cor. cor. 13 are, together, equal to four right an- (n) If two (or more) parallels are cut by gles cor. 5 any number of straight lines, which (e) Angles, which have the sides of the pass through the same point, they shall one parallel, or perpendicular, or be similarly divided ; and, if two paequally inclined to the sides of the rallels are similarly divided, the other, in the same order, are equal 14 straight lines which join the corre sponding points of division, pass, or (C) Of parallel Straight Lines. may be produced to pass, all of them Difficulty in the theory, sch. 11 through the same point 58 (a) Straight lines which are perpendicu- (6) If the legs of an angle cut two lar to the same straight line are pa- parallel straight lines, the intercepted rallel; and conversely 11 parts of the parallels shall be to one (6) A parallel to a given straight line another as the parts which they cut off may be drawn through any given from either of the legs cor. 59 point without it; but through the same point there cannot be drawn (C) Of Straight Lines, which are not in the more than one parallel to the same same plane. given straight line 11 (a) Straight lines, to which the same (c) Straight lines which make equal straight line is parallel, although not angles with the same straight line in one plane with it, are parallel to towards the same parts are parallel ; one another 130 and conversely 12 (b) The shortest distance of two straight (d) If a straight line falls upon two lines, which are not in the same plane, other straight lines, so as to make the is a straight line, which is at right alternate angles equal to one another, angles to each of them, and is equal or the exterior angle equal to the to the perpendicular which is drawn interior and opposite upon the same from the vertex to the hypotenuse of a side, or the two interior angles upon the right-angled triangle, whose sides are same side together equal to two right the perpendiculars drawn to one of the angles, those two straight lines are straight lines from the two points in parallel which the other is cut by any two (e) And conversely, if a straight line planes passing through the first at falls upon two parallel straight lines, right angles to one another cor. 155 19, 20 (a) A X (B + C + D) = A B + AC (f) If a straight line falls upon two +AD. other straight lines, so as to make the (b) (A + B) x B= AB + B2. (A+B) + (A - B) = 2 (A? meet one another upon that side + B2). cor. 13 (9) (A + B) X (A – B) = A– B?. (9) Straight lines, to which the same straight line is parallel, are parallel to (E) Of Straight Lines which are proporone another 13 tionals. (h) Parallel straight lines are every (a) If four straight lines are proporwhere equidistant 13 tionals, the rectangle under the ex(i) Parallel straight lines intercept equal tremes is equal to the rectangle under parts of parallel straight lines cor. 13 the means; and conversely, if two (k) The straight lines which join the rectangles are equal to one another, extremities of equal and parallel their sides are proportionals, the sides 15 63 (1) If 'two straight lines, which pass (6) If A, B are two straight lines, and through the same point, are cut by A', B' other two, the rectangle AXA' two parallel straight lines, their parts shall be to the rectangle B x B' in the terminated in that point shall be pro- ratio which is compounded of the portional 57 ratios of A to B and A' to B' 63 (m) If two (or more) straight lines (c) If A, B and A', B' are proportionals, which pass through the same point, AXA' is to Bx B'as A2 to B2 cor. 63 are cut by any number of parallel (d) If A, B, C are proportionals, A is to straight lines, they shall be similarly Cas A2 to B2 cor. 63 divided by them (e) If A, B, C, D, and also A, B, C, cor. 58 cor. 13 cor. 63 . D', are proportionals, the rectangles vertex through the middle point of the A X A', B x B', C x C', and D X D' base, and the straight line which is are also proportionals cor. 63 drawn through the vertex parallel to The squares of proportional straight the base are harmonicals cor. 70 lines are proportionals; and con- (1) The two sides of a triangle, and the versely lines which bisect the vertical and (9) If A, B are two straight lines, A', B’ exterior vertical angles, are harmoni. other two, and A", B” other two, the cals cor. 71 rectangular parallelopipeds AX A' XA", B x B'XB” shall be to one ano. (G) Problems relating to Straight Lines. ther in the ratio which is compounded (a) Through a given point to draw a of the ratios of A to B, A' to B', and straight lineA" to B" 143 1. Perpendicular to a given straight [(h) If the ratios of A to B, A' to B', and line; several methods 24, 25 A" to B” are all equal, A X A' X A" 2. Parallel to a given straight line is to B x B' x B" as Ao to BP.] 26 (0) If A, B, C, D are in continued pro When many parallels are to be portion, A is to D as A3 to B?.] drawn, See II. 49. [(k) If A, B, C, D, and also A’, Bo, C',D', 3. Which shall pass through a point and also A", B”, C", D”, are propor which is the point of concourse tionals, the rectangular parallelopipeds of two given straight lines, but is A X A X A", B X B' x B", CXC without the limits of the draught xC”, D X D' x D”, are also propor (See “Centrolinead.”) 75 tionals.] 4. So that its parts intercepted by (1) The cubes of proportional straight two given straight lines shall be lines are proportionals ; and con to one another in a given ratio versely 144 74 5. Which shall form with the parts (F) Of Straight Lines harmonically divided. cut off by it from two given (a) If A B, AC, A D are harmonical straight lines a triangle equal to progressionals in the same straight a given triangle 77 sine, and A B the least, DC, DB, DA shall likewise be harmonical progres. (6) To divide a given straight line 1. Into two or more equal parts 26 sionals 68 2. Similarly to (6) The same being supposed, and the a given divided straight line 71 mean AC being bisected in K, KB, 3. In extreme and mean ratio 72 KC, KD are proportionals; and, 4. In any given ratio 72 conversely, if KB, KC,K D are pro 5. So that the rectangle under the portionals, and if K A is taken in the opposite direction equal to KC, AB, parts shall be equal to a given rectangle 72 AC, A D shall be in harmonical progression 68 (c) To produce a given straight line(c) The same being supposed, D A, DK, 1. In a given ratio 72 D B, D C are proportionals 69 2. So that the rectangle under the (d) The harmonical mean between two whole line produced and the part straight lines is a third proportional produced shall be equal to a given to the arithmetical and geometrical rectangle 72 cor. 69 (d) To find a straight line, (e) If any four straight lines pass 1. Which shall be a mean proporthrough the same point, and lie in the tional between two given straight same plane, to whichsoever of the four lines. 71 a parallel is drawn, its parts intercepted 2. Which shall be a third proporby the other three shall be to one an tional to two given straight lines other in the same ratio 70 71 (f) If a parallel is drawn to any one of 3. Which shall be a fourth proporfour harmonicals, equal parts of such tional to three given straight parallel are intercepted by the other lines. 71 three; and, conversely, if four straight 4. Which shall be an harmonical lines pass through the same point, mean between two given straight and if a parallel drawn to one of them lines 72 has equal parts of it intercepted by 5. Which shall be a third harmothe other three, the four are harmo nical progressional to two given nicals straight lines 72 (9) Harmonicals divide every straight 6. To which a given straight line line, which is cut by them, harmoni A shall have the ratio, which is cally 70 compounded of the given ratios (h) The two sides of a trianglè, the of A to B, A2 to B2, A3 to B3, and straight line which is drawn from the A4 to B4 . means cor. 70 75 cor. 147 156 34 (e) To find two straight lines, there being tour upon the other side, in the same pergiven the pendicular to the line (or plane), and at the 1. Sum, and difference. same distance from it. 2. Sum of squares, and difference Synthesis, geometrical. See “ Analysis.” of squares. 3. Sum, and sum of squares. Tangent, of any curve. See “ Touch.” 4. Difference, and sum of squares. No straight line can be drawn be5. Sum, and difference of squares. tween a curve and its tangent, from 6. Difference, and difference of the point of contact, so as not to cut squares. the curve sch. 212 7. Ratio, and rectangle. Tangent of a circle. (See “Circle.") def. 8. Sum, and ratio. Tangent of a conic section. See “ Conic Sec9. Difference, and ratio. tion.” 10. Sum, and rectangle. Tangent plane. See " Touch.” 11. Difference, and rectangle. Terms of a ratio. (See “Numerical Ratio.") 12. Sum of squares, and ratio. def. 32 13. Difference of squares, and ratio. Tetrahedron, or triangular pyramid def. 126 14. Sum of squares, and rectangle. (a) Is contained by the least number of 15. Difference of squares, and rect- faces possible—viz. four 126 angle 123 (6) Is equal to one-sixth of a parallelo(f) To draw the shortest distance be- piped, which has three of its edges tween two given straight lines which coincident with, and equal to, three do not lie in the same plane 154 edges of the tetrahedron Subcontrary section of an oblique cone 229 (c) To inscribe a sphere within a given of an oblique cylinder tetrahedron 232 (d) To circumscribe a sphere about a Subiluplicate, one ratio said to be of another 34 given tetrahedron 156 Subtriplicate, one ratio said to be of another Regular. See “ Regular Polyhedron.” Third proportional (see“ Straight Line”def.33 Supplement, or supplementary, one angle said Third harmonical progressional def. 68 to be of another note 5 [Touch ; (a) a straight line is said to touch a Supplementary triangle, a name sometimes curve in any point, when it meets, but does given to the polar triangle note 188 not cut the curve, in that point*. Such a Surface, (See * Convex,” “ Plane.") def. 1 line is called a tangent. Surface of revolutio is the surface of a solid (6) One curve is said to touch another of revolution. See “ Solid of Revolution.” in any point when they have the Symmetrical, spherical triangles said to be same tangent at that point. 185 (c) A plane is said to touch a curved [Symmetrical. Any two solids are said to be surface in any point or line, when it symmetrical, when they can be placed upon meets, but does not cut, the surface in the two sides of a plane, so that for every that point or line: such a plane is point in the surface of the one, there is a called a tangent-plane, and the point or point in the surface of the other, in the line in which it touches the surface, is same perpendicular to the plane, and at the called the point or line of contact. same distance from it; also, the solids, (d) One curved surface is said to touch when so placed, are said to be symmetri- another in any point or line, when they cally situated, with regard to the plane*. have the same tangent plane at that Of such solids it may be demonstrated point or at every point of that line. 1. That if any polyhedron what- (e) The following examples may be ever be inscribed in, or circum given of the contact of surfacesscribed about the one, a polyhe 1. A plane touches dron symmetrical with it may be The surface of a sphere in a point, inscribed in, or circumscribed the convex surface of a cylinder about the other. in a straight line which is parallel 2. That they are equal to one an. to the axis, the convex surface of other, and have equal surfaces.) a cone in a slant side. Symmetrically divided ; a plane (or solid) 2. A spherical surface touches figure is said to be symmetrically divided Aspherical surface, whether exterby a straight line (or plane), when, for nally or internally, in a point; the every point in the contour of the figure convex surface of a cylinder or which is upon one side of that line (or plane), there is another point in the con- 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, * According to a similar description, it is evident there are always one or more points of contrary that two plane figures may be said to be symmetri- fexure, i. c. points where the curvature which had cal, and symmetrically situated with regard to a before been towards one side changes to the other straight line which is in the same plane with them; side. Through such points no straight line can be but such figures are also similar, and may be made drawn so as not to cut the curve : every other point, to coincide, which is not the case with symmetrical however, admits of a straight line being drawn, solids. which meets and does not cut the curye in that point. cor, 7 con cor. 8 def. 2 14 cor. 14 cone externally in a point, inter- (k) If one side of a triangle be greater nally in one or two points or in than another, the opposite angle shall the circumference of a circle.] likewise be greater than the angle Transverse axis, of an ellipse or hyperbola, opposite to that other; and def. 217 7 Trapezoid, (also its “ Altitude") def. 2 (1) Any two of the sides are together (a) Is equal to the rectangle under its greater than the third side, and any altitude, and half the sum of its pa- side is greater than the difference of rallel sides 17 8 [(6) Half the sum of the parallel sides of (m) If two triangles stand upon the a trapezoid is equal to the straight line same base, and if the vertex of the one which joins the middle points of the falls within or upon a side of the other, other two sides.] that triangle has a less perimeter than Triangle, rectilineal (also its “Vertex,” the other “Sides," - Base"). (n) Of triangles which have the two When said to be equilateral, isosceles, sides of the one equal to the two sides scalene 2 of the other, each to each, that which When said to be right-angled, oblique- has the greater vertical angle has the angled, obtuse-angled, acute-angled 2 greater base; and conversely 9 (0) The three angles of a triangle are (A) of the first Properties, and of Triangles together equal to two right angles; which are equal in all respects. and if one of the sides be produced, (a) Any one of the angles of a triangle the exterior angle is equal to the two is less than two right angles cor. 5 interior and opposite angles (6) Triangles which have two sides and (p) In a right angled trianglethe included angle of the one equal to 1. One of the angles, viz. the right two sides, and the included angle of angle, is equal to the sum of the the other, each to each, are equal in other two all respects 5 2. The straight line which joins (c) Triangles which have two angles one of the angles, viz. the right and the interjacent side of the one angle, with the middle point of equal to two angles and the interja the opposite side is equal to half cent side of the other, each to each, that side ; and conversely, if a triare equal in all respects 6 angle has either of these proper(d) If one side of a triangle be equal to ties, it is a rightangled triangle another, the opposite angle is likewise cor. 14 equal to the angle opposite to the [(9) Any two triangles are equal to one other; and conversely 6 another in all respects, when they have (e) In an isosceles triangle, 1. Two sides and the included angle 1. The angles at the base are equal of the one equal to two sides and to one another; and if the equal included angle of the other, each sides are produced, the angles to each (6). upon the other side of the base 2. Two angles and the interjacent are likewise equal cor. 6 2. The following lines, viz. the line 3. The three sides ($). which bisects the vertical angle, 4. Two angles, and a side opposite the line which is drawn from the to one of them vertex to the middle point of the 5. An angle of the one equal to an base, and the line which is drawn angle of the other, and the sides from the vertex perpendicular to about two other angles, each to the base, coincide with one ano each, and the remaining angles ther cor. 7 of the same affection, or one of 3. The straight line which bisects them a right angle (see II. 33.)) the base at right angles passes (r) Two right-angled triangles are equal through the vertex, and bisects to one another in all respects, when the vertical angle cor. 7 they have Triangles which have the three 1. The hypotenuse and a side of sides of the one equal to the three the one equal to the hypotenuse sides of the other, each to each, are and a side of the other, each to equal in all respects 7 10 (9) Any two angles of a triangle are 2. The hypotenuse and an acute together less than two right angles 7 angle 10 (h) A triangle cannot have more than 3. The two sides (I. 4.) one right angle, or more than one 4. A side and the adjoining acute obtuse angle cor. 7 (i) If one side of a triangle be produced, angle (1. 5.) 5. A side, and the opposite acute the exterior angle is greater than either of the interior and opposite angle (I. 19. and I. 4.) (6) If in two right-angled triangles, the angles hypotenuse of the one is equal to the side (c). cor. 14 . hypotenuse of the other, but a side exterior vertical angle be bisected by of the one greater than a side of the a straight line which cuts the base, or other; the angle which is opposite to the base produced, the base or base the side of the first shall be greater produced shall be divided in the ratio than the angle which is opposite to of the sides : also the square of the the side of the other lem. 131 bisecting line shall be equal to the difference of the rectangles under the (B) Of the mutual Relations of the Sides. sides and the segments of the base, (a) In a right-angled triangle, the or base produced 70, 89 square of the hypotenuse is equal to (i) In every triangle, if the base is the squares of the two sides 21 equally produced both ways, so that (b) In every triangle, the square of the the base produced is a third proporside which is opposite to a given angle tional to the base and sum of the is greater or less than the squares of sides, the sides of the triangle are to the sides containing that angle, by one another as the corresponding segtwice the rectangle contained by ments of the base produced sch. 64 either of these sides, and that part of (k) In every triangle, if the base is it which is intercepted between the equally reduced both ways, so that perpendicular, let fall upon it from the base reduced is a third proportional the opposite angle, and the acute to the base and the difference of the angle; greater if the given angle is sides, the sides of the triangle are to greater than a right angle, and less one another as the corresponding segif it is less 22 ments of the base reduced sch. 65 (c) Any angle of a triangle is equal to or (1) In an isosceles triangle, if the equal greater or less than a right angle, angles are each of them double of the according as the square of the opposite vertical angle, the sides and base are side is equal to or greater or less than in extreme and mean ratio. See the squares of the containing sides “ Errata." cor. 21, 22 (d) In every triangle, if a perpendicular (C) Of the Area of a Triangle, and of Trian be drawn from the vertex to the base, gles which are not equal in all respects. is double of the triangle 17 sides as the difference of the sides (6) Every triangle is equal to the half of to the difference of the segments of a rectangle which has the same base the base or sum of the segments of and the same altitude; i.e. to half the the base produced 22 product of its base and altitude * (e) In a right-angled triangle, if a per pendicular be drawn from the right Every triangle is a mean proportional between two tenuse, and the square of either side rectangles, the sides of which are equal to the semiis equal to the rectangle under the perimeter of the triangle and the excesses of the hypotenuse and the segment adjoining semiperimeter above the three sides. The demonstration of this is briefly as follows:to it; i. e. the perpendicular is a Let a circle be described within the triangle A B C, mean proportional between the seg and a circle upon the other side ments of the hypotenuse, and either of BC, the one touching B C, AC, and A B, in the points D, E, F, side is a mean proportional between and the other BC and A C, A B the hypotenuse and the segment ad produced in the points G, H, K; joining to it cor. 21 and 61 then the centres L, M of these (f) In an isosceles triangle, if a straight circles lie in the straight line which bisects the angle at A, and line is drawn from the vertex to any if B L, B M be joined, the angle point in the base, or in the base pro LBM will be a right angle, beduced, the square of that straight line cause it is half the sum of the shall be less or greater than the square angles A BC, C B K; also, beв. cause A K and A Hare togeof the side by the rectangle under the ther equal to A B, B G and AC, segments of the base, or of the base CG, that is to the perimeter of produced 23 the triangle A B C, each of them is equal to the semiperineter. (9) In every triangle, the squares of the Again, because F B and B K (i.e. BD and BG) are two sides are together double of the together equal to E C and C H (i. e. to C D'and squares of half the base, and of the CG), taking away GD from each, 2 B G is equal to straight line which is drawn from the 2C D, and consequently B G to CD); therefore FB vertex to the middle point of the base is equal to C H, and B'K to E C. A F, therefore, is the semiperimeter, B K is the excess above the 23 side A B, B F the excess above the side A C, and () In every triangle, if the vertical or A F the excess above the side B C. cor. 17 rem G K к Н M |