PARA L L ELOI'l P ED (raps:Mpe Thu doe, parallel-plane solid) more correctly written paralldepiped, is the name given to a solid contained by six parallelograms, which are equal and parallel, two and two. It is in fact a quadrangular prism.
When all the parallelograms are rectangles, we have one of the figures to which our eyes are most accustomed, as in the case of a die, a box, a plank, a room, he kc. Persons not acquainted with mathe matics would hardly believe that English mathematicians seldom express this moat simple and elementary of all solids in less than ten syllables, as follows:— rect-an-gulair A more simple term might easily be obtained, and one perfectly con sistent with analog. namely ri!.ht solid. Thus a right line might be conceived as generated by the most simple motion of a point ; a right surface (or rectangle) by the most simple motion of a right line; and a right solid (or rectangular parallelopiped) by the most simple motion of a right surface. We shall consider the properties of a right solid in the article Ilec-rAsete.
When the adjacent rectangles of a right solid are squares, tho solid is a Cunt, for which fortunately there is a shorter term than equi lateral rectangular parallelopiped.
The number of cubic units in a parallelopiped is found by multiply ing the number of square units in either base by the number of linear units in the perpendicular distance between that base and the opposite One. The diagonals A 0, BH, c13, DF, meet in the same point, which bisects them all, and the sum of their squares is equal to the sum of the squares of all the twelve sides of the solid.
l'ARALLELS (rapcbanha, by the side of each other), the name given by.the Greek geometers to lines in the same plane having that relation of situation of which it is one of the most obvious pro perties that such lines never meet, however far they may be produced or lengthened.
If we examine the properties of lines experimentally, it will be easy to satisfy ourselves of the existence of such pairs as A B, c n, which neither diverge nor converge, and to which conunon perpendiculars, such ? 3 and r Q. all of the same length, can be drawn through any point of either. Moreover the angles a a n and It T D made by the same line with both, will be found to be the same. If then we take the notion of permanence of direction which always accompanies that of straight ness [Dtaturios], and also the notion of differing directions, which is suggested by two lines which make an angle, we may readily see that the relation of situation which, adopting Euclid's term, may be called Is really that which would be also conveyed by the words nenren of direction ; so that if two lines A and n be parallel, A may be substituted fors or n for a, in any proposition which involves relations of direction only, without affecting the truth of that proposi tion. If true, or its falsehood, if false.
Geometry, as every beginner known, depends upon a small number of self-evident truths, or rather of propositions the truth of which (with one r.rception) is so soon and tin easily perceived, that no one doubts of them when stated with ordinary attention to clearness of ex preasion. The exception alluded to appears for the first time in and has been the occasion of a controversy which has lasted from his time to the present.
It will be remarked that the definition of parallel lines is purely negative : it describes what they are not, not what they are : if lines which meet, or which will meet if produced, be called intersectors, parallels are som-intersectors. Those who would found geometry upon definitions entirely, may think that the difficulty of the theory of parallels arises from insufficient definition : but those who believe it to be deducible from real and positive conceptions, having nothing arbi tary about them, must suspect that, in this purely Negative definition of parallels, we have not sufficiently described that very obvious relation of position which distinguishes parallelism from convergence, — however short the lines we image to ourselves, or however little we think of what will take place if they are produced. Euclid, - proceeding upon axioms the admission of which is not con sidered to be a question connected with the present difficulty, establishes the following proposition :—If the two lines s Band T D make the angles I'S T and s T D equal, or n s n and S T D equal, or n s T and s T D together equal to two right angles (all which amount to the same thing), then s and T D are non-intersectors. But before any further step can be made, it must either be proved or assumed that in every other case they are intersectors, and Euclid, being unable to prove it, assumes it directly. That is to say, he requires it to be granted that if n s T and s T D be together less than two right angles, n and T will meet., if produced, and on that side on which they make with s T the angles less than two right angles. The last clause is not a necessary part of the axiom, since it can be shown, independently of the present theory, that two lines which meet must make angles together less than two right angles with any line which cuts them internally on the side of meeting.