Interminate Ex. 1ft.J-Divide.5376 by 7. 2d.] Divide.83 by 5. Decimals. 7.).5376(-76,095238, .5)843(.1686

In thefe accounts the quotient is never finite. It may repeat, if the dividend repeats; or, if the dividend circulate, it may circulate an equal number of places, often more, and never fewer. The greateft poffible extent of the circle is found by multiplying the divifor into the number of places in the circle of the dividend. Thus, a circulate of 3 places, divided by 3, quotes a circulate of 3 times 3, or 9 places. 3

CASE II." When the divifor is interminate, the "multiplications and fubtractions must be performed, according to the directions given for repeating and circulating decimals"

66

Ex. ft.] Divide .37845 by 8 2).37845(.68121

333333

45116

44444

672

55%

118

The foregoing method is the only one which pro- Extraction perly depends on the principles of decimal arithmetic; of Roots. but it is generally fhorter to proceed by the following

rule.

"Reduce the divifor to a vulgar fraction, multiply "the dividend by the denominator, and divide the pro"duct by the numerator."

Ex 1ft.] Divide .37845 by 8:

9

5)3.40605(.68121

2d.] Divide .3784% by 6=} 3

2)1.13536(.567683

Note 1. Divifion by g triples the dividend, and divifion by increases the dividend one-half.

Note 2. When the divifor circulates, the denomina-. tor of the vulgar fraction confills of 9's, and the multiplication is fooner performed by the contraction explained p. 295. col. 1. par. 1. It may be wrought in the fame way, when the divifor repeats, and the deno.. minator, of confequence, is 9.

Note 3. If a repeating dividend be divided by a repeating or circulating divifor; or, if a circulating di-. vidend be divided by a fimilar circulating dividend, or, if the number of places in the circle of the divifor be a multiple of the number in the dividend; then the product of the dividend multiplied by the denominator of the divifor will be terminate, fince like figures are fubtracted from like in the contracted multiplication, and confequently no remainder left. The form of the quotient depends on the divifor, as explained at large, p. 316. col. 1. par. 1.-p. 318. col. 2. par. 3.

Note 4. In other cafes, the original and multiplied. dividend are fimilar, and the form of the quotient is the fame as in the cafe of a finite divifor. See p. 322. col. 2. par. ult. &c.

Note 5. If the terms be fimilar, or, extended till they become fo, the quotient is the fame as if they were finite, and the operation may be conducted ac-. cordingly; for the quotient of vulgar fractions that have the fame denominator is equal to the quotient of

ed.] Divide .245892 by 2,18, .2,18,).245892(1.127005 218181,81,

27710,18,

21818,18,

5892,00,

4363,63,

1528,36,

1527,27,

1090,90 1090,90

CHAP. XI. OF THE EXTRACTION or ROOTS.

THE origin of powers by involution has already been explained under the article ALGEBRA. There now remains therefore only to give the most expeditious methods of extracting the fquare and cube roots; the reafons of which will readily appear from what is faid under that article. As for all powers above the cube, unless fuch as are multiples of either the fquare and cube, the extraction of their roots admits of no deviation from the algebraic canon which must be always conftructed on purpose for them."

If the root of any power not exceeding the feventh power, be a fingle digit, it may be obtained by infpection, from the following TABLE of powers.

33

1

power or root.

2d power or

fquare.
3d power or
cube.
4th power or
biquadrate.
5th power or
furfolid.

6th power or cube fquared.

7th power.

2 div. 725) 3625 refolvend.

3625 product.

2d.] Required the fquare root of 72, to eight de

cimal places.

72.00000000 (8.48528137 root.

64

66

164)800 656

1688)14400

13504

16965)89600

8

64

512409632768|262144|2097152 729 656159049531441 4782969

84825 169702)477500.

Sect. i. EXTRACTION of the SQUARE ROOT.

RULE I. Divide the given number into periods "of two figures, beginning at the right hand in integers, and pointing toward the left. But in deci"mals, begin at the place of hundreds, and point to"ward the right. Every period will give one figure "in the root."

II. Find by. the table of powers,. or by trial, "the nearest leffer root of the left-hand period, place. "the figure fo found in the quot, fubtract its fquare "from the faid period, and to the remainder bring "down the next period for a dividual or refolvend."

III. "Double the quot for the first part of the "divifor; inquire how often this first part is contain"ed in the whole refolvend, excluding the units place; "and place the figure denoting the anfwer both in the quot and on the right of the first part; and you have "the divifor complete."

following period for a new refolvend, and then pro"ceed as before."

Note 1. If the first part of the divifor, with unity fuppofed to be annexed to it, happen to be greater than the refolvend, in this cafe place o in the quot, and alfo on the right of the partial divifor; to the refolvend bring down another period; and proceed to divide as before.

Note 2. If the product of the quotient-figure into the divifor happen to be greater than the refolvend, you muft go back, and give a leffer figure to the quot.

Note 3. If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation, by annexing periods or pairs of cyphers, till there be no remainder, or till the decimal part of the quot repeat or circulate, or till you think proper to limit it.

339404

169704)138096

135763

2333

1697

636

509

127

1-18

(9)

After getting half of the decimal places, work by contracted divifion for the other half; and obtain them with the fame accuracy as if the work had been at large.

3d.] Required the fquare root of .2916.

.2916(.54 root. 25 104) 416 416

If the fquare root of a vulgar fraction be required, find the root of the given numerator for a new numera tor, and find the root of the given denominator for a new denominator. Thus, the fquare root of is, and the root of is; and thus the root of (=61) is =2.

But if the root of either the numerator or denominator cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root, as in Ex. 3d. above.

Sect. ii. EXTRACTION of the CUBE ROOT RULE I. "Divide the given number into periods tegers, and pointing toward the left. But in deci"of three figures, beginning at the right hand in in

mals, begin at the place of thousands, and point to"ward the right. The number of periods shows the "number of figures in the root."

II. Find by the table of powers, or by trial, the "neareft leffer root of the left-hand period; place the "figure fo found in the quot; fubtract its cube from "the faid period; and to the remainder bring down: "the next period for a dividual or refolvend.”

The divifor confifts of three parts, which may be - found as follows.

35

III." The first part of the divifor is found thus: Multiply the fquare of the quot by 3, and to the pro"duct annex two cyphers; then inquire how often this "first part of the divifor is contained in the refolvend, "and place the figure denoting the answer in the quot." IV. Multiply the former quot by 3, and the pro"duct by the figure now put in the quot; to this last "product annex a cypher; and you have the fecond "part of the divifor. Again, fquare the figure now

66

put in the quot for the third part of the divifor; "place these three parts under one another, as in ad“dition; and their fum will be the divifor complete."

V." Multiply the divifor, thus completed, by the "figure laft put in the quot, fubtract the product from "the refolvend, and to the remainder bring down the "following period for a new refolvend, and then pro"ceed as before."

Note 1. If the first part of the divifor happen to be equal to or greater than the refolvend, in this cafe, place o in the quot, annex two cyphers to the faid first part of the divifor, to the refolvend bring down an other period, and proceed to divide as before.

Note 2, If the product of the quotient-figure into the divifor happen to be greater than the refolvend, you must go back, and give a leffer figure to the quot.

Note 3. If, after every period of the given number is brought down, there happen at laft to be a remainder, you may continue the operation by annexing periods of three cyphers till there be no remainder, or till you have as many decimal places in the root as you judge neceffary.

Ex. ift. Required the cube root of 12812904.

Cube number 12812904(234 foot.

If the cube root of a vulgar fraction be required, find the cube root of the given numerator for a new numerator, and the cube root of the given denomina- · tor for a new denominator. Thus, the cube root of

is, and the cube root of is: and thus the cube root of (=15}) is {=2}.

But if the root of either the numerator or denomina→ tor cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the

root.

Extraction of Roots,

ARI

ARIUS, a divine of the fourth century, the head and founder of the ARIANS, a fect which denied the eternal divinity and fubftantiality of the Word. He was born in Libya, near Egypt. Eufebius bishop of Nicomedia, a great favourite of Conftantia fifter of the emperor Conftantine and wife of Licinius, became a zealous promoter of Arianifm. He took Arius under his protection, and introduced him to Conftantia; fo that the fect increased, and several bishops embraced it openly. There arofe, however, fuch difputes in the cities, that the emperor, in order to remedy thefe diforders, was obliged to affemble the council of Nice, where, in the year 325, the doctrine of Arius was condemned, Arius was banished by the emperor, all his books were ordered to be burnt, and capital punishment was denounced against whoever dared to keep them. After five years banishment, he was recalled to

ARI

Conftantinople, where he prefented the emperor with a confeffion of his faith, drawn up fo artfully, that it fully fatisfied him. Notwithstanding which, Athanafius, now advanced to the fee of Alexandria, refused to admit him and his followers to communion. This so enraged them, that, by their intereft at court, they procured that prelate to be depofed and banished. But the church of Alexandria ftill refufing to admit Arius into their communion, the emperor fent for him to Conftantinople; where, upon delivering in a fresh confeffion of his faith in terms lefs offenfive, the emperor commanded Alexander, the bishop of that church, to receive him the next day into his communion: but that very evening Arius died. The manner of his death was very extraordinary: as his friends were conducting him in triumph to the great church of Conftantinople,, Arius, preffed by a natural neceffity, ftepped afide to eafe

him.

himfelf; but expired on the spot, his bowels gufhing

out.

But the herefy did not die with the herefiarch: his party continued ftill in great credit at court. Athanafius, indeed, was foon recalled from banishment, and as soon removed again; the Arians being countenanced by the government, and making and depofing bifhops as it beft ferved their purposes. In short, this feet continued with great luftre above 300 years: it was the reigning religion of Spain for above two centuries; it was on the throne both in the eaft and weft; it prevailed in Italy, France, Pannonia, and Africa; and was not extirpated till about the end of the 8th century.

This herefy was again fet on foot in the weft by Servetus, who, in 1531, wrote a little treatife against the mystery of the Trinity. After his death, Arianifm got footing in Geneva; from whence it removed into Poland; but, at length, degenerated, in a great meafure, into Socinianifm. Erafmus feems to have aimed at reviving Arianifm, in his commentaries on the New Teftament; and the learned Grotius feems to lean a little that way.

With regard to the ftate of Arianifmn in England, it may be fufficient to obferve, that from the numerous publications of that caft which are daily making their appearance, it feems to be rather a growing, than exploded, doctrine there.

ARK, or Noah's ARK, a floating veffel built by Noah for the prefervation of his family and the feveral fpecies of animals during the deluge.

The ark has afforded feveral points of curious inquiry among the critics and naturalifts, relating to its form, capacity, materials, &c.

The wood whereof the ark was built is called in the Hebrew Gopher-wood, and in the Septuagint Square timbers. Some tranflate the original cedar, others pine, others box, &c. Pelletier prefers cedar on account of its incorruptibility, and the great plenty of it in Afia; whence Herodotus and Theophraftus relate, that the kings of Egypt and Syria built whole fleets thereof, inftead of deal.

The learned Mr Fuller, in his Mifcellanies, has obferved, that the wood whereof the ark was built was nothing but that which the Greeks call nuragicos, or the cypress-tree; for, taking away the termination, kupar and gopher differ very little in found. This obfervation the great Bochart has confirmed, and fhown very plainly that no country abounds fo much with this wood as that part of Affyria which lies about Babylon.

In what place Noah built and finished his ark is no lefs made a matter of difputation. But the most probable opinion is, that it was built in Chaldea, in the territories of Babylon, where there was fo great a quan tity of cyprefs in the grores and gardens in Alexander's time, that that prince built a whole fleet out of it for want of timber. And this conjecture is confirmed by the Chaldean tradition, which makes Xithurus (another name for Noah) fet fail from that country.

The dimenfions of the ark, as given by Mofes, are 300 cubits in length, 50 in breadth, and 30 in height; which fome have thought too feanty, confidering the number of things it was to contain; and hence an argument has been drawn against the authority of the relation. To folve this difficulty many of the ancient

fathers, and the modern critics, have been put to very miferable shifts: But Buteo and Kircher have proved geometrically, that, taking the common cubit of a foot and a half, the ark was abundantly fufficient for all the animals fuppofed to be lodged in it. Snellius computes the ark to have been above half an acre in area. Father Lamy fhows, that it was 110 feet longer than the church of St Mary at Paris, and 64 feet narrower: and if fo, it must have been longer than St Paul's church in London, from weft to east, and broader than that church is high in the infide, and 54 feet of our measure in height; and Dr Arbuthnot computes it to have been 81062 tons.

The things contained in it were, befides eight perfons of Noah's family, one pair of every fpecies of unclean animals, and feven pair of every species of clean animals, with provifions for them all during the whole year. The former appears, at firft view, almost infinite; but if we come to a calculation, the number of fpecies of animals will be found much less than is generally imagined; out of which, in this cafe, are excepted fuch animals as can live in the water; and bishop Wilkins fhows that only 72 of the quadruped kind needed a place in the ark.

By the defcription Mofes gives of the ark, it appears to have been divided into three ftories, each ten cubits or 15 feet high; and it is agreed on, as moft probable, that the loweft flory was for the beafts, the middle for the food, and the upper for the birds, with Noah and his family; each story being fubdivided into different apartments, ftalls, &c. though Jofephus, Philo, and other commentators, add a kind of fourth ftory under all the reft; being, as it were, the hold of the veffel, to contain the ballaft and receive the filth and feces of so many animals: but F. Calmet thinks, that what is here reckoned a flory, was no more than what is called the keel of fhips, and ferved only for a confervatory of fresh water. Drexelius makes 300 apartments; F. Fournier, 333; the anonymous author of the Queftions on Genefis, 400; Buteo, Temporarius, Arias Montanus, Hoftus, Wilkins, Lamy, and others, fuppofe as many partitions as there were different forts of animals. Pelletier makes only 72, viz. 36 for the birds, and as many for the beafts. His reafon is, that if we fuppofe a greater number, as 333 or 400, each of the eight perfons in the ark must have had 37, 41, or 50 ftalls to attend and cleanfe daily, which he thinks impoffible to have been done. it is obferved, that there is not much in this: to diminish the number of ftal's without a diminution of animals is vain; it being perhaps more difficult to take care of 300 animals in 72 ftalls than in 300. As to the number of animals contained in the ark, Buteo computes that it could not be equal to 500 horfes; he even reduces the whole to the dimenfions of 56 pair of oxen. F. Lamy enlarges it to 64 pair of oxen, or 128 oxen; fo that, fuppofing one ox equal to two horfes, if the ark had room for 256 horfes, there must have been room for all the animals. But the same autor demonftrates, that one floor of it would fuffice for 500 horfes, allowing nine fquare feet to a horfe.

But

As to the food in the fecond ftory, it is obferved by Buteo from Columella, that 30 or 40 pounds of hay ordinarily fuffices for an ox a-day; and that a folid cubit of hay, as ufually preffed down in our hay-ricks,

floating

on the waters of the Deluge?