F=

the only solution of which in whole numbers is T= 3; F= 12; which gives E = 30, 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

4T

8 - 2T. the only solution of which is T = 3, F= 6; which gives E = 12, S = 8. This is the cube. (Geom. p. 161.) (69.) For regular solids composed entirely of triangles, we have

4T

6 - T
This gives the following results. (Geom. p. 161.)

T F S Name of the solid.
5 20 30 12 Icosahedron
4 8 12 6 Octohedron

3 4 6 4 Tetrahedron (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.

F=

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. (O). Assume tan. x = \tan. a tan. b cos. C. Then cos. A cos. b

Or assume tan. X = tan, a cos. C; cos.? x

COS.C

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.

LONDON: Printed by WILLIAM CLOWES,

Duke-street, Lambeth.

CONTENT S.

.

PART I.

APPLICATION OF ALGEBRA TO PLANE GEOMETRY.

CHAPTER I.

INTRODUCTION,

Art.

Page

1 1 2 3 4

4 4

1. Object of the Treatise
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
8. The solution of an equation leads either to numerical calculations, or to Geo-

metrical Constructions
9, 10, 11. The Geometrical Construction of the quantities

ab a Ib, Nab, Jab + cd, Na? +6, Na? +62 +c?, 12, NE

c
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

Na, Na + b, &c., the numerical unit is understood, and must be
ū
expressed prior to construction

4

5

7

a

7

CHAPTER II.

DETERMINATE PROBLEMS.

.

14. Geometrical Problems may be divided into two classes, Determinate and Inde.

terminate : 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

9

9 Art.

Page 19. Through a given point M equidistant from two perpendicular straight lines, to

11 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

a

13

CHAPTER III.

THE POINT AND STRAIGHT LINE.

14

14 15

15

16 17

18

19 19 20 21 22 22

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-6)+ (x-2) = 0 26. Consideration of the negative sign as applied to the position of points in

Geometry 27, 28. The position of points on a plane, and examples 29. To find the distance between two points

D? =(a - a')? + (6 - b)2 30. The distance between two points referred to oblique axes

D2 = =(a – a')2 + (6 – 61)2 + 2(a – a') (b — b') cos. w 31, 33. The locus of the equation y=ax + b proved to be a straight line 34. Various positions of the locus corresponding to the Algebraic signs of a and b 35. The loci of the equations y=Ib, and y = 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

9 - 91=(2 – 31) 41. The equation to a straight line through two given points is

yı Y - Y1 =

- X2 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+6 be a given straight line, the straight line parallel to it is

y = ax + 6 44. The co-ordinates of the intersection of two given lines y = ax + b, and

Y
y = ax + V, are

-6 all - ab
X=
car

- a
If a third line, whose equation is y=x" x + 6", passes through the point of in.
tersection, then

(«V-b) - (b"-4" b) + (du - 1) = 0. 45. If & and d) are the angles which two lines make with the axis of x,

ca!

ituá
tan. ( - ) =

1!
cos.(-) =

wilt«) (1 tá?)