REG 



II E G 



The term is now chiefly ufed, as it is defciibed by ft.it. 

 5 & 6 Edw. VI. cap. 14. to denote one that buys corn, 

 Of other dead victuals, in any market, and fells them 

 again, in the fame market, or within four miles of the 

 place. 



Regrating is an offence againft the public, and is liable, 

 hy the ftatute juft cited, to the fame penalty with engroiling 

 and forellalling. 



Reguatou is alfo ufed for a perfon who furbi{hes up 

 old moveables, to make them pafs for new. See FiiiP- 



rjiiiv. 



Among mafons, &c. to regrate is to take off the outer 

 I'lnface of an old hewn ftone, with the hammer and ripe, 

 ill order to whiten and make it look frefh again. 



REGRESSION, or Retrogradation of curves, &c. 

 See Retroguapation. 



REGUINY, in Geography, a town of France, in the 

 department of the Morbihan ; five miles N.W. of Joffelin. 

 REGULA. See Rule. 

 Regui.a, in Architedure. See Reglet. 

 REGULAR, Regulauis, denotes the relation of any 

 thing that is agreeable or conformable to the rules of art. 



In this fenle, the word Hands oppofed to irregular, or 

 anomalous. 



Thus we fay, a regular proceeding, a regular building, 

 regular poem, regular verb, &c. 



Regular Figure, in Geometry, is a figure which is both 

 equilateral and equiangular ; i. e. whofe fides, and confe- 

 quently its angles, are all equal. 



The equilateral triangle and fquare are regular figures. 

 All other regular figures, confiding of more than four fides, 

 are called regular polygons. 



Every regular figure may be infcribed in a circle ; which 

 fee. 



For the dimenfions, properties, &c. of regular figures, 

 fee Polygon. 



Regular Body, called alfo Platonic body, is a folid ter- 

 minated on all fides by regular and equal planes, and whofe 

 folid angles are all equal. 



The regular bodies are five in number ; i>i%. the cube, 

 vvliich coniifts of fix equal fquares ; the tetrahedron, or re- 

 gular triangular pyramid, having four equal triangular 

 faces ; the oBahedron, having eight ; the dodecahedron, hav- 

 ing twelve pentagonal faces ; and the icofahedron, having 

 twenty triangular faces. See each under its proper article. 

 Befides thefe five, there can be no other regular bodies in 

 nature. 



To meafure the furface and Jolidity, life, of the Jive regular 

 bodies. — The folidity, &c. of the cube is (hewn under the 

 article Cube. The tetrahedron being a pyramid, and the 

 oftahedron a double pyramid ; and the icofahedron con filling 

 of twenty triangular pyramids ; and the dodecahedron of 

 twelve quinquangular ones, whofe bafes are in the furface 

 of the icofaliedron and dodecahedron, and their vertices meet- 

 ing in a centre ; the folidities of thefe bodies are all found 

 from what we have fhewn under Pyra.mid. 



I. Their furface is had by finding the area of one of the 

 planes, from the lines that bound it ; and multiplying the 

 area thus found by the number from which the body is de- 

 nominated : e. gr. for the tetrahedron, by 4 ; for the hexahe- 

 dron, or cube, by 6 ; for the oftahedron, by 8 ; for the 

 dodecahedron, by 12; and for the ifocahedron, by 20. 

 The product is the fuperficial area. 



Or, t!ie fuperficial contents of any of the five Platonic 

 bodies may be had by the following proportion ; as i is to 

 the fquare of the fide of the given Platonic body, 



i5 



f i.732C5o8'l I* tetrahedroB. 



3.4641016 I . ,, r r • I otlahedron. 

 r • I J ^ ^ I to the fuperficial ■ , , 



lo IS <^ 6.0000000 > . . r .1 { hexaliedron. 

 , o rt- \ content 01 the ' . r ■ ■ 



I 0.6602540 I I icolahedron. 



L 20.6457788 J (^dodecahedron. 



Hence we have the following rule : multiply the proper 

 tabular area, taken from the preceding table, by the fquare 

 of the fide of the given folid, for the fuperficies. 



2. Tlie diameter of a fpherc being given, to find the fide 

 of any of the Platonic bodies, thut may be cither infcribed 

 in the fphere, or circumfcribed about the fphere, or that is 

 equal to the fphere. 



As I is to the number in the follov.-ing table, refpecting 

 the thing fought, fo is the diameter of the given fpherc to 

 the fide of the Platonic body fought. 



3 . The fide of any of the five Platonic bodies being given, 

 to find the diameter of the fphere, that may be infcribed in that 

 body, or circumfcnbed about it, or that is equal to it. As 

 the refpeftive number, in the above table, under the title, 

 infcribed, circumfcribed, or equal, is to i, fo is the fide of the 

 given Platonic body to the diameter of its infcribed, circum- 

 fcribed, or equal fphere, in lolidity. 



4. The fide of any of the five Platonic bodies being given, 

 to find the fide of either of the Platonic bodies, which are 

 equal in folidity to that of the given body. As the numbet 

 under the title equal, againlt the given Platonic body, is to 

 the number under the fame title, againft the body whofe 

 fide is fought, fo is the fide of the given Platonic body to the 

 fide of the Platonic body fought. 



5. To find the folid contents of any of the five Platonic 

 bodies. As i is to the cube of the fide of any of thefe 

 bodies, fo is o. 11 785 13 to the folid content of a tetrahe- 

 dron, 0.4174045 to that of the octahedron, i.oooooooto 

 that of the hexahedron, 2.1816950 to that of the icofa- 

 hedron, and 7.663 1 1 89 to the folid content of the dode- 

 cahedron. 



Hence we have the following rule : multiply the tabular 

 folidity by the cube of the fide or hnear edge, for the foHd 

 content. The demonftration of this rule, and that for the 

 fuperficies above given, is as follows : 



The tabular numbers denote the furface and folidity of 

 each body, when its fide or edge is one ; and, becaufe, in 

 iimilar bodies, the furfaces are as the fquares of the hnear 

 edges, and the folidities as the cubes of the fame, the truth 

 of the rules is manifeft. 



If one of thefe bodies be required to be cut out of the 

 fphere of any diameter, let dr (P/afcXII. Geometry, jig.<^.^ 

 be the diameter of any fphere, and d a one-third of it, =: 

 ab -^br. Erefl the perpendiculars a f, c f, and^_f ; and 

 draw d e, df, e r, f r, and g r ; then will (l) re be the 

 fide of the tetrahedron ; (2) df, the fide of the hexahedron ; 

 (3) d e, the fide of the oftahedron ; (4) and cutting d e m 

 extreme and mean proportion in h, d h wiU be the fide of 

 the dodecahedron ; (5) fetting the diameter d r up perpendi- 

 cular at r, from the centre c, to its top, draw the line eg, 



cutting 



