Page:The New International Encyclopædia 1st ed. v. 04.djvu/863

CIRCLE. 750 CIRCULATING DECIMALS. tosiether wifli S ami Uie ciivuiiicoiitre of the tri- aii-ile, lie on a circle called the seven-points or Broeard circle. P, P' are called the Brocard points. Consult: McClelland, Geometry of ihe Circle (London. ISOU ; Casey, Hequel to Euclid (Dublin. 18S8) ; Catalan, Thioremes et -pro- hUmif de giumelrie iUnientaire (Paris, 1870). CIRCLE, ]NLG[C. A space in Aiiich sorcerers were wont, aecordin"; to the ancient popular be- lief, to protect themselves from the fury of the evil spirits they had raised. This circle was usually formed on a piece of ground from seven to nine feet square, in the midst of some dark forest, in a churchyard, vault, or other lonely and dismal spot. The circle was described at midnight in certain conditions of the moon and weather. Inside the outer circle was another somewhat less, in the centre of which the sorcerer had his seat. The spaces between the circles, as well as between the parallel lines that inclosed the larger one, were filled 'with all the holy names of God,' and a variety of other characters supposed to lie potent against the powers of evil. Without the protection of this circle, the magician, it was believed, would have been carried off by the spirits, as he would have been had he by chance got out of the charmed space. CIRCLE, Meridian. See Meridian Cibcle. CIRCLE, :MrR.L. See Mural Circle. CIRCLEVILLE. A city and the county-seat of Pickaway County, Ohio, 30 miles south of Columbus : on the Scioto River, the Ohio and Erie Caiial, and the Cincinnati and Muskingum Valley and the Norfolk and Vestern railroads (Map: Ohio. E (5). It w.as laid out in 1810, on the site of a prehistoric circular fortification (described in Howe's History of Ohio), from which the name is taken. Pork-packing is an important industry, and there are also exten- sive manufactures of straw-boards, furniture, farming implements, canned goods, flour, and corn-meal. Settled in 1S06, Circleville was in- corporated in 1814, and is governed at present under a revised charter of 1852, which provides for a mavor, elected biennallv, and a city council. Population, in 1800, 0.5.56; in I'JOO, (WSl. CIRCUIT (Fr. circuit, from Lat. circuitus, from circiiire, circumire, to go around; from circum, around -4- ire, to go). A territorial division, within which a court of justice is to l)e held, at stated times and places. The practice of dividing England into circuits, and assigning to each division judges whose duty it is to hold court therein at prescribed places and terms, be- came settled during the reign of Edward I. A3 early as the first Magna Charta. in 1215, it was provided that certain assizes (q.v.) for the re- <overy of lands should always be taken in the counties in which the lands were situated, and that two justices of the King should be sent to each county four times a year for that purpose. But the evils arising from the remote and ir- regular sittings of the King'.s Court (curia regis) were not obviated luitil the end of that century. Tliis early orningement of circiiita has lieen modified from time to time, and it is now regulated by the .Tuilicature Act of 1875, and an order in council of 187G. In the United States there are two classes of circuits — one belonging to the judicial system of the Federal Oovernmenl. (he other to tluit of the State. For a description of the several courts desig- nated as circuit courts, see Court. CIRCULAR NOTES. See Credit, Letters OF. CIRCULAR NUMBERS. Numbers whose powers end with the same figure as do the num- bers themselves: e.g. 0, 1, 5, (>, for 0"=0, 11 = 1, 5° being a multiple of 5, ends in 5. ()-=3(i, ti'=216, etc. CIRCULAR PARTS. The five elements in- A'olved in a rule for solving riglit- angled spherical triangles, formu- lated by John Na- pier (q.v.). The five parts, c, a, complement of A, complement of C, and complement of h. as indicated in the triangle, when co-J arranged on the '^ circumference of a circle, admit of the following selection: Any part, as co-C, has two adjacent parts, as eo-h, a, ^^^ and two opposite — parts, as co-.l, c. The rules of Na- . pier connect these "^""y >"• by the following mnemonic : The sine of the middle part equals the product of the tan- gents of the adja- ^ cent parts, and the sine of the middle part equals the product of the cosines of the opposite parts. CIRCULATING DECIMALS. Decimals in which one or more figures are continually re- peated in the same order ; e.g. 0.333 0.25G()G . . . . , 0.3172172 are circulating decimals. These are sometimes called repeating decimals, and the figure or set of figures repeated is called the repeieiid. If the repetend begins at the deci- mal point, the decimal is called a pure circulate; otherwise the decimal is called a mi.Kcd circulate; e g. 0.2727 .... is a pure circulate, but 0.25999 .... is mi.xed. If the repetend contains but one figure, it is called simple — otherwise, compound. If the first fi,gurcs of repetcnds are of th<' same order, the repetends are said to be similar; and if they end with figures of the same order, they are said to be conterminous; e.g. 0.C39292 .... and 0.253232 .... are both similar and conter- minous. Periods over the first and last figures of the repetend sen-c to indicate that a decimal is a circulate: thus, 0.273 = 0.27373 .... Opera- tions with circulating decimals may be performed in the usual way. or the circulates may be re- duced to common fractions. This is usually done by applying the fonnula for the sum of an infinite geometric progression. (See Series.) Thus. 0.35 is the same as 0.35 -f 0.0035 + 0.000035 + , in which the first term is 0.35 and the rate 0.01 ; hence, the sum is 0.3.5 _ 0.35 _ 35 1-0.01 ~ 0.99 ~ 99