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Book III.

PRO P. XXI.

E

VERY sphere is two thirds of the circumscri

bing 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 a described by BD, the cone and the a 18. 3.Supo hemisphere together will exceed the cylinder by W. But

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that cylinder is equal to the sum of all the cylinders described by the rectangle Hh, Gg, Ff, Hs, Gr, Fq, DNb; therefore b 20.3. Sup. the hemisphere and the cone added together exceed the sum of all these cylinders by the given folid W; which is absurd, for it has been thewn , that the hemisphere and the c19. 3. Sup. -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 circumfcribing cylinder, C. E. D.

END OF THE SUPPLEMENT TO THE ELEMENTS.

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E L E M E N T S

OF

PLANE

AND

SPHERICAL

TRIGONOMETRY.

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ELEMENTS

OF

PLANE TRIGONOMETRY.

RIGONOMETRY is the application of Arithmetic to Geo

metry; 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 exprefled thus ; (A+B)C, (A+B)(C+D), or sometimes thus, A+BxC, and A+B XC+D. The quotient of one number A, divided by another B, is

A written thus,

B The fign N is used to fignify the square root : Thus NM is the square root of M, or it is a number which, if multitiplied into itself, will produce M. So also, N MP+N2 is the square root of Ma+N’, &c. The elements of Plane Trigo. nometry, 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 trigono

metrical

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