1011 



REGULAR FIGURES, tc. 



RELATION. 



101J 



f=4, r = t, or four triangles; we have here the regular tetrahedron, or 

 triangular pyramid : (3) when = 4, giving = 3, m = 4, f=12,/=8, 

 e=6, or eight triangles; we have here the regular octahedron : (4) 

 whenw = 5, giving = 8, w = 5, =30, /=20, c-12, or 20 triangle*; 

 we hare here the regular icosahedron. 



Let = 4,or e= 8w (8-2m). This Ua whole number (1) when = 2; 

 which reject: (2) when m = 3, giving = 4, m=8, < --= 12, /= 6, e=8,or 

 ii iquaret ; we have here the regular hexahedron, or cube, the only 

 on of its kind. 



Let = 5, or <=10m-i-(10 8m). This is a whole number (1) when 

 1=2; which reject: (2) when w=3,giving = 5,i = 3, =80,/=12, 

 e=20, or 12 pentagons ; we have here the regular dodecahedron, the 

 only one of its kind 



We hare thw the five regular solids, and have shown that there can 

 be no others. 



When the rfaMtl. 



The centre of a regular polyhedron is obviously the point of inter 

 section of lines drawn from the corners, each inclined at the same 

 angle to all the edges which meet it. The radius of the circumscribed 

 sphere is the line drawn from any corner to the centre ; that of the 

 inscribed sphere is the perpendicular let fall from the centre upon 

 any of the faces. 



The following table answers to that for polygons, and ia taken from 



the same source : 



WTien the Side = 1. 



When Rodvu of Ctrmmteribed Sphere = 1. 



When Radius of Inicribtd Sphere = 1. 



When t>* Superfcie* = 1. 



REGULAR POLYGON, REGULAR SOLI1JS. [It ::KS.] 



REGULATORS OF MOTION. Fly-wheels are the means umally 

 employed to render the movements of machines an nearly as po 

 uniform ; and the nature as well as the applications of these v 

 described under WHEELS, and under STEAM-ENGINE. Pendulums, as 

 regulators of motion for clock-work, are described under I' 

 and the (iorernor, by which the supply of steam is regulated, will be 

 described under STEAM-ENGINE. Some account of air-vessels for regu- 

 lating motion in the tread-wheels, which are employed in prisons, has 

 been given under AIR-VESSEL, and it may be added that a particular 

 kind of fly has been occasionally used for the like purpose. This con- 

 sists of a vertical rod or shaft about 20 feet high, carrying at its * 

 extremity, on opposite sides, a long rectangular frame, which is pro- 

 vided with shutters turning on hinges ; by the revolution of the shaft 

 these frames turn round horizontally ; and the shutters being connected 

 with two governor-balls by means of wheel-work, when the motion of 

 the tread-wheel becomes too rapid, the diverging balls cause the 

 shutters to close, and thus the resistance of the air diminishes the 

 velocity. Should the movement of the tread-wheel become too slow, 

 the balls collapsing allow the shutters to open, when the resistance of 

 the air is diminished and the velocity of revolution increases. 



RE'GULUS. A line drawn from the pole-star, not through the two 

 pointers, but between them and the five secondary stars of the Great 

 Bear, which lie near them, will pass through the bright star called 

 a Leonis, or Cor Leonis (the lion's heart). By Ptolemy and other 

 Greeks it was called f}curi\iirKos, whence comes the Latin name Regulus, 

 a word which is the diminuti 



REINSCH'S TEST FOR ARSENIC. [AESE.MC. DETECTION OF.] 

 REJOINDER. [PLEADING.] 



RELAPSING FEVER. [BLOOD, DISEASES OF THE.] 

 RELATION (Mathematics). What we here mean by this word 

 would have been explained in the article EQUATION, if we had confined 

 ourselves to the explanation of arithmetical algebra ; but having in the 

 articles ALGEBRA and OPERATION endeavoured to give higher vie 

 are induced to insert the present article by remembering that the diffi- 

 culties of such a subject are of very different kinds to different persons, 

 insomuch that any point of view may be usefully taki-ii with reference 

 to some minds, and any detail upon a fundamental notion may remove 

 misapprehension in one quarter or another. 



All reasoning is the discovery of relations which are not evident from 

 those which are; or rather, since thu proposed result is sometimes 

 evident iu itself, reasoning is the establishment of one relation as a 

 necessary consequence of others. The term relation would be difficult 

 to define in a manner satisfactory to all ; it is enough for our pi 

 purpose to say a relation exists between any two objects, whether of 

 sense or intellect, whenever they have anything in common ; that is to 

 say, the common point, whatever it may be, may be made the me 

 referring one to the other, or bringing our thoughts from one to the 

 other, so as to think of both at the same time, and to compare the two. 

 All the manifold senses of the word may be derived from this one : the 

 relationship of blood implies a common ancestry ; the relationship of 

 office, common duties. In mathematics, the relation of greater, equal, 

 or less, implies thaf one of the magnitudes U the same as to quantity 

 with part or all of the other, and so on. Sameness in cvi-ry i 

 would constitute identity ; sameness in one or more respects, relation. 

 The triangles in Euclid, i. 4, are by hypothesis related in a given 

 manner in three particulars : a change of place shows that they can be 

 made identical ; that is, their difference before the change of place was 

 difference of position only, not at all of form ; in all that can distinguish 

 one triangle from another, except its position in space, they are iden- 

 tical. We do not quarrel with the phrase that they are the same 

 triangles differently placed, because sameness is understood with a 

 reservation, and the preceding means that they are the same except in 

 difference of place. 



In an algebraical expression we may have to consider its meaning, form, 

 magnitude, source, mode of derivation, and properties. The meaning 

 depends upon the fundamental definitions which are employed and the 

 form ; the form, upon the arrangement of the symbols ; the magni- 

 tude, when magnitude is signified, upon the form and the particular 

 values given to the symbols; so that these various sources of re I 

 are closely connected with one another. The fundamental meaning o 

 the sign = implies equality of quantity or magnitude, and some insist 

 that it shall always retain this meaning. There can be no objection to 

 any one insisting on this point for himself ; but the learner who, if 

 he be wise, will learn all languages with the majority, even though he 

 should afterwards teach with the minority must make himself accus- 

 tomed to various uses of this sign, as follows : 

 1. The sign = means that on one side we hmvo tin operation to be 



