which is parallel to ABC or abc, and cuts the axis Oo in E. The section P Q R shall be a circle having the centre E. Let P be any point in the curve P QR; join PE; through P draw PA parallel to E O, and, consequently (V. def. 1.), lying in the convex surface of the cylinder, to meet the circumference A B C in A, and join O A. Then, because the parallels PA, EO are intercepted between parallel planes, they are equal to one another (IV. 13.); and, because PA and E O are both equal and parallel, E P is equal to OA (I. 21.), that is, to the radius of the circle AB C. And, in the same manner, it may be shown that the straight line drawn from E to any other point Q of PQR is equal to the same radius. Therefore, the point E is at the same distance from every point of PQR; and, consequently, PQR is a circle having the centre E. Therefore, &c. Cor. The radius of every circular section of a cylinder, which is made by a plane parallel to its base, is equal to the radius of the base. PROP. 26. A aa' A' in the straight line P P' (IV. 2.), take any point Q in the curve P Q R, and through Q draw the plane D Q D' R parallel to the base of the cylinder (IV. 43.), and let this plane cut the plane A a a' A' in the straight line D ED' and the plane PQR in the straight line QR; then DQD' is a circle having the centre E (25.). And, because the planes D Q D', PQR are each of them perpendicular to the plane A a a' A', their common section Q R is perpendicular to the same plane (IV. 18.), and consequently to the straight lines DD', PP', which meet it in that plane (IV. def. 1.). Now, because, by the supposition, the planes PQR, DQD' are equally inclined to the straight lines A a, A' a' respectively, the angles of inclination N P D, N D'P' are equal to one another; but N D'P' is equal to N D P, because A'a' is parallel to A a (I. 15.); therefore, the angle N PD is likewise equal to N D P, and consequently (I. 6.) the side N P is equal to the side ND. And, for the like reasons, N P' is equal to N D'. Therefore, the rectangle P N x N P' is equal to the rectangle DN × ND'. But, because D E D' is the diameter of the circle D Q D', and is perpendicular to the chord QR at the point N, QR is bisected in N (III. 3.), and Q N2 is equal to DN × ND' (III. 20.). Therefore, Q N2 is equal to PN ×N P'. Therefore, because PP' bisects every straight line QR which is drawn perpendicular to it from a point Q of the section PQR, and that the square QN of the half of such straight line is equal to the rectangle PNXN P under the segments of PP', the section PQR is a circle having P P for its diameter (III. 3. and III. 20.). Also, the middle point of PP' is the centre of the circle. But, because Oo, Aa and A'a' are parallel, and that A'O is equal to OA, P' F is equal to FP (II. 29.), that is, F is the middle point of PP'. There fore, F is the centre of the circle P Q R. Therefore, &c. In an oblique cylinder, if Aa and Aa' are the parallel straight lines in which the surface of the cylinder is cut by a plane passing through the axis Oo perpendicular to the base, and if the cylinder be cut by a plane PQR which is perpendicular to the plane A a a' A', and is inclined to either of the parallel straight lines, Aa, at the same angle at which the base is inclined to the other A' a', the section made by the plane PQR shall be a circle having its centre in the axis of the cylinder; or, in other words, every subcontrary section of an oblique cylinder is a circle having its centre in the axis of the cylinder. Let the plane PQR cut the plane Cor. The radius of every subcontrary section of an oblique cylinder is equal to the radius of the base of the cylinder. PROP. 27. Every plane section of a cylinder which is neither parallel to the axis* nor parallel to the base, nor subcontrary, is an ellipse having its centre in the * A plane which is parallel to the axis of a cylin der, cuts the convex surface in two straight lines which are parallel to the axis. ABA', nor subcontrary: PQR shall be an ellipse having for its centre the point C in which its plane cuts the axis of the cylinder. Through O draw a plane parallel to the plane PQR (IV. 43.), and let it cut the plane ABA' in the straight line OB; draw the diameter AA' perpendicular to OB, and let the plane AOo cut the convex surface of the cylinder in the parallel straight lines A a, A' a', and the plane P Q R in the straight line PCP: in the curve PQR take any two points Q, Q', and through these points draw the planes KQLR, K'Q' L'R' parallel to the base A B A' (IV. 43.) and cutting the plane PQR in the straight lines QNR, Q'N'R' and the plane A a a' A' in the straight lines K N L, K'N' L' respectively (IV. 2.). Then, because 2 QR is the common section of two planes which are parallel respectively to the two passing through O B, QR is parallel to OB (IV. 12. Cor.); and, for the like reason, Q'R' is parallel to O B or Q R. Also, because KL and A' A are the common sections of parallel planes by the same plane A aa' A', K L is parallel to A'A (IV. 12.); and, for the like reason, K'L' is parallel to A'A or K L. But A'OA is at right angles to OB. Therefore, QR is at right angles to KL, and Q'R' is at right angles to K'L' (IV. 15.). And, because the diameter KL of the circle K QLR is at right angles to the chord Q R, it bisects Q R in the point N (III. 3.); and, for the like reason, Q'R' is bisected in N'. Therefore (III. 20.) Q N2 is equal to KN × NL, and Q'N' to K'N'× N'L'. But, because the triangles P' K N, PN L are similar to the triangles P' K'N', PN' L' respectively (I.15.) KN: NP':: K'N': N'P', and NL: PN :: N' L' : P N' (II. 31.), and, consequently, KN×N L or QN PNXN P:: K'N'x N' L' or Q'N'2 : PN'× N'P' (II. 37. Cor. 3.). Therefore, alternando, QN: Q'N'2 :: PN x N P PN' x N'P'; and consequently (19.), P Q R is an ellipse having the diameter P P', and the tangent at P parallel to O B. Also, because A A' is bisected in O, and that A a, Oo and A' a' are parallel to one another, P P' is bisected in C (II. 29.). Therefore, C is the centre of the ellipse PQ R. Therefore, &c. *143 2 25 1 26 26 1 52 28 1 28 2 45 2 * 46 1 *46 1 46 1 33 equal to read is equal to the points D E read the points D, E triangle A B C read triangle abc 3 from bottom, for H C read AC 16 for [10]. Cor. read [11]. Cor. 2 5 and 2 from bottom, for A': B':: A: B read A': B:: A: B' 4 and 3 from bottom, for A read A', and for A' read A 1 from bottom, for B read B', and for B' read B 1 for B' read B, omitting because it is supposed to be greater than B, which is greater than Q * 46 2 47 1 13 of note, for and d read c and d 45 for def. 7 read def. [7] 12 48 6 64 1 29 64 2 33 70 2 39 A, B, A', B read A, B, A', B' 10. Cor. read 11. Cor. 2 homologous, and read homolo gous; and 145 1 3 from bottom, for themselves read 33 for altitude GH read altitude CH 16 ABCD read ABCDE *146 1 27 omit the reference (II. 12) *150 1 in the figure the line AE is dotted by mistake 30 for I. 12 read I. 12. Cor. 1. 152 2 154 2 157 1 16 4 - G H, being read G H being AB, AC read O B, OC 1 - ABCD read ABCDE 4 from bottom, for FL read E L I in the lower figure, for u read U 19 for right angle read right-angled 21 1 Cor. 2 read 2 Cor. 2 IV. 32 read IV. 32. Cor. 2 12 from bottom, for pyramid read cone 13 for 5. Cor. 1 read 7. Cor. 1 26 pyramid read cone 174 1 in the figure, the letter C is wanting at the 174 1 175 2 Euc. I. 17 read Enc. VI. 17 I. 38 read I. 34 181 1 *182 1 1.5 read I. 6 182 1 182 2 75 78 1 1 19 2 78 2 in the figure of the Scholium, for C' 1 83 1 in the figure of the Scholium, for the C 92 2 22. nearest to A read c 1 99 1 101 2 109 2 17 is greater read is, greater 1 from bottom, for impracticable read impracticable in the way of calculation 2 from bottom, for II. 17 read II. 13 41 for as may easily be shown read (11.23) 25 for P C D read B C D 110 1 *110 2 31 PA2 X A G read P A2 X EG *111 1 16 and 17 for (produced if necessary) read produced *111 2 5 for B C D read B C D and the point A *111 2 read line 19 immediately before line 18 117 1 56 for of given read of the given 120 1 31 - 12. Cor. 1 read 11. Cor. 41 12. Cor. 1 read 11. Cor. *122 2 in the figure the letter C is wanting between N and B Page 93, col. 2, after Book III. prop. 28, add, Cor. In an isosceles triangle, which has each of the equal angles double of the vertical angle, the sides and base are in extreme and mean proportion; and conversely. In such a triangle, each of the equal angles is four-fifths of a right angle (I. 19.) As we are enabled, therefore, to describe such a triangle by II. 59, we can thus divide a right angle into five equal parts, as was observed in the scholium of p. 26. And generally, if a regular polygon of n sides can be inscribed in a circle, (as, in the present instance, the regular polygon of five sides,) a right angle may be divided into n equal parts, by taking for one of those parts (I. 46. Cor.) a fourth of the angle which the side of the polygon subtends at the centre of the circle. (I. 3. Cor.) INDE X. The theorems and problems of Plane Geometry will be found under the heads Straight Line, Angle, Triangle, Square, Rhombus, Rectangle, Parallelogram, Quadrilateral, Rectilineal Figure, Circle; those of Solid Geometry under Plane, Dihedral Angle, Solid Angle, Tetrahedron, Cube, Rectangular Parallelopiped, Parallelopiped, Prism, Pyramid, Polyhedron, Regular Polyhedron, Cylinder, Cone, Sphere, Solid of Revolution; of Spherical Geometry under that head; of Ratios and Proportion under those respective heads; and so of Projection and the Conic Sections. The parts included in unciæ are additions; having been made, either with the view of supplying such connecting links as seemed wanting in the present digest of the whole work, as in "Circle" (E) and (G); or of completing what had been left imperfect, as in the notes on "Proportion" and "Rectangle;" or of extending and generalising where only partial views had been given, as under the heads "similar," "symmetrical," " touch;" or of adding whatever of use or interest had been inadvertently omitted, as in " Annulus," "Lunes," and the note on "Centrolinead." ABSCISSA (Lat., a part cut off) of a conic section def. 220 Acute, (Lat., pointed,) a term applied to angles, whether plane, dihedral, or spherical, which are less than right angles. Adjacent angles are those which one straight line, or plane, or spherical arc makes with another upon one side of it. Affection, angles said to be of the same, or of different affections note 62 Algebra, its signs +, -, x, &c., borrowed with advantage by geometry Alternando, a rule in Proportion. See "Proportion." 20 Alternate, certain angles said to be, which are made by two straight lines (or planes), with a third straight line (or plane) 13 Altitude, of any figure, is a perpendicular drawn to the base from the vertex, line, or plane, opposite to it. Analysis, (Gr., undoing, or taking to pieces,) in geometry, is that mode of demonstrating a theorem, or solving a problem, which searches into the thing proposed, and takes it (as it were) to pieces, in order to discover the more simple truths and constructions upon which it is built: the reverse process is called Synthesis, (Gr, putting together,) and proceeds in a didactic form, by the putting together of truths and constructions already established, to do, or establish the certainty of the thing proposed 107 Angle, dihedral. See "Dihedral Angle." Angle, rectilineal def. 1 When said to be right, oblique, acute, obtuse def.2 angular note 5 85 85 More complete definition of magnitude (a) The magnitude of an angle is independent of the extent of its legs def. 1 (b) Equal angles may be made to coincide ax. 4 (c) All right angles are equal to one another 4 (d) Every angle is measured by the cir cular arc, which is described about the angular point with a given radius, and is included between the legs, sch. 85 (e) By continued bisections, a given angle may be divided into 2, 4, 8, 16, &c. equal parts; but the division of an angle (in general) into any other number of equal parts is impracticable by a plane construction, i. e. with the right line and circle only sch. 26 25 (f) To bisect a given angle (g) At a given point in a given straight line to make an angle equal to a given angle See "Straight Line." 26 Angle, solid. (See "Solid Angle.") def. 125 Angle, spherical. (See "Spherical Geometry.") def. 185 Angle in a segment. (See "Circle.”) def. 79 [Annulus (Lat., a ring,) a name given to the space which is the difference of two concentric circles-two annuli are said to be similar, when the radii of the interior and exterior circumferences are to one another in the same ratio. See "Circle."] Antecedent, of a ratio, is the leading term: the antecedents of a proportion are the first and third terms def. 32, 33 Apothem of a regular polygon def. 91 Approximation to the area of a circle, when the radius is given; or, to the radius when the circumference is given; or, to the radius when the area is given. (See "Circle.") sch. 97, 98 Arc (Lat., a bow) of a circle. (See "Circle.") def. 78 Arc, spherical. (See "Spherical Geometry.") def. 184 Area (Lat., a floor) means, sometimes, the same as surface; but is more properly applied to signify the number of times any surface contains the superficial unit sch. 18 Area of a triangle. See "Triangle.” Area of the circle. See "Circle." Arithmetical mean, between two magnitudes, is a magnitude which exceeds the lesser of the two by as much as it falls short of the greater. (b) Is greater than the geometrical mean cor. 42 (c) Arithmetical means being continually taken between two magnitudes, between the new mean and the last, and so on; to arrive at the approximate result immediately, after a certain number have been taken note 98 Arithmetical progression. Magnitudes A, B, C, D, &c. are said to be in arithmetical progression, when their successive differences are equal, i. e. when A is as much greater or less than B, as B is than C, as C is than D, and so on. Arithmetical theory of proportion. (See "Proportion.") In what respects inadequate to the purposes of geometry note 57 Arris of a solid. (See "Polyhedron.") def. 126 As. Peculiar use of the word, in expressing certain proportions Asymptote of a curve. tions," and "Projection.") Axiom (Gr.), a self-evident truth Axioms of equality and inequality Axis of a circle of the sphere def. 184 Of a cone or cylinder def. 166, 167 Of a regular prism or pyramid def. 127 Axis of a figure of revolution (and in this sense of a sphere) is the straight line about which the revolution is supposed to take place. Axis of a conic section def. 217 def. 217 def. 226 men. * A more simple instrument, for the same purpose, consisting of three rulers, which are stiffly moveable about a common joint in the same, or in parallel planes, is likewise of frequent use among draughtsIts form is not so convenient, neither from the want of the additional rulers is its adjustment susceptible of the same accuracy as Mr. Nicholson's. It derives its use, however, from the same principle, and the manner of applying it may be explained as follows: Let A B and CD be two given lines, converging to the distant point O, and let it be required to draw through any given point P a straight line, which shall pass through the same point O. In AB take any point A, and in CD any point C, and join AC; Arc, chord, segment, sector, tangent def. 78, 79 Similar arcs, similar segments, similar sectors (A.) First Properties, III. § 1. def. 91 (a) If a straight line meets a circle in two points, it cuts the circle in those points, and the part between them falls within the circle 79 (b) A straight line cannot meet a circle in more than two points cor. 79 (c) A circle is every where concave towards its centre cor. 79 79 (d) The straight line which is drawn at (B). Of the Diameter and other Chords, III. §1. (a) The diameters of the same circle are equal to one another let QA, QP, QC, represent the three rulers, having the common joint Q, and let the angles PQA, PQC be set (or made equal) to the angles A CD and BAC respectively; fix two pins, one at A, the other at C, and move the rulers AQ, CQ, along these pins, until QP passes through the point P; let the ruler Q P be then steadily fixed, and the line QP drawn; QP is the line required. For, if a circle be described through the points A, C, O, the point Q will always lie in its circumference, because AQC is equal to the sum (fig. 1) or the difference (fig. 2) of PQA and PQC, i. e. of BAC and ACD, i. e. to the supplement of the angle O (fig. 1), or to the angle O (fig. 2); there fore, since the angle PQC is equal to BAC, QP produced passes through O. |