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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 theorem II.
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.
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
Ist. If surf. p = surf. Þ', P> p'.
Þ', surf. p < surf. p. For, let c and c' be right cones 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 c= c', surf. c < surf. c'; and, if surf. c = surf. c',c>c (th. 33).
this being a maximum its square must be so likewise (Flux, art. 53), that is, atr?- 2та?
-, or, rejecting the denominator, as constant, aʻq? 21214 must 9 be a maximum. This, in Auxions, is 2a*ri -- Sma2x32 = 0; whence we bave R? - 4*22 = 0, and consequently I =N; and a? = 472% Substituting this value of as for it, in the value of . above given, there results,
31. 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 bare, as 2 to 1.
But, Ist. surf. P : surf. c :: surf. Þ' : surf. c'; whence, if surf. P = surf. p', surf. c'= surf. c'; therefore c> c. But p:c:: P':c'. Therefore p > P'.
2dly. Pic::p':c'. Theref. if p=p', c=c': consequently surf. c < surf. c. But, surf. P : surf. c:: surf. Þ': 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 maximum capacity with the same surface, relatively to all right pyramids with equilateral triangular bases, and, a fortiori, relatively to every other triangular pyramid.
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 centre of the sphere and through all the edges of the body, the body will be divided inte 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. l. 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.
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 out of 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 therefore 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.
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.
Q. E. D.
Ex. 1. Find the areas of an equilateral triangle, a square, a hexagon, a dodecagon, and a circle, the perimeter of each being 36. - Er. 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
Ex.3 can be constituted with two given sides 8 and 11: and what will be the length of its third side?
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 ?
Ex. 8. The whole surface of a right cone is 201.061952: what is its solidity when a maximum ?
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 solidities 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?
Er. 12. The solidity of a sphere, equilateral cone, and Archimedean cylinder, are each 500: what are the surfaces and respective dimensions of each ?
Ex. 13. If the surface of a sphere be represented by the number 4, the circunscribed 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.
PLANE TRIGONOMETRY CONSIDERED ANALYTICALLY.
ART. 1. There are two methods which are adopted by mathematicians in investigating the theory of Trigonometry: the one Geometrical, the other Algebraical. In the former, the various relations of the sines, cosines, tangents, &c, of single or multiple arcs or angles, and those of the sides and angles of triangles, are deduced immmediately from the
figures to which the several enquiries are referred; each individual case requiring its own particular method, and resting on evidence peculiar to itself. In the latter, the nature and properties of the linear-angular quantities (sines, tangents, &c,) being first defined, some general relation of these quantities, or of them in connection with a triangle, is expressed by one or more algebraical equations; and then every other theorem or precept, of use in this branch of science, is developed by the simple reduction and transformation of the primitive equation. Thus, the rules for the three fundamental cases in Plane Trigonometry, which are deduced by three independent geometrical investigations, in the second volume of this Course of Mathematics, are obtained algebraically, by forming, between the three data and the three unknown quantities, three equations, and obtaining, in expressions of known terms, the value of each of the unknown quantities, the others being exterminated by the usual processes. Each of these general methods has its peculiar advantages. The geometrical method carries conviction at every step; and by keeping the objects of enquiry constantly before the eye of the student, serves admirably to guard him against the admission of error: the algebraical method, on the contrary, requiring little aid from first principles, but merely at the commencement of its career, is more properly mechanical than mental, and requires frequent checks to prevent any deviation from truth. The geometrical method is direct, and rapid, in producing the requisite conclusions at the outset of trigonometrical science; but slow and circuitous in arriving at those results which the modern state of the science requires : while the algebraical method, though sometimes circuitous in the developement of the mere elementary theorems, is very rapid and fertile in producing those curious and interesting formulæ, which are wanted in the higher branches of pure analysis, and in mixed mathematics, especially in Physical Astronomy. This mode of developing the theory of Trigonometry is, consequently, well suited for the use of the more advanced student: and is therefore introduced here with as much brevity as is consistent with its nature and utility.
2. To save the trouble of turning very frequently to the 2d volume, a few of the principal definitions, there given, are here repeated, as follows:
The SINE of an arc is the perpendicular let fall from one of its extremities upon the diameter of the circle which passes through the other extremity.