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The SPHEROID, in geometry, is generated by the entire revolution of a semi-elipsis about its for z its value " = pbsri x

by aris. When the revolution is made round the largest axis, the spheroid is called prolate; and substituting for yy its value 66 x when round the shortest, oblate. This last is the figure of the earth, and probably of all the pfreix

by putting for abs its planets.

To obtain the solid dimensions of a spheroid, value ?f; and hence the fluent pfror x .multiply continually together the fixed axis, the ad —

Q — } rx

3 aa - Ir or $ pfrr x

will be square of the revolving axis, and the number •52359877, or 4 of 3.14159, and the last product the value of the frustum KACL; which, when will be the solidity; i. e. spttc = the oblate, ES or r becomes SI or a, gives jpfrr for the and picc = the oblong spheroid, where p = value of the semi-spheroid KSL; or the whole 3:14159, t = the transverse, and c = the con- spheroid & P FR R, putting F and R for the jugate axis of the generating ellipsis. Or, mul- whole fixed and revolving axes. Q. E. D. tiply the area of the generating ellipse by i of

Cor. 1.-From the foregoing demonstration it the revolving axis, and the product will be the appears that the value of the general frustum content of the spheroid ; i.e. tA= the oblate, KAECL is expressed by i Pfirrix and | A = the oblong spheroid ; where A is

3 aa II the area of the ellipse. E. g. Required the content of an oblate, and of an oblong spheroid, the axes being 50 and 30. Thus, 50 X 30 X

And if for fr be substituted its value abs, the

same frustum will also be expressed by porss -78539816 = 1178.09724 = the area of the

Заа ellipse. And 1178·09724 X 1 X 30=23561.9448

х -- the oblong spheroid; and 1178.09724 x x 50 = 39269.908 = the oblate one. Dr. Hutton has demonstrated the rule above

Also, if for a a be put its value given in the following manner. Put j = BI ihe fixed semi-axis, o = IM the revolving last expression will become prsxx

b

" z instead of its value

6'

gives fpsr x (267 + yz) for the value of the frustun, viz. the sum of the area of the less end, and twice that of the greater, drawn into one-third of the altitude or distance of the ends.

And out of this last expression may be expunged any one of the four quantities 6, 7, y, 2, by means of the proportion b:r::y: 2.

When the ends of the frustum are perpendicuzemi-axis of the spheroid, a = SI any semi- lar to the fixed axis, then « is = f, and the diameter of the section N BM, 6 = 1K its value of the frustum becomes t 'proix semi-conjugate, y = A E an ordinate to the 3.f.f rr

for the value of the frustum whose diameter Š I, or a semi-axis of the elliptic fj section AF Ć parallel to K L, and 2 = EF ends are perpendicular to the fixed axis, its altiits other semi-axis, also r = EI, s = the sine tude being a. of the angle A ES, or of the angle K IS, to

And, when the ends of the frustum are paralle' the radius 1, and p = 3.14159.

to the fixed axis, a is = 1, and the expression for Then, by the property of the ellipse KSL, such a frustum becomes & pf x x

3rr - II ca;66:: 24 – x3;66 x

=yY;

Cor. 2.-If to or from & pfrr, the value of But the fluxion of the the semi-spheroid be added or subtracted,

3 aa-81 tp.frrrx

the value of the gesolid K AC L is psysi=psyyi

b

, by writing neral frustum K AC L, there will result sg frokh VOL. XXI.- Part 1.

8

or t psex (2br + "%"); which, by writing

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Cor. 9.-And if about the two axes of an for the value of a general segment, ellipse be generated two spheres and two either greater or less than the semi-spheroid, spheroids, the four solids will be continual prowhose height, taken upon the diameter passing portionals, and the common ratio will be that through its vertex and centre of its base, is h = of the two axes of the ellipse; that is, as the a + r.

greater sphere, or the sphere upon the greater When a coincides with f; the above expres- spheroid to the oblong spheroid, so is the oblong

axis, is to the oblate spheroid, so is the oblate

3f - h sion becomes $ prohh x for the va- spheroid to the less sphere, and so is the trans

verse axis to the conjugate. For these four lue of a segment whose base is perpendicular to

bodies will be as T, T? C, TCP, C', where each the fixed axis. And here if we put R for the term is to the consequent one as T to C. radius of the segment's base, and for re its value To find the content of a universal spheroid, RRff

or a solid conceived to be generated by the the said segment will become 2 fh-h

revolution of a semi-ellipse about its diameter,
h.

whether that diameter be one of the axes of the
ellipse or not.

1. Divide the square of the proAnd when a coincides with r, the general ex

duct of the axes of the ellipse by the axis of

3r- - h the solid, or the diameter about which the semipression will become $ pfhh x for ellipse is conceived to revolve; multiply the the value of the segment whose base is parallel «quotient by ·5236, and the product will be the

Т? С? to the fixed axis. And if we put P, R, for the content required. That is,

X 5236 = two semi-axes of the elliptic base of this sergo the content ; T and C being the transverse and

d ment, respectively corresponding or parallel to f, r, the semi-axes of the generating ellipse, conjugate axes of the ellipse, and d the axis of when parallel to the base of the segment, and for the solid. - and r substitute their values and

F RR+hh,

Or, 2. The continual product of •5236, the

diameter about which the revolution is made, R

2 h the said frustum will be expressed by s p Fh x

the square of its conjugate diameter, and the 3RR +hh

square of the sine of the angle made by those in which the dimensions of itself diameters, the radius being 1, will be the con2 R

That is, dcess X •5236 = the content; only are concerned.

c being the conjugate diameter to d, and s the Cor. 3.—A semi-spheroid is equal to f of a' sine of the angle made by the diameters. For cylinder, or to double a cone of the same base the demonstration of this rule see Hutton, ubi and height; or they are in proportion as the infra. Hence, if d = T, the rule becomes íp T numbers 3, 2, 1. For the cylinder is = 4n frr C for the oblong spheroid : and, if d=C, it will =nfrr, the semi-spheroid = {nfrr, and be pCT for the oblate spheroid : and if T, C, the cone = : infrr.

and d, be all equal, the rule will be įpd for the Cor. 4.-When f = r, the spheroid becomes sphere. See Hutton's Mensuration. a sphere, and the expression fnrr for the Dr. Halley has shown, that in a sphere, Mercasemi-spheroid becomes f n ry for the semi-sphere. tor's nautical meridian line is a scale of logaAnd, in like manner, f and « being supposed rithmic tangents of the half complements of the equal to each other in the values of the frustums latitudes. But, as the earth has been found to be and segments of a spheroid, in the preceding a spheroid, this figure will make some alteration corollaries, will give the values of the like parts in the numbers resulting from Dr. Halley's theoof a sphere.

Mr. Maclaurin has therefore given us a Cor. 5.-All spheres and spheroids are to each rule, by which the meridional parts to any other as the fixed axes drawn into the squares of spheroid may be found with the same exactness the revolving axes.

as in a sphere. Cor. 6.-Any spheroids and spheres, of the SPHERUS, a Greek philosopher, a disciple same revolving axis, as also their like or corres- of Zeno of Cyprus, who flourished about A.A.C. ponding parts cut off by planes perpendicular to 243. He came to Sparta in the reign of Agis the said common axis, are to one another as III. and Cleomenes III., and opened a school their other or fixed axes. This follows from the for philosophy. Plut. Diod. foregoing corollaries.

SPHEX, ichneumon wasp, or savage, a genus Cor. 7.—But if their fixed axes be equal, and of insects belonging to the order of hymenoptera. their revolving axes unequal, the spheroids and See ENTOMOLOGY. The mouth is armed with spheres, with their like parts terminated by entire jaws, but contains no tongue; the manplanes perpendicular to the common fixed axis, dibles are horny, crooked, dentated; the lip will be to each other as the squares of their re- horny, the apex membranaceous. The palpi or volving axes.

feelers are four. The antennæ have from ten to Cor. 8.-An oblate spheroid is, to an oblong sixteen joints. The wings of both sexes are exspheroid generated from the same ellipse, as tended without folds, and laid horizontally on the longer axis of the ellipse is to the shorter. the back. The sting is sharp, and concealed For if T be the transverse axis, and C the con- within the abdomen. There are ninety-seven jugate; the oblate spheroid will be = f n T°C, species. The manner of living is different in and the oblong = inCT; and these quantities the various species, and so is the general form are in the ratio of T to C.

of the body and their haunts; but though the

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method of life be utterly different, yet the same mities upon a stick. See Jocasta and Oedipus. manners appear innate and inherent in all. The sphinx, upon this explanation, dashed her They agree in being the fiercest of all flies: head against a rock, and expired. Among the they will attack insects much larger than them- Egyptians the sphinx was the symbol of religion, selves, and this whether they be defenceless or by reason of the obscurity of its mysteries ; armed, as they are provided with a sting. The and on the same account the Romans placed a strength in all this savage kind is great ; their sphinx in the pronaos or porch of their temples. jaws are hard and sharp, and in their sting lies Sphinxes were used by the Egyptians to show the a poison suddenly fatal to the creatures with beginning of the water's rising in the Nile; with whom they engage. The savage seizes bardily this view, as it had the head of a woman and body on the animal he attacks, and gives a stroke of of a lion, it signified that the Nile began to swell amazing force; after which he falls down as if in July and August, when the sun passes through himself were killed, but it is to rest from his fa- the signs of Leo and Virgo. There are several of tigue, and enjoy his victory. He keeps a steady these still to be seen; one in particular, near the eye on the creature he has struck till it dies, pyramids, much spoken of by the ancients, being which happens in a few minutes, and then drags of a prodigious size, and cut out of the rock; the it to the nest to feed the young. The number of head and neck appear only at present, the rest of other insects they destroy is scarcely to be con- the body being hid in the sand. This, according ceived; the mouth of their cave is like that of a to Thevenot, is twenty-six feet high, and fifteen giant in the days of yore, strewed with the re- feet from the ear to the chin; but Pliny assures mains of prey. The eyes, the filament that serves us, the head was no less that 102 feet in circumas a brain, and a small part of the contents of ference, and sixty-two feet high from the belly, the body, are all the savage eats, and he will and that the body was 143 feet long, and was kill fifty for a meal. Of this numerous genus thought to be the sepulchre of king Amasis. See only two are natives of Britain and Ireland, viz. PYRAMIDS. The learned Mr. Bryant (in his 1. S. cribraria is black, with yellow ringlets on Ancient Mythol. vol. iii. p. 532), observes that the abdomen ; the antennæ are short, and turned the sphinx seems to have been originally a vast backwards; the fore legs are broad, with an ap- rock of different strata; which, from a shapeless pendix like a shield. 2. S. viatica is black; the mass, the Egyptians fashioned into an object of antennæ are short and thick; the first three seg- beauty and veneration. The Egyptians used ments of the abdomen red brown; the pedicle is this figure in their buildings; from them the short; the length half an inch.

Greeks derived it, and afterwards improved it SPHINCTER, in anatomy, a term applied to into an elegant ornament. It is also frequently a kind of citcular muscles, or muscles in form used in modern architecture. The sphinx of the of rings, which serve to close and draw up seve- Egyptians is said in the Asiatic Researches, vol. ral orifices of the body, and prevent the excretion ii. p. 334, to have been found in India. Colonel of the contents. See ANATOMY.

Pearce was told by Murari Pandit, a man of SPHINX, n. s. Gr. opeys. Defined below. learning among the Hindoos, that the sphinx,

The sphinr was a famous monster in Egypt, that there called singh, is to appear at the end of the remained by conjoined Nilus, having the face of a world, and as soon as he is born will prey on an virgin, and the body of a lion.

elephant: he is therefore figured seizing an elePeacham on Drawing.. phant in his claws; and the elephant is made Sphinx, or Sphynx, in the mythology, a mon- small

, to show that the singh, even a moment ster which had the head and breasts of a woman, after his birth, will be very large in proportion to the body of a dog, the tail of a serpent, the it. But in opposition to this account, given by wings of a bird, the paws of a lion, and a hu- Murari Pandit

, the late Sir William Jones, the man voice. It sprang from the union of Ortho3 learned and illustrious president of the Asiatic with the Chimæra, or of Typhon with Echidna. Society, was assured by several Brahmins that The sphinx had been sent into the neighbourhood the figure taken for a sphinx was a representation of Thebes by Juno, who wished to punish the of a lion seizing a young elephant. family of Cadmus, which she persecuted with Sphinx, in entomology, hawk-moth; a genus immortal hatred; and it laid this part of Beotia of insects belonging to the order of lepidopteræ. under continual alarms, by proposing enigmas, The antennæ are shaped somewhat like a prism, and devouring the inhabitants if unable to explain and are more slender at each end than at the them. In the midst of their consternation, the middle. The tongue is generally thrust out: Thebans were told by the oracle that the sphinx the two palpi are bent back, and the wings dewould destroy herself as soon as one of the flexed. The name sphinx is given to this genus enigmas she proposed was explained. In this on account of the singular attitudes of their caterenigma she wished to know what animal walked pillars, who apply the hinder part of their body on four legs in the morning, two at noon, and to a branch of a tree, holding the rest of it erect, three in the evening. Upon this Creon king of like the fabulous sphinx. Most of them spin Thebes promised his crown and his sister Jocasta, their cod under ground, making them up with the widow of king Laius, in marriage to him whó small parcels of earth and grains of corn intercould deliver his country from the monster by a woven with threads. The sphinxes fly either successful explanation of the enigma. It was at early in the morning or after sunset in the last explained by Edipus, who observed that evening. They fly heavily and sluggishly, otten man walked on his hands and feet in the morning emitting a kind of sound. There are about 165 of life, at the noon of life he walked erect, and species already discovered, of which ten are in the evening of his days he supported his infir- found in Great Britain and Ireland : viz.:

cera.

It

The an

1. S. atropos, jessamine hawk-moth. The gray. Caterpillar green, shagreened, with a poswings are entire ; the truuk long, spiral. Above, terior horn. first wings brown, clouded with gray and yellow, 10. S. tipulifornis, small bee-moth. The and a yellowish spot in the centre; second, yellow, thorax is yellow beneath ; the wings are short, with iwo waved transverse stripes. The abdo- with black veins. The abdomen black, bearded, men is yellow, with seven black brown belts. The yellow at the extremity. Caterpillar on the lonithorax marked like a Death's head, whence the name, from Atropos, the third and last of the Sphinx, in zoology. See Simia. Fatal Sisters, who cuts the thread of life. The SPHERULITE, in mineralogy. Colors brown length is two inches. Caterpillar very large, yel- and gray. In imbedded roundish balls and grains. low, with six green and orange oblique belts, and Glimmering. Fracture even, splintery. Opaque. a posterior horn.

Scratches quartz with difficulty. Brittle. Spe2. S. convolvuli, unicorn, or bindweed hawk- cific gravity 2.4 to 2.5. Nearly infusible. moth. The antennæ are long and thick; the occurs in pearlstone and pitchstone porphyries, trunk very long and spiral. Above, body marked in the vicinity of Glasshütte near Schemnitz; with black and red belts; wings entire, brown- and in the pitchstone of Meissen. gray, with black zig zag transverse lines. The SPHONDYLIUM, in anatomy, one of the verbreadth three inches. Caterpillar smooth, green, tebræ of the back. with a posterior horn.

SPHRAGIDIUM, a famous cave of Bæotia, 3. S. elpedor, elephant moth. The wings are in Mount Citheron.-Paus. ix. c. 3. angular, entire. Above, first wings striped SPI'AL, n. s. Fr. espial. A spy; scoui ; a transversely with red and green : second, black watcher. Obsolete. at the base, and red outwards. The body red

His ears be as spials, alarum to crie. and green. Caterpillar smooth, brown and yel

Tusser's Husbandry. low, with a posterior horn, and a snout like a He privy spials placed in all his way, hog. It lives on vines, convolvulus, &c. To weet what course he takes, and how he fares. 4. S. filipendulæ, burnet moth. The antenne,

Spenser. legs, and body, are black. Second wings red, Their trust towards them hath rather been as to with a greenish border. First wings bluish-green, good spials and good whisperers, than good magiswith six red spots, in pairs. Length eight lines.

trates and officers.

Bacon. Caterpillar yellow, with black spots. It lives on

For he by faithful spiul was assured

That Egypt's king was forward on his way. grass.

Fairfax. 5. S. ligustri, privet hawk-moth. tennæ are long, thick, and brown. Trunk long,

SPICA, in botany. See BOTANY spiral. First wings two inches long, narrow, en- SPICE, n. s. & v.a.) Fr. espices; Italian tire, brown; second, short, red, with black bars. Spi'cer, n. s. specei, of Lat. speci. The abdomen is red, with black rings. Cater- Spi'cERY.

À vegetable producpillar smooth, yellow-green, with a posterior tion, fragrant to the smell, and of pungent tasie; horn.

an aromatic substance used in sauces: to season 6. S. ocellata, eyed willow hawk-moth. There with spice : a spicer is a dealer in spice : spicery is no trunk; the wings are indented. Above, is the commodity or depository of spices. first wings dark and light brown, marbled;

Dangerous rocks, second, red, with a large yellow black eye. Be

Which, touching but by gentle vessel's side, neath, a large red triangle from the base of the Would scatter all the spices on the stream. first wings. The breadth one inch and a half.

Shakspeare. Caterpillar smooth, green, with oblique white Is not manhood, learning, gentleness, and virtue, lines on the sides, and a posterior horn. The the spice and salt that seasons a man? eggs are green; it lives on willows.

Id. Troilus and Cressida. 7. S. populi, poplar hawk-moth. The wings His mother was a votress of my order, are scalloped, bluish-gray, and wared with dark And in the spiced Indian air by night lines. On the first wings a long white spot,

Full often she hath gossip'd by my side. and the base of the second red brown. Wings

Shakspeare. reversed. Length one inch. A long spiral trunk and myrrh.

Their camels were loaden with spicery, and balm, caterpillar, green, smooth, with oblique white

Raleigh's History of the World.

Names have been derived from occupations, as spots, and a posterior horn. It lives on poplars Salter and Spicer.

Camden. and willows.

She in whose body 8. S. stellatarum, large bee-moth. The an- The western treasure, eastern spicery, tennæ are thick, towards the ends brown. The Europe and Africk, and the unknown rest, trunk is spiral; the wings are short and entire; Were easily found.

Donne, the body is thick, brown, and hairy. First wings These hymns may work on future wits, and so are brown, waved ; second, red brown. It re- May great-grandchildren of thy praises grow; sembles a large bee. Caterpillar smooth, with a

And so, though not revive, embalm and spice posterior blue horn, tipt with red. It lives on

The world, which else would putrify with vice. Id.

With a festivall gallium. 9. S. tiliæ, lime hawk-moth. No trunk; the She'll first receive thee ; but will spice thy bread

With flowrie poysons.

Chapman. wings are scalloped; the antennæ are white on So in the wicked there 's no vice, the upper side, yellow on the under. Above, first of which the saints have not a spice. Hudibras. wings gray-brown, with two irregular rge green What though some have a fraught spots; second wings orange. Beneath, greenish. Of cloves and nutmegs, and in cinnamon sail,

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