generated by the rectangle HB, that is, by a solid less than W, for the cylinder generated by HB is less than W. In the same manner, it may be demonstrated, that the sum of the cylinders circumscribing the cone ICE is greater than 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 cylinders 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. Q. E. D, PROP. XX. 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 DCB 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 revolution of the rectangles Hs, Gr, Fq, and DN are equal to the cylinder described by the revolution of the rectangle DB. Let L be the point in which GO meets the circle ADB, 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 distances 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, Еe Fq, DN, are together equal to the cylinder described by DB, that is, to the cylinder having the same base and altitude with the hemisphere Q. E. D. PROP. XXI. 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 hl A I B 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 shewp (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. Q. E. D. END OF THE SUPPLEMENT TO THE ELEMENTS. T NRIGONOMETRY 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. Thus, the product of two 'numbers A and B, is either denoted by A.B or AXB; and the products of two or more into one, or into more than one, as of A+B into C, or of A+B into C+D, are expressed thus : (A+B) C, (A+B) (C+D), or sometimes thus, A+B x C, and A+B x C+D. The quotient of one number A, divided by another B, is written A thus, B The sign is used to signify the square root : Thus M is the square root of M, or it is a number which, if multiplied into itself, will produce M. So also, v_M3 + N2 is the square root of Mu+N2, &c. 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 arch 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 AČ : the angle ABC is to four right angles as the arch AC to the whole circumference ACF. |