236 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 note 98 number have been taken 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 sch. 62 (See "Conic Sec def. 211 4 4 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 Principal or transverse def. 217 def. 217 def. 226 Base of a triangle, def. 2—of a pyramid, 127 -of a cone, 167—of a spherical segment, 179-of a spherical sector, 179. Bases of a parallelopiped, def. 126-of a prism, 127-of a cylinder, 166. Centre of a circle, def. 3-of a regular polygon, 91-of a sphere, 127-of a regular polyhedron, 161-of an ellipse, or hyperbola Centrolinead, an instrument for drawing converging lines 217 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 draughts men. Its 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 A B take any point A, and in CD any point C, and join AC; (A.) (d) The straight line which is drawn at cor. 80 (h) Tangents at the extremities of the same diameter are parallel cor. 80 (B). Of the Diameter and other Chords, III. § 1. (a) The diameters of the same circle are equal to one another def. 3 (b) Every diameter divides the circle and its circumference into two equal note 78 parts (c) The diameter is the greatest straight line in a circle; and, of others, that which is nearer to the centre is greater than the more remote; also, the greater is nearer to the centre than the less 80 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 ACD 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. (d) Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre are equal to one another cor. 81 cor. 80 (e) If a diameter cuts any other chord at right angles, it bisects it; and conversely, if a diameter bisects any other chord, it cuts it at right angles 80 (f) Two chords of a circle cannot bisect one another, except they both pass through the centre (g) The straight line which bisects any chord at right angles, passes through the centre of the circle cor. 80 (h) If two circles have a common chord, it shall be bisected at right angles by the straight line joining their centres cor. 80 (1) A diameter bisects all chords which are parallel to the tangent at either of its extremities cor. 80 [(k) Every diameter divides the circle symmetrically. See "Symmetrically divided."] (7) If a point be taken, from which to the circumference of a circle there fall more than two equal straight lines, that point is the centre of the circle 81 (m) From any other point than the centre there cannot be drawn to the circumference of a circle more than two straight lines that are equal to one another, whether the point be within or without the circle cor. 81 (n) If a point be taken within a circle which is not the centre, of all the straight lines which can be drawn from that point to the circumference, the greatest is that which passes through the centre, and the other part of that diameter is the least; also, of any others, that which is nearer to the greatest is greater than the more remote, and any two, which are equally distant from the greatest upon either side of it, are equal to one another sch. 83 (0) If a point be taken without a circle, and straight lines be drawn from it to the circumference, of those which fall upon the concave circumference, the greatest is that which passes through the centre, and of the rest that which is nearer to the greatest is greater than the more remote, and any two which are equally distant from the greatest upon either side of it are equal to one another; also, of those which fall upon the convex circumference, the least is that between the point without the circle and the diameter, and of the rest, that which is nearer to the least is less than the more remote, and any two which are equally distant from the least upon either side of it are equal to one another sch. 83 81 cor. 81 (c) Circles cannot cut one another in more than two points 82 (d) If two circles meet one another in a point which is not in the straight line joining their centres, or in that straight line produced, they shall meet one another in a second point upon the other side of that straight line, and shall cut one another 82 (e) If two circles meet one another in a point which is in the straight line joining their centres, or in that straight line produced, they meet in no other point; the circumference of the circle which has the greater radius falls wholly without the circumference of the other, and the circles touch one another 83 (f) Circles which cut one another meet in two points, [and the distance between the two is bisected at right angles by the straight line which joins their centres] cor. 83 (g) If two circles cut one another, the straight line which joins their centres is less than the sum, and greater than the difference of their radii cor. 83 (h) Circles which touch one another meet in one point only; and the straight line which joins their centres, or that straight line produced, passes through the point of contact cor. 82 (i) If two circles touch one another, the distance of their centres is equal to the sum or to the difference of their radii; the sum, if they touch externally; the difference, if they touch internally cor. 82 (k) If the circumferences of two circles do not meet one another in any point, the distance between their centres shall be greater than the sum, or less than the difference of their radii, according as each of the circles is without the other, or one of them within the other; [and the circles approach nearest to one another in the straight line joining their centres, or in that straight line produced] 83 (7) If the distance between the centres of two circles be at once less than the sum, and greater than the difference of their radii, the circles will cut one another; if that distance be equal to the sum, or to the difference of the radii, the circles will touch one another; and, if that distance be greater than the sum, or less than the difference of the radii, the circles will not meet one another cor. 84 (D) Of Arcs and Angles in a Circle, III. $2. (a) In the same, or in equal circles, 85 86 (i) If a triangle and a circular segment stand upon the same base and upon the same side of it, the vertex of the triangle will fall without, or within, or upon the arc of the segment, according as the vertical angle is less than, or greater than, or equal to, the angle in the segment cor. 87 87 (h) If any chord be drawn in a circle, the angles contained in the two opposite segments shall be, together, equal to two right angles (1) The opposite angles of a quadrilateral in a circle are, together, equal to two right angles cor. 87 (m) If at one extremity of a chord a tangent be drawn, the angles which it makes with the chord shall be equal to the angles in the alternate segments; and conversely, if at one extremity of a chord a straight line be drawn such that the angles it makes with the chord are equal to the angles in the alternate segments, such straight line is a tangent 87 (E) if the arc between a chord and tangent be bisected by the point of contact, the chord and tangent shall be parallel cor. 88 (p) If any two chords meet one another, the angle contained by them is measured by half the sum, or by half the difference of the intercepted arcs, according as the point in which they meet is within or without the circle 88 (2) If a chord meet a tangent, the angle contained by them is measured by half the difference of the intercepted arcs [and the same measure obtains when two tangents meet one another.] cor. 88 Of Rectangles under the Segments of Chords, III. § 3. [(a) If a diameter bisects any chord, the square of half the bisected chord is equal to the rectangle under the segments of the diameter (I. 36, and 1. 34.)] [(b) The tangents at the extremities of a chord meet one another in the same point T of the bisecting diameter, and that in such a manner, that if C is the centre of the circle, N the point of bisection, and CA the radius, CN, CA, and CT, are proportionals (II. 34.)] (c) If any two chords cut one another, the rectangles under their segments shall be equal, whether they cut one another within or without the circle 88 (d) If two straight lines A B, C D, cut one another in a point E, and if the points A, B, and C, D, are taken, (the two first upon the same side of E, and the two last likewise upon the same side; or the two first upon opposite sides of E, and the two last likewise upon opposite sides,) so that the rectangle under A E, EB, shall be equal to the rectangle under CE, ED, the points A, B, C, D, shall lie in the circumference of the same circle cor. 89 (e) If a chord meet a tangent, the square of the tangent shall be equal to the rectangle under the segments of the chord 89 (f) If two straight lines, A B and CE, meet one another in a point E, and if the points A, B, and C, are so taken, that, A and B being upon the same side of E, the square of E C is equal to the rectangle under AE, EB, the straight line EC shall touch the circle which passes through the points A, B, C cor. 89 (g) If a triangle be inscribed in a circle, and if a perpendicular be drawn from the vertex to the base, the rectangle under the two sides shall be equal to the rectangle under the perpendicular and the diameter of the circle (h) If triangles are inscribed in the same, or in equal circles, the rectangles 90 (k) If a quadrilateral be inscribed in a circle, the rectangle under its diagonals shall be equal to the sum of the rectangles under its opposite sides 90 (If from any point without a circle, two straight lines are drawn to touch the circle, every straight line which is drawn through that point to cut the circle shall be harmonically divided by the circumference, and the chord joining the points of contact; and the tangents at the points in which every such straight line cuts the circumference shall meet one another in the chord produced lem. 220 (m) If through any point taken within or without a circle, there are drawn any number of straight lines, each cutting the circle in two points, and if at every such two points tangents are drawn intersecting one another in a point P, the locus of the points P shall be a straight line; and every straight line which is drawn through the point taken to cut the circle shall be harmonically divided by that straight line and the circle cor. 112 (F) Of regular Polygons, inscribed and circumscribed, III. § 4. 91 (a) If any two adjoining angles of a regular polygon be bisected, the intersection of the bisecting lines will be the common centre of two circles, the one circumscribed about, the other inscribed in, the polygon (b) If the circumference of a circle be divided into any number of equal parts, the chords joining the points of division shall include a regular polygon inscribed in the circle; and the tangents drawn through those points shall include a regular polygon of the same number of sides circumscribed about the circle 92 (c) If any regular polygon be inscribed in a circle, a similar polygon may be circumscribed about the circle by drawing tangents through the angular points of the former; and, conversely cor. 92 (d) If any regular polygon be inscribed in a circle, and if a tangent be drawn parallel to one of its sides, and be terminated both ways by radii passing through the extremities of that side, such terminated tangent shall be a side of a similar polygon, circumscribed cor. 92 about the circle (G) (e) The side of a regular hexagon is equal to the radius of the circle in which it is inscribed; the side of a regular decagon is equal to the greater segment of the radius divided medially; and the side-square of a regular pentagon *is greater than the square of the radius by the side-square of a regular decagon inscribed in the same circle 93 (f) If K and L represent two regular polygons of the same number of sides, the one inscribed in, and the other circumscribed about the same circle, and if M and N represent the inscribed and circumscribed polygons of twice the number of sides; M shall be a geometrical mean between K and L, and N an harmonical mean between L and M 96 (g) If k and represent the radii of the circles which are inscribed in any regular polygon, and circumscribed about it, and if m and n represent these radii for a regular polygon which has twice as many sides as the former, and an equal perimeter; m shall be an arithmetical mean between k and 1, and n a geometrical mean between and m note 98 (h) If k and represent the radii of the circles which are circumscribed about any regular polygon and inscribed in it, and m an arithmetical mean between them; and if k' and represent these radii for a regular polygon which has twice as many sides as the former and an equal area, k' shall be a mean proportional between k and , and a mean proportional between and m note 98 (i) The area of any regular polygon is equal to half the rectangle under its perimeter and the radius of the inscribed circle 93 Of the Circumference and Area of the Circle, III. § 4, 5. [(a) The circumference of a circle is greater than the perimeter of any inscribed polygon, and less than that of any circumscribed polygon; so also The diagonals, also, of a regular pentagon are severally parallel to its sides, and greater than them in extreme and mean ratio. Note, 159. 240 (e) The circumferences of circles are as the radii, and their areas are (in the duplicate ratio, or) as the squares of 96 the radii (f) The ratio of the circumference to the diameter is 3.1415926535 &c.; to which number we may approximate by any of the theorems (f), (g), (h), sch. 97 in the last division For rules to abridge the calculation, see note 97 (g) The circumference and diameter have been demonstrated to be incommensurable, so that their approximasch. 99 tion has no limit (h) If R represents the radius of a circle, and the number 3.1415 &c. the circumference = 2 R, and the area sch. 99 =R2 (i) The circle is a mean proportional between any circumscribed polygon and a similar polygon which has the same perimeter with the circle (k) The circle is greater than any (regular polygon, or any rectilineal figure, or any other) plane figure whatever which has the same perimeter; and has a less perimeter than any other plane figure whatever which has the 102, 105 same area note 102 (i) To inscribe in or circumscribe about a given circle 1. An equilateral triangle. 2. A square. 3. A regular pentagon. 4. A regular hexagon. 5. A regular decagon. 6. A regular pentedecagon 120 (k) To inscribe in or circumscribe about a given circle a regular figure of any required number of sides, very nearly sch. 120 (1) To describe a circle, in which four given straight lines, of which every one is less than the other three, shall together subtend the whole circumfer. 91 ence • Describe a circle about the same centre, touching a chord which is drawn from any point of the cir cumference to cut off a segment containing the given angle, and through the given point draw a tangent to this circle. |