The terms Tangent and Secant are often restricted in meaning;-the first to the line which by one extremity touches an extremity of the diameter at right angles to it, and which has its other extremity terminated by a straight line, the Secant, from the centre of the circle across the circumference; the second to the line from the centre, across the circumference, and terminated by the tangent. Thus BC is the tangent and DB the secant of the arc CH, or of the angle CDB measured by the arc; the name Cosine being given to the space DK cut off between the centre D and the sine; and Versed Sine to the space KC between the sine and the tangent point C. The term Sine denotes a perpendicular to a diameter from the point where the secant crosses the circumference; as HK. The terms Tangent, Secant, Sine, &c., thus restricted, were of continual use in Trigonometry; and with a widely extended meaning are now constantly employed. The process is instructive, by which extension has been given to Trigonometrical Symbols, and may thus be briefly stated; 1. Sine, cosine, &c., at first denoted lines so named drawn in and about a circle, with reference to an angle at the centre, and measured by its arc, each angle having a different sine, &c., according as the radius of the circle was increased or diminished in length. 2. To avoid these continual diversities, that radius was supposed always to be a unit, or rather the unit of measurement for the other lines; and the secant, sines, &c., to be multiples or fractional parts of that unit; thus the sine of 60°, being equal to the radius, was unity, or 1, and the sine of 30° was, or 5. The names sines, cosines, &c., in this way lost their first meaning; they denoted, not lines, but the numerical ratios of those lines to the radius, and were abstract numbers. 3. Another step was to represent the angle itself by an abstract number. Degrees and minutes had been the measure of the central angle, that angle was measured by its arc, and the arc bore a numerical ratio to the unit of measurement, the radius. 4. A fourth step made the process perfect. Hitherto the sum of the angles could not exceed four right angles, but this limit also was to be passed. The idea of a line revolving round a point, and continuing its rotation after a revolution had been completed, originated the method of using angles consisting of more than four right angles. Thus angles, sines, &c., were all represented by numbers; and though the old names were retained, Trigonometry which at first was a simple application of Geometrical truths, and which still rests on Geometry for its foundation, became a branch of the higher Arithmetic, and has its operations conducted on arithmetical and algebraical principles. 3. Circles are said to touch one another, which meet, but do not cut one another. Thus the circle of which L is the centre, touches EFC in E, and circle G touches it in F. 4. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal; thus EF and GH are equally distant from C, when perp. CA = perp. CB. E K D B 5. And the straight line on which the greater perpendicular falls, is said to be farther from the centre; thus IK is farther from C than G H is, because the perp. CD > perp. C B. F G As the distance from the vertex of a triangle to its base is measured by a perpendicular, so the distance of a straight line from the centre of a circle is the perpendicular drawn to it from the centre. A Proposition analagous to Props. 7 and 8, Book III., would explain "why the perpendicular from a point on a straight line is called the distance from that line." 6. A segment of a circle is the figure contained by a straight line, and the circumference which it cuts off; as the fig. ABCA. "A figure included by an arc and its chord is The The straight line of a segment, as AB, is named 7. The angle of a segment is that which is contained by the straight line and the circumference; as angle ABC contained by AB and the arc BCA. 8. An angle in a segment is the angle contained by two straight lines drawn from any point of the circumference of the segment, to the extremities of the straight line which is the base of the segment; as ACB, or ▲ ADB. 9. An angle is said to insist, or, stand upon the circumference intercepted between the straight lines that contain the angle; as ZACB on arc AEB. 10. A sector of a circle is the figure contained by two straight lines drawn from the centre, and by the circumference between them; as ACB. Sectors are equal when they have equal radii and equal angles; for by superposition their boun- Certain Sectors of a definite size are known by 11. Similar segments of circles are those in which the angles are equal, or which contain equal angles; as ▲ ACB = / DF E. B DD The idea of similarity here introduced belongs to A 12. Concentric circles are such as have a common centre; thus, circle A and circle B have the same centre C. A B AXIOM A. "If the distance of a point from the centre of a circle be less than the radius of the circle, the point is within the circle; and if the distance of a point from the centre of a circle is greater than the radius, the point is without the circle."-Hose, p. 300. See also ScH. 2, Pr. 1, III. PROPOSITIONS. PROP. 1.-PROB. To find the centre of a given circle. SOL.-Pst. 1. Let it be granted that a st. line may be drawn from any one point to any other point. 10, I. To bisect a given finite st. line. 11, I. To draw a st. line at rt.s to a given st. line from a given point in the same. Pst. 2. A terminated line may be produced to any length in a st. line. DEM.-Def. 15, I. A is a plane figure contained by one line, which is called the circumference, and is such that all st. lines drawn from a certain point within the figure to the Oce are equal to one another. 8, I. If two As have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one, shall be equal to the angle contained by the two sides equal to them of the other. E. 1 Def. 10, I. When a st. line standing on another st. line makes the adj. Dat. D. 1 C.1,Def.15,I| In As ADG, BDG DA=DB, GA=GB COR.-If in a the cen. of the & GD com. .. / ADG=/ BDG, .. BDG is a rt. . an impossibility .. G is not the cen. .. any other point in CE is not the cen., ABC. Q. E. F. a st. line, CE, bisects another, AB, at rt. ≤8, is in the line, CE, which bisects the other. SCH.-1. When the G' is taken in the diam. CE, the demonstration holds good only if G' coincides with F; should this not be the case, it is evident that G'CG'E.. G' is not the cen. 2. The rigour of the reasoning would have been greatly promoted, if Euclid, previously to the above Problem, had established the following proposition; Any point, D, fig. 1, F, fig. 2. being assumed within a O, a rt. line, HD or HF, drawn through it and produced indefinitely in both directions, will meet the in two points, and not in more; and every point of the line between these two points of intersection will be within the O, and every point beyond them without it. LARDNER'S Euclid. p. 91 I. Let HD through D, also pass through the cen. C. C 1 Pst. 3. & 2 Cor. 3, I. D 3 Ax. A. III... the s K & L are without the O. 4 3. I. 5 Ax. A.III. 6 3. I. 7 Def. 15, I. Also from C make CD, CH each the ⚫s H & D are within the O. .. the s A & B on the Oce. II. Let H F through F not pass through the cen. C. From C draw CDFH; FIG. 1. C 1 11 I. |