 ,in triangle #e1 if n o is the base1 which side is the perpendicular8 ,,lesson ,,twenty-,,second4 ,,review4 ,name four acute-angled triangles4 7,,diagram #ag47 ,why is the triangle #h acute-angled8 ,what is an acute-angled triangle8 ,name three obtuse-angled triangles4 ,why is the triangle #i an obtuse-angled triangle8 ,what is an obtuse-angled triangle8 ,name four right-angled triangles4 ,why is the triangle #f a right-angled triangle8 ,what is a right-angled triangle8 ,in the triangle #f1 which side is the hypothenuse8 ,why8 ,what is the hypothenuse8 ,what two sides of the triangle #f may be regarded as the base8 ,if q r be considered the base1 what do you call the side q s8 ,read the hypothenuse of each of the triangles #c1 #e1 #f1 and #aa4 ,illustration3 ,diagram #ah4 ,,lesson ,,twenty-,,third4 ,,triangles4 7,continued47 ,,isosceles ,,triangles4 ,of the triangle #a1 which two sides are equal to each other8 ,then it is called an isosceles triangle4 ,an isosceles triangle is one that has two equal sides4 ,name eight isosceles triangles4 ,why is the triangle #b an isosceles triangle8 ,what kind of a triangle is it on account of its angles8 ,then it is an acute-angled isosceles triangle4 ,name four acute-angled isosceles triangles4 ,what kind of a triangle is ,fig4 #d on account of the angle k j l8 ,what kind on account of its equal sides8 ,then it is called an obtuse-angled isosceles triangle4 ,name one other obtuse-angled isosceles triangle4 ,what kind of a triangle is ,fig4 #f on account of the angle q p r8 ,what kind on account of its equal sides8 ,then it is called a right-angled isosceles triangle4 ,name one other right-angled isosceles triangle4 ,why is ,fig4 #ab a right-angled triangle8 ,why isosceles8 ,,equilateral ,,triangles4 ,which of the isosceles triangles has all its three sides equal to each other8 ,it is called an equilateral triangle4 8,equi0 means 8equal40 8,latus0 means a 8side40 ,an equilateral triangle is one that has its three sides equal to each other4 ,what kind of a triangle is ,fig4 #g on account of its three equal sides8 ,what kind on account of its two equal sides s t1 s u1 or t s1 t u1 or u s1 u t8 ,then must not every equilateral triangle be also isosceles8 ,what kind of a triangle is ,fig4 #b on account of its equal sides d e1 d f8 ,if the side e f is longer than either of the other two sides1 is it an equilateral triangle8 ,then is every isosceles triangle also equilateral8 ,name another isosceles triangle that is not equilateral4 ,name one that is equilateral4 ,in any equilateral triangle the three angles are equal to each other4 ,on account of its equal angles1 it is also called an equiangular triangle4 ,what is ,fig4 #h called on account of its three equal sides8 ,on account of its three equal angles8 ,every equilateral triangle is also equiangular4 ,every equiangular triangle is also equilateral4 ,name a triangle that has no two sides equal to each other4 ,it is called a scalene triangle4 ,what kind of a triangle is ,fig4 #e on account of its right angle8 ,what kind on account of its three unequal sides8 ,then it is a right-angled scalene triangle4 ,what name can you give ,fig4 #aa on account of the angle g e f8 ,on account of its three unequal sides8 ,then what may it be called8 ,illustration3 ,diagram #ai4 ,,lesson ,,twenty-,,fourth4 ,,review4 ,name eight isosceles triangles4 7,,diagram #ai47 ,why is ,fig4 #b an isosceles triangle8 ,what is an isosceles triangle8 ,name two right-angled isosceles triangles4 ,name five acute-angled isosceles triangles4 ,name one obtuse-angled isosceles triangle4 ,name two isosceles triangles that are also equilateral4 ,are all isosceles triangles equilateral8 ,name six isosceles triangles that are not equilateral4 ,what does 8equi0 mean8 8,latus08 ,what are equilateral triangles called on account of their equal angles8 ,are all equilateral triangles equiangular8 ,are all equiangular triangles equilateral8 ,what are equilateral triangles8 ,name four scalene triangles4 ,name two right-angled scalene triangles4 ,why is ,fig4 #c a right-angled triangle8 ,why scalene8 ,what is a scalene triangle8 ,name one obtuse-angled scalene triangle4 ,name one acute-angled scalene triangle4 ,,problems4 ,from the same point draw two straight lines of any length1 making an acute angle with each other4 ,make them equal to each other by measuring4 ,join their ends4 ,what kind of a triangle is it on account of its angles8 ,on account of its two equal sides8 ,write its two names inside of it4 ,draw an isosceles triangle whose equal sides shall each be less than the third side4 ,write its two names within it4 ,draw an oblique straight line twice as long as any short measure or unit4 ,at one end draw a straight line perpendicular to it1 and three times as long as the same measure4 ,connect the ends of the two lines by a straight line4 ,what kind of an angle is that opposite the last line drawn8 ,are any two of its sides equal8 ,write its two names under it4 ,draw a horizontal straight line of any length4 ,at one end draw a vertical line of equal length4 ,complete the triangle1 and write two names inside4 ,draw a right-angled triangle whose base is of any length1 and its perpendicular twice as long4 ,draw a right-angled triangle whose base is three times as long as any short measure1 and its perpendicular five times as long as the same measure or unit4 ,illustration3 ,diagram #bj4 ,,quadrilaterals4 ,how many sides has the figure a b d c8 ,what is it called on account of the number of its sides8 ,name three other quadrilaterals whose vertices are marked4 ,name seven by numbers4 ,quadrilaterals are sometimes named by means of two opposite vertices4 ,the quadrilateral a b d c1 or c d b a1 may be read a d1 or b c1 or c b1 or d a4 ,name the quadrilateral1 g h f e1 four ways4 ,how many angles has each figure8 ,on account of the number of their angles they are called quadrangles4 ,has the quadrilateral a d any two sides parallel to each other8 ,then it is called a trapezium4 ,a trapezium is a quadrilateral that has no two sides parallel4 ,name two other trapeziums4 ,why is ,fig4 #g a trapezium8 ,has the quadrilateral e h any two sides parallel8 ,which two8 ,are the other two sides parallel8 ,it is called a 8trapezoid40 8,oid0 means like4 ,what does 8trapezoid0 mean8 ,a trapezoid is a quadrilateral that has only one pair of sides parallel4 ,name another trapezoid4 ,why is ,fig4 #f a trapezoid8 ,how many pairs of parallel sides has the quadrilateral i l8 ,name the horizontal parallels4 ,name the oblique parallels4 ,it is called a 8parallelogram40 ,a parallelogram is a quadrilateral whose opposite sides are parallel4 ,name five other parallelograms4 ,why is ,fig4 #d a parallelogram8 ,why is not ,fig4 #f a parallelogram8 ,why is not e h a parallelogram8 ,what two names may you give to ,fig4 #e8 ,why is it a quadrilateral8 ,why a trapezium8 ,what two names may we give to ,fig4 #f8 ,why is it a quadrilateral8 ,why a trapezoid8 ,what two names may we give to ,fig4 #c8 ,why is it a parallelogram8 ,why a quadrilateral8 ,,lesson ,,twenty-,,fifth4 ,,review4 ,how many quadrilaterals in the diagram4 7,,diagram #bj47 ,why is ,fig4 a d a quadrilateral8 ,what is a quadrilateral8 ,on account of the number of its angles1 what may it be called8 ,name all the quadrilaterals4 ,name three trapeziums4 ,why is ,fig4 #e a trapezium8 ,what is a trapezium8 ,name two trapezoids4 ,why is ,fig4 #f a trapezoid8 ,name its parallel sides4 ,what is a trapezoid8 ,name six parallelograms4 ,why is ,fig4 #d a parallelogram8 ,name its two pairs of parallel sides4 ,what is a parallelogram8 ,what two names can you give to ,fig4 #d8 ,why the first8 ,why the second8 ,what two names may be given to ,fig4 #g8 ,why the first8 ,why the second8 ,what two to ,fig4 #f8 ,why the first8 ,why the second8 ,illustration3 ,diagram #ba4 ,,lesson ,,twenty-,,sixth4 ,,kinds ,,of ,,parallelograms4 ,,rhomboids4 ,how many quadrilaterals in the diagram8 7,,diagram #ba47 ,how many parallelograms8 ,has the parallelogram a d any right angle8 ,it is called a 8rhomboid40 ,a rhomboid is a parallelogram which has no right angle4 ,name five other rhomboids4 ,what three names may be given to ,fig4 #b8 ,why is it a quadrilateral8 ,why a parallelogram8 ,why a rhomboid8 ,,rhombs4 ,are the four sides of the rhomboid a d equal to each other8 ,are the four sides of the rhomboid e h equal to each other8 ,if a triangle has its three sides equal to each other1 what do you call it8 ,then when a rhomboid has its sides equal to each other1 what may it be called8 ,an equilateral rhomboid is called a rhombus4 ,a rhombus is an equilateral rhomboid4 ,see ,note ,d1 ,appendix4 ,name two other rhombuses1 or rhombs4 ,what four names can you give to ,fig4 e h8 ,why a quadrilateral8 ,why a parallelogram8 ,why a rhomboid8 ,why a rhombus8 ,,rectangles4 ,has the parallelogram i l any right angles8 ,how many8 ,it is called a 8rectangle40 ,a rectangle is a right-angled parallelogram4 ,name four other rectangles4 ,what three names may be given to ,fig4 i l8 ,why a quadrilateral8 ,why a parallelogram8 ,why a rectangle8 ,,squares4 ,has the rectangle i l its four sides equal8 ,has the rectangle m p its four sides equal8 ,it is called a 8square40 ,a square is an equilateral rectangle4 ,name another 8square40 ,what four names may be given to ,fig4 m p8 ,why a quadrilateral8 ,why a parallelogram8 ,why a rectangle8 ,why a square8 ,,lesson ,,twenty-,,seventh4 ,,review4 ,name six rhomboids4 7,,diagram #ba47 ,what three names may be given to ,fig4 #c8 ,why a quadrilateral8 ,why a parallelogram8 ,why a rhomboid8 ,what is a quadrilateral8 ,parallelogram8 ,rhomboid8 ,name three rhombs4 ,what four names may you give ,fig4 #e8 ,why a quadrilateral8 ,why a parallelogram8 ,why a rhomboid8 ,why a rhomb8 ,what is a rhomboid8 ,a rhomb8 ,name five rectangles4 ,what three names may be given to ,fig4 #a8 ,why a quadrilateral8 ,why a parallelogram8 ,why a rectangle8 ,what is a rectangle8 ,name two squares8 ,by what four names may ,fig4 #g be called8 ,why by the first8 ,by the second8 ,by the third8 ,by the fourth8 ,what is a square8 ,what is a rectangle8 ,what is a parallelogram8 ,what is a quadrilateral8 ,illustration3 ,diagram #bb4 ,,lesson ,,twenty-,,eighth4 ,,comparison ,,and ,,contrast4 ,,trapezium ,,and ,,trapezoid4 ,in what respect are ,figs4 ,a and ,b alike8 ,on this account1 what name may be given to each8 ,how does ,fig4 ,b differ from ,fig4 ,a8 ,what particular name may you give to ,fig4 ,b8 ,what one to ,fig4 ,a8 ,,rhomboid ,,and ,,rectangle4 ,in what two respects are ,figs4 ,c and ,d alike8 ,on account of the number of their sides1 what may each be called8 ,because their opposite sides are parallel1 what may each be called8 ,in what respect do they differ8 ,what particular name may be given to ,fig4 ,c8 ,what one to ,fig4 ,d8 ,what three names may you give to the figure with right angles8 ,what three to the one without right angles8 ,,rhomboid ,,and ,,rhombus4 ,in what three things are ,figs4 ,e and ,f alike8 ,what three names may be given to each8 ,how do they differ from each other8 ,what particular name may you give to ,fig4 ,f8 ,what four names has ,fig4 ,f8 ,,rectangle ,,and ,,square4 ,in what three things are ,figs4 ,g and ,h alike8 ,on account of the number of their sides1 what may each be called8 ,because their opposite sides are parallel1 what may each be called8 ,because they have right angles1 what may they be called8 ,in what respect is ,fig4 ,h different from ,fig4 ,g8 ,on this account1 what particular name may be applied to ,fig4 ,h8 ,what three names may be applied to ,fig4 ,g8 ,what four to ,fig4 ,h8 ,,rhombus ,,and ,,square4 ,in what three things are ,figs4 ,f and ,h alike8 ,on account of the number of their sides1 what name may be given to each8 ,because their opposite sides are parallel1 what name may be given to each8 ,because both are parallelograms1 and both have their sides equal1 what name may be given to each8 ,what particular name has ,fig4 ,f8 ,what particular name has ,fig4 ,h8 ,what four names may be given to ,fig4 ,f8 ,what four to ,fig4 ,h8 ,,lesson ,,twenty-,,ninth4 ,,review4 ,what two names may be given to ,fig4 ,a4 7,,diagram #bb47 ,to ,fig4 ,b8 ,in what are they alike8 ,in what do they differ8 ,by what three names may ,fig4 ,c be called8 ,by what three names may ,fig4 ,d be called8 ,in what two things are they alike8 ,in what one thing do they differ8 ,what particular name has ,c8 ,what one has ,d8 ,what three names may be applied to ,fig4 ,e8 ,what four to ,fig4 ,f8 ,what property has ,f that ,e has not8 ,what particular name has it on that account8 ,what three names has ,fig4 ,g8 ,what four has ,fig4 ,h8 ,what property has ,fig4 ,h that ,g has not8 ,what particular name has it in consequence8 ,what four names may you give to ,fig4 ,f8 ,what four to ,fig4 ,h8 ,what three names may be applied to either8 ,in what three things are they alike8 ,in what respect do they differ8 ,what particular name has ,fig4 ,f8 ,what particular name has ,fig4 ,h8 ,illustration3 ,diagram #bc4 ,,lesson ,,thirtieth4 ,,measurement ,,of ,,surfaces4 ,in ,fig4 #a 7,,diagram #bc47 call the line a b a unit4 ,rectangle #a is how many units long8 ,how many high8 ,because its sides are equal1 what is it called8 ,rectangle #b is how many units long8 ,how many high or wide8 ,how many squares does it contain8 ,rectangle #c is how many units long8 ,how many wide8 ,how many squares does it contain8 ,if it were four units long and one wide1 how many squares would it contain8 ,if it were five long and one wide8 ,six long8 c4 ,rectangle #d is how many units long8 ,how many wide8 ,how many squares does it contain8 ,how many squares in that part which is two units long1 m n1 and one unit wide1 m l8 ,on account of the second unit in width1 l k1 how many times two squares are there8 ,if the width were one unit more1 how many times two squares would there be8 ,rectangle #e is how many units long8 ,how many units wide or high8 ,how many squares does it contain8 ,how many squares in that part which is three units long1 o p1 and one unit wide1 o t8 ,the second unit in width1 t q1 gives how many more squares8 ,how many times three squares8 ,if another unit were added to the width1 how many more squares would be made8 ,how many times three squares8 ,if it were four units wide1 how many times three squares would there be8 ,rectangle #f is how many units long8 ,how many units high or wide8 ,how many squares in that part which is four units long and one high8 ,how many times four squares in that part which is four long and two high8 ,how many times four squares when it is four long and three high8 ,if another unit were added to the height1 how many more squares would be added8 ,how many times four squares would there be8 ,if a rectangle were five units long and one unit wide1 how many square units would it contain8 ,if it were two units wide1 how many times five square units would it contain8 ,if it were three units wide8 ,four8 c4 ,if your ruler is ten inches long and only one inch wide1 how many square inches are there in it8 ,if it were two inches wide1 how many times ten square inches would it contain8 ,if your arithmetic-cover is seven inches long and five inches wide1 how many square inches are there in it8 ,if a wall of this room is twenty feet long1 how many square feet are there in that part which is one foot high8 ,two high8 ,three high8 ,four high8 ,if the same wall is sixteen feet high1 how many square feet in it8 ,fig4 #e has how many times three squares8 ,fig4 #g has how many times two squares8 ,which has the greater number of squares8 ,what difference is there between two times three squares and three times two squares8 ,,lesson ,,thirty-,,first4 ,,review4 ,draw a rectangle of any width whose length is three times the width4 ,how many squares has it if the width be taken as the unit8 ,make it twice as wide as before4 ,how many squares has it now8 ,what two numbers multiplied together will give the number of squares8 ,make it three times as wide4 ,how many squares has it now8 ,what two numbers multiplied together will give the number of squares8 ,the cover of a geography is one foot long and one foot wide1 how many square feet in it8 ,how many inches long is the same cover8 ,how many wide8 ,how many square inches does it contain8 ,how many square inches are equal to one square foot8 ,a table is one yard long and one yard wide1 how many square yards in it8 ,how many feet long is the same table8 ,how many feet wide8 ,how many square feet does it contain8 ,one square yard equals how many square feet8 ,draw a square whose side is a unit of any length4 ,draw another whose side is two units of the same length4 ,the second square is how many times as large as the first one8 ,how many squares in half the second square8 ,which is greater1 two square inches1 or two inches square8 ,two inches square is how many times two square inches8 ,draw a square whose side is three inches4 ,how many square inches does it contain8 ,how many times as many squares as the square of one inch8 ,how many square inches in the bottom row8 ,how many in all8 ,which is greater1 three inches square1 or three square inches8 ,three inches square is how many times three square inches8 ,,problems4 ,an equilateral triangle has each of its sides one inch long1 what is its perimeter8 ,if each side were two inches long1 what would be its perimeter8 ,an isosceles triangle has its two equal sides each three inches long1 and its third side five inches long1 what is its perimeter8 ,a right-angled isosceles triangle has its base five inches1 and its hypothenuse seven inches long1 what is its perimeter8 ,a square geography-cover is nine inches long on one side1 how long all round8 ,how many square inches in it8 ,a slate is sixteen inches long and twelve wide1 how many inches all round it8 ,a rectangle is five inches long and three wide1 how long all round8 ,how many square inches in it8 ,a slate is one foot long and eight inches wide1 what is its perimeter8 ,a room is twenty-four feet long and twenty-one feet wide1 how many feet all round it8 ,how many square feet in the floor8 ,how many pieces of paper each a foot square would exactly cover it8 ,a yard of carpet is two feet wide1 how many square feet in it8 ,charles and ,henry start from the same place1 and walk in opposite directions2 ,charles goes twenty yards1 and ,henry fifteen1 how many yards apart are they8 ,if they start from opposite ends of a straight walk twenty-five feet long1 and walk towards each other1 how many feet will ,charles have to walk to meet ,henry who has walked fifteen feet8 ,a lot is forty rods long and thirty wide1 how long must the fence be8 ,what length of fence will divide it into four equal parts8 ,illustration3 ,diagram #bd4 ,,lesson ,,thirty-,,second4 ,,the ,,circle ,,and ,,its ,,lines4 ,if the straight line c a were a string made fast at c1 with a sharp pencil-point at the other end a1 and the pencil-point were moved towards d1 what line would be drawn8 ,what kind of a line would it be8 ,if the pencil-point continued to move in the same direction until it returned to the starting-point a1 what curved line would be drawn1 naming it by all the points in it which are marked8 ,the plane figure bounded by this curve is called a 8circle40 ,what point is at the centre of this figure8 ,a circle is a plane figure bounded by a curved line1 all points of which are equally distant from the centre4 ,the curved line is called a 8circumference40 ,the circumference of a circle is the curve which bounds it4 ,name a straight line that joins two points in the circumference4 ,it is called a 8chord40 ,a chord is a straight line that joins two points of a circumference4 ,read six chords in the diagram4 ,which two of these chords pass through the centre8 ,they are called 8diameters40 ,a diameter is a chord that passes through the centre4 ,name a line that joins the centre with a point of the circumference4 ,it is called a 8radius40--7,plural1 radii47 ,a radius is a straight line that joins the centre to a point of the circumference4 ,read five radii4 ,which is farther from the centre1 the point a or the point d8 ,can the radius c d be greater than the radius c a8 ,or greater than c v1 or c o8 ,then all radii of the same circle are equal to each other4 ,what do we call the lines o d1 c d1 c o8 ,what part of the diameter o d is the radius o c8 ,name a chord that is produced without the circle4 ,it is called a 8secant40 ,a secant is a chord produced4 ,name two secants4 ,if the chord d i were made a secant1 would it become longer or shorter8 ,in how many points does the straight line l m touch the circumference8 ,it is called a 8tangent40 ,a tangent is a straight line that touches a circumference in only one point4 ,name three tangents4 ,,lesson ,,thirty-,,third4 ,,review4 ,read six chords4 7,,diagram #bd47 ,why is i d a chord8 ,what is a chord8 ,name two diameters4 ,why is a j a diameter8 ,what is a diameter8 ,is every chord a diameter8 ,is every diameter a chord8 ,name five radii4 ,why is c a a radius8 ,what is a radius8 ,a diameter is equal to how many radii8 ,are all radii equal to each other8 ,are all chords equal to each other8 ,are all diameters equal to each other8 ,name two secants4 ,why is either one a secant8 ,what is a secant8 ,name three tangents4 ,why is a b a tangent8 ,what is a tangent8 ,is a tangent inside of a circle or outside of it8 ,is a chord inside or outside of a circle8 ,is a secant within or without a circle8 ,if the radius is three inches1 how long is the diameter8 ,illustration3 ,diagram #be4 ,,lesson ,,thirty-,,fourth4 ,,arcs ,,and ,,degrees4 ,what small part of the circumference of circle #a 7,,diagram #be47 is marked8 ,it is called an 8arc40 ,an arc is any part of a circumference4 ,read five arcs that are marked4 ,which is longer1 the arc e d1 or the arc e f8 b d1 or b d e8 a b d1 or a b d e8 ,name an arc which is half of the circumference4 ,it is called a 8semi-circumference40 8,semi0 means 8half40 ,a semi-circumference is half of a circumference4 ,read three arcs1 each of which is one-fourth of the circumference4 ,if the whole circumference were divided into three hundred and sixty equal arcs1 would each arc be large or small8 ,each of these arcs would be called a 8degree40 ,degrees are marked 774 ,a degree of a circumference is a three hundred and sixtieth part of it4 ,how many degrees in a semi-circumference8 ,how many degrees in one-fourth of a circumference8 ,if a fourth of a circumference were divided into three equal parts1 how many degrees would there be in each part8 ,into how many parts would each third of a quarter have to be again divided to make single degrees8 ,is an arc of ninety-one degrees greater or less than one-fourth of a circumference8 ,is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference8 ,can there be more than three hundred and sixty degrees in a circumference8 ,if the circumference of circle #a were divided into degrees1 each degree would be so small an arc that it would look like a dot4 ,if a degree were divided into sixty equal parts1 each part would be called a minute4 ,if a minute were divided into sixty equal parts1 each part would be called a second4 ,how many degrees in the large circle of ,fig4 #b8 ,how many in the smaller one8 ,has a large circle any more degrees than a small circle8 ,in the large circle how many degrees from a to b8 ,in the small circle how many from a to b8 ,which is greater1 an arc of ninety degrees of the large circle1 or one of ninety degrees of the small one8 ,which is greater1 an arc of a degree of the large circle1 or one of a degree of the small one8 ,the angle a o b has its vertex at what part of the larger circle8 ,at what part of the smaller circle8 ,on how many degrees of the larger circle does the angle stand8 ,on how many degrees of the smaller circle does it stand8 ,then it is said to be an angle of #ij4 ,if the angle a o f is an angle of #cj1 how many degrees must there be in the arc a f8 ,if the arc f e is an arc of #fj1 what is the size of the angle f o e8 ,an angle of #aj stands upon an arc of how many degrees8 ,of #h8 ,of #a8 ,the angle a o b is what kind of an angle8 ,upon how many degrees does it stand8 ,then a right angle is an angle of how many degrees8 ,if an angle stand upon less than #ij1 what kind of an angle is it8 ,if an angle stand upon more than #ij1 what kind of an angle is it8 ,can an angle have as many degrees as a hundred and eighty8 ,,lesson ,,thirty-,,fifth4 ,,review4 ,read nine arcs whose ends are marked4 7,,diagram #bf47 ,read three arcs each of which is one-fourth of a circumference4 ,read two arcs each of which is one-half of a circumference4 ,why is e g an arc8 ,what is an arc8 ,how many degrees in the arc f h8 ,in e h8 ,if the arc f h were divided into three equal parts1 how many degrees would there be in each8 ,how many degrees in a circumference8 ,in a semi-circumference8 ,how many more degrees in a large circumference than in a small one8 ,if the arc i f is #dj1 what is the size of the angle f o i8 ,if the angle f o g is an angle of #acj1 what is the size of the arc f i h g8 ,how many degrees in each of the adjacent angles f o h1 h o e8 ,when two adjacent angles are equal to each other1 what is each called8 ,how many degrees in a right angle8 ,illustration3 ,diagram #bf4 ,,lesson ,,thirty-,,sixth4 ,,parts ,,of ,,the ,,circle4 ,the part of the circle bounded by the chord a b and the arc a b is called a segment4 ,read three segments1 each less than half a circle1 thus1--the segment bounded by the chord a d and the arc a b d4 ,a segment is a part of a circle bounded by an arc and a chord4 ,read two segments that are each half a circle4 ,what is the chord called8 ,what is the arc called8 ,a segment bounded by a diameter and a semi-circumference is a 8semicircle40 ,a semicircle is half a circle4 ,read four segments each larger than a semicircle4 ,the part of the circle between the two radii o f1 o i1 and the arc f i1 is called a 8sector40 ,read four sectors each less than one-fourth of a circle4#b ,footnote #b3 ,thus1 a sector bounded by the two radii o g1 o h1 and the arc g h4 ,a sector is a part of a circle bounded by two radii and an arc4 ,what part of the whole circle is the sector f o h8 ,it is called a 8quadrant40 ,a quadrant is a sector which is one-fourth of a circle4 ,read a sector which is greater than a quadrant4 ,if the chord e f be regarded a diameter1 what do you call the semicircle below it8 ,if it be regarded as two radii1 what is the semicircle called8 ,then a semicircle is both a segment and a sector4 ,,lesson ,,thirty-,,seventh4 ,,review4 ,name ten segments4 7,,diagram #bf47 ,what is a segment8 ,of the segments named1 which are less than a semicircle8 ,which are greater8 ,which two are semicircles8 ,which two are on the chord a f8 ,name nine sectors4 ,why is g o i a sector8 ,what is a sector8 ,which four of the sectors named are each less than a quadrant8 ,which three are quadrants8 ,which two are greater than a quadrant8 ,what part of the circle is both a segment and a sector8 ,how many quadrants in a circle8 ,how many semicircles8 ,,part ,,second4 ,,axioms ,,and ,,theorems4 ,,axioms ,,illustrated4 ,,axiom #a4 ,the triangle ,a is equal to the triangle ,c4 ,the triangle ,b is also equal to the triangle ,c4 ,what do you think of the two triangles ,a and ,b8 ,why8 ,illustration ,if two things are separately equal to the same thing1 they are equal to each other4 ,,axiom #b4 ,the square ,a is equal to the square ,b4 ,to the rectangle ,c add the square ,a1 and we have an ,l pointing in what direction8 ,to the same rectangle ,c add the square ,b1 and we have an ,l pointing in what direction8 ,illustration ,which is larger1 the ,l pointing to the left1 or that pointing to the right8 ,to what same thing did you add two equals8 ,what two equals did you add to it8 ,what was the first sum8 ,the second8 ,what do you think of the two sums8 ,if equals be added to the same thing1 the sums will be equal4 ,,axiom #c4 ,illustration ,the square ,a is equal to the square ,b4 ,from the inverted ,t take away the square ,a1 and we have an ,l pointing in what direction8 ,from the same ,fig4 ,t take away the square ,b1 and we have an ,l pointing in what direction8 ,which is larger1 the ,l pointing to the right1 or that pointing to the left8 ,what two equal things did we take away from the same thing8 ,from what same thing did we take them away8 ,what did we find true of the two remainders8 ,if equals be taken from the same thing1 the remainders will be equal4 ,,axiom #d4 ,illustration ,the rectangle #a #b is equal to the rectangle #a #c4 ,from the rectangle #a #b take away the square ,a1 and what rectangle remains8 ,from the rectangle #a #c take away the same square ,a1 and what rectangle remains8 ,which is greater1 the rectangle ,b1 or the rectangle ,c8 ,what same thing did we take away from equals8 ,from what did we first take it8 ,what remained8 ,from what did we next take it8 ,what remained8 ,what did we find true of the two remainders8 ,if the same thing be taken from equals1 the remainders will be equal4 ,,axiom #e4 ,if equals be added to equals1 the sums will be equal4 ,,axiom #f4 ,if equals be subtracted from equals1 the remainders will be equal4 ,,axiom #g4 ,if the halves of two things are equal1 the wholes will be equal4 ,,axiom #h4 ,every ,whole is equal to the sum of all its parts4 ,,axiom #i4 ,from one point to another only one straight line can be drawn4 ,,axiom #aj4 ,a straight line is the shortest distance between two points4 ,,axiom #aa4 ,if two things coincide throughout their whole extent1 they are equal4 ,,theorems ,,illustrated4 ,illustration3 ,diagram #bi4 ,,development ,,lesson4 ,do the angles ,blue1 ,red1 take up all the space on the line a b8 ,do the angles ,blue1 ,yellow1 ,red1 take up all the space on the line8 ,do the angles ,blue1 ,yellow1 ,green1 ,red1 take up all the space on the line8 ,is there room between any two of the angles to put in another angle8 ,then are not the angles ,blue1 ,yellow1 ,green1 ,red1 equal to all the space on the line a b8 ,,note4--,the word space1 as here used1 means angular space2 and it is indispensable that the teacher impress this fact upon the learner4 ,by means of former lessons1 the pupil has learned positively1 that an angle is the difference between the directions of two lines2 and1 impliedly1 that the included space has nothing to do with the size of the angle4 ,there cannot1 therefore1 be much danger that the pupil will imbibe any erroneous notion from this style of expression1 which is very much more simple than to say that the difference of direction of two given lines is equal to the difference of direction of two other given lines1 which style will be used somewhat later in these lessons4 ,illustration3 ,diagram #cj4 ,,proposition ,i4 ,,theorem4 ,,development ,,lesson4 ,are the adjacent angles ,green1 ,red1 equal to all the angular space on the line a b8 ,place a paper square corner or right angle on the line a b at the left of c d with its vertex at c4 ,it will cover all the angle ,green and part of the angle ,red up to the line c d4 ,now place another square corner on the line a b to the right of the line c d1 and with its vertex at the point c4 ,it will cover the remaining part of the angle ,red1 and two edges of the square corners will meet along the line c d4 ,are the two right angles equal to all the angular space on the line a b8 ,then if the two adjacent angles ,green1 ,red1 are equal to all the angular space on the line a b1 and the two right angles are also equal to the same space1 what do you infer concerning the adjacent angles and the two right angles8 ,what axiom do you apply when you say that the adjacent angles are equal to the two right angles8 ,to what same thing did you find two things separately equal8 ,what did you first see equal to it8 ,what did you next see equal to it8 ,then what did you find true8 ,if the angle ,red were smaller1 and the angle ,green larger1 would the adjacent angles still be equal to two right angles8 ,then1-- ,any two adjacent angles are equal to two right angles4 ,if we draw the straight line c d where the edges of the square corners come together1 what kind of angles will a c d1 d c b1 be8 ,see now if you can understand the following demonstration3-- ,,demonstration4 ,we wish to prove that ,any two adjacent angles are equal to two right angles4 ,let the two straight lines a b1 m n1 intersect each other in the point c4 7,,diagram #cj47 ,then will any two adjacent angles1 as ,green1 ,red1 be equal to two right angles8 ,for1 from the point c1 draw the straight line c d so as to make the angles a c d1 d c b1 right angles4 ,the adjacent angles ,green1 ,red1 are equal to all the angular space on the line a b4 ,the right angles a c d1 d c b1 are also equal to all the angular space on the line a b4 ,therefore the adjacent angles ,green1 ,red1 are equal to two right angles4 ,,test ,,questions4 ,to what same thing did you find two things equal8 ,what did you first see equal to it8 ,what did you next see equal to it8 ,then what new thing did you find true8 ,what axiom did you make use of8 ,illustration3 ,diagram #ca4 ,,test ,,lesson4 ,by means of ,fig4 ,a1-- #a4 ,prove that the adjacent angles ,green1 ,red1 are equal to two right angles4 #b4 ,prove that the adjacent angles ,blue1 ,yellow1 are equal to two right angles4 ,by means of ,fig4 ,b1-- #c4 ,prove that the adjacent angles ,green1 ,red1 are equal to two right angles4 #d4 ,prove that the adjacent angles ,yellow1 ,blue1 are equal to two right angles4 ,by means of ,fig4 ,c1-- #e4 ,prove that the adjacent angles ,red1 ,blue1 are equal to two right angles4 #f4 ,prove that the adjacent angles ,green1 ,yellow1 are equal to two right angles4 #g4 ,give the preceding demonstrations again1 but name the angles by their letters instead of by their colors4 ,illustration3 ,diagram #cb4 ,,test ,,lesson4 ,by means of ,fig4 ,a prove1-- #a4 ,that the adjacent angles a c m1 m c b1 are equal to two right angles4 #b4 ,that the adjacent angles a c n1 n c b1 are equal to two right angles4 ,by means of ,fig4 ,b prove1-- #c4 ,that the adjacent angles a c n1 n c b1 are equal to two right angles4 #d4 ,that the adjacent angles a c m1 m c b1 are equal to two right angles4 ,by means of ,fig4 ,c prove1-- #e4 ,that the adjacent angles a c m1 m c b1 are equal to two right angles4 #f4 ,that the adjacent angles a c n1 n c b1 are equal to two right angles4 ,by means of ,fig4 ,d prove1-- #g4 ,that the adjacent angles a c n1 n c b1 are equal to two right angles4 #h4 ,that the adjacent angles b c m1 m c a1 are equal to two right angles4 ,illustration3 ,diagram #cc4 ,,proposition ,,ii4 ,,theorem4 ,,development ,,lesson4 ,what kind of angles are ,p and ,s8 ,how do the adjacent angles ,yellow1 ,blue1 compare with the right angles ,p1 ,s8 ,how do the adjacent angles ,blue1 ,red1 compare with the two right angles8 ,then if the adjacent angles ,yellow1 ,blue1 are equal to two right angles1 and the adjacent angles ,blue1 ,red1 are also equal to two right angles1 what do you think of the two pairs of adjacent angles1 ,yellow1 ,blue1 and ,blue1 ,red8 ,if1 from the adjacent angles ,yellow1 ,blue1 we take away the angle ,blue1 what remains8 ,if1 from the adjacent angles ,blue1 ,red1 we take away the same angle ,blue1 what remains8 ,then1 since the same angle ,blue has been taken from equal pairs of adjacent angles1 what do you think of the two remainders1 ,yellow1 ,red8 ,suppose the lines a b and m n were so drawn that the angles ,yellow1 ,red1 were larger or smaller1 would they still be equal to each other8 ,then1-- ,all vertical angles are equal to each other4 ,illustration3 ,diagram #cd4 ,,demonstration4 ,we wish to prove that ,all vertical angles are equal to each other4 ,let the straight lines a b1 m n1 intersect each other at the point c1 then will any two vertical angles1 as ,yellow1 ,red1 be equal to each other4 ,for the adjacent angles ,yellow1 ,blue1 are equal to two right angles4#c ,footnote #c3 ,when this comparison is made1 let the pupil look at the right angles ,p and ,s4 ,the adjacent angles ,blue1 ,red1 are also equal to two right angles4 ,therefore the adjacent angles ,yellow1 ,blue1 are equal to the adjacent angles ,blue1 ,red4 ,if1 from the adjacent angles ,yellow1 ,blue1 we take away the angle ,blue1 we shall have left the angle ,yellow4 ,if1 from the adjacent angles ,blue1 ,red1 we take away the same angle ,blue1 we shall have left the angle ,red4 ,therefore the vertical angles ,yellow1 ,red1 are equal to each other4 ,,test ,,questions4 ,when you say that the adjacent angles ,yellow1 ,blue1 are equal to two right angles1 do you know it because you see it1 or because you have proved it8 ,how do you know that the adjacent angles ,blue1 ,red1 are equal to two right angles8 ,when you say the adjacent angles ,yellow1 ,blue1 are equal to the adjacent angles ,blue1 ,red1 what axiom do you use8 ,what same thing do you take away from equals8 ,from what equals do you take it away8 ,when you take the angle ,blue from the adjacent angles ,yellow1 ,blue1 what is the remainder8 ,when you take the same angle ,blue from the adjacent angles ,blue1 ,red1 what is the remainder8 ,what do you find true of the two remainders8 ,what axiom do you use8 ,illustration3 ,diagram #ce4 ,,other ,,methods ,,of ,,demonstration4 ,the adjacent angles ,yellow1 ,green1 are equal to what8 ,the adjacent angles ,green1 ,red1 are equal to what8 ,then what do you know of the two pairs of adjacent angles ,yellow1 ,green1 and ,green1 ,red8 ,from the adjacent angles ,yellow1 ,green1 take away the angle ,green4 ,what remains8 ,from the adjacent angles ,green1 ,red1 take the same angle ,green4 ,what remains8 ,what do you know of the two remainders8 ,why8 ,what axiom do you use8 ,in the last lesson1 when you proved the vertical angles ,yellow1 ,red1 equal to each other1 you made use of the angle ,blue2 now prove the same two angles equal by means of the angle ,green4 ,the adjacent angles ,blue1 ,red1 are equal to what8 ,the adjacent angles ,red1 ,green1 are equal to what8 ,then what do you know of the two pairs of adjacent angles ,blue1 ,red1 and ,red1 ,green8 ,from the adjacent angles ,blue1 ,red1 take away the angle ,red4 ,what remains8 ,from the adjacent angles ,red1 ,green1 take away the same angle ,red4 ,what remains8 ,then what do you know of the two remainders1 ,blue1 ,green8 ,now apply the preceding demonstration to the vertical angles ,blue1 ,green4 ,prove the vertical angles ,blue1 ,green1 equal to each other by means of the angle ,yellow4 ,illustration3 ,diagram #cf4 ,,test ,,lesson4 ,by means of ,fig4 ,a1-- #a4 ,prove that the vertical angles ,yellow1 ,red1 are equal to each other1 using the angle ,green4 #b4 ,prove the same thing1 using the angle ,blue4 #c4 ,prove that the vertical angles ,blue1 ,green1 are equal to each other1 using the angle ,yellow4 #d4 ,prove the same thing1 using the angle ,red4 ,by means of ,fig4 ,b1-- #e4 ,prove the vertical angles ,yellow1 ,red1 equal to each other1 using the angle ,green4 #f4 ,prove the same thing1 using the angle ,blue4 #g4 ,prove the vertical angles ,green1 ,blue1 equal by means of the angle ,red4 #h4 ,prove the same thing by means of the angle ,yellow4 ,go through the preceding eight demonstrations again1 calling the angles by their letters instead of by their colors4 ,by means of ,fig4 ,c1 prove that #i4 a c n equals m c b1 by means of a c m4 #aj4 a c n equals m c b1 by means of b c n4 #aa4 a c m equals n c b1 by means of a c n4 #ab4 a c m equals n c b1 by means of m c b4 ,by means of ,fig4 ,d1 prove that #ac4 m c a equals b c n1 by means of a c n4 #ad4 m c a equals b c n1 by means of m c b4 #ae4 m c b equals a c n1 by means of m c a4 #af4 m c b equals a c n1 by means of b c n4 ,illustration3 ,diagram #cg4 ,,proposition ,,iii4 ,,theorem4 ,,development ,,lesson4 ,in the above diagram1 the lines a b1 c d1 are parallel1 and are intersected by the line e f at the points m and n4 ,the angle ,red measures the difference of direction between the line m b and what other line8 ,the angle ,yellow measures the difference of direction between the line n d and what other line8 ,then1 as the lines m b and n d are parallel1 must there not be the same difference of direction between them and the line e f8 ,then can there be any difference between the angles which measure those equal directions8 ,then what do you think of the opposite exterior and interior angles ,red1 ,yellow8 ,,demonstration4 ,we wish to prove that ,opposite exterior and interior angles are equal to each other4 ,let the straight line e f intersect the two parallel straight lines a b1 c d1 at the points m and n4 ,then will any two opposite exterior and interior angles1 as ,red1 ,yellow1 be equal to each other4 ,for the angle ,red measures the difference of direction of the lines m b and e f4 ,and the angle ,yellow measures the difference of direction of the lines n d and e f4 ,but because the lines m b1 n d1 are parallel1 these differences are equal4 ,therefore the angles which measure them are equal2 that is1 ,the opposite exterior and interior angles ,red1 ,yellow1 are equal to each other4 ,illustration3 ,diagram #ch4 ,,test ,,lesson4 ,by means of ,fig4 ,a1-- #a4 ,prove that the opposite exterior and interior angles ,green1 ,blue1 are equal to each other4 #b4 ,prove that the opposite exterior and interior angles ,red1 ,yellow1 are equal to each other4 #c4 ,prove the opposite exterior and interior angles c n e1 a m n1 equal4 #d4 ,prove the opposite exterior and interior angles e n d1 n m b1 equal4 ,by means of ,fig4 ,b1-- #e4 ,prove the opposite exterior and interior angles e m a1 m n d1 equal4 #f4 ,prove the opposite exterior and interior angles a m n1 d n f1 equal4 #g4 ,prove the opposite exterior and interior angles e m b1 m n c1 equal4 #h4 ,prove the opposite exterior and interior angles b m n1 c n f1 equal4 ,illustration3 ,diagram #ci4 ,,proposition ,,iv4 ,,theorem4 ,,development ,,lesson4 ,what do you know of the opposite exterior and interior angles ,red1 ,yellow8 ,what do you know of the vertical angles ,red1 ,green8 ,then if the interior alternate angles ,green1 ,yellow1 are separately equal to the angle ,red1 what new fact do you know8 ,what axiom do you employ8 ,to what same thing did you find two things equal8 ,what two things did you find equal to it8 ,,demonstration4 ,we wish to prove that ,any two interior alternate angles are equal to each other4 ,let the straight line e f intersect the two parallel straight lines a b1 c d1 in the points m and n4 ,then will any two interior alternate angles1 as ,green1 ,yellow1 be equal to each other4 ,for the opposite exterior and interior angles ,red1 ,yellow1 are equal4 ,the vertical angles ,red1 ,green1 are also equal4 ,then because the interior alternate angles ,green1 ,yellow1 are separately equal to the angle ,red1 they are equal to each other4 ,illustration3 ,diagram #dj4 ,,test ,,lesson4 ,what do you know of the vertical angles ,green1 ,red1 in ,fig4 ,a8 ,what do you know of the opposite exterior and interior angles ,red1 ,yellow8 ,then if the interior alternate angles ,green1 ,yellow1 are separately equal to the angle ,red1 what do you infer8 ,by means of ,fig4 ,a1-- #a4 ,prove that the interior alternate angles ,green1 ,yellow1 are equal1 using the angle ,red4 #b4 ,prove the same angles equal1 using the angle ,blue4 #c4 ,go through the same demonstrations again1 calling the angles by their letters instead of by their colors4 ,by means of ,fig4 ,b1-- #d4 ,prove the interior alternate angles ,red1 ,blue1 equal1 using the angle ,yellow4 #e4 ,prove the same angles equal1 using the angle ,green4 #f4 ,go through the same two demonstrations again1 naming the angles by their letters instead of by their colors4 ,by means of ,fig4 ,c1-- #g4 ,prove the interior alternate angles c n m1 n m b1 equal1 using the angle f n d4 #h4 ,prove the same1 using the angle a m e4 #i4 ,prove the interior alternate angles a m n1 m n d1 equal1 using the angle e m b4 #aj4 ,prove the same1 using the angle c n f4 ,illustration3 ,diagram #da4 ,,proposition ,v4 ,,theorem4 ,,development ,,lesson4 ,what do you know of the opposite exterior and interior angles ,red1 ,yellow8 ,what do you know of the vertical angles ,yellow1 ,green8 ,then if the exterior alternate angles ,red1 ,green1 are separately equal to the angle ,yellow1 what new thing do you know to be true8 ,what axiom do you employ8 ,to what same thing did you know two things to be equal8 ,what two things did you know to be equal to it8 ,then what new thing did you find to be true8 ,,demonstration4 ,we wish to prove that ,any two exterior alternate angles are equal to each other4 ,let the straight line e f intersect the two parallel straight lines a b1 c d1 at the points m and n4 ,then will any two exterior alternate angles1 as ,red1 ,green1 be equal4 ,for the opposite exterior and interior angles ,red1 ,yellow1 are equal to each other4 ,and the vertical angles ,yellow1 ,green1 are also equal to each other4 ,then because the exterior alternate angles ,red1 ,green1 are separately equal to the angle ,yellow1 they are equal to each other4 ,illustration3 ,diagram #db4 ,,test ,,lesson4 ,what do you know of the opposite exterior and interior angles ,yellow1 ,red8 ,what do you know of the vertical angles ,red1 ,blue8 ,then if the exterior alternate angles ,yellow1 ,blue1 are separately equal to the angle ,red1 what do you know of them8 ,by means of ,fig4 ,a1-- #a4 ,prove that the exterior alternate angles ,yellow1 ,blue1 are equal1 using the angle ,red4 #b4 ,prove the same thing1 using the angle ,green4 #c4 ,go through the same demonstrations1 calling the angles by their letters4 #d4 ,prove the exterior alternate angles e m b1 c n f1 equal1 using the angle a m n4 #e4 ,prove the same1 using the angle m n d4 ,by means of ,fig4 ,b1-- #f4 ,prove that the exterior alternate angles c m e1 f n b1 are equal1 using the angle n m d4 #g4 ,prove the same1 using the angle a n m4 #h4 ,prove the exterior alternate angles e m d1 a n f1 equal1 using the angle c m n4 #i4 ,prove the same1 using the angle m n b4 ,illustration3 ,diagram #dc4 ,,proposition ,,vi4 ,,theorem4 ,,development ,,lesson4 ,what do you know of the interior alternate angles ,yellow1 ,red8 ,if to the angle ,green you add the angle ,yellow1 what is the sum8 ,if to the same angle ,green you add the equal angle ,red1 what is the sum8 ,then1 having added equals to the same thing1 what do you think of the two sums1--the adjacent angles ,green1 ,yellow1 and the interior opposite angles ,green1 ,red8 ,what do you know of the adjacent angles ,green1 ,yellow1 and the right angles ,p1 ,s8 ,then if the interior opposite angles ,green1 ,red1 and the two right angles ,p1 ,s1 are separately equal to the adjacent angles ,green1 ,yellow1 what new thing do you know8 ,,demonstration4 ,we wish to prove that ,any two interior opposite angles are equal to two right angles4 ,let the straight line e f intersect the two parallel straight lines a b1 c d1 in the points m and n4 ,then will any two interior opposite angles be equal to two right angles4 ,for the interior alternate angles ,yellow1 ,red1 are equal4 ,if to the angle ,green we add the angle ,yellow1 we shall have the adjacent angles ,green1 ,yellow4 ,if to the same angle ,green we add the equal angle ,red1 we shall have the interior opposite angles ,green1 ,red4 ,then the adjacent angles ,green1 ,yellow1 are equal to the interior opposite angles ,green1 ,red4 ,but the adjacent angles ,green1 ,yellow1 are equal to two right angles4 ,then because the interior opposite angles ,green1 ,red1 and two right angles1 are separately equal to the two adjacent angles ,green1 ,yellow1 they are equal to each other4 ,illustration3 ,diagram #dd4 ,,test ,,lesson4 ,by means of ,fig4 ,a1-- #a4 ,prove the interior opposite angles ,green1 ,yellow1 equal to two right angles1 using the angle ,red4 #b4 ,prove the same1 using the angle ,blue4 #c4 ,prove the same1 using the angle e g b4 #d4 ,prove the same1 using the angle f h d4 #e4 ,go through the same demonstrations again1 naming the angles by their letters instead of by their colors4 #f4 ,prove the interior opposite angles ,red1 ,blue1 equal to two right angles1 using the angle ,yellow4 #g4 ,prove the same1 using the angle ,green4 #h4 ,prove the same1 using the angle e g a4 #i4 ,prove the same1 using the angle c h f4 #aj4 ,go through the same demonstrations again1 calling the angles by their letters instead of by their colors4 ,by means of ,fig4 ,b1-- #aa4 ,prove the interior opposite angles a g h1 g h c1 equal to two right angles1 using the angle g h d4 #ab4 ,prove the same1 using the angle c h f4 #ac4 ,prove the same1 using the angle a g e4 #ad4 ,prove the interior opposite angles b g h1 g h d1 equal to two right angles1 using the angle a g h4 #ae4 ,prove the same1 using the angle e g b4 #af4 ,prove the same1 using the angle f h d4 ,compare the angles ,yellow1 ,green1 each with its exterior opposite angle1 and see if you can prove that the exterior opposite angles e g b1 f h d1 are also equal to two right angles4 ,illustration3 ,diagram #de4 ,,proposition ,,vii4 ,,theorem4 ,,development ,,lesson4 ,suppose we do not know whether the lines a b1 c d1 are parallel1 or not2 ,but1 by measuring1 we find that the interior angles ,blue1 ,yellow1 on the same side of the secant#d line e f1 are equal to two right angles3 ,footnote #d3 8,secant0 means 8cutting40 ,the adjacent angles ,blue1 ,red1 are equal to what8 ,then1 if the interior angles ,blue1 ,yellow1 are equal to two right angles1 ,and the adjacent angles ,blue1 ,red1 are also equal to two right angles1 ,what do you infer8 ,from the interior angles ,blue1 ,yellow1 take away the angle ,blue3 what remains8 ,from the adjacent angles ,blue1 ,red1 take away the same angle ,blue3 what remains8 ,what do you know of the two remainders8 ,the angle ,red measures the direction of the line g b from what line8 ,the equal angle ,yellow measures the direction of the line h d from what line8 ,then if the lines g b1 h d1 have the same direction from the line e f1 what do you call them8 ,illustration3 ,diagram #df4 ,,demonstration4 ,we wish to prove1 that1 ,if a straight line intersects two other straight lines so that two interior angles on the same side of the intersecting line are equal to two right angles1 the two lines are parallel4 ,let the straight line e f intersect the two straight lines a b1 c d1 in the points g and h1 so that the angles ,red1 ,blue1 are equal to two right angles4 ,then will the lines a b1 c d1 be parallel4 ,for the angles ,red1 ,blue1 are supposed equal to two right angles4 ,the adjacent angles ,red1 ,green1 are known to be also equal to two right angles4 ,then the interior angles ,red1 ,blue1 are equal to the adjacent angles ,red1 ,green4 ,if from the interior angles ,red1 ,blue1 we take away the angle ,red1 we have left the angle ,blue4 ,if from the adjacent angles ,red1 ,green1 we take the same angle ,red1 we shall have left the angle ,green4 ,then the angle ,blue is equal to the angle ,green4 ,but the angle ,blue measures the direction of the line h d from the line e f4 ,and the angle ,green measures the direction of the line g b from the line e f4 ,then the lines g b1 h d1 have the same direction1 and are parallel4 ,,test ,,lesson4 #a4 ,prove the same without the colors4 #b4 ,prove the same1 using the angle f h d4 #c4 ,prove the same1 supposing the angles a g h1 g h c1 equal to two right angles1 and using the angle a g e4 #d4 ,prove the same1 using the angle c h f4 ,see ,note ,e1 ,appendix4 ,,proposition ,,viii4 ,,theorem4 ,the following demonstration is very easy4 ,read it once1 and see if you can go through it without a second reading3-- ,,demonstration4 ,illustration ,we wish to prove that ,the sum of any two sides of a triangle is greater than the third side4 ,let the figure a b c be a triangle1 then will the sum of any two sides1 as a c1 c b1 be greater than the third side a b4 ,for the straight line a b is the shortest distance between the two points a and b1 and is therefore less than the broken line a c b4 ,,proposition ,,ix4 ,,problem4 ,the following solution is so easy that you will understand it at once3-- ,we wish ,to construct an equilateral triangle on a given straight line4 ,illustration ,,solution4 ,let a b be the given line4 ,with the point a as a centre1 and a b as a radius1 draw the circumference of the circle1 or a part of one4 ,with the point b as a centre1 and the same radius a b1 draw another circumference1 or a part of one4 ,from the point c1 in which the circumferences or arcs intersect1 draw the straight lines a c and b c4 ,now1 because the lines a b and a c are radii of the same circle1 they are equal4 ,and1 because the lines a b and b c are radii of the same circle1 they are also equal4 ,then1 because the two lines a c1 b c1 are separately equal to the line a b1 they are equal to each other1 and the triangle is equilateral4 ,illustration ,,proposition ,x4 ,,theorem4 ,,development ,,lesson4 ,let the figure a b c be a triangle4 ,produce the side a c to d4 ,we have now another angle1 b c d1 and we wish to find out if it is equal to any of the angles of the triangle4 ,from the point c draw the line c e parallel to a b4 ,because the straight line a d intersects the two parallels a b1 c e1 the angle a is equal to what other angle8 ,because the straight line b c intersects the two parallels a b1 c e1 the angle b is equal to what other angle8 ,then the angles a and b are equal to what two angles8 ,how does the angle b c d compare with the angles b c e1 e c d8 ,then1 if the angles a and b1 on the one hand1 and the angle b c d1 on the other1 are separately equal to the angles b c e1 e c d1 ,what have you found out8 ,what axiom have you just employed8 ,to what same thing have you found two other things equal8 ,what two things did you find equal to it8 ,,demonstration4 ,we wish to prove1 that1 ,if any side of a triangle be produced1 the new angle formed will be equal to the sum of the angles that are not adjacent to it4 ,let a b c be a triangle4 ,produce the side a c to d2 then will the new angle b c d be equal to the sum of the angles a and b4 ,for from the point c draw c e parallel to a b4 ,then1 because the straight line a d intersects the two parallels a b1 c e1 in the points a and c1 ,the opposite exterior and interior angles a and e c d are equal to each other4 ,and because the straight line b c intersects the same parallels in the points b and c1 ,the interior alternate angles b and b c e are equal4 ,then the angles a and b of the triangle are equal to the angles b c e and e c d4 ,but the new angle b c d is equal to the angles b c e1 e c d4 ,then because the new angle b c d1 and the angles a and b are separately equal to the angles b c e1 e c d1 they are equal to each other4 ,illustration ,,proposition ,,xi4 ,,theorem4 ,,development ,,lesson4 ,let the figure a b c be a triangle4 ,produce the side a c to d4 ,by the last theorem1 the angle b c d is equal to what angles of the triangle8 ,what angle must we add to these angles to make up the three angles of the triangle8 ,if we add the same angle to the angle b c d1 what adjacent angles do we get8 ,then the three angles of the triangle1 a1 b1 and c1 are equal to what two angles8 ,but the adjacent angles a c b and b c d are equal to what8 ,then1 because the three angles of the triangle1 a1 b1 and c1 and two right angles1 are separately equal to the two adjacent angles c and b c d4 ,what new thing have you found out8 ,,demonstration4 ,we wish to prove that ,the three angles of any triangle are equal to two right angles4 ,let the figure a b c be a triangle2 then will the sum of the angles a1 b1 and c1 be equal to two right angles4 ,for1 produce the side a c to d4 ,the new angle b c d is equal to the sum of the angles a and b4 ,if to the angles a and b we add the angle c1 we shall have the three angles of the triangle4 ,if to the angle b c d we add the same angle c1 we shall have the adjacent angles c and b c d4 ,then the three angles of the triangle a1 b1 c1 are equal to the adjacent angles c and b c d4 ,but the adjacent angles c and b c d are equal to two right angles4 ,then1 because the three angles of the triangle are equal to the adjacent angles c and b c d1 they are equal to two right angles4 ,illustration ,,proposition ,,xii4 ,,theorem4 ,,development ,,lesson4 ,let the ,fig4 ,a ,b ,c ,d be a parallelogram4 ,produce the side ,c ,d to ,f4 ,because the straight line ,b ,d intersects the parallels ,a ,b and ,c ,f1 the angle ,b is equal to what other angle8 ,because the straight line ,c ,f intersects the parallels ,a ,c and ,b ,d1 the angle ,c is equal to what other angle8 ,then what follows from this8 ,to what angle did you find two others equal8 ,what two angles did you find equal to it8 ,what axiom do you think of8 ,see if you can go through the demonstration without reading it even once4 ,,demonstration4 ,we wish to prove that ,the opposite angles of a parallelogram are equal to each other4 ,let the ,fig4 ,a ,b ,c ,d be a parallelogram4 ,then will any two opposite angles1 as ,b and ,c1 be equal to each other4 ,for produce the line ,c ,d to ,f4 ,because the straight line ,b ,d meets the two parallels ,a ,b and ,c ,f1 ,the interior alternate angles ,b and ,e are equal to each other4 ,because the straight line ,c ,f meets the two parallels ,b ,d and ,a ,c1 ,the opposite exterior and interior angles ,c and ,e are equal to each other4 ,then1 because the angles ,b and ,c are separately equal to the angle ,e1 they are equal to each other4 99 99 99 99 99 #a4 ,prove the same by producing the line ,a ,b towards the left4 #b4 ,prove the same by producing the line ,b ,d downwards4 #c4 ,prove the angles ,a and ,d equal to each other by producing the line ,c ,d towards the left4 #d4 ,prove the same by producing the line ,d ,b upwards4 #e4 ,see if you can prove the same by drawing a diagonal through the points ,a and ,d4 ,illustration ,,proposition ,,xiii4 ,,theorem4 ,,development ,,lesson4 ,in these two triangles we have tried to make the side a b of the one equal to the side d e of the other2 the side a c of the one equal to the side d f of the other2 and the included angle b a c of the one equal to the included angle e d f of the other4 ,we now wish to find out if the third side b c of the one is equal to the third side e f of the other1 and if the two remaining angles b and c of the one are equal to the two remaining angles e and f of the other4 ,suppose we were to cut the triangle d e f out of the page1 and place it upon the triangle a b c1 so that the line d e should fall upon the line a b1 and the point d upon the point a4 ,as the line d e is equal to the line a b1 upon what point will the point e fall8 ,if the angle e d f were less than the angle b a c1 would the line d f fall within or without the triangle8 ,if the angle e d f were greater than the angle b a c1 where would the line d f fall8 ,since the angle a is equal to d1 where1 then1 must the line d f fall8 ,as the line d f is equal to the line a c1 upon what point will the point f fall8 ,then1 if the point e falls upon the point b1 and the point f upon the point c1 where will the line e f fall8 ,now1 because the three sides of the triangle d e f exactly fall upon the three sides of the triangle a b c1 we say the two magnitudes coincide throughout their whole extent1 and are therefore equal4 ,what three parts of the triangle a b c did we suppose to be equal to three corresponding parts of the triangle d e f before we placed one upon the other4 ,what line of the one do we find equal to a line in the other8 ,what two angles of the one do we find equal to two angles in the other8 ,what do you think of the areas of the triangles8 ,illustration ,,demonstration4 ,we wish to prove1 that1 ,if two triangles have two sides1 and the included angle of the one equal to two sides and the included angle of the other1 each to each1 the two triangles are equal in all respects4 ,let the triangles a b c and d e f have the side a b of the one equal to the side d e of the other2 the side a c of the one equal to the side d f of the other2 and the included angle b a c of the one equal to the included angle e d f of the other1 each to each2 then will the two triangles be equal in all their parts4 ,for1 place the triangle d e f upon the triangle a b c1 so that the line d e shall fall upon the line a b1 with the point d upon the point a4 ,because the line d e is equal to the line a b1 the point e will fall upon the point b4 ,because the angle e d f is equal to the angle b a c1 the line d f will fall upon the line a c4 ,because the line d f is equal to the line a c1 the point f will fall upon the point c4 ,then1 because the point e is on the point b1 and the point f on the point c1 the line e f will coincide with the line b c1 and the two triangles will be found equal in all their parts2 ,that is1 the angle e is found to be equal to the angle b1 the angle f to the angle c1 the line e f to the line b c1 and the area of the triangle a b c to the area of the triangle d e f4 ,illustration ,,proposition ,,xiv4 ,,theorem4 ,,development ,,lesson4 ,in these two triangles we have tried to make the angle b of the one equal to the angle e of the other2 the angle c of the one equal to the angle f of the other2 and the included side b c of the one equal to the included side e f of the other4 ,we now wish to find out if the remaining angle a of the one is equal to the remaining angle d of the other1 and if the two remaining sides a b and a c of the one are equal to the two remaining sides d e and d f of the other4 ,suppose we were to cut the triangle d e f out of the page and place it upon the triangle a b c1 so that the line e f shall fall upon the line b c1 with the point e upon the point b4 ,because the line e f is equal to the line b c1 upon what point will the point f fall8 ,because the angle e is equal to the angle b1 where will the line e d fall8 ,because the angle f is equal to the angle c1 where will the line d f fall8 ,then1 if the line d e falls upon the line a b and the line d f upon the line a c1 where will the point d fall8 ,now because the three sides of the triangle d e f exactly fall upon the three sides of the triangle a b c1 we say the two magnitudes coincide throughout their whole extent1 and are therefore equal4 ,suppose the angle e were greater than the angle b1 would the line e d fall within or without the triangle8 ,if it were less1 where would the line fall8 ,why does the line d e fall exactly upon the line a b8 ,illustration ,,demonstration4 ,we wish to prove that1 ,if two triangles have two angles1 and the included side of the one equal to two angles and the included side of the other1 each to each1 the two triangles are equal to each other in all respects4 ,let the triangles a b c and d e f have the angle b of the one equal to the angle e of the other2 the angle c of the one equal to the angle f of the other2 and the included side b c of the one equal to the included side e f of the other1 each to each2 then will the two triangles be equal in all their parts4 ,for place the triangle d e f upon the triangle a b c1 so that the line e f shall fall upon the line b c1 with the point e upon the point b4 ,because the line e f is equal to the line b c the point f will fall upon the point c4 ,because the angle e is equal to the angle b1 the line e d will fall upon the line b a1 and the point d will be somewhere in the line b a4 ,because the angle f is equal to the angle c1 the line f d will fall upon the line c a1 and the point d will be somewhere in the line c a4 ,then1 because the point d is in the two lines1 b a and c a1 it must be in their intersection1 or upon the point a4 ,and1 as the two triangles coincide throughout their whole extent1 they are equal in all their parts4 ,that is1 the angle a is found to be equal to the angle d2 the side b a to the side e d2 the side c a to the side f d2 and the area of the triangle a b c to the area of the triangle d e f4 ,illustration ,,proposition ,,xv4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,the opposite sides of any parallelogram are equal4 ,let the figure a b c d be a parallelogram2 then will the sides a b and c d be equal to each other2 likewise the sides a d and b c4 ,for1 draw the diagonal b d4 ,because the figure is a parallelogram1 the sides a b and d c are parallel1 and the interior alternate angles n and o are equal4 ,because the figure is a parallelogram1 the interior alternate angles r and m are equal4 ,then the two triangles a d b1 b d c1 have two angles and the included side of the one equal to two angles and the included side of the other1 each to each1 and are therefore equal2 ,and the side a b opposite the angle m is equal to the side c d opposite the equal angle r2 ,and the side a d opposite the angle n is equal to the side b c opposite the equal angle o4 ,,test4 ,prove the same by drawing a diagonal from a to c4 ,illustration ,,proposition ,,xvi4 ,,theorem4 ,,development ,,lesson4 ,suppose ,a ,b to be a straight line1 and ,c any point out of it4 ,from the point ,c draw a perpendicular ,c ,f to ,a ,b4 ,let us see if this perpendicular is not shorter than any other line we can draw from the same point to the same line4 ,draw any other line from ,c to ,a ,b as ,c ,e4 ,now1 as ,c ,e is any line whatever other than a perpendicular1 if we find that the perpendicular ,c ,f is shorter than it we must conclude that it is the shortest line that can be drawn from ,c to ,a ,b4 ,produce ,c ,f until ,f ,d is equal to ,c ,f1 and then join ,e and ,d4 ,in the triangles ,e ,f ,c1 ,e ,f ,d1 what two sides were drawn equal8 ,what line is a side to each8 ,how great an angle is ,c ,f ,e8 ,what is a right angle8 ,then how do the angles ,c ,f ,e and ,e ,f ,d compare with each other8 ,if the two triangles ,e ,f ,c1 ,e ,f ,d1 have the side ,c ,f of the one equal to the side ,f ,d of the other1 the side ,e ,f common to both1 and the included angle ,e ,f ,c of the one equal to the included angle ,e ,f ,d of the other1 each to each1 what do you infer8 ,then what third side of the one have you found equal to a third side of the other8 ,c ,e is what part of the broken line ,c ,e ,d8 ,c ,f is what part of the line ,c ,d8 ,which is shorter1 the straight line ,c ,d1 or the broken line ,c ,e ,d8 ,then how does the half of ,c ,d or ,c ,f compare with the half of ,c ,e ,d or ,c ,e8 ,if ,c ,e is any line whatever other than a perpendicular1 what may we now say of the perpendicular from the point ,c to the straight line ,a ,b8 ,illustration ,,demonstration4 ,we wish to prove that ,a perpendicular is the shortest distance from a point to a straight line4 ,let ,a ,b be a straight line1 and ,c ,a point out of it2 then will the perpendicular ,c ,e be the shortest line that can be drawn from the point to the line4 ,for draw any other line from ,c to ,a ,b1 as ,c ,f4 ,produce ,c ,e until ,e ,d equals ,c ,e1 and join ,f ,d4 ,the two triangles ,f ,e ,c1 ,f ,e ,d1 have the side ,c ,e of the one equal to the side ,e ,d of the other1 the side ,f ,e common1 and the included angle ,f ,e ,c of the one equal to the included angle ,f ,e ,d of the other1 they are therefore equal1 and the side ,c ,f equals the side ,f ,d4 ,but the straight line ,c ,d is the shortest distance between the two points ,c ,d2 therefore it is shorter than the broken line ,c ,f ,d4 ,then ,c ,e1 the half of ,c ,d1 is shorter than ,c ,f1 the half ,c ,f ,d4 ,and1 as ,c ,f is any line other than a perpendicular1 the perpendicular ,c ,e is the shortest line that can be drawn from ,c to ,a ,b4 ,illustration ,,proposition ,,xvii4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,a tangent to a circumference is perpendicular to a radius at the point of contact4 ,let the straight line ,a ,b be tangent at the point ,d to the circumference of the circle whose centre is ,c4 ,join the centre ,c with the point of contact ,d1 the tangent will be perpendicular to the radius ,c ,d4 ,for draw any other line from the centre to the tangent1 as ,c ,f4 ,as the point ,d is the only one in which the tangent touches the circumference1 any other point1 as ,f1 must be without the circumference4 ,then the line ,c ,f1 reaching beyond the circumference1 must be longer than the radius ,c ,d1 which would reach only to it2 therefore ,c ,d is shorter than any other line which can be drawn from the point ,c to the straight line ,a ,b2 therefore it is perpendicular to it4 ,,proposition ,,xviii4 ,,theorem4 ,,demonstration4 ,illustration ,we wish to prove that ,in any isosceles triangle1 the angles opposite the equal sides are equal4 ,let the triangle ,a ,b ,c be isosceles1 having the side ,a ,b equal to the side ,a ,c2 then will the angle ,b1 opposite the side ,a ,c1 be equal to the angle ,c1 opposite the equal side ,a ,b4 ,for draw the line ,a ,d so as to divide the angle ,a into two equal parts1 and let it be long enough to divide the side ,b ,c at some point as ,d4 ,now the two triangles ,a ,d ,b1 ,a ,d ,c1 have the side ,a ,b of the one equal to the side ,a ,c of the other1 the side ,a ,d common to both1 and the included angle ,b ,a ,d of the one equal to the included angle ,c ,a ,d of the other2 therefore the two triangles are equal in all respects1 and the angle ,b1 opposite the side ,a ,c1 is equal to the angle ,c1 opposite the side ,a ,b4 ,illustration ,,proposition ,,xix4 ,,theorem4 ,,demonstration4 ,we wish to prove that1 ,if two triangles have the three sides of the one equal to the three sides of the other1 each to each1 they are equal in all their parts4 ,let the two triangles ,a ,b ,c1 ,a ,d ,c1 have the side ,a ,b of the one equal to the side ,a ,d of the other2 the side ,b ,c of the one equal to the side ,d ,c of the other1 and the third side likewise equal2 then will the two triangles be equal in all their parts4 ,for place the two triangles together by their longest side1 and join the opposite vertices ,b and ,d by a straight line4 ,because the side ,a ,b is equal to the side ,a ,d1 the triangle ,b ,a ,d is isosceles1 and the angles ,a ,b ,d1 ,a ,d ,b1 opposite the equal sides are equal4 ,because the side ,b ,c is equal to the side ,d ,c1 the triangle ,b ,c ,d is isosceles1 and the angles ,c ,b ,d1 ,c ,d ,b1 opposite the equal sides are equal4 ,if to the angle ,a ,b ,d we add the angle ,d ,b ,c1 we shall have the angle ,a ,b ,c4 ,and if to the equal of ,a ,b ,d1 that is1 ,a ,b ,d1 we add the equal of ,d ,b ,c1 that is1 ,b ,d ,c1 we shall have the angle ,a ,d ,c4 ,therefore the angle ,a ,b ,c is equal to the angle ,a ,d ,c4 ,then the two triangles ,a ,b ,c1 ,a ,d ,c1 have two sides1 and the included angle of the one equal to two sides and the included angle of the other1 each to each1 and are equal in all their parts2 that is1 the three angles of the one are equal to the three angles of the other1 and their areas are equal4 ,illustration ,,proposition ,,xx4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,an angle at the circumference is measured by half the arc on which it stands4 ,let ,b ,a ,d be an angle whose vertex is in the circumference of the circle whose centre is ,c2 then will it be measured by half the arc ,b ,d4 ,for through the centre draw the diameter ,a ,e1 and join the points ,c and ,b4 ,the exterior angle ,e ,c ,b is equal to the sum of the angles ,b and ,b ,a ,c4 ,because the sides ,c ,a1 ,c ,b1 are radii of the circle1 they are equal1 the triangle is isosceles1 the angles ,b and ,b ,a ,c opposite the equal sides are equal1 and the angle ,b ,a ,c is half of both4 ,then1 because the angle ,b ,a ,c is half of ,b and ,b ,a ,c1 it must be half of their equal ,e ,c ,b4 ,but ,e ,c ,b1 being at the centre1 is measured by ,b ,e2 then half of it1 or ,b ,a ,c1 must be measured by half ,b ,e4 ,in like manner1 it may be proved that the angle ,c ,a ,d is measured by half ,e ,d4 ,then1 because ,b ,a ,c is measured by half ,b ,e1 and ,c ,a ,d by half ,e ,d1 the whole angle ,b ,a ,d must be measured by half the whole arc ,b ,d4 ,,second ,,case4 ,suppose the angle were wholly on one side of the centre1 as ,f ,a ,b4 ,draw the diameter ,a ,e and the radius ,b ,c as before4 ,prove that the angle ,b ,a ,e is measured by half the arc ,b ,e4 ,draw another radius from ,c to ,f1 and prove that ,f ,a ,e is measured by half the arc ,f ,e4 ,then1 because the angle ,f ,a ,e is measured by half the arc ,f ,e1 and the angle ,b ,a ,e is measured by half the arc ,b ,e1 ,the difference of the angles1 or ,f ,a ,b1 must be measured by half the difference of the arcs1 or half of ,f ,b4 ,illustration ,,proposition ,,xxi4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,parallel chords intercept equal arcs of the circumference4 ,let the chords ,a ,b1 ,c ,d1 be parallel2 then will the intercepted arcs ,a ,c and ,b ,d be equal4 ,for draw the straight line ,b ,c4 ,because the lines ,a ,b and ,c ,d are parallel1 the interior alternate angles ,a ,b ,c1 ,b ,c ,d1 are equal4 ,but the angle ,a ,b ,c is measured by half the arc ,a ,c2 ,and the angle ,b ,c ,d is measured by half the arc ,b ,d3 ,then1 because the angles are equal1 the half arcs which measure them must be equal1 and the whole arcs themselves must be equal4 ,illustration ,,proposition ,,xxii4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,the angle formed by a tangent and a chord meeting at the point of contact is measured by half the intercepted arc4 ,let the tangent ,c ,a ,b and the chord ,a ,d meet at the point of contact ,a2 then will the angle ,b ,a ,d be measured by half the intercepted arc ,a ,d4 ,for draw the diameter ,a ,e ,f4 ,because ,a ,b is a tangent1 and ,a ,e a radius at the point of contact1 the angle ,b ,a ,f is a right angle1 and is measured by the semicircle ,a ,d ,f4 ,because the angle ,f ,a ,d is at the circumference1 it is measured by half the arc ,d ,f4 ,then the difference between the angles ,b ,a ,f and ,d ,a ,f1 or ,b ,a ,d1 must be measured by half the difference of the arcs ,a ,d ,f and ,d ,f1 or ,a ,d2 ,that is1 the angle ,b ,a ,d is measured by half the arc ,a ,d4 ,illustration ,,proposition ,,xxiii4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,a tangent and chord parallel to it intercept equal arcs of the circumference4 ,let ,a ,b be tangent to the circumference at the point ,d1 and let ,c ,f be a chord parallel to the tangent2 then will the intercepted arcs ,c ,d and ,d ,f be equal4 ,for from the point of contact ,d1 draw the straight line ,d ,c4 ,because the tangent and chord are parallel1 the interior alternate angles ,a ,d ,c and ,d ,c ,f are equal4 ,but the angle ,a ,d ,c1 being formed by the tangent ,d ,a and the chord ,d ,c1 is measured by half the intercepted arc ,d ,c2 ,and the angle ,d ,c ,f1 being at the circumference1 is measured by half the arc on which it stands1 ,d ,f3 ,then1 because the angles are equal1 the half arcs which measure them are equal1 and the arcs themselves are equal4 ,illustration ,,proposition ,,xxiv4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,the angle formed by the intersection of two chords in a circle is measured by half the sum of the intercepted arcs4 ,let the chords ,a ,b and ,c ,d intersect each other in the point ,e2 then will the angle ,b ,e ,d or ,a ,e ,c be measured by half the sum of the arcs ,a ,c1 ,b ,d4 ,for from the point ,c draw ,c ,f parallel to ,a ,b4 ,because the chords ,a ,b and ,c ,f are parallel1 the arcs ,a ,c1 ,b ,f1 are equal4 ,add each of these equals to ,b ,d1 and we have ,b ,d plus ,a ,c equal to ,b ,d plus ,b ,f2 that is1 the sum of the arcs ,b ,d1 ,a ,c1 is equal to the arc ,f ,d4 ,because the chords ,a ,b1 ,c ,f1 are parallel1 the opposite exterior and interior angles ,d ,e ,b1 ,d ,c ,f1 are equal4 ,but ,d ,c ,f is an angle at the circumference1 and is therefore measured by half the arc ,f ,d4 ,then the equal angle ,d ,e ,b must be measured by half of the arc ,f ,d1 or its equal ,b ,d1 plus ,a ,c4 ,illustration ,,proposition ,,xxv4 ,,theorem4 ,,demonstration4 ,we wish to prove that ,the angle formed by two secants meeting without a circle is measured by half the difference of the intercepted arcs4 ,let the secants ,a ,b1 ,a ,c1 intersect the circumference in the points ,d and ,e2 then will the angle ,b ,a ,c be measured by half the difference between the arcs ,b ,c and ,d ,e4 ,for from the point ,d draw the chord ,d ,f parallel to ,e ,c4 ,because ,a ,c and ,d ,f are parallel1 the opposite exterior and interior angles ,b ,d ,f and ,b ,a ,c are equal4 ,because the chords ,d ,f1 ,e ,c1 are parallel1 the arcs ,d ,e and ,f ,c are equal4 ,if from the arc ,b ,c we take the arc ,d ,e1 or its equal ,f ,c1 we shall have left the arc ,b ,f2 ,but the angle ,b ,d ,f1 being at the circumference1 is measured by half the arc ,b ,f3 ,then the equal of ,b ,d ,f1 or ,b ,a ,c1 must be measured by half the arc ,b ,f1 or half the difference between the intercepted arcs ,b ,c and ,d ,e4 ,,appendix4 ,,note ,a4--,to those teachers who think that the line should be derived from a surface1 and the surface from a solid1 the author would say1 that1 according to his experience1 children apprehend the ideas conveyed by the terms line and surface as readily as they do any ideas whatever2 and that1 therefore1 there seems to be no necessity for extraordinary care in this case to avoid giving wrong impressions4 ,still1 if it be considered desirable in this manner to derive lines and surfaces1 it will be apparent that all that can be done in the matter is to give such instruction only by way of a preliminary lesson4 99 99 99 99 99 ,,note ,b4--,crooked and curved lines are here treated of before straight lines1 because the first two are defined by means of an affirmative property1--they do change direction2 while the last is defined by means of the absence of one1--they do not change direction4 ,it is easier for a child to comprehend what is than what is not4 99 99 99 99 99 ,,note ,c4--,if the pupils are old enough1 they may be shown that vertical lines cannot be parallel1 but only seem so on account of their shortness and nearness to each other4 99 99 99 99 99 ,,note ,d4--,this definition may be considered objectionable because rhomboid means like a rhomb4 ,that the more general figure1 the rhomboid1 has been named from the more restricted one1 the rhomb1 is unfortunate1 because it interferes with the symmetry of the nomenclature4 ,the rhomb possesses all the properties of the rhomboid1 and should1 therefore1 when these are considered1 be called by the same name2 its additional property entitles it to a name which should comprehend the other names4 ,if the rectangle had been called a squaroid1 the difficulty would have been repeated4 99 99 99 99 99 ,,note ,e4--,if teachers consider it desirable1 they may require the class to prove1 by way of corollary1 such propositions as assert the parallelism of the lines when the interior alternate angles are equal1 when the opposite exterior and interior angles are equal1 c41 in continuation of what has already been done4 ---------------------------------------- -------------------------------- ,,townsend',s ,,analysis ,,of ,,the ,,constitution ,,of ,,the ,,united ,,states ,a ,,chart of #eb pages on one roller1 designed to hang in ,class-,rooms4 ,an exposition of the ,constitution2 making it one of the most interesting as it is the most important study for our young men4 ,,townsend',s ,,analysis ,,of ,,civil ,,government4 ,a ,,text-,,book ,,to ,,accompany ,,the ,,chart ,,of ,,the ,,constitution ,,wells' ,,scientific ,,series4 ,,containing ,,the ,,latest ,,researches ,,in ,,physical ,,sciences1 ,,and ,,their ,,practical ,,application ,,to ,,every-,,day ,,life4 ,science of ,common ,things4 ,natural ,philosophy4 ,principles of ,chemistry4 ,first ,principles of ,geology4 ,,gray',s ,,botanical ,,series1 ,by ,,asa ,,gray1 ,fisher ,professor of ,natural ,science in ,harvard ,university4 ,the leading ,botanist in ,america4 ,they have no equal4 8,how ,plants ,grow40 ,lessons in ,botany4 ,manual of ,botany 7,new ,edition74 ,gray's ,lessons and ,manual in #a vol4 ,gray's ,field1 ,forest and ,garden ,botany4 ,gray's ,school and ,field ,book of ,botany4 ,consisting of ,gray's ,lessons1 and ,field1 ,forest and ,garden ,botany1 bound in #a vol4 ,manual1 with ,mosses4 ,structural and ,systematic ,botany4 ,,bryant ,,stratton',s ,,book-,,keeping ,,series4 ,the best ,,text-,,books ever prepared on this subject4 ,common ,school ,edition4 ,high ,school ,edition4 ,counting-,house ,edition4 ,,wilson',s ,,school ,,histories4 ,primary ,american ,history4 ,history of the ,united ,states4 ,american ,history1 ,school ,edition4 ,outlines of ,general ,history4 ,parley's ,universal ,history4 ---------------------------------------- -------------------------------- ,,transcriber',s ,,notes #a4 ,moved advertisements from before title page to the end4 #b4 ,silently corrected typographical errors4 #c4 ,retained anachronistic and non-standard spellings as printed4 #d4 ,enclosed italics font in underscores4