cor. 205 205 (1) Through two given points and a third (not lying in the circumference of a great circle) to describe two equal and parallel small circles (m) To describe a spherical triangle which shall be equal to a given spherical polygon, and shall have a side and adjacent angle the same with a given side and adjacent angle of the polygon (n) Two spherical arcs being given, which are together less than a semicircumference; to place them so, that with a third not given they may contain the greatest surface possible 206 (0) Through a given point to describe a great circle which shall touch two given equal and parallel small circles 206 206 (d) The square of 2 M is 4 M2, of 3 M is cor. 18 9 M2, &c. (e) The square of 5 M is equal to the square of 4 M together with the square of 3 M cor. 18 (f) Squares are to one another in the duplicate ratio of their sides (g) The squares of proportional straight lines are proportionals (h) To describe a square, cor. 63 cor. 63 28 1. Upon a given straight line 2. Which shall be equal to a given rectangle, or other rectilineal figure 30 3. Which shall be equal to the difference of two given squares, or other rectilineal figures 30 4. Which shall be equal to the sum See "Rectangle," Straight Line (A) (B) def. 1 2 When said to be perpendicular to another straight line When parallel (b) If at a point in a straight line two other straight lines upon opposite sides of it make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line (c) A straight line is the shortest distance between two points sch. 8 (d) A perpendicular may be drawn to a given straight line from any given point, whether without or in the straight line; but from the same point there cannot be drawn more than one perpendicular to the same straight line (e) From a point to a straight line, the perpendicular is the shortest distance, and of other straight lines which are drawn from the same point, such as are equal to one another are at equal distances from the foot of the perpendicular, and conversely; and the greater is always further from the perpendicular, and conversely cor. 10 cor. 5 are, together, equal to four right angles (e) Angles, which have the sides of the one parallel, or perpendicular, or equally inclined to the sides of the other, in the same order, are equal 14 (C) of parallel Straight Lines. Difficulty in the theory sch. 11 11 12 cor. 13 (a) Straight lines which are perpendicular to the same straight line are parallel; and conversely (b) A parallel to a given straight line inay be drawn through any given point without it; but through the same point there cannot be drawn more than one parallel to the same given straight line cor. 11 (c) Straight lines which make equal angles with the same straight line towards the same parts are parallel; and conversely (d) If a straight line falls upon two other straight lines, so as to make the alternate angles equal to one another, or the exterior angle equal to the interior and opposite upon the same side, or the two interior angles upon the same side together equal to two right angles, those two straight lines are parallel (e) And conversely, if a straight line falls upon two parallel straight lines, it makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite upon the same side, and the two interior angles upon the same side together equal to two right angles cor. 13 (f) If a straight line falls upon two other straight lines, so as to make the two interior angles upon the same side together less than two right angles, those two straight lines are not parallel, but may be produced to meet one another upon that side cor. 13 (g) Straight lines, to which the same straight line is parallel, are parallel to one another (h) Parallel straight lines are every where equidistant (1) Parallel straight lines intercept equal parts of parallel straight lines cor. 13 (k) The straight lines which join the extremities of equal and parallel straight lines towards the same parts, are equal and parallel 13 13 15 57 (1) If two straight lines, which pass through the same point, are cut by two parallel straight lines, their parts terminated in that point shall be proportional (m) If two (or more) straight lines which pass through the same point, are cut by any number of parallel straight lines, they shall be similarly divided by them cor. 58 (C) (n) If two (or more) parallels are cut by any number of straight lines, which pass through the same point, they shall be similarly divided; and, if two parallels are similarly divided, the straight lines which join the corresponding points of division, pass, or may be produced to pass, all of them through the same point 58 (0) If the legs of an angle cut two parallel straight lines, the intercepted parts of the parallels shall be to one another as the parts which they cut off from either of the legs cor. 59 Of Straight Lines, which are not in the same plane. 130 (a) Straight lines, to which the same straight line is parallel, although not in one plane with it, are parallel to one another (b) The shortest distance of two straight lines, which are not in the same plane, is a straight line, which is at right angles to each of them, and is equal to the perpendicular which is drawn from the vertex to the hypotenuse of a right-angled triangle, whose sides are the perpendiculars drawn to one of the straight lines from the two points in which the other is cut by any two planes passing through the first at right angles to one another cor. 155 (b) If A, B are two straight lines, and A', B' other two, the rectangle A× A' shall be to the rectangle BX B' in the ratio which is compounded of the ratios of A to B and A' to B' . 63 (c) If A, B and A', B' are proportionals, AXA' is to BX B' as A2 to B2 cor. 63 (d) If A, B, C are proportionals, A is to C as A2 to B2 cor. 63 (e) If A, B, C, D, and also A, B, C, D', are proportionals, the rectangles A x A', B x B', C x C', and DX D' are also proportionals cor. 63 cor. 63 (f) The squares of proportional straight lines are proportionals; and conversely (9) If A, B are two straight lines, A', B' other two, and A", B" other two, the rectangular parallelopipeds Ax A′ XA", BxB'x B" shall be to one another in the ratio which is compounded of the ratios of A to B, A' to B', and A" to B" 143 [(h) If the ratios of A to B, A' to B', and A" to B" are all equal, A × A' × A” is to B x B x B" as A3 to B3.] [() If A, B, C, D are in continued proportion, A is to D as A3 to B3.] [(k) If A, B, C, D, and also A', B', C', D', and also A", B", C", D", are proportionals, the rectangular parallelopipeds AX A' x A", B x B′ x B", CX C' × C", D X D' × D'', are also proportionals.] (The cubes of proportional straight lines are proportionals; and conversely 144 (F) Of Straight Lines harmonically divided. (a) If A B, AC, AD are harmonical progressionals in the same straight line, and A B the least, DC, DB, ĎA shall likewise be harmonical progressionals 68 (b) The same being supposed, and the mean AC being bisected in K, KB, KC, KD are proportionals; and, conversely, if KB, KC, K D are proportionals, and if K A is taken in the opposite direction equal to K C, AB, AC, A D shall be in harmonical progression (c) The same being supposed, DA, DK, DB, D C are proportionals (d) The harmonical mean between two straight lines is a third proportional to the arithmetical and geometrical cor. 69 means 68 69 70 (e) If any four straight lines pass through the same point, and lie in the same plane, to whichsoever of the four a parallel is drawn, its parts intercepted by the other three shall be to one another in the same ratio (f) If a parallel is drawn to any one of four harmonicals, equal parts of such parallel are intercepted by the other three; and, conversely, if four straight lines pass through the same point, and if a parallel drawn to one of them has equal parts of it intercepted by the other three, the four are harmonicals cor. 70 (9) Harmonicals divide every straight line, which is cut by them, harmonically 70 (h) The two sides of a triangle, the straight line which is drawn from the (e) To find two straight lines, there being given the 1. Sum, and difference. 2. Sum of squares, and difference 3. Sum, and sum of squares. 7. Ratio, and rectangle. 123 9. Difference, and ratio. 10. Sum, and rectangle. 11. Difference, and rectangle. 12. Sum of squares, and ratio. 13. Difference of squares, and ratio. 14. Sum of squares, and rectangle. 15. Difference of squares, and rectangle (f) To draw the shortest distance between two given straight lines which do not lie in the same plane 154 Subcontrary section of an oblique cone 229 of an oblique cylinder 232 Subduplicate, one ratio said to be of another 34 Subtriplicate, one ratio said to be of another 34 Supplement, or supplementary, one angle said to be of another note 5 Supplementary triangle, a name sometimes given to the polar triangle note 188 Surface, (See Convex," "Plane.") def. 1 Surface of revolution, is the surface of a solid of revolution. See "Solid of Revolution." Symmetrical, spherical triangles said to be 185 [Symmetrical. Any two solids are said to be symmetrical, when they can be placed upon the two sides of a plane, so that for every point in the surface of the one, there is a point in the surface of the other, in the same perpendicular to the plane, and at the same distance from it; also, the solids, when so placed, are said to be symmetrically situated, with regard to the plane*. Of such solids it may be demonstrated 1. That if any polyhedron whatever be inscribed in, or circumscribed about the one, a polyhedron symmetrical with it may be inscribed in, or circumscribed about the other. 2. That they are equal to one an other, and have equal surfaces.] Symmetrically divided; a plane (or solid) figure is said to be symmetrically divided by a straight line (or plane), when, for every point in the contour of the figure which is upon one side of that line (or plane), there is another point in the con According to a similar description, it is evident that two plane figures may be said to be symmetrical, and symmetrically situated with regard to a straight line which is in the same plane with them; but such figures are also similar, and may be made to coincide, which is not the case with symmetrical solids. tour upon the other side, in the same perpendicular to the line (or plane), and at the same distance from it. Synthesis, geometrical. See "Analysis." Tangent, of any curve. See "Touch." No straight line can be drawn between a curve and its tangent, from the point of contact, so as not to cut the curve sch. 212 Tangent of a circle. (See "Circle.") def. Tangent of a conic section. See "Conic Section." Tangent plane. See "Touch." Terms of a ratio. (See "Numerical Ratio.") 126 def. 32 Tetrahedron, or triangular pyramid def. 126 (a) Is contained by the least number of faces possible-viz. four (b) Is equal to one-sixth of a parallelopiped, which has three of its edges coincident with, and equal to, three edges of the tetrahedron cor. 147 Regular. See "Regular Polyhedron." Third proportional (see "Straight Line" def.33 Third harmonical progressional def. 68 [Touch; (a) a straight line is said to touch a curve in any point, when it meets, but does not cut the curve, in that point*. Such a line is called a tangent. in (b) One curve is said to touch another any point when they have the same tangent at that point. (c) A plane is said to touch a curved surface in any point or line, when it meets, but does not cut, the surface in that point or line: such a plane is called a tangent-plane, and the point or line in which it touches the surface, is called the point or line of contact. (d) One curved surface is said to touch another in any point or line, when they have the same tangent plane at that point or at every point of that line. (e) The following examples may be given of the contact of surfaces 1. A plane touches The surface of a sphere in a point, the convex surface of a cylinder in a straight line which is parallel to the axis, the convex surface of a cone in a slant side. 2. A spherical surface touches A spherical surface, whether externally or internally, in a point; the convex surface of a cylinder or * Some curves are of a winding or serpentine form, having the curvature or bending now towards the one side, now towards the other. In such curves, there are always one or more points of contrary flexure, i. c. points where the curvature which had before been towards one side changes to the other side. Through such points no straight line can be drawn so as not to cut the curve: every other point, however, admits of a straight line being drawn, which meets and does not cut the curve in that point. cor. 5 (a) Any one of the angles of a triangle is less than two right angles (b) Triangles which have two sides and the included angle of the one equal to two sides, and the included angle of the other, each to each, are equal in all respects 5 (e) Triangles which have two angles and the interjacent side of the one equal to two angles and the interjacent side of the other, each to each, are equal in all respects (d) If one side of a triangle be equal to another, the opposite angle is likewise equal to the angle opposite to the other; and conversely (e) In an isosceles triangle, 6 6 1. The angles at the base are equal to one another; and if the equal sides are produced, the angles upon the other side of the base are likewise equal cor. 6 2. The following lines, viz. the line which bisects the vertical angle, the line which is drawn from the vertex to the middle point of the base, and the line which is drawn from the vertex perpendicular to the base, coincide with one another. cor. 7 3. The straight line which bisects the base at right angles passes through the vertex, and bisects the vertical angle cor. 7 (f) Triangles which have the three sides of the one equal to the three sides of the other, each to each, are equal in all respects 7 cor. 7 (9) Any two angles of a triangle are together less than two right angles 7 (h) A triangle cannot have more than one right angle, or more than one obtuse angle (i) If one side of a triangle be produced, the exterior angle is greater than either of the interior and opposite cor, 7 angles [(9) Any two triangles are equal to one another in all respects, when they have 1. Two sides and the included angle of the one equal to two sides and included angle of the other, each to each (b). 2. Two angles and the interjacent side (c). 3. The three sides (ƒ). 4. Two angles, and a side opposite 5. An angle of the one equal to an angle of the other, and the sides about two other angles, each to each, and the remaining angles of the same affection, or one of them a right angle (see II. 33.)] (r) Two right-angled triangles are equal to one another in all respects, when they have 1. The hypotenuse and a side of the one equal to the hypotenuse and a side of the other, each to each 10 |