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3. Vertical 'angle, sum of sides and area;
4. Base, sum of sides, and area.
1. Let AB
be the given base, AC the given sum of the two sides, and D the given vertical angle. Upon
A B (60.) describe a segment AEB, containing an angle equal to the half of D. With the centre A and radius AC describe a circle cutting the arc A E B in E. Join A E, B E, and at the point B make the angle EBF equal to BEA: the triangle F A B shall be the triangle required. For, because the angle FEB is equal to FB E, the side F B is equal to FE (I. 6.), and the two sides AF, FB together are equal to AE, that is, to AC, the given sum of the sides. Again, because the angle AFB (I. 19.) is equal to the sum of the angles at E and B, and that these angles are equal to one another, the angle AFB is equal to twice the angle at E, that is, to the given angle D. And the triangle is described upon the given base A B. Therefore, &c. 2. Let A B be the given base, upon which let. there be described the rectangle ABCD, contain
ing an area equal to twice the given area (I. 57.), and a segment AEB containing an angle equal to the given angle (60.). Then if the arc A E B cut the side CD in E, and EA, EB be joined, E A B will evidently be the triangle required.
3. Let A be the given vertical angle, and let the triangle ABC (II.69.) contain an equal to the given area: and let D be the given sum of the two sides. Divide D into two parts, such that their rect
angle may be equal to the rectangle under AB, AC (II.56.). Take AE equal to one, and AF equal to the other of these parts, and join EF. Then, because the triangles ABC, AEF have the common angle A, they are to one another (II.40. Cor.) as the rectangles under the con
D draw DG parallel to AB (I. 48.) to meet FG in G, and join G A. Through E and C draw E H and C K parallel to FG, to meet GA and GA produced in the points H, K, so that K G, KH, KA will be proportionals (II. 29.): from the centre K, with the radius KH, describe a circle cutting G D in L, and join L A, LB: LAB shall be the triangle required.
Produce G K to M, so that KM may be equal to KH; and from L draw LŃ perpendicular to AB. Then, because KÁ, KH, KG are proportionals, MA, MH, MG are in harmonical progression (II. 46.); but the point L is in the circumference of a circle upon the_mean MH; therefore (51. Cor.) LA: LG:: AH: HG; but LG is (I. 22.) equal to NF, and AH: HG::AE: EF (II. 29.), that is, since CA, CE, CF are proportionals (II. 22. Cor. 1.):: CA: CE; therefore LA is to N F as CA to CE, or in the subduplicate ratio of C A to CF; and (II. 38. Schol. Lem. 1. Cor.) if Bf be taken equal to A F, LB is to Nf in the same ratio. Therefore the sum of LA, LB is to the whole line Ff in the same ratio, or, if Ce be taken equal to CE, as E e to F f: therefore, the sum of LA, LB is equal (II. 11. Cor. 1.) to E e, that is, to the given sum, and LAB is the triangle required. Therefore, &c.*
If the difference of the sides be supposed given instead of the sum in cases 1, 3 and 4, solutions of the same character may be obtained; viz. in case 1, by describing upon the given base AB a segment which shall contain an angle exceeding by a right angle half the given angle D; in case 3, by dividing the difference D produced, so that the rectangle under the segments may be equal to A BXA C; and in case 4, by making use of the following corollary to
PROP. 65. Prob. 12.
To find two straight lines, there being assumed any two out of the six following data; viz. their sum, their difference, the sum of their squares, the difference of their squares, their ratio, and their rectangle.
The cases of this proposition are fifteen in number, and may be arranged as follows:
1. Sum, and difference.
2. Sum of squares, and difference of squares.
3. Sum, and sum of squares.
and CD in F, and take E G a third proportional to EB and C F. From the centre E, with the radius EA, describe a semicircle A H B; and from the point G (I. 44.) draw GH perpendicular to A B, to meet the circumference in H. Join AH, HB: they shall be the straight lines required.
3. Let AB be the given sum, and the square of AC the given sum of the squares. Bisect AC in D, (I. 43.) and from D (I.44.) draw DE perpendicular to À C. From the centre D
with the radius DA or DC describe a circle cutting DE in E; and from the centre E, with the radius EA or EC describe the circle C FA: lastly, from the centre A with the radius A B describe an arc cutting CFA in F. Join AF, let AF cut the circumference AEC in G: and join G C. AG and GC shall be the lines required.
4. Let A B be the given difference, and the square of A C the given sum of the squares. Bisect AC in D (I, 43.), and from D (I. 44.) draw DE perpendicular to AC from the centre D with the radius DA or DC describe in E; and from the a circle cutting DE
centre E with the radius EA or EC de scribe the circle CFA; lastly, from the centre A with the radius AB describe let AF produced cut the circumference an arc cutting CFA in F: join AF, and AECG in G, and join GC. AG and GC shall be the straight lines required.
5. Let A B be the given sum, and the square of AC the given difference of the squares: take AD a third pro
14. Let the square of AB be the given sum of the squares, and let the rectangle under A B and C be the given rectangle : divide A B in D (II. 56.), so that the rectangle under AD, DB may be equal to the square of C; from D (II. 56.) draw DE at_right angles to
AB; and upon AB as a diameter describe a circle cutting DE in E: AE, EB shall be the straight lines required. For it is evident, that the square of ED being, by construction, equal to the rectangle under AD, DB (II. 34.Cor.), ED is equal to C; and the rectangle under AE, EB is equal to the rectangle under AB, ED, that is, to the rectangle under AB and C.
15. Let the square of AB be the given difference of the squares, and let the rectangle under A B and C be the given rectangle ; produce A B to D (II. 56.), so that the rectangle under AD, DB may be equal to the square of C, from B (I. 44.) draw BE at right angles to AD, and
upon A D describe a circle cutting BE in E: A E, EB shall be the straight lines required. For it is evident that the square of ED being (as in the last case) by the construction equal to the rectangle under AD, DB (II. 34. Cor.), ED is equal to C; and, because the triangles ABE, EBD are similar (II. 34.), AB is to AE as EB to ED, and therefore the rectangle under AE, EB is (II. 38.) equal to the rectangle under AB, ED, that is to the rectangle under AB and C.
2. x2+ y2=a2 x2 - y2 = b2
4. x - y = a x2+y2 = b2
6. x―y =α x2 + y2 = b2
9.x-y = a
y = a xy=t&
y C 14. x2+y2=a2 = b2 xy = ba In the construction of these and other problems of the foregoing Sections, the data have always been supposed such that the problem in question be not impossible. For, as we have already had occasion to observe, many of them are possible, only so long as the mutual relations of the data are confined within certain limits. Thus, if it be required to find two lines, such that their squares may together contain 9 square feet, it is evident that the sum of the lines in question must not be less than 3 feet, nor must their difference exceed 3 feet, (II. 56. N. B.). The solution, therefore, of a problem, which should require the sum of the two to fall short of this quantity, or their difference to exceed it,
would be impossible. In this manner does one of the conditions frequently set limits to the other-frequently, but not in every case:-thus, if two lines be required, which shall contain a given rectangle, their ratio may be any whatever, and a problem which should require them to be to one another in any ratio, how great or how small soever, would be possible. The limits of possibility, when there are any, are commonly indicated by the construction, if the problem be solved geometrically, as they are, if algebraically, by the form of the final equation. See the cases of Prop. 64., where the vertex of the triangle sought is determined by the intersection of a straight line and circle, or of two circles: if the data be such that no intersection can take place, the construction fails, and the problem becomes impossible.
§1. Of Lines perpendicular, or inclined, or parallel to planes.-§ 2. Of Planes which are parallel, or inclined, or perpendicular to other Planes.-§ 3. Of Solids contained by Planes. § 4. Problems.
SECTION 1.-Of Lines perpendicular, or inclined, or parallel to Planes.
IN the preceding books our attention has been confined to lines which lie in one and the same plane, the intersection of such lines, and the figures contained by them; we are now to consider lines which lie in different planes, planes which intersect one another, and solids which are contained by plane or other surfaces. In other words, we have been hitherto engaged with Plane Geometry; we are now to enter upon Solid Geometry.
Def. 1. (Euc. xi. def. 3.) A straight line is said to be perpendicular (or at right angles) to a plane, when it makes right angles with every straight line meeting it in that plane, (see Prop 3.). Also, conversely, in this case the plane is said to be perpendicular to the straight line.
The foot of the perpendicular is the point* in which it meets the plane.
It is evident that a straight line cannot meet a plane in more than one point, unless it lies altogether in the plane; and in like manner that one plane cannot meet another plane in a portion of sur
3. A straight line is said to be parallel to a plane, when it cannot meet the plane, to whatever extent both be produced. Also, conversely, in this case the plane is said to be parallel to the straight line.
4. If two planes ABC, ABD intersect one another in a line as A B (see Prop. 2.), they are said to form at that line
a dihedral angle CABD.
The magnitude of a dihedral angle does not depend upon the extent of the containing planes, but upon the opening between them. Thus, the dihedral angle CABD is greater than dral angle C ABE. the dihedral angle EA BD by the dihe
5. When one plane standing upon dral angles equal to one another, each another plane makes the adjacent dihe
of them is called a right dihedral angle; and the plane which is said to be perpenstands upon the other dicular (or at right angles) to it.
A dihedral angle is also said to be acute or obtuse, according as it less or greater than a right angle.
6. (Euc. xi. def. 8.) Planes, which do not meet one another, though produced to any extent, are said to be parallel.
7. (Euc. xi. def. 9.) If three or more planes pass through a point as A, they are
face common to both, unless they coincide altogether. (See Prop. 1.) Therefore a straight line cuts a plane in a point; and a plane cuts a plane in a line, which line (see Prop. 2.) is a straight line.
8. A polyhedron is a solid figure in cluded by any number of planes, which are called its faces: if it have four faces only, which is the least number possible, it is called a tetrahedron; if six, a hexahedron; if eight, an octahedron; if twelve, a dodecahedron; if twenty, an icosahedron; and so on.*
The intersections of the faces of a
polyhedron are called arrises or edges, and the points of the solid angles vertices or angular points. The diagonals of a polyhedron are the straight lines which join any two vertices not lying in the same face..
The surfaces of the polyhedrons here treated of are supposed to be convex, that is, such that the same straight line can cut them in two points only.
9. A polyhedron is said to be regular, when its faces are similar and equal regular polygons, and its solid angles equal to one another. There are only five such figures. (See Prop. 20. Cor.) 10. Two polyhedrons are said to be similar, when they are contained by simi
lar faces similarly situated, and forming equal dihedral and solid angles.†
The Greek word for "seat" being in all cases annexed to the Greek numeral which indicates the number of seats, or faces, on which the figure may be seated. A solid figure may be contained by any number of faces above three, in the same manner as a plane figure may be contained by any number of sides above two, the numbers 4, 6, 8, 12, and 20 being here specified only because the other solids (of 5, 7, &c. faces) are less frequently subjects of consideration.
The same observation may be made here as at Book II. def. 14, viz. that in this definition there are some things assumed which have not been as yet demonstrated. These are,
1. If all the plane angles but one which contain two solid angles be equal, each to each, in order, and make with one another equal dihedral angles, the remaining plane angle of the one shall be equal to the remaining plane angle of the other, and the two remaining dihedral angles of the one equal respectively to the two remaining dihedral angles of the other. This may be proved by coincidence.
11. (Euc. xi. def. A.) A parallelopiped is a solid figure having six faces, of which every opposite two are parallel.
Such a figure may be formed by taking
any solid angle A, which is formed by three plane angles, assuming any points B, C, D in the three edges, and passing through those points planes parallel to the planes AC D, A BD, ABC respectively.
The faces of a parallelopiped are sometimes distinguished by naming any two opposite faces the bases of the parallelopiped, and the other four the sides in which case the altitude of the
parallelopiped is the perpendicular dis
tance between the two bases.
12. A rectangular parallelopiped is that which has one of its solid angles contained by three right (see Prop. 17.) every angles, and therefore face at right angles to those which are adjoining to it. piped which has the three 13. A cube is a rectangular paralleloedges terminated in one of the solid angles equal to
one another. The cube of the cube of which A B is any straight line AB, is an edge.
14. (Euc. xi. def. 13.) A prism is a solid figure having any number of faces, two of which are similar and equal rectilineal figures, so placed as to have their corresponding sides parallel, and the rest parallelograms.
Such a figure may be formed by
2. If all the planes but one which form a convex surface be similar and similarly situated to all the planes but one which contain a solid figure, each to each, and if the dihedral angles which every adjoining two of the first make with one another be equal to the dihedral angles which every corresponding two of the latter make with one another, each to each, the remaining edges of the surface (viz. those which are not common to adjoining planes) shall lie in one plane, and shall inclose a rectilineal figure similar to the last face of the solid, and making equal dihedral angles with the corresponding faces adjoining to it. This may be demonstrated by making any two of the equal solid angles coincide. (See Prop. 14. Cor.)
It is evident, also, that the definition would be complete without mentioning the equality of the solid angles, for the several plane and dihedral angles of the one being equal and similarly situated with the corresponding plane and dihedral angles of the other, it is evident that any two corresponding solid angles may be said to coincide.