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BC, CD, DE, amount to two right angles, which would no longer be the case, were the radius made either greater or less ; as the triangles have their vertices at the centre, and contained by the radii and the sides AB, BC, &c., would evidently have the magnitudes of their angles changed.
It is plain that the area of the polygon will be the same in whatever order the sides are arranged; as, however they are placed, they will cut off equal segments, and the polygon is the part of the semicircle remaining, when those segments are taken away.
PROP. XXII. THEOR.
Or quadrilaterals and polygons which have their sides equal, each to each, the greatest is that which can be inscribed in a circle.
In the rectilineal figures ABCDE, FGHKL, let the sides AB, BC, CD, DE, EA be severally equal to FG, GH, HK, KL, LF, and let AD be inscribed in a circle, but FK not be capable of being inscribed in one: AD is greater than FK.
Draw the diameter CM, and join AM, EM ; construct the triangle FNL, having the sides FN, NL respectively equal to AM, ME, so that (I. 8.) the triangles themselves are equal ; and join HN. Then, by the preceding proposition, the figures CBAM, CDEM are respectively greater than HGFN, HKLN; and therefore (I. ax. 4.) the polygon ABCDEM is greater than FGHKLN; and, by taking away the equal triangles AME, FNL, there remains the figure ABCDE greater than FGHKL.
Cor. From this proposition and the twentieth it follows, that of polygons of the same number of sides and of equal perimeters, the regular polygon is the greatest ; and that of quadrilaterals of equal perimeters, the greatest is the square.
Of regular polygons which have equal perimeters, that which has the greater number of sides is the greater.
Let AB be half the side of the polygon which has the less
number of sides, and BC a perpendicular to it, which will evidently pass through the centre of its inscribed or circumscribed circle: let C be that centre, and join AC. Then, ACB will be the angle at the centre subtended by the half side AB. Make BCD equal to the angle subtended at the centre of the other polygon by half its side, and from C as centre, with CD as radius, describe an arc cutting AC in E, and CB produced in F. Then, it is plain, that the angle ACB is to four right angles, as AB to the common perimeter ; and four right angles are to DCB, as the common perimeter to the half of a side of the other polygon, which, for brevity, call S: then, ex æquo, the angle ACB is to DCB, as AB to S. But (VI. 33.) the angle ACB is to DCB, as the sector ECF to the sector DCF; and consequently (V. 11.) the sector ECF is to DCF, as AB to S, and, by division, the sector ECD is to DCF, as AB — Sto S. Now the triangle ACD is greater than the sector CED, and DCB is less than DCF. But (VI. 1.) these triangles are as their bases AD, DB; therefore AD has to DB a greater ratio, than AB - S to S. Hence AB, the sum of the first and second, has to DB, the second, a greater ratio than AB, the sum of the third and fourth, has to S the fourth ;* and therefore (V. 10.) S is greater than DB. Let then BG be equal to S, and draw GH parallel to DC, meeting FC produced in H. Then, since the angles GHB, DCB are equal, BH is the perpendicular drawn from the centre of the polygon having the greater number of sides to one of the sides ; and since this is greater than BC, the like perpendicular in the other polygon, while the perimeters are equal, it follows that the area of that which has the greater number of sides is greater than that of the other,
PROP. XXIV. THEOR.
A circle is greater than any regular polygon of the same perimeter.
For, if a polygon P similar to the one Q proposed for comparison, be described about the circle C; since the area of the circumscribed polygon is evidently equal to the rectangle under its perimeter and half the radius, and (App. I. 39.) the area of the circle equal to the rectangle under its perimeter and half its radius; therefore (VI. 1.) P is to C, as the perimeter of P to the
This may be proved by dividing the first term by the second, and the third by the fourth, and adding unity to each quotient. See the Supplement to Book V.
perimeter of C, or of Q: and (VI. 20.) P is to Q in the duplicate ratio of a side of P to a side of Q, or (VI. 20. cor. 4.) in the duplicate ratio of the perimeter of P to the perimeter of Q. Hence P is to a mean proportional between P and Q, as the perimeter of P to the perimeter of Q; and therefore (V. 11.) P is to C, as P to a mean proportional between P and Q ; that is, (V.9.) C is a mean proportional between P and Q ; and (V. 14.) being less than P, it is greater than Q, the polygon of equal perimeter.
Schol. Every thing here proved will evidently hold equally regarding a circle C, as compared with any polygon Q of equal perimeter, which is such that a circle can be inscribed in it: for then a polygon P, similar to Q, can be described about C.
Cor. Hence it appears also, that a circle is a mean proportional between any rectilineal figure described about it, and a similar one of equal perimeter with the circle.
ELEMENTS OF PLANE TRIGONOMETRY.*
1. Two perpendicular diameters divide a circle into four parts called quadrants. These are evidently all equal to one another. +
2. If the arc of a quadrant be divided into 90 equal parts, each of the parts is called a degree. If a degree be divided into 60 equal parts, each is called a minute ; and if a minute be divided into 60 equal parts, each of them is called a second. I
Hence the arc of a semicircle contains 180 degrees, and the entire circumference 360 degrees.
Degrees, minutes, seconds, are usually denoted by the marks °, ',". Thus, 23° 27' 38" is read 23 degrees, 27 minutes, 38 seconds.
Cor. Since (App. I. 38.) the circumferences of circles are proportional to their radii; degrees, minutes, and seconds in different circles are likewise proportional to the radii of those circles. Thus, let C, R, and D denote respectively the circumference of a circle, its radius, and an arc of the circumference of one degree : and let c, r, and d respectively denote the like lines in another circle. Then R:r::C:c, and (V. 15.) D:d :: C:c: whence (V. 11.) R:1::D:d. Hence, alternately, and by inversion, D: R::d:r, or (V. Sup. 2.) R=Let now n denote any number of degrees and parts of a degree; and by multiplying
nd nd by it, we obtain "R=: whence it appears, that if, in different circles, there be arcs containing the same number of degrees and parts of a degree, the quotients obtained by dividing the arcs by their respective radii are equal.
Plane Trigonometry, in its primitive meaning, is that branch of mathematical science which determines by computation, certain sides or angles of a plane triangle, by means of others that are given in numbers. In its extended and improved state, it is a powerful instrument of investigation in many of the higher speculations in mathematics. The short tract here given will be confined to the former object.
+ As the circumference of a circle is sometimes called the circle, so the arc of a quadrant is often called a quadrant.
I In several modern French works, the arc of the quadrant is divided into 100 equal parts called degrees, the degree into 100 minutes, and the minute into 100 seconds.
3. An arc of a circle described from the vertex of an angle as centre, is assumed as the measure of an angle : and the angle is said to contain as many degrees as there are degrees in the arc.
Thus, if ABC be a right angle, the arc AC is evidently that of a quadrant: the right angle is therefore said to be an angle of 90°. If, again, the arc AD contain 32° 30', the angle ABD is said to be an angle of 32° 30'.
Schol. That the arc of a circle described from the vertex of an angle as centre, is the proper measure of the angle, is plain from the 27th proposition of the third book, or from the 33d of the sixth book : as it is shown in the former, that angles at the centre, which stand on equal arcs, are equal ; and in the latter, that angles at the centre are proportional to the arcs on which
4. The difference between an arc and a quadrant, or between an angle and a right angle, is called the complement of that arc or angle.
5. The straight line drawn from one extremity of an arc, perpendicular to the diameter passing through the other extremity, is called the sine of that arc, or of the angle measured by it: and the part of that diameter intercepted between the sine and the arc, is called the versed sine of the arc, or of the angle which it measures.
6. If a straight line touch a circle at one extremity of an arc, the part of it intercepted between that extremity and the diameter produced, passing through the other, is called the tangent of the arc, or of the angle which it measures ; and the straight line drawn from the centre to the remote extremity of the tangent, is called the secant of the arc or angle.
7. The cosine of an arc or angle is the sine of its complement. In like manner, the coversed sine, cotangent, and cosecant of an arc or angle, are respectively the versed sine, tangent, and secant of its complement.
The sine, versed sine, tangent, and secant may be denoted by the abbreviated expressions, sin, versin, tan, and sec; and the cosine, coversed sine, cotangent, and cosecant, by cos, coversin, cotan, and
For the sake of simplicity, the radius of the circle employed for comparing different angles, is generally taken in investigations as unity: when this is not done, it is denoted by its initial letter R.
The sides of a triangle are often conveniently denoted by the small letters corresponding to the capital ones placed at the opposite angles. Thus, a denotes the side opposite to the angle A, &c. To prevent ambiguity, we may read A, B, C, angle A, angle B, angle C: while a, b, c, may be called side a, side b, side c.
To illustrate the foregoing definitions, let C be the centre of a circle, and AB, DE two diameters perpendicular to each other. Through any point F in the circumference draw the diameter FL; draw FG perpendicular to AB, and FI to DE; through A