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V. 21

11-11-52 MFP

11-7-52

LONDON

THE

ENCYCLOPÆDIA.

The SPHEROID, in geometry, is generated by the entire revolution of a semi-ellipsis about its

axis. When the revolution is made round the largest axis, the spheroid is called prolate; and when round the shortest, oblate. This last is the figure of the earth, and probably of all the planets.

To obtain the solid dimensions of a spheroid, multiply continually together the fixed axis, the square of the revolving axis, and the number 52359877, or of 3.14159, and the last product will be the solidity; i. e. pttc the oblate, and pt cc the oblong spheroid, where p 3-14159, the transverse, and c the conjugate axis of the generating ellipsis. Or, multiply the area of the generating ellipse by of the revolving axis, and the product will be the content of the spheroid; i.e. tA= the oblate, and c A the oblong spheroid; where A is 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 × 30 × -78539816 = 1178-09724 the area of the ellipse. And 1178·09724 × 3 × 30—23561·9448

the oblong spheroid; and 1178-09724 × 3 × 50 39269-908 the oblate one.

Dr. Hutton has demonstrated the rule above given in the following manner. Put ƒ BI the fixed semi-axis, IM the revolving

N

B

semi-axis of the spheroid, a = SI any semidiameter of the section N BM, b = IK its semi-conjugate, y = AE an ordinate to the diameter S I, or a semi-axis of the elliptic section AFC parallel to K L, and z = EF its other semi-axis, also r E I, s the sine of the angle A E S, or of the angle K IS, to the radius 1, and p = 3.14159.

Then, by the property of the ellipse KSL,

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VOL. XXI.-PART 1.

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pfrri X , by putting for abs its value rf; and hence the fluent pfrrr × " a — } x x or pfrr x will be the value of the frustum K ACL; which, when EJ or r becomes SI or a, gives pfrr for the value of the semi-spheroid K SL; or the whole spheroid=p FRR, putting F and R for the whole fixed and revolving axes. Q. E. D.

Cor. 1.-From the foregoing demonstration it appears that the value of the general frustum KA ECL is expressed by pfrrr x

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z instead of its value, gives psr × (2 br

+y) for the value of the frustum, 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 b, r, y, z, by means of the proportion br:: y : z.

When the ends of the frustum are perpendicular to the fixed axis, then a is = f, and the value of the frustum becomes prr X 3.f.ƒ — x x for the value of the frustum whose ff ends are perpendicular to the fixed axis, its altitude being .

And, when the ends of the frustum are paralle' to the fixed axis, a isr, and the expression for such a frustum becomes pƒr ×

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by writing neral frustum K ACL, there will result pfrrkh

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for the value of a general segment, either greater or less than the semi-spheroid, whose height, taken upon the diameter passing through its vertex and centre of its base, is h= a + x.

When a coincides with f, the above expression becomes p r r h h × 3f-h for the vaff lue of a segment whose base is perpendicular to the fixed axis. And here if we put R for the radius of the segment's base, and for rr its value RRff the said segment will become 2fh-hh'

pR Rh x

3ƒ— h. 2f-h

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And when a coincides with r, the general ex3 r h pression will become pfhh x for the value of the segment whose base is parallel to the fixed axis. And if we put F, R, for the two semi-axes of the elliptic base of this seg ment, respectively corresponding or parallel to f, r, the semi-axes of the generating ellipse, when parallel to the base of the segment, and for f F RR+hh, andr substitute their values and R 2 h the said frustum will be expressed by p Fh x 3 RR+hh in which the dimensions of itself 2 R only are concerned.

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Cor. 3.-A semi-spheroid is equal to of a cylinder, or to double a cone of the same base and height; or they are in proportion as the numbers 3, 2, 1. For the cylinder is 4nfrr nfrr, the semi-spheroid §n frr, and the conen frr.

=

Cor. 4.-When fr, the spheroid becomes a sphere, and the expression fnrr for the semi-spheroid becomes n for the semi-sphere. And, in like manner, ƒ and r being supposed equal to each other in the values of the frustums and segments of a spheroid, in the preceding corollaries, will give the values of the like parts of a sphere.

Cor. 5.-All spheres and spheroids are to each other as the fixed axes drawn into the squares of the revolving axes.

Cor. 6.-Any spheroids and spheres, of the same revolving axis, as also their like or corresponding parts cut off by planes perpendicular to the said common axis, are to one another as their other or fixed axes. This follows from the foregoing corollaries.

Cor. 7.-But if their fixed axes be equal, and their revolving axes unequal, the spheroids and spheres, with their like parts terminated by planes perpendicular to the common fixed axis, will be to each other as the squares of their revolving axes.

Cor. 8.-An oblate spheroid is, to an oblong spheroid generated from the same ellipse, as the longer axis of the ellipse is to the shorter. For if T be the transverse axis, and C the conjugate; the oblate spheroid will be = }n T2 C, and the oblong=n CT; and these quantities are in the ratio of T to C.

Cor. 9. And if about the two axes of an ellipse be generated two spheres and two spheroids, the four solids will be continual proportionals, and the common ratio will be that of the two axes of the ellipse; that is, as the greater sphere, or the sphere upon the greater spheroid to the oblong spheroid, so is the oblong axis, is to the oblate spheroid, so is the oblate spheroid to the less sphere, and so is the transverse axis to the conjugate. For these four bodies will be as T3, T C, T C2, C3, where each term is to the consequent one as T to C.

To find the content of a universal spheroid, or a solid conceived to be generated by the revolution of a semi-ellipse about its diameter, whether that diameter be one of the axes of the

ellipse or not. 1. Divide the square of the product of the axes of the ellipse by the axis of the solid, or the diameter about which the semiellipse is conceived to revolve; multiply the quotient by 5236, and the product will be the content required. That is, X 5236 = the content; T and C being the transverse and d conjugate axes of the ellipse, and d the axis of the solid.

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Or, 2. The continual product of 5236, the diameter about which the revolution is made, the square of its conjugate diameter, and the square of the sine of the angle made by those diameters, the radius being 1, will be the content. That is, dccss x 5236 the content; c being the conjugate diameter to d, and s the sine of the angle made by the diameters. For the demonstration of this rule see Hutton, ubi infra. Hence, if d=T, the rule becomes ip T C for the oblong spheroid : and, if d=C, it will be p CT for the oblate spheroid: and if T, C, and d, be all equal, the rule will be pd3 for the sphere. See Hutton's Mensuration.

Dr. Halley has shown, that in a sphere, Mercator's nautical meridian line is a scale of logarithmic tangents of the half complements of the latitudes. But, as the earth has been found to be a spheroid, this figure will make some alteration in the numbers resulting from Dr. Halley's theorem. Mr. Maclaurin has therefore given us a rule, by which the meridional parts to any spheroid may be found with the same exactness as in a sphere.

SPHERUS, a Greek philosopher, a disciple of Zeno of Cyprus, who flourished about A. A.C. 243. He came to Sparta in the reign of Agis III. and Cleomenes III., and opened a school for philosophy.-Plut. Diod.

SPHEX, ichneumon wasp, or savage, a genus of insects belonging to the order of hymenoptera. See ENTOMOLOGY. The mouth is armed with entire jaws, but contains no tongue; the mandibles are horny, crooked, dentated; the lip horny, the apex membranaceous. The palpi or feelers are four. The antennæ have from ten to sixteen joints. The wings of both sexes are extended without folds, and laid horizontally on the back. The sting is sharp, and concealed within the abdomen. There are ninety-seven species. The manner of living is different in the various species, and so is the general form of the body and their haunts; but though the

method of life be utterly different, yet the same manners appear innate and inherent in all. They agree in being the fiercest of all flies: they will attack insects much larger than themselves, and this whether they be defenceless or armed, as they are provided with a sting. The strength in all this savage kind is great; their jaws are hard and sharp, and in their sting lies a poison suddenly fatal to the creatures with whom they, engage. The savage seizes hardily on the animal he attacks, and gives a stroke of amazing force; after which he falls down as if himself were killed, but it is to rest from his fatigue, and enjoy his victory. He keeps a steady eye on the creature he has struck till it dies, which happens in a few minutes, and then drags it to the nest to feed the young. The number of other insects they destroy is scarcely to be conceived; the mouth of their cave is like that of a giant in the days of yore, strewed with the remains of prey. The eyes, the filament that serves as a brain, and a small part of the contents of the body, are all the savage eats, and he will kill fifty for a meal. Of this numerous genus only two are natives of Britain and Ireland, viz. 1. S. cribraria is black, with yellow ringlets on the abdomen; the antennæ are short, and turned backwards; the fore legs are broad, with an appendix like a shield. 2. S. viatica is black; the antennæ are short and thick; the first three segments of the abdomen red brown; the pedicle is short; the length half an inch.

SPHINCTER, in anatomy, a term applied to a kind of circular muscles, or muscles in form of rings, which serve to close and draw up several orifices of the body, and prevent the excretion of the contents. See ANATOMY.

SPHINX, n. s. Gr. opy. Defined below. The sphinx was a famous monster in Egypt, that remained by conjoined Nilus, having the face of a virgin, and the body of a lion.

Peacham on Drawing. SPHINX, OF SPHYNX, in the mythology, a monster which had the head and breasts of a woman, the body of a dog, the tail of a serpent, the wings of a bird, the paws of a lion, and a human voice. It sprang from the union of Orthos with the Chimera, or of Typhon with Echidna. The sphinx had been sent into the neighbourhood of Thebes by Juno, who wished to punish the family of Cadmus, which she persecuted with immortal hatred; and it laid this part of Boeotia under continual alarms, by proposing enigmas, and devouring the inhabitants if unable to explain them.

In the midst of their consternation, the Thebans were told by the oracle that the sphinx would destroy herself as soon as one of the enigmas she proposed was explained. In this enigma she wished to know what animal walked on four legs in the morning, two at noon, and three in the evening. Upon this Creon king of Thebes promised his crown and his sister Jocasta, the widow of king Laius, in marriage to him who could deliver his country from the monster by a successful explanation of the enigma. It was at last explained by Edipus, who observed that man walked on his hands and feet in the morning of life, at the noon of life he walked erect, and in the evening of his days he supported his infir

mities upon a stick. See JOCASTA and OEDIPUS. The sphinx, upon this explanation, dashed her head against a rock, and expired. Among the Egyptians the sphinx was the symbol of religion, by reason of the obscurity of its mysteries; and on the same account the Romans placed a sphinx in the pronaos or porch of their temples. Sphinxes were used by the Egyptians to show the beginning of the water's rising in the Nile; with this view, as it had the head of a woman and body of a lion, it signified that the Nile began to swell in July and August, when the sun passes through the signs of Leo and Virgo. There are several of these still to be seen; one in particular, near the pyramids, much spoken of by the ancients, being of a prodigious size, and cut out of the rock; the head and neck appear only at present, the rest of the body being hid in the sand. This, according to Thevenot, is twenty-six feet high, and fifteen feet from the ear to the chin; but Pliny assures us, the head was no less that 102 feet in circumference, and sixty-two feet high from the belly, and that the body was 143 feet long, and was thought to be the sepulchre of king Amasis. See PYRAMIDS. The learned Mr. Bryant (in his Ancient Mythol. vol. iii. p. 532), observes that the sphinx seems to have been originally a vast rock of different strata; which, from a shapeless mass, the Egyptians fashioned into an object of beauty and veneration. The Egyptians used this figure in their buildings; from them the Greeks derived it, and afterwards improved it into an elegant ornament. It is also frequently used in modern architecture. The sphinx of the Egyptians is said in the Asiatic Researches, vol. ii. p. 334, to have been found in India. Colonel Pearce was told by Murari Pandit, a man of learning among the Hindoos, that the sphinx, there called singh, is to appear at the end of the world, and as soon as he is born will prey on an elephant he is therefore figured seizing an elephant in his claws; and the elephant is made small, to show that the singh, even a moment after his birth. will be very large in proportion to it. But in opposition to this account, given by Murari Pandit, the late Sir William Jones, the learned and illustrious president of the Asiatic Society, was assured by several Brahmins that the figure taken for a sphinx was a representation of a lion seizing a young elephant.

SPHINX, in entomology, hawk-moth; a genus of insects belonging to the order of lepidopteræ. The antennæ are shaped somewhat like a prism, and are more slender at each end than at the middle. The tongue is generally thrust out: the two palpi are bent back, and the wings deflexed. The name sphinx is given to this genus on account of the singular attitudes of their caterpillars, who apply the hinder part of their body to a branch of a tree, holding the rest of it erect, like the fabulous sphinx. Most of them spin their cod under ground, making them up with small parcels of earth and grains of corn interwoven with threads. The sphinxes fly either early in the morning or after sunset in the evening. They fly heavily and sluggishly, often emitting a kind of sound. There are about 165 species already discovered, of which ten are found in Great Britain and Ireland: viz.

The

1. S. atropos, jessamine hawk-moth. wings are entire; the trunk long, spiral. Above, first wings brown, clouded with gray and yellow, and a yellowish spot in the centre; second, yellow, with two waved transverse stripes. The abdomen is yellow, with seven black brown belts. The thorax marked like a Death's head, whence the name, from Atropos, the third and last of the Fatal Sisters, who cuts the thread of life. The length is two inches. Caterpillar very large, yellow, with six green and orange oblique belts, and a posterior horn.

2. S. convolvuli, unicorn, or bindweed hawkmoth. The antennæ are long and thick; the trunk very long and spiral. Above, body marked with black and red belts; wings entire, browngray, with black zig zag transverse lines. The breadth three inches. Caterpillar smooth, green, with a posterior horn.

3. S. elpenor, elephant moth. The wings are angular, entire. Above, first wings striped transversely with red and green: second, black at the base, and red outwards. The body red and green. Caterpillar smooth, brown and yellow, with a posterior horn, and a snout like a hog. It lives on vines, convolvulus, &c.

4. S. filipendulæ, burnet moth. The antennæ, legs, and body, are black. Second wings red, with a greenish border. First wings bluish-green, with six red spots, in pairs. Length eight lines. Caterpillar yellow, with black spots. It lives on

grass.

The an

5. S. ligustri, privet hawk-moth. tennæ are long, thick, and brown. Trunk long, spiral. First wings two inches long, narrow, entire, brown; second, short, red, with black bars. The abdomen is red, with black rings. Caterpillar smooth, yellow-green, with a posterior horn.

6. S. ocellata, eyed willow hawk-moth. There is no trunk; the wings are indented. Above, first wings dark and light brown, marbled; second, red, with a large yellow black eye. Beneath, a large red triangle from the base of the first wings. The breadth one inch and a half. Caterpillar smooth, green, with oblique white lines on the sides, and a posterior horn. The eggs are green; it lives on willows.

7. S. populi, poplar hawk-moth. The wings are scalloped, bluish-gray, and waved with dark lines. On the first wings a long white spot, and the base of the second red brown. Wings reversed. Length one inch. A long spiral trunk caterpillar, green, smooth, with oblique white spots, and a posterior horn. It lives on poplars and willows.

8. S. stellatarum, large bee-moth. The antennæ are thick, towards the ends brown. The trunk is spiral; the wings are short and entire; the body is thick, brown, and hairy. First wings are brown, waved; second, red brown. It resembles a large bee. Caterpillar smooth, with a posterior blue horn, tipt with red. It lives on gallium.

9. S. tiliæ, lime hawk-moth. No trunk; the wings are scalloped; the antennæ are white on the upper side, yellow on the under. Above, first wings gray-brown, with two irregular large green spots; second wings orange. Beneath, greenish

gray. Caterpillar green, shagreened, with a posterior horn. The

10. S. tipuliformis, small bee-moth. thorax is yellow beneath; the wings are short, with black veins. The abdomen black, bearded, yellow at the extremity. Caterpillar on the loni

cera.

SPHINX, in zoology. See SIMIA.

SPHORULITE, in mineralogy. Colors brown and gray. In imbedded roundish balls and grains. Glimmering. Fracture even, splintery. Opaque. Scratches quartz with difficulty. Brittle. Specific gravity 24 to 2.5. Nearly infusible. It occurs in pearlstone and pitchstone porphyries. in the vicinity of Glasshütte near Schemnitz; and in the pitchstone of Meissen.

SPHONDYLIUM, in anatomy, one of the vertebræ of the back.

SPHRAGIDIUM, a famous cave of Baotia, in Mount Citheron.-Paus. ix. c. 3. SPI'AL, n. s. Fr. espial. A spy; scout; a watcher. Obsolete.

His ears be as spials, alarum to crie.

Tusser's Husbandry.

He privy spials placed in all his way, To weet what course he takes, and how he fares. Spenser. Their trust towards them hath rather been as to

good spials and good whisperers, than good magis

trates and officers.

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Bacon.

Fairfax.

Fr. espices; Italian specei, of Lat. speci. Á vegetable produc

tion, fragrant to the smell, and of pungent taste; an aromatic substance used in sauces: to season with spice: a spicer is a dealer in spice: spicery is the commodity or depository of spices. Dangerous rocks, Which, touching but by gentle vessel's side, Would scatter all the spices on the stream.

Shakspeare. Is not manhood, learning, gentleness, and virtue, the spice and salt that seasons a man?

Id. Troilus and Cressida.

His mother was a vot'ress of my order, And in the spiced Indian air by night Full often she hath gossip'd by my side.

Shakspeare.

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