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1. A folid angle is that which is made by three or more plane angles, which meet each other in the fame. point.

2. Similar folids, contained by plane figures, are such as have all their folid angles equal, each to each, and are bounded by the fame number of fimilar planes.

3. A prifm is a folid whofe ends are parallel, equal, and like plane figures, and its fides parallelograms.

4. A parallelepipedon is a prifm contained by fix parallelograms, every oppofite two of which are equal, alike, and parallel.

5. A rectangular parallelepipedon is that whose bounding planes are all rectangles, which are perpendicular to each other.

6. A cube is a prism, contained by fix equal fquare fides, or faces.

7. A pyramid is a solid whose base is any right lined plane figure, and its fides triangles, which meet each other in a point above the bafe, called the vertex.

8. A cylinder is a folid generated by the revolution of a right line about the circumferences of two equal and parallel circles, which remain fixed.

9. The axis of a cylinder is the right line joining the centres of the two parallel circles, about which the figure is described.

10. A cone is a folid generated by the revolution of a right line about the circumference of a circle, one end of which is fixed at a point above the plane of that circle.

11. The axis of a cone is the right line joining the vertex, or fixed point, and the centre of the circle about which the figure is defcribed.

12. Similar cones and cylinders are fuch as have their altitudes and the diameters of their bafes proportional.

13. A fphere is a folid generated by the revolution of a femi-circle about its diameter, which remains fixed.

14. The axis of a fphere is the right line about which the femi-circle revolves; and the centre is the fame as that of the femi-circle.

15. The diameter of a sphere is any right line paffing through the centre, and terminated both ways by the furface.

PRO P. I. LEMMA.

If from the greater of two magnitudes, there be taken more than its half; and from the remainder, more than its half; and fo on there will at length remain a magnitude less than the leaft of the propofed magnitudes.

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Let AB and c be any two magnitudes, of which AB is the greater; then, if from AB there be taken more than

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its half; and from the remainder more than its half; and fo on there will at length remain a magnitude lefs than c.

For fince AB and c are each finite magnitudes, it is evident that be taken fuch a number of times as at

may

length to become greater than AB.

Let, therefore, DE be fuch a multiple of c as is greater than AB, and divide it into the parts DF, FG, GE, each equal to c.

Alfo from AB take вH greater than its half; and from the remainder AH, take HK greater than its half, and fo on, till there be as many divifions in AB as there are

in DE.

Then because DE is greater than AB, and вн, taken from AB, is greater than its half, but EG, taken from DE, is not greater than its half; the remainder GD will be greater than the remainder HA.

And, again, because GD is greater than HA, and нK, taken from HA, is greater than its half, but GF, taken from GD, is not greater than its half; the remainder FD will be greater than the remainder AK.

But FD is equal to c by conftruction, whence c is greater than AK; or, which is the fame thing, AK is lefs than c, as was to be shewn.

PRO P. II. THEOREMS.

Similar polygons infcribed in circles, are to each other as the fquares of the diameters of those circlès.

B

M

Let ABCDE, FGHKL be two fimilar polygons, inscribed in the circles ABD, FGK: then will the polygon ABCDE be to the polygon FGHKL as the fquare of the diameter BM is to the fquare of the diameter GN.

For join the points B, E and A, M, G, L and F, N: Then, because the polygon ABCDE is fimilar to the polygon FGHKL (by Hyp.), the angle BAE is equal to the angle GFL, and BA is to AE, as GF is to FL (VI. Def. 1.)

And, fince the angle BAE, of the triangle ABE, is equal to the angle GFL, of the triangle FGL, and the fides about thofe angles are proportional, the angle AEB will also be equal to the angle FLG (VI. 5.)

But the angle AEB is equal to the angle AMB, and the angle FLG to the angle FNG (III. 15.), confequently the angle AMB is also equal to the angle FNG.

And fince thefe angles are equal to each other, and the angles BAM, GFN are each of them right angles (II. 16.), the angle MBA will alfo be equal to the angle NGF (I. 28. Gar.), and BM will be to GN as BA is to GF (VI. 5.)

But

But the polygon ABCDE is to the polygon FGHKL as the fquare of BA is to the fquare of GF (VI. 17.), therefore the polygon ABCDE is also to the polygon FGHKL as the fquare of BM is to the fquare of GN.

PROP. III. THEOREM.

Q. E. D.

A polygon may be infcribed in a circle that fhall differ from it by less than any affigned magnitude whatever.

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Let ABCD be a circle, and s any given magnitude whatever; then may a polygon be infcribed in the circle ABCD that shall differ from it by less than the magnitude s.

For, let AC, EG be two fquares, the one defcribed in the circle ABCD, and the other about it (IV. 6, 7.); and bifect the arcs AB, BC, CD, DA, in the points m, n, r and s (III. 23.); and join Am, m3, вn, nc, cr, rD, DS and SA:

Then fince the fquare AC is half the fquare EG (I. 32.), and the fquare EG is greater than the circle ABCD, the fquare AC will be greater than half the circle ABCD.

In like manner, if tangents be drawn to the circle through the points m, n, r, s, and parallelograms be de

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