the cone by a solid less than the cylinder generated by the rectangle DN, that is, by a solid less than W. Therefore, since the sum of the cylinders inscribed in the hemisphere, together with a solid less than W, is equal to the hemisphere; and, since the sum of the cylinders described about the cone is equal to the cone together with a solid less than W; adding equals to equals, the sum of all these cylinders, together with a solid less than W, is equal to the sum of the hemisphere and the cone together with a solid less than W. Therefore, the difference between the whole of the cylin ders and the sum of the hemisphere and the cone, is equal to the difference of two solids, which are each of them less than W; but this difference must also be less than W, therefore the difference between the two series of cylinders and the sum of the hemisphere and cone is less than the given solid W. PROP. XX. THEOR. The same things being supposed as in the last proposition, the sum of all the cylinders inscribed in the hemisphere, and described about the cone, is equal to a cylinder, having the same base and altitude with the hemisphere. Let the figure BCD be constructed as before, and supposed to revolve about CD; the cylinders inscribed in the hemisphere, that is, the cylinders described by the revolution of the rectangles Hh, Gg, Ff, together with those described about the cone, that is, the cylinders described by the revo lution of the rectangles Hs, Gr, Fq, and DN are equal to the cylinder de scribed by the revolution of the rectangle BD. Let L be the point in which GO meets the circle ABD, then, because CGL is a right angle if CL be joined, the circles described with the distances CG and GL are equal to the circle described with the distance CL (2. Cor. 6. 1 Sup.) or GO; now, CG is equal to GR, because CD is equal to DE, and therefore also, the circles described with the distance GR and GL are together equal to the circle described with the distance GO, that is, the circles described by the revolution of GR and GL about the point G, are together equal to the circle described by the revolution of GO about the same point G; therefore also, the cylinders that stand upon the two first of these circles, having the common altitudes GH, are equal to the cylinder which stands on the remaining circle, and which has the same altitude GH. The cylinders described by the revolution of the rectangles Gg, and Gr are therefore equal to the cylinder described by the rectangle GP. And as the same may be shewn of all the rest, therefore the cylinders described by the rectangles Hh, Gg, Ff, and by the rectangles Hs, Gr, Fq, DN, are together equal to the cylinder described by BD, that is, to the cylinder having the same base and altitude with the hemisphere. PROP. XXI. THEOR. Every sphere is two-thirds of the circumscribing cylinder. Let the figure be constructed as in the two last propositions, and if the hemisphere described by BDC be not equal to two-thirds of the cylinder described by BD, let it be greater by the solid W. Then, as the cone described by CDE is one-third of the cylinder (18. 3. Sup.) described by BD, the cone and the hemisphere together will exceed the cylinder by W. But that cylinder is equal to the sum of all the cylinders described by the rectangles Hh, Gg, Ff, Hs, Gr, Fq, DN (20. 3. Sup.); therefore the hemisphere and the cone added together exceed the sum of all these cylinders by the given solid W, which is absurd; for it has been shewn (19. 3. Sup.), that the hemisphere and the cone together differ from the sum of the cylinders by a solid less than W. The hemisphere is therefore equal to two-thirds of the cylinder described by the rectangle BD; and therefore the whole sphere is equal to two-thirds of the cylinder described by twice the rectangle BD, that is, to two-thirds of the circumscribing cylinder. END OF THE SUPPLEMENT TO THE ELEMENTS. ELEMENTS OF PLANE TRIGONOMETRY, TRIGONOMETRY is the application of Arithmetic to Geometry: or, more precisely, it is the application of number to express the relations of the sides and angles of triangles to one another. It therefore necessarily supposes the elementary operations of arithmetic to be understood, and it borrows from that science several of the signs or characters which peculiarly belong to it. The elements of Plane Trigonometry, as laid down here, are divided into three sections: the first explains the principles; the second delivers the rules of calculation; the third contains the construction of trigonometrical tables, together with the investigation of some theorems, useful for extending trigonometry to the solution of the more difficult problems. SECTION I. LEMMA I. An angle at the centre of a circle is to four right angles as the arc on which it stands is to the whole circumference. Let ABC be an angle at the centre of the circle ACF, standing on the circumference AC: the angle ABC is to four right angles as the arc AC to the whole circumference ACF. Produce AB till it meet the circle in E, and draw DBF perpendicular to AE. Then, because ABC, ABD are two angles at the centre of the circle ACF, the angle ABC is to the angle ABD as the arc AC to the arc AD, (33. 6.); and therefore also, the angle ABC is to four times the angle ABD as the arc AC to four times the arc AD (4. 5.). But ABD is a right angle, and therefore four times the arc AD is equal to E H K B A the whole circumference ACF; therefore the angle ABC is to four right angles as the arc AC to the whole circumference ACF. COR. Equal angles at the centres of different circles stand on arcs which have the same ratio to their. circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arcs AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC to four right angles; and the arc HG is to the whole circumference of the circle GHK in the same ratio. DEFINITIONS. 1. Ir two straight lines intersect one another in the centre of a circle, the arc of the circumference intercepted between them is called the Measure of the angle which they contain. Thus the arc AC is the measure of the angle ABC. 2. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a Degree; and if a degree be divided into 60 equal parts, each of these is called a Minute; and if a minute be divided into 60 equal parts, each of them is called a Second, and so on. And as many degrees, minutes, seconds, &c. as are in any arc, so many degrees, minutes, seconds, &c. are said to be in the angle measured by that arc. COR. 1. Any arc is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it is to the number 360. And any angle is to four right angles as the number of degrees and parts of a degree in the arc, which is the measure of that angle, is to 360. COR. 2. Hence also, the arcs which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree. For the number of degrees and parts of a degree contained in each of these arcs has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.). The degrees, minutes, seconds, &c. contained in any arc or angle, are usually written as in this example, 49°. 36'. 24". 42′′; that is, 49 de grees, 36 minutes, 24 seconds, and 42 thirds. 3. Two angles, which are together equal to two right angles, or two arcs which are together equal to a semicircle, are called the Supplements of one another. 4. A straight line GD drawn through C, one of the extremities of the aro 5. The segment DA of the diameter passing through A, one extremity of the arc AC, between the sine CD and the point A, is called the Versed sine of the arc AC, or of the angle ABC. 6. A straight line AE touching the circle at A, one extremity of the arc AC, and meeting the diameter BC, which passes through C the other extremity, is called the Tangent of the arc AC, or of the angle ABC COR. The tangent of half a right angle is equal to the radius. 7. The straight line BE, between the centre and the extremity of the tangent AE is called the Secant of the arc AC, or of the angle ABC. COR. to Def. 4, 6, 7, the sine, tangent and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF. It is manifest, from Def. 4. that CD is the sine of the angle CBF. Let CB be produced till it meet the circle again in I; and it is also manifest, that AE is the tangent, and BE the secant, of the angle ABI, or CBF, from Def. 6. 7. COR. to Def. 4, 5, 6, 7. The sine, versed sine, tangent, and secant of an arc, which is the measure of any gi ven angle ABC, is to the sine, versed sine, tangent and secant, of any other arc which is the measure of the same angle, as the radius of the first arc is to the radius of the second. B OMD A Let AC, MN be measures of the angle ABC, according to Def. 1.; CB the sine, DA the versed sine. AE the tangent, and BE the secant of the arc AC, according to Def. 4, 5, 6, 7, NO the sine, OM the versed sine, MP the tangent, and BP the secant of the arc MN, according to the same definitions. Since CD, NO, AE, MP are parallel, CD : NO :: rad. CB: rad. NB, and AE : MP :: rad. AB: rad. BM, also BE BP :: AB: BM; likewise because BC: BD :: BN: BQ, that is, BA: BD :: BM: BO, by conversion and alternation, AD: MO ;; AB: MB. Hence the corollary is manifest. And |