For the capacities of such cones vary as their surfaces (th. 29).

THEOREM XXXIII.

Of all right cones of equal whole surface, the greatest is that whose side is triple the radius of its base: and reciprocally, of all right cones of equal capacity, that whose side is triple the radius of its base has the least surface.

For, by th. 29, the capacity of a right cone is in the compound ratio of its whole surface and the radius of its inscribed sphere. Therefore, the whole surface being given, the capacity is proportional to the radius of the inscribed sphere and consequently is a maximum when the radius of the inscribed sphere is such; that is, (th. 32) when the side of the cone is triple the radius of the base*.

Again, reciprocally, the capacity being given, the surface is in the inverse ratio of the sphere inscribed: therefore, it

Here again a similar result may easily be deduced from the method of fluxions. Let the radius of the base be denoted by x, the slant side of the cone by z, its whole surface by a', and 3.141593 by . Then the circumference of the cone's base will be 21, its area x', and the con. vex surface maz. The whole surface is, therefore, *I2+1: and

=

-X. But the altitude of the cone is

equal to the square root of the difference of the squares of the side and

a4 2a2

of the radius of the base; that is, it is V(

And this mul

22

tiplied into 3 of the area of the base, viz. by Jπz2, gives πza v ( 21 ) — 24

for the capacity of the cone. Now, this being a maximum, its square

a'x-2=a2x1
9

must be so likewise (Flux. art. 58), that is, or rejecting the denominator, as constant, a12-2-a21 must be a maximum. This, in fluxions, is 2a xi - Sa2x'x = 0; whence we have a2 — 4x2 = 0, and consequently x = √

-

a2

4T

; and a2 4mx2. Substituting this value of

for it, in the value of above given, there results

-x=4x -x=3x. Therefore, the side of the cone is triple the radius of its base. Or, the square of the altitude is to the square of the radius of the base, as 8 to 1, or, to the square of the diameter of the base, as 2 to 1.

is the smallest when that radius is the greatest; that is (th. 32) when the side of the cone is triple the radius of its base.

Q. E. D.

THEOREM XXXIV.

The surfaces, whether total or lateral, of pyramids circumscribed about the same right cone, are respectively as their solidities. And, in particular, the surface of a pyramid circumscribed about a cone, is to the surface of that cone, as the solidity of the pyramid is to the solidity of the cone ; and these ratios are equal to those of the surfaces or the perimeters of the bases.

For, the capacities of the several solids are respectively as their bases; and their surfaces are as the perimeters of those bases: so that the proposition may manifestly be demonstrated by a chain of reasoning exactly like that adopted in theo. rem 11.

THEOREM XXXV.

The base of a right pyramid being given in species, the capacity of that pyramid is a maximum with the same surface, and, on the contrary, the surface is a minimum with the same capacity, when the height of one face is triple the radius of the circle inscribed in the base.

P

Let p and P' be two right pyramids with similar bases, the height of one lateral face of P being triple the radius of the circle inscribed in the base, but this proportion not obtaining with regard to p': then

1st. If surf. P = surf. P', P > P'.
2dly. If. P=

P', surf. P< surf. r'.

For, let c and c' be right comes inscribed within the pyramids P and P': then, in the cone c, the slant side is triple the radius of its base, while this is not the case with respect to the cone c'. Therefore, if cc', surf. c < surf. c'; and, if surf. c surf. c', c> c' (th. 33).

But, 1st. surf. P: surf. c:: surf. P': surf. c'; whence, if surf. P surf. p', surf. c = surf. c'; therefore cc. But P c:: P: c'. Therefore P>P'.

2dly. PC:: P': c'. Theref. if PP', c=c': consequently surf. c<surf. c'. But surf. P: surf. c: surf. r': surf. c. Whence, surf. p < surf. P'.

Cor. The regular tetraedron possesses the property of the minimum surface with the same capacity, and of the maxi

mum capacity with the same surface, relatively to all right pyramids with equilateral triangular bases, and, a fortiori, relatively to every other triangular pyramid.

THEOREM XXXVI.

A sphere is to any circumscribing solid, bounded by plane surfaces, as the surface of the sphere to that of the circumscribing solid.

For, since all the planes touch the sphere, the radius drawn to each point of contact will be perpendicular to each respective plane. So that, if planes be drawn through the cen. tre of the sphere and through all the edges of the body, the body will be divided into pyramids whose bases are the respective planes, and their common altitude the radius of the sphere. Hence, the sum of all these pyramids, or the whole circumscribing solid, is equal to a pyramid or a cone whose base is equal to the whole surface of that solid, and altitude equal to the radius of the sphere. But the capacity of the sphere is equal to that of a cone whose base is equal to the surface of the sphere, and altitude equal to its radius. Consequently, the capacity of the sphere, is to that of the circumscribing solid, as the surface of the former to the surface of the latter both having, in this mode of considering them, a common altitude.

Q. E. D.

Cor. 1. All circumscribing cylinders, cones, &c. are to the sphere they circumscribe, as their respective surfaces.

For the same proportion will subsist between their indefinitely small corresponding segments, and therefore between their wholes.

Cor. 2. All bodies circumscribing the same sphere, are respectively as their surfaces.

THEOREM XXXVII.

The sphere is greater than any polyedron of equal surface.

For, first it may be demonstrated, by a process similar to that adopted in theorem 9, that a regular polyedron has a greater capacity than any other polyedron of equal surface. Let P, therefore, be a regular polyedron of equal surface to a sphere s. Then P must either circumscribe s, or fall partly within it and partly without it, or fall entirely within it. The first of these suppositions is contrary to the hypothesis of the proposition, because in that case the surface of P could not

be equal to that of s. Either the 2d or 3d supposition there. fore must obtain; and then each plane of the surface of p must fall either partly or wholly within the sphere s : whichever of these be the case, the perpendiculars demitted from the centre of s upon the planes, will be each less than the radius of that sphere and consequently the polyedron P must be less than the sphere s, because it has an equal base, but a less altitude.

Q. E. D.

Cor. If a prism, a cylinder, a pyramid, or a cone, be equal to a sphere either in capacity, or in surface; in the first case, the surface of the sphere is less than the surface of any of those solids; in the second, the capacity of the sphere is greater than that of either of those solids.

The theorems in this chapter will suggest a variety of practical examples to exercise the student in computation. A few such are given below.

EXERCISES.

Er. 1. Find the areas of an equilateral triangle, a square, a hexagon, a dodecadon, and a circle, the perimeter of each being 36.

Ex. 2. Find the difference between the area of a triangle whose sides are 3, 4, and 5, and of an equilateral triangle of equal perimeter.

Er. 3. What is the area of the greatest triangle which can be constituted with two given sides 8 and 11; and what will be the length of its third side ?

Ex. 4. The circumference of a circle is 12, and the perimeter of an irregular polygon which circumscribes it is 15: what are their respective areas?

Ex. 5. Required the surface and the solidity of the greatest parallelopiped, whose length, breadth, and depth, together make 18?

Ex. 6. The surface of a square prism is 546: what is its solidity when a maximum ?

Ex. 7. The content of a cylinder is 169-645968: what is its surface when a minimum?

Er. 8. The whole surface of a right cone is 201·061952: what is its solidity when a maximum?

Er. 9. The surface of a triangular pyramid is 43.30127: what is its capacity when a maximum ?

Ex. 10. The radius of a sphere is 10. Required the soVOL. I.

72

lidities of this sphere, of its circumscribed equilateral cone, and of its circumscribed cylinder.

Ex. 11. The surface of a sphere is 28-274337, and of an irregular polyedron circumscribed about it 35: what are their respective solidities?

Ex. 12. The solidity of a sphere, equilateral cone, and Archimedean cylinder, are each 500: what are the surfaces and respective dimensions of each?

Er. 13. If the surface of a sphere be represented by the number 4, the circumscribed cylinder's convex surface and whole surface will be 4 and 6, and the circumscribed equilateral cone's convex and whole surface, 6 and 9 respectively. Show how these numbers are deduced.

Ex. 14. The solidity of a sphere, circumscribed cylinder, and circumscribed equilateral cone, are as the numbers 4, 6, and 9. Required the proof.

PRACTICAL EXERCISES IN MENSURATION.

QUEST. 1. WHAT difference is there between a floor 28 feet long by 20 broad, and two others, each of half the dimensions; and what do all three come to at 45s. per square, or 100 square feet?

Ans. dif. 280 sq. feet.

Amount 18 guineas.

3 inches long, and I

QUEST. 2. An elm plank is 14 feet would have just a square yard slit off it; at what distance from the edge must the line be struck ?

Ans. 7 inches. QUEST. 3. A ceiling contains 114 yards 6 feet of plastering, and the room is 28 feet broad; what is the length of it? Ans. 36 feet.

QUEST. 4. A common joist is 7 inches deep, and 2} thick; but I want a scantling just as big again, that shall be 3 inches thick; what will the other dimension be?

Ans. 11 inches.

QUEST. 5. A wooden trough cost me 3s. 2d. painting within, at 6d. per yard; the length of it was 102 inches, and the depth 21 inches; what was the width ?

Ans. 27 inches.

QUEST. 6. If my court-yard be 47 feet 9 inches square, and I have laid a foot-path with Purbeck-stone, of 4 feet wide, along one side of it; what will paving the rest with flints come to, at 6d. per square yard? Ans. 51. 16s. Old.

QUEST. 7. A ladder, 36 feet long, may be so planted,