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PROP. XXI.

Book III.

E

VERY fphere is two thirds of the circumfcri-
bing cylinder.

Let the figure be constructed as in the two laft propofitions, and if the hemifphere defcribed by BDC be not equal to two thirds of the cylinder defcribed by BD, let it be greater by the folid W. Then, as the cone defcribed by CDE is one third of the cylinder a described by BD, the cone and the a 18. 3.Supi hemifphere together will exceed the cylinder by W. But

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that cylinder is equal to the fum of all the cylinders defcribed by the rectangle Hh, Gg, Ff, Hs, Gr, Fq, DN b; therefore b 20.3 Sup. the hemifphere and the cone added together exceed the fum

of all these cylinders by the given folid W; which is ab

furd, for it has been fhewn, that the hemifphere and the c19. 3. Sup.

cone together differ from the fum of the cylinders by a folid

lefs than W. The hemifphere 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 defcribed by twice the rectangle BD, that is, to two thirds of the circumfcribing cylinder. Q. E. D.

END OF THE SUPPLEMENT TO THE ELEMENTS.

ELEMENTS

OF

PLANE

AND

SPHERICAL

TRIGONOMETRY.

From here to the end

paging is wrong.

of the book the

ELEMENTS

OF

PLANE TRIGONOMETRY.

RIGONOMETRY is the application of Arithmetic to Geometry; or, more precifely, it is the application of number to express the relations of the fides and angles of triangles to one another. It therefore neceffarily fuppofes the elementary operations of arithmetic to be understood, and it borrows from that science feveral of the figns 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 expreffed thus (A+B)C, (A+B)(C+D), or sometimes thus, A+B×C, and A+B XC+D.

The quotient of one number A, divided by another B, is A written thus,

B

The fignis used to fignify the fquare root: Thus M N is the fquare root of M, or it is a number which, if multitiplied into itself, will produce M. So alfo, /M2+N2 is the fquare root of M2+N2, &c. The elements of Plane Trigonometry, as laid down here, are divided into three fections; the first explains the principles; the second delivers the rules of calculation; the third contains the conftruction of trigonometrical

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