SYLLABUS OF PROPOSITIONS in GEOMETRY SYLLABUS OF PROPOSITIONS IN GEOMETRY INTENDED FOR USE IN PREPARING STUDENTS FOR HARVARD COLLEGE AND THE LAWRENCE SCIENTIFIC SCHOOL CAMBRIDGE IPublisbeb b- 1barvartb ntipersitp 1899 TO THE TEACHER T HIS Syllabus is a collection of propositions in Elementary Geometry which may properly be regarded as forming the pupil's working outfit in this subject. They constitute the ' book work" of the requirement in Geometry for admission to the Freshman Class of Harvard College and of the Lawrence Scientific School.* Owing to the variety of ways in which the axioms and postulates may be assumed, it has seemed best not to include these in the Syllabus, but to leave this matter in the hands of the teacher. Attention should however be called to the desirability of introducing the Method of Limits before the propositions regarding the ratio and measure of angles and arcs (Book II) are taken up, and of using the method for the proof of these propositions. So large a proportion of the present text-books agree in their division of the subject matter of Plane and Solid Geometry into books that their division has been here adopted. The order of propositions belonging to one and the same Book is not prescribed; but it is not expected that a proposition of a given Book shall be proved by the aid of propositions appearing in later Books. Should the candidate however have used a textbook in which the division into Books is inconsistent with the division of this Syllabus, and should he prefer to follow the order of propositions with which he is familiar, he will be allowed to do so on stating in his examination book the name of the text-book he has used. The propositions of each Book have been arranged in one of the many possible logical orders. But no attempt has been * A complete statement of the requirement, together with some suggestions to the teacher, will be found in the Harvard University Catalogue. ii * * made to obtain a desirable teaching order. In fact, teachers may find even the separation of the Solid from the Plane Geometry an unfortunate one for purposes of instruction, when the excellent recommendations of the Committee of Ten regarding the early study of Concrete Geometry shall have been more generally adopted in the schools. These recommendations contemplate early training in the formation of space conceptions,* so that the pupil will be in possession of a knowledge of the more important facts both of Plane and of Solid Geometry, which will have been studied together, before he begins the formal study of these subjects. It may then be desirable to use many of the simpler propositions of Solid Geometry as exercises in Plane Geometry, and to complete the earlier topics in Solid Geometry before the later ones in Plane Geometry have been taken up. For the sake of conciseness, such expressions as: "the product of two lines," " the product of a line and a surface," etc., have been used in the sense of: "the product of the lengths of two lines," " the product of the length of a line and the area of a surface," etc. Candidates will be provided with a copy of the Syllabus for use at the admission examination in Geometry. They may refer to the propositions of the Syllabus by Book and number, instead of writing them out at length. * Aids to such training are (1) the accurate drawing, by means of ruler and compass, of plane figures (e. g. the medial lines of a triangle, or a regular hexagon) and (2) the use of models and carefully drawn diagrams of solid figures. The pupil should provide himself with a sphere on which he can draw, and a hemispherical cup to fit the sphere, by means of which he can draw great circles. Models of the regular bodies are readily constructed out of card-board. PLANE GEOMETRY BOOKS I TO V. BOOK I. ANGLES, TRIANGLES, AND PERPENDICULARS. THEOREM I. If two triangles have two sides and the included angle of one respectively equal to two sides and the included angle of the other, the triangles are equal. THEOREM II. If two triangles have a side and the two adjacent angles of one respectively equal to a side and the two adjacent angles of the other, the triangles are equal. THEOREM III. In an isosceles triangle the angles opposite the equal sides are equalo Conversely, if two angles of a triangle are equal, the triangle is isosceles. THEOREM IV. If two angles of a triangle are unequal, the side opposite the greater angle is greater than the side opposite the less angle. If two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side. 2 THEOREM V. If two triangles have two sides of one respectively equal to two sides of the other, and the included angles unequal, the triangle which has the greater included angle has the greater third side. If two triangles have two sides of one respectively equal to two sides of the other, and the third sides unequal, the triangle which has the greater third side has the greater included angle. THEOREM VI. If two triangles have the three sides of one respectively equal to the three sides of the other, the triangles are equal. THEOREM VII. If straight lines are drawn from a point within a triangle to the extremities of a side, their sum is less than the sum of the other two sides of the triangle. THEOREM VIII. At a given point in a straight line but one perpendicular to the line can be drawn. THEOREM IX. The two adjacent angles which one straight line makes with another are together equal to two right angles. Conversely, if the sum of two adjacent angles is two right angles, their exterior sides are in the same straight line THEOREM X. If two straight lines intersect each other, the opposite (or vertical) angles are equal. 3 THEOREM XI. From a given point without a straight line one perpendicular can be drawn to the line, and but one. The perpendicular is the shortest line that can be drawn from a point to a straight line. THEOREM XII. If two oblique straight lines drawn from a point to a straight line meet the line at equal distances from the foot of the perpendicular drawn from the point to the line, they are equal. If they meet the line at unequal distances from the foot of the perpendicular, the more remote is the greater. THEOREM XIII. If two right triangles have the hypotenuse and a side of one respectively equal to the hypotenuse and a side of the other, the triangles are equal. THEOREM XIV. If a perpendicular is erected at the middle of a straight line, then every point in the perpendicular is equally distant from the extremities of the line, and every point not in the perpendicular is unequally distant from the extremities of the line; that is, the locus of points equidistant from the extremities of a line is a line bisecting that line at right angles. THEOREM XV. Every point in the bisector of an angle is equally distant from the sides of the angle; and every point not in the bisector is unequally distant from the sides of the angle; that is, the bisector of an angle is the locus of the points within the angle and equally distant from its sides. 4 PARALLELS AND PARALLELOGRAMS. THEOREM XVI. Two straight lines perpendicular to the same straight line are parallel. THEOREM XVII. When two straight lines are cut by a third, if the alternate interior angles are equal, the two straight lines are parallel. Corollary I. When two straight lines are cut by a third, if a pair of corresponding angles are equal, the lines are parallel. Corollary II. When two straight lines are cut by a third, if the sum of two interior angles on the same side of the secant line is equal to two right angles, the two lines are parallel. Corollary III. Two straight lines parallel to the same straight line are parallel to each other. THEOREM XVIII. If two parallel lines are cut by a third straight line, the alternate interior angles are equal. Corollary I. If two parallel lines are cut by a third straight line, any two corresponding angles are equal. Corollary II. If two parallel lines are cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles. Corollary III. If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other. THEOREM XIX. The sum of the three angles of any triangle is equal to two right angles. Corollary. If one side of a triangle is extended, the exterior angle is equal to the sum of the two interior opposite angles. 5 THEOREM XX. The sum of the angles of a polygon of n sides is 2n - 4 right angles. THEOREM XXI. If the sides of one angle are perpendicular respectively to the sides of another, the angles are either equal or supplementary. THEOREM XXII. The opposite sides of a parallelogram are equal, and the opposite angles are equal. THEOREM XXIII. The diagonals of a parallelogram bisect each other. THEOREM XXIV. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. THEOREM XXV. If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram. BOOK II. THE CIRCLE AND THE MEASURE OF ANGLES. THEOREM I. The diameter of a circle is greater than any other chord; and it bisects the circle and its circumference. THEOREM II. A straight line can intersect a circumference in only two points. THEOREM III. Through three points not lying in a straight line one circumference, and only one, can be drawn. THEOREM IV. Two circumferences can intersect each other in only two points. THEOREM V. In the same circle or in equal circles, equal angles at the centre intercept equal arcs on the circumference. Conversely, in the same circle or in equal circles, equal arcs subtend equal angles at the centre. THEOREM VI. If, in the same circle or in equal circles, two arcs are equal, the chords subtending them are equal. Conversely, if in the same circle or in equal circles, two chords are equal, the arcs subtended by them are equal. 7 THEOREM VII. In the same circle or in equal circles, the greater of two unequal arcs, neither of which exceeds a semi-circumference, is subtended by the greater chord. Conversely, in the same circle or in equal circles, the greater of two unequal chords subtends the greater arc, if neither arc exceeds a semi-circumference. THEOREM VIII. The diameter perpendicular to a chord bisects the chord and the arcs which the chord subtends. Corollary I. A line bisecting a chord at right angles passes through the centre of the circle. Corollary II. When two circumferences intersect each other, the straight line joining their centres bisects at right angles their common chord. THEOREM IX. In the same circle or in equal circles, equal chords are equally distant from the centre; and of two unequal chords the less is at the greater distance from the centre. Conversely, in the same circle or in equal circles, chords equally distant from the centre are equal; and of two chords unequally distant from the centre, that is the greater whose distance from the centre is the less. THEOREM X. A straight line tangent to a circle is perpendicular to the radius drawn to the point of contact. Corollary. A perpendicular to a tangent at the point of contact passes through the centre of the circle. THEOREM XI. When two tangents to the same circle intersect each other, the distances from their point of intersection to their points of contact are equal. 8 THEOREM XII. Two parallel straight lines intercept equal arcs on a circumference. THEOREM XIII. In the same circle or in equal circles, two angles at the centre are in the same ratio as their intercepted arcs. THEOREM XIV. An angle inscribed in a circumference is measured by one half its intercepted arc. Corollary. An angle inscribed in a semi-circumference is a right angle. THEOREM XV. If a quadrilateral is inscribed in a circle, the sum of two opposite angles is two right angles; and conversely, if the sum of two opposite angles of a quadrilateral is two right angles, the quadrilateral can be inscribed in a circle. THEOREM XVI. An angle formed by a tangent and a chord is measured by one half the intercepted arc. THEOREM XVII. An angle formed by two chords intersecting each other within a circumference is measured by one half the sum of the arcs intercepted between its sides and between the sides of its vertical angle. THEOREM XVIII. An angle formed by two secants intersecting each other without a circumference, by two tangents, or by a tangent and a secant, is measured by one half the difference of the intercepted arcs. BOOK III. SIMILAR POL YG ONS. THEOREM I. A straight line parallel to the base of a triangle divides the other two sides proportionally. Conversely, if a straight line divides two sides of a triangle proportionally, it is parallel to the third side. THEOREM II. If two triangles have their angles respectively equal, the triangles are similar. Corollary. If two triangles have two angles of one respectively equal to two angles of the other, the triangles are similar. THEOREM III. If two triangles have an angle of one equal to an angle of the other and the sides including these angles proportional, the triangles are similar. THEOREM IV. If two triangles have their sides respectively proportional, the triangles are similar. THEOREM V. The bisector of an angle of a triangle divides the opposite side into segments proportional to the sides of the angle. THEOREM VI. If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. 10 Conversely, if two polygons are similar, they can be decomposed into the same number of triangles, similar each to each and similarly placed. THEOREM VII. The perimeters of two similar polygons are in the same ratio as any two corresponding sides. THEOREM VIII. If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse: I. The two triangles thus formed are similar to each other and to the whole triangle. II. The perpendicular is a mean proportional between the segments of the hypotenuse. II. Each leg of the right triangle is a mean proportional between the hypotenuse and the segment adjacent to that leg. THEOREM IX. The product of the segments of a chord that passes through a fixed point within a circle is the same for all directions of the chord. THEOREM X. If from a fixed point without a circle a secant is drawn, the product of the whole secant and its external segment is the same for all directions of the secant. Corollary. If a tangent and a secant intersect, the tangent is a mean proportional between the whole secant and its external segment. BOOK IV. AREA S OF POL YG ONS. THEOREM I. Parallelograms having equal bases and equal altitudes are equivalent. THEOREM II. The areas of two rectangles having equal altitudes are to each other as their bases. THEOREM 111. The areas of two rectangles are to each other as the products of their bases and their altitudes. Corollary. The area of a rectangle is equal to the product of its base and its altitude. THEOREM IV. The area of a parallelogram is equal to the product of its base and its altitude. THEOREM V. The area of a triangle is equal to half the product of its base and its altitude. Corollary. Two triangles having equal bases and equal altitudes are equivalent. THEOREM VI. The area of a trapezoid is equal to the product of its altitude and half the sum of its parallel sides. 12 THEOREM VII. The areas of two similar triangles are to each other as the squares of any two corresponding sides. Corollary. The areas of two similar polygons are to each other as the squares of any two corresponding sides; and also as the squares of their perimeters. THEOREM VIII. The square described on the hypotenuse of a right triangle is equivalent to the sum of the squares described on the other two sides. BOOK V. REGULAR POLYGONS AND THE MEASURE OF THE CIRCLE. THEOREM I. An equilateral polygon inscribed in a circle is a regular polygon. THEOREM II. A circle may be circumscribed about any regular polygon, and a circle may also be inscribed in it. THEOREM III. If the circumference of a circle be divided into any number of equal parts, the chords joining the successive points of division form a regular polygon inscribed in the circle; and the tangents drawn at the points of division form a regular polygon circumscribed about the circle. Corollary I. If the vertices of a regular inscribed polygon are joined with the middle points of the arcs subtended by the sides of the polygon, the joining lines will form a regular inscribed polygon of double the number of sides. Corollary II. If at the middle points of the arcs joining adjacent points of contact of the sides of a regular circumscribed polygon tangents are drawn, a regular circumscribed polygon of double the number of sides will be formed. THEOREM IV. Regular polygons of the same number of sides are similar. THEOREM V. The perimeters of regular polygons of the same number of sides are to each other as the radii of the circumscribed circles, or as the radii of the inscribed circles; and their areas are to each other as the squares of these radii. 14 THEOREM VI. The area of a regular polygon is equal to half the product of its perimeter and its apothem. THEOREM VII. The circumference of a circle is less than the perimeter of any polygon circumscribed about it; it being assumed that the length of an arc of a circle is less than the length of any broken line that envelopes it and has the same extremities. THEOREM VIII. If the number of sides of a regular polygon inscribed in a circle be increased indefinitely, the apothem of the polygon will approach the radius of the circle as its limit. THEOREM IX. The circumference of a circle is the limit which the perimeters of regular inscribed and circumscribed polygons approach when the number of their sides is increased indefinitely; and the area of the circle is the limit of the areas of these polygons. THEOREM X. The lengths of the circumferences of two circles are to each other as the radii, and the areas as the squares of the radii. Corollary. The length of a circumference of a circle is equal to the product of the radius and twice the constant number,r. C 27r. THEOREM XI. The area of a circle is equal to half the product of its circumference and its radius. Corollary. The area of a circle is equal to the square of its radius multiplied by the constant number 7r. S - 7r2. SOLID GEOMETRY. BOOKS VI TO IX. BOOK YI. PLANES AND LINES IN SPACE. THEOREM I. If a straight line is perpendicular at its point of intersection with a plane to each of two lines lying in the plane and passing through this point, it is perpendicular to every line of the plane that passes through the point. Corollary I. At a given point of a straight line one plane can be drawn perpendicular to the line, and but one. Corollary II. Through a given point without a straight line one plane can be drawn perpendicular to the line, and but one. Corollary III. At a given point of a plane one perpendicular to the plane can be drawn, and but one. Corollary IV. All the perpendiculars that can be drawn to a straight line at a given point lie in a plane perpendicular to the line at the point. THEOREM II. If two straight lines are parallel, every plane passed through one of them and not coincident with the plane of the parallels is parallel to the other. Corollary I. Through a given straight line a plane can be passed parallel to any other given straight line; and if the lines are not parallel, only one such plane can be drawn. Corollary II. Through a given point a plane can be passed parallel to any two given straight lines in space; and if the lines are not parallel, only one such plane can be drawn. Corollary III. Through a given point one plane, and but one, can be passed parallel to a given plane. 16 THEOREM III. Planes perpendicular to the same straight line are parallel to each other. Conversely, if two planes are parallel, a straight line perpendicular to one of the planes is perpendicular to the other. Corollary. From a given point without a plane, one perpendicular to the plane can be drawn, and but one. THEOREM IV. The perpendicular is the shortest line that can be drawn from a point to a plane. THEOREM V. Two oblique straight lines drawn from a point to a plane are equal when, and only when, they meet the plane at equal distances from the foot of the perpendicular drawn from the point to the plane; and of two oblique lines drawn from a point to a plane, and meeting the plane at unequal distances from the foot of the perpendicular, the more remote is the greater. THEOREM VI. The intersections of two parallel planes with any third plane are parallel. THEOREM VII. If two angles, not in the same plane, have their sides respectively parallel and lying in the same direction, they are equal and their planes are parallel. THEOREM VIII. If one of two parallel lines is perpendicular to a plane, the other is also perpendicular to that plane. Corollary. Two straight lines perpendicular to the same plane are parallel to each other. 17 THEOREM IX. All plane angles of the same diedral angle are equal. THEOREM X. Two diedral angles are equal if their plane angles are equal. THEOREM XI. Two diedral angles are in the same ratio as their plane angles. THEOREM XII. If a straight line is perpendicular to a plane, every plane passed through the line is perpendicular to the plane. THEOREM XIII. If two planes are perpendicular to each other, a straight line drawn in one of them, perpendicular to their intersection, is perpendicular to the other. Corollary I. If two planes are perpendicular to each other, a straight line drawn through any point of their intersection perpendicular to one of the planes will lie in the other. Corollary II. If two planes are perpendicular, a straight line let fall from any point of one plane perpendicular to the other will lie in the first plane. THEOREM XIV. If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane. THEOREM XV. Through any given straight line a plane can be passed perpendicular to any given plane. 18 THEOREM XVI. Two straight lines not lying in a plane have one and only one common perpendicular; and this line is the shortest line that can be drawn between the two lines. THEOREM XVII. The locus of points equally distant from the extremities of a straight line is a plane perpendicular to that line at its middle point. THEOREM XVIII. The locus of points equally distant from the sides of a diedral angle is the plane bisecting that angle. THEOREM XIX. The projection of a straight line upon a plane is a straight line. THEOREM XX. The acute angle which a straight line makes with its own projection upon a plane is the least angle it makes with any line of that plane. THEOREM XXI. The sum of any two face angles of a triedral angle is greater than the third face angle. THEOREM XXII. The sum of the face angles of any convex polyedral angle is less than four right angles. THEOREM XXIII. If two triedral angles have the three face angles of one respectively equal to the three face angles of the other, the corresponding diedral angles are equal. BOOK VIIL PRISMS AND PYRAMIDS. THEOREM I. The lateral area of a prism is equal to the product of the perimeter of a right section of the prism and of a lateral edge. Corollary. The lateral area of a right prism is equal to the product of the perimeter of its base and of its altitude. THEOREM II. Two prisms are equal, if three faces including a triedral angle of one are respectively equal to three faces similarly placed including a triedral angle of the other. Corollary. Two right prisms are equal if they have equal bases and equal altitudes. THEOREM III. An oblique prism is equivalent to a right prism whose base is a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism. THEOREM IV. An oblique parallelopiped is equivalent to a rectangular parallelopiped of the same altitude and an equivalent base. THEOREM V. The plane passed through two diagonally opposite edges of a parallelopiped divides it into two equivalent triangular prisms. THEOREM VI. The volumes of two rectangular parallelopipeds having equal bases are to each other as their altitudes. 20 THEOREM VII. The volumes of two rectangular parallelopipeds are to each other as the products of their three dimensions. THEOREM VIII. The volume of a rectangular parallelopiped is equal to the product of its three dimensions. THEOREM IX. The volume of any parallelopiped is equal to the product of the area of its base and its altitude. THEOREM X. The volume of a triangular prism is equal to the product of its base and its altitude. Corollary. The volume of any prism is equal to the product of its base and its altitude. THEOREM XI. If a pyramid is cut by a plane parallel to its base, 1st, the edges and the altitude are divided proportionally; 2d, the section is a polygon similar to the base; 3d, the area of the section is to the area of the base as the square of its distance from the vertex is to the square of the altitude of the pyramid. Corollary. If two pyramids have equal altitudes and equivalent bases, sections made by planes parallel to their bases and at equal distances from their vertices are equivalent. THEOREM XII. The lateral area of a regular pyramid is equal to the product of the perimeter of its base and half its slant height. THEOREM XIII. If the altitude of any given triangular pyramid is divided into equal parts, and through the points of division planes are 21 passed parallel to the base of the pyramid, and on the sections made by these planes as upper bases prisms are described having their edges parallel to an edge of the pyramid and their altitudes equal to one of the equal parts into which the altitude of the pyramid is divided, the total volume of these prisms will approach the volume of the pyramid as its limit, as the number of parts into which the altitude of the pyramid is divided is indefinitely increased. THEOREM XIV. Two triangular pyramids having equivalent bases and equal altitudes are equivalent. THEOREM XV. The volume of a triangular pyramid is one third the volume of a triangular prism of the same base and altitude. Corollary. The volume of a triangular pyramid is equal to one third of the product of its base and its altitude. THEOREM XVI. The volume of any pyramid is equal to one third of the product of its base and its altitude. BOOK VIII. THE CYLINDER. THEOREM I. Every section of a cylinder made by a plane passing through an element is a parallelogram. Corollary. Every section of a right cylinder made by a plane perpendicular to its base is a rectangle. THEOREM II. The bases of a cylinder are equal. Corollary. All the sections of a circular cylinder parallel to its bases are equal circles, and the straight line joining the centres of the bases passes through the centres of all the parallel sections. THEOREM III. A right circular cylinder may be generated by the revolution of a rectangle about one of its sides as an axis. THEOREM IV. A plane passing through a tangent to the base of a circular cylinder and the element drawn through the point of contact is tangent to the cylinder. Corollary. If a plane is tangent to a circular cylinder, its intersection with the plane of the base is tangent to the base. THEOREM V. If a prism, the base of which is a regular polygon be inscribed in or circumscribed about a given circular cylinder, the volume of the prism will approach the volume of the cylinder as its limit, and the area of its lateral surface will approach the area of the lateral surface of the cylinder as its limit, as the number of sides of the base is indefinitely increased. 23 THEOREM VI. The lateral area of a cylinder is equal to the product of the perimeter of a right section of the cylinder by the length of an element of the surface. Corollary I. The lateral area of a cylinder of revolution is equal to the product of the circumference of the base and the altitude. S- 27rrh. Corollary II. The lateral areas of similar cylinders of revolution are to each other as the squares of the altitudes, or as the squares of the radii of the bases. THEOREM VII. The volume of a circular cylinder is equal to the product of its base and its altitude. V =- 7r"2h. Corollary. The volumes of similar cylinders of revolution are to each other as the cubes of the altitudes, or as the cubes of the radii of the bases. THE CONE. THEOREM VIII. Every section of a cone made by a plane passing through the vertex is a triangle. THEOREM IX. Every section of a circular cone made by a plane parallel to its base is a circle, the centre of which is the intersection of the plane with the axis. THEOREM X. A right circular cone may be generated by the revolution of a right triangle about one of its sides as an axis. 24 THEOREM XI. A plane passing through a tangent to the base of a circular cone and the element drawn through the point of contact is tangent to the cone. Corollary. If a plane is tangent to a circular cone, its intersection with the plane of the base is tangent to the base. THEOREM XII. If a pyramid, the base of which is a regular polygon, be inscribed in or circumscribed about a given circular cone, the volume of the pyramid will approach the volume of the cone as its limit, and the lateral area of the pyramiid will approach the area of the convex surface of the cone as its limit, as the number of sides of the base is indefinitely increased. THEOREMA XIII. The lateral area of a cone of revolution is equal to the product of the circumference of the base and half the slant height. S - rrl. Corollary. The lateral areas of similar cones of revolution are to each other as the squares of the slant heights, or as the squares of the altitudes, or as the squares of the radii of the bases. THEOREM XIV. The volume of a circular cone is equal to one third the product of its base and its altitude. V-3- 7 'r, h. Corollary. Similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. BOOK IX. THE SPHERE. THEOREM I. Every section of a sphere made by a plane is a circle. Corollary I. All great circles of the same sphere are equal. Corollary II. Every great circle divides the sphere into two equal parts. Corollary III. Any two great circles on the same sphere bisect each other. Corollary IV. An are of a great circle may be drawn through any two given points on the surface of a sphere, and, unless the points are the opposite extremities of a diameter, only one such arc can be drawn. Corollary V. Through any three points on the surface of a sphere one and only one circle can be drawn. THEOREM II. All the points in the circumference of a circle on a sphere are equally distant from either of its poles. Corollary I. All the arcs of great circles drawn from a pole of a circle to points in its circumference are equal. Corollary II. The polar distance of a great circle is a quadrant. Corollary III. If a point on the surface of a sphere is at a quadrant's distance from each of two given points of the surface which are not opposite extremities of a diameter, it is the pole of the great circle passing through them. THEOREM III. A sphere may be generated by the revolution of a semicircle about its diameter. 26 THEOREM IV. Through four points not lying in a plane one sphere, and only one, can be drawn. THEOREM V. A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact. Conversely, a plane perpendicular to a radius of a sphere at its extremity is tangent to the sphere. THEOREM VI. The intersection of two spheres is a circle the plane of which is perpendicular to the straight line joining the centres of the spheres, and the centre of which is in that line. THEOREM VII. The angle formed by two arcs of great circles is equal to the angle between the planes of the circles, and is measured by the arc of a great circle described from its vertex as a pole and included between its sides (produced if necessary). Corollary. All arcs of great circles drawn through the pole of a given great circle are perpendicular to its circumference. SPHERICAL TRIANGLES AND POLYGONS. THEOREM VIII. If the first of two spherical triangles is the polar triangle of the second, then, reciprocally, the second is the polar triangle of the first. THEOREM IX. In two polar triangles, each angle of one is measured by the supplement of the side lying opposite to it in the other. 27 THEOREM X. Two triangles on the same sphere are either equal or symmetrical when two sides and the included angle of one are respectively equal to two sides and the included angle of the other. THEOREM XI. Two triangles on the same sphere are either equal or symmetrical when a side and the two adjacent angles of one are respectively equal to a side and the two adjacent angles of the other. THEOREM XII. Two triangles on the same sphere are either equal or symmetrical when the three sides of one are respectively equal to the three sides of the other. THEOREM XIII. If two triangles on the same sphere are mutually equiangular, they are mutually equilateral, and are either equal or symmetrical. THEOREM XIV. Any side of a spherical triangle is less than the sum of the other two. THEOREM XV. The sum of the sides of a convex spherical polygon is less than the circumference of a great circle. THEOREM XVI. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles. THEOREM XVII. Two symmetrical spherical triangles are equivalent. 28 THEOREM XVIII. If two arcs of great circles intersect on the surface of a hemisphere, the sum of the opposite spherical triangles which they form is equivalent to a lune whose angle is the angle between the arcs in question. THEoREM XIX. A lune is to the surface of the sphere as the angle of the lune is to four right angles. THEOREM XX. The area of a spherical triangle is equal to its spherical excess. THEOREM XXI. The shortest line that can be drawn on the surface of a sphere between two points is the arc of a great circle, not greater than a semi-circumference, joining the two points. THE MEASUREMENT OF THE SPHERE. THEOREM XXII. The area of the surface generated by a straight line revolving about an axis in its plane is equal to the length of the projection of the line on the axis multiplied by the circumference of the circle the radius of which is the perpendicular erected at the middle of the line and terminated by the axis. THEOREM XXIII. The area of a zone is equal to the product of its altitude and the circumference of a great circle. S - 2rrh. 29 THEOREM XXIV. The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle. S - 27rr X 2r- 47rr2. Hence the surface of a sphere is equivalent to four great circles. Corollary. The surfaces of two spheres are to each other as the squares of the diameters, or as the squares of the radii. THEOREMI XXV. The volume of a sphere is equal to the area of the surface multiplied by one third of the radius. V - r1.3. Corollary. The volumes of two spheres are to each other as the cubes of the diameters, or as the cubes of the radii. THEOREM XXVI. The volume of a spherical sector is equal to the area of the zone which forms the base multiplied by one third the radius of the sphere. PROBLEMS IN CONSTRUCTION. The pupil is expected to be able to make the following constructions by the aid of straight lines and circles (ruler and compasses) and to prove their correctness:1. To bisect a given straight line. 2. To bisect a given angle. 3. At a given point in a straight line to erect a perpendicular to that line. 4. From a given point to let fall a perpendicular upon a given straight line. 5. At a given point in a straight line to construct an angle equal to a given angle. 6. Through a given point to draw a parallel to a given straight line. 7. To find the centre of a given circular arc. 8. At a given point in a given circumference to draw a tangent. 9. Through a given point without a given circumference to draw a tangent to the circumference. 10. To inscribe a circle in a given triangle. 11. To draw a circumference through three given points not lying in the same straight line. 12. On a given straight line as a base to construct a circular segment in which a given angle can be inscribed. 13. To divide a given straight line into any given number of equal parts. 14. To divide a given straight line into parts proportional to two given straight lines. 15. To find a fourth proportional to three given straight lines. - ~ ~ — -- - - - -- -I 1-`-r`- ~-'^-~- -- -- -`-d'' ~^ ""' n 31 16. To find a mean proportional between two given straight lines. 17. To divide a given straight line in extreme and mean ratio. 18. To construct a triangle equivalent to a given polygon. 19. To construct a square equivalent to a given triangle. 20. To inscribe a square in a given circle. 21. To inscribe a regular hexagon in a given circle. 22. To inscribe a regular decagon in a given circle. RATIO AND PROPORTION. The pupil is assumed to be familiar with the following propositions: 1. If a, b, c, d are numbers and a: b = c: d, then (1) ad = be, (2) a: c b: d, (3) ka: b = 7c: d, (4) a=tc: b= d - a: b. (5) a2: b2 = 2: d2. 2. If the numerical measures of four quantities A, B, C, D form a proportion, and if A and B are of the same kind and C and D of the same kind, then A: B - C: D.