the only solution of which in whole numbers is T = 3; F = 12; which gives E30, and S = 20. This solid is the regular dodecahedron. (Geom. p. 161, &c.) (68.) Similarly the regular solids all whose faces are squares are determined from the equation 4 T the only solution of which is T3, F6; which gives E 12, S=8. This is the cube. (Geom. p. 161.) (69.) For regular solids composed entirely of triangles, we have (70.) The materials of this treatise have been for the most part collected from Puissant, Traité de Géodésie, Delambre, Traité d'Astronomie (3 vols. 4to), and Legendre, Traité de Géométrie (Brewster's translation) to all of which works we refer the reader. The following transformations, which, though not always possible, may often be used with advantage, have been suggested by a Member of the Committee. They may be very easily demonstrated. The formulæ are referred to, as in the work. (01). Assume tan. x = √tan. a tan. b cos. C. Then LIBRARY OF USEFUL KNOWLEDGE. A TREATISE ON ALGEBRAICAL GEOMETRY. BY THE REV. S. W. WAUD, M.A. F. AST. S. FELLOW AND TUTOR OF MAGDALENE COLLEGE, CAMBRIDGE. PUBLISHED UNDER THE SUPERINTENDENCE OF THE SOCIETY FOR THE DIFFUSION OF USEFUL KNOWLEDGE. LONDON: BALDWIN AND CRADOCK, PATERNOSTER-ROW. MDCCCXXXV. 2. The method of expressing the length of a straight line by Algebra 3. The method of expressing the size of an area 4. The method of expressing the volume of a solid 5. General signification of an equation when referring to Geometry 6. Particular cases where the equation refers to areas and surfaces 7. Equations of the second and third order refer to some Geometrical Theorem 9, 10, 11. The Geometrical Construction of the quantities ab a±b, a, Jab, √ab+cd, √a2±b3, Noa2+b2+c3, √12, NË 12. Method of uniting the several parts of a construction in one figure 13. If any expression is not homogeneous with the linear unit, or is of the form a —-, Ja, √a2 + b, &c., the numerical unit is understood, and must be . CHAPTER II. DETERMINATE PROBLEMS. 14. Geometrical Problems may be divided into two classes, Determinate and Indeterminate an example of each. 15. Rules which are generally useful in working Problems 16. To describe a square in a given triangle 17. In a right-angled triangle the lines drawn from the acute angles to the points of bisection of the opposite sides are given, to find the triangle 18. To divide a straight line, so that the rectangle contained by the two parts may be equal to a given square. Remarks on the double roots Art. 19. Through a given point M equidistant from two perpendicular straight lines, to draw a straight line of given length: various solutions 20. Through the same point to draw a line so that the sum of the squares upon the two portions of it shall be equal to a given square 21. To find a triangle such that its three sides and perpendicular on the base are in a continued progression CHAPTER III. THE POINT AND STRAIGHT LINE. 22. Example of an Indeterminate Problem leading to an equation between two quantities x and y. Definition of a locus 23. Division of equations into Algebraical and Transcendental 24. Some equations do not admit of loci 25. The position of a point in a plane determined. Equations to a point, x=a, y = b; or (y—b)2 + (x− a)2 = 0 14 26. Consideration of the negative sign as applied to the position of points in 30. The distance between two points referred to oblique axes 31, 33. The locus of the equation yax+b proved to be a straight line =0 36. Examples of loci corresponding to equations of the first order 37, 39. Exceptions and general remarks 40. The equation to a straight line passing through a given point is 41. The equation to a straight line through two given points is 42, To find the equation to a straight line through a given point, and bisecting a finite portion of a given line 43. If y = ax+b be a given straight line, the straight line parallel to it is y=ax+b 44. The co-ordinates of the intersection of two given lines yux+b, and If a third line, whose equation is y = a"x+b", passes through the point of intersection, then (a b' — a' b) — (« b'" — all b) + (a' bll — aꞌꞌ b') = 0. 45. If and are the angles which two lines make with the axis of x, 24 25 25 26 |