11 a bus of a course In ^ne An'O.ytlc Geometry . by -erly UNIVERSITY OF CALIFORNIA AT LOS ANGELES SYLLABUS OF A COURSE IN PLANE ANALYTIC GEOMETRY. INTRODUCTORY. BOSTON: PUBLISHED BY GINN, HEATH, & CO. 1884. COPYRIGHTED BY GINN, HEATH, & Co., 1888. MATHEMATICS. THE following volumes in Wentworth's Mathematical ^Series are now ready for delivery : INTHOD. PRICE. Elements of Algebra $1.12 Complete Algebra 1.40 Plane Geometry .75 Plane and Solid Geometry Plane and Solid Geometry, and Plane Trigonometry. . 1.40 Plane Trigonometry. Paper . . . . . .30 Plane Trigonometry and Tables. Paper Plane and Spherical Trigonometry Plane and Spherical Trigonometry, Surveying, and Navigation 1.12 Plane and Spherical Trigonometry, and Surveying. With Tables . . ' 1.25 Surveying. Paper ....... Trigonometric Formulas (Two Charts., each 30 X 40 indies) 1.00 "Wentworth & Hill's Five-Place Logarithmic and Trigo- nometric Tables. (Seven Tables) . . . . .50 Wentworth & Hill's Five-Place Logarithmic and Trigo- nometric Tables. Complete Edition . . . 1.00 Wentworth & Hill's Practical Arithmetic . . . 1.00 Wentworth & Hill's Examination Manual. I. Arithmetic .35 Wentworth & Hill's Examination Manual. II. Algebra . Wentworth & Hill's Exercise Manual. II. Algebra . .35 (The last two may be had in one volume.) Sample copies sent to teachers on receipt of Introduction Price.* ANNOUNCEMENTS. EXERCISE MANUAL OF ARITHMETIC. In Press. EXERCISE MANUAL OF GEOMETRY. In Press. WENTWORTH & MCLELLAX'S UNIVERSITY ALGEBRA. Ready in June. n'* GRAMMAR SCHOOL Ai'ITIOiTTI' . O PRliVrATn SCHOOL ARITHMETIC , , , fteady in December. HEATH, &> CO., Publishers. I>QhTO>', yE,%;YVlRK, ^ND CHICAGO. QA 55-*. STLLABUS INTEODUCTOEY COUESE IN PLANE ANALYTIC GEOIETEY, 1. Describe the object of Analytic Geometry. Show that, on account of the inherent difference between Algebra and Geometry, two important difficulties present themselves at the outset in our attempt to apply Algebra to Geometiy, the geo- metrical interpretation of the distinction between positive and negative quantities, and the algebraic representation of position. 2. Show that the first difficulty has been already met and overcome in Trigonometry. The rule that opposite signs are interpreted geometrically by opposite directions is adopted in Analytic Geometry. Illustrations. [1] AB = -BA. 3. Show that the position of a point is known when we know its distances from two given intersecting lines. Explain rectangular coordinates. Define axes, origin, abscissa, ordinate, coordinates, axis of abscissas, axis of ordinates. Give the rule concerning the signs of the coordinates of a point. Explain the different ways of writing the coordinates of a point. 462617 4. It is necessary to devise means of performing algebraically the various operations required in Geometry. We begin with the length of a given line. Obtain a formula for the distance between two given points / and ajj, 2 . [2] D = Show, by considering different positions of the points, that the formula [2] is entirely general. Examples. Geometrical problems. 5. Find the coordinates of a point bisecting a given line. 1 J 2 Find the coordinates of a point dividing a given line in a given ratio m x : m 2 . _ m 2 Xi -f mi x 2 _ m 2 y i + m l ?/ 2 m 2 + OTI ' m 2 + m i Examples. Geometrical problems. 6. Define what is meant in Geometry by a locus. Examples and problems in loci. If the given conditions can be expressed by an equation be- tween the coordinates of a point, the curve generated by a point so moving that its coordinates always satisfy the equation is called the locus of the equation. Examples. Numerous prob- lems in forming equations of simple loci. Give a second definition of the locus of an equation. Show that every equation between x and y has a locus. Explain the method of constructing the locus of an equation by finding points of the required curve. Examples. Define the intercepts of a curve on the axes, and show how they may be found from the equation. Explain the method of finding the points of intersection of two curves whose equations are given. Examples. THE STRAIGHT LINE. 7. Find the equation of a line when the intercepts on the axes are given. Show that [5] is correct wherever the point (#,?/) may be taken on the line, and whatever the signs of a and b. Find the equation of a line when two points of the line are given. [6] x ~ x i _. yy* When the intercept b on the axis of ordinates and the angle y made with the axis of abscissas are given, [7] y = lx-\-b, where 1 = tan y. When the coordinates of a point through which the line passes and the angle made with the axis of abscissas are given, [8] y 7/1 = l(x XT) . 8. Show that every equation of the first degree is the equation of a straight line, and prove that TO * *=-, =_ 9. Show how to obtain the equation of a line through two given points by determining the coefficients of Ax+By-\-C = ? so that the equation shall be satisfied by the given coordinates. 10. Find from a geometrical figure the condition that two lines whose equations are given shall be parallel, [10] Z 1 = Z, or, |j = |; that they shall be perpendicular, [U] ^-1, or, | ; = -f Problems. 11. Find the angle between two straight lines whose equa- tions are given. + a - Obtain formulas [10] and [11] from [12]. 12. Explain the method of finding the equation of a line passing through a given point and parallel to a given line ; perpendicular to a given line. Use undetermined coefficients. Geometrical problems. 13. Devise a method for finding the distance from a given point to a given line. Obtain by it the formula [13] D = 14. Find the area of a triangle when the coordinates of its vertices are given. [14] M= 15. Describe the general method of finding the equation of any given geometrical locus. Numerous problems. TRANSFORMATION OF COORDINATES. 16. Obtain formulas for transforming from one system of coordinates to another when the new axes are respectively parallel to the old. [15] x = X -\-x'', y = y + y'. 17. Obtain formulas for transforming from one system of coordinates to another having the same origin. [16] a; = ;c'cos0 y' siu6 ; y = x'sinO-\- y' cosO. 18. Explain polar coordinates. Obtain formulas for trans- forming from rectangular to polar coordinates. [17] x = Problems in polar coordinates. THE CIRCLE. 19. Find the equation of the circle in the form [18] (x-a)* + (y-b)* = i*. When the centre is at the origin, this becomes [19] 20. Show, by expanding [18], that the equation of any circle may be written in the form [20] x 2 + y 2 +Dx+Ey+F=0. Explain the method of finding the centre and radius of a circle whose equation is given in expanded form. 21. Give two methods of finding the equation of a circle passing through three given points. 22. Find the equation of a tangent to the circle [19] at a given point on the circumference. [21] 23. Define a normal to a curve. Find the equation of the normal at a given point of the circle. [19]. f~OO~l f\ L J 2/1 *^ ~"~ *^i 2/ ""~" * 24. Find the locus of the middle points of a system of parallel chords. [23] a + tan0.?/= 0. Such a line is called a diameter. Prove that every chord through the centre is a diameter. Problems on the circle. THE COKIC SECTIONS. 25. Define the ellipse, the hyperbola, the parabola, and find their equations in the forms [24] - + ^ = i _tf = \ y 2 =2mx. 1 J a?^W ' a? W Prove that in the ellipse and the hyperbola every chord through the centre is bisected by the centre. 26. Find the equations of the tangent and normal to each of these curves. Prove that the tangents at the opposite extremi- ties of a chord through the centre in the ellipse and the hyper- bola are parallel. [25] Tangents. ? + M = i, ^-M = i, a- b 2 a- V a 2 b 2 a 2 b 2 [26] Normals, x -- y = a 2 b 2 , x-\ y = a 2 -f-6 2 , x i y\ x i y\ y\ 27. Prove that the tangent and normal at any point of the ellipse or the hyperbola bisect the angle between the focal radii drawn to the point. Show how this property can be used in drawing a tangent to the curve. Prove that con-focal conies intersect at right angles. 28. Prove that if on the major axis of an ellipse as a diam- eter a circle be described, the ordinate of any point of the ellipse will be to the ordinate of the point of the circle having the same abscissa as b : a. Prove that the area of an ellipse is -n-ab. 29. Define the as3*mp totes of an hyperbola and find then* equations. Prove that a point moving along the hyperbola away from the centre approaches indefinitely near to the asymp- tote but never reaches it. 30. Prove that the angle between the focal radius of a point on the parabola and a line through the point parallel to the principal axis is bisected by the normal at the point. 462G17 8 31. Find for the ellipse the equation of the diameter bisecting a given set of parallel chords ; for the hyperbola. [27] Z/ 2 z + a 2 tan0.2/=0, b*x a*tan0.?/ = 0. Prove that every chord through the centre is a diameter. 32. Find the equation of the diameter bisecting a given set of parallel chords in the parabola. [28] tau0.y = m. 33. Describe the method of finding the centre of an ellipse or an hyperbola when an arc of the curve is given. 34. Show how to find out whether a given arc known to belong to a conic section is part of an ellipse, of an hyperbola, or of a parabola. 35. Prove that if of two diameters in a central conic the first bisects the chords parallel to the second, the second will bisect the chords parallel to the first. Such a pair of diameters are called conjugate. Find the equation of the diameter conjugate to the diameter through a given point (a^, 2/1) of an ellipse. Show that the tangent at a given point of a central conic can be drawn by the aid of a pair of conjugate diameters. 36. Find polar equations for the circle, the ellipse, the hyper- bola, and the parabola. W. E. BYERLY, Professor of Mathematics in Harvard University. Books on English Literature. INTROD. PRICE. Allen Reader's Guide to English History .... $ .25 History Topics 25 Arnold English Literature 1.50 Carpenter .... Anglo-Saxon Grammar 60 English of the XlVth Century 90 Church Stories of the Old World 40 ((.'lassies for Children.) Craik English of Shakespeare 90 Three Vol. Shakespeare, per vol 1.25 Text-Book of Poetry 1.25 Text-Book of Prose 1.25 Pamphlet Selections, Prose and Poetry . . .20 Classical English Reader 1.00 Hudson < Lamb . Merchant of Venice 25 (Classics fcr Children.) Hunt Exodus and Daniel 60 Lambert .... Robinson Crusoe 35 (Classics for Children.) Memory Gems 30 Lounsbury . . . Chaucer's Parlament of Foules 50 Minto Manual of English Prose Literature . . . 2.00 Sprague Selections from Irving j gird's . . . . . !s5 Two Books of Paradise Lost, and Lycidas . .45 Thorn Two Shakespeare Examinations 45 Yonge Scott's Qaentin Durward 40 (Classics for Children.) Copies sent to TeacJiers for Examination, with a vieiv to Introduction, on receipt of Introduction Price. GINN, HEATH, & CO., Publishers. BOSTON. NEW YORK. CIIIC.UJO. IMVhKsriY OF CALIFORNIA, LOS ANGFLF.S THE UNIVERSITY LIBRARY This book is DUE on the last date stamped below 552 Byerly - B98sy Syllabus of a 1884 course in plane analytic geometry . 552 B98sy 1884