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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
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
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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
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
—-, Ja, √a2 + b, &c., the numerical unit is understood, and must be
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
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
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
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
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,