* 



PARALLKLOPll-KP. 



PARALLELS. 



HO 



Oft** (M Urn* at A and ) r together equal to two right angle*, or 



m mffli " of each other. 



X The di*ooal* UP and c)b4ect one another. 



4. The *um of the quare* on the four aide* U equal to the turn of 

 the aquar** on UM diagonal*. 



5. The area of a parallelogram depend* only on the aide (a* c D) and 

 the perpendicular distance (pi)) between that and the oppocite side ; *o 

 that lanHeV^umi on equal lae* and between the aamo parallel! are 



0. If the point (5) be taken on a diagonal (en) and mule the v. -rt.-x 

 of a pair o( parlW-^nun ( A 1 .'. and D452) lying in th- <s,iul tri- 

 aagokr halve* of the parallelogram, theee parallelogram* will be equal 

 to oat another in area. 



I'AKALLKI.Ol'IPED (rvXXiX-w(or, parallel plane solid) 

 more correctly written paraiirltplped, i* the name given to a solid 

 ~t.i.^l by tux parallelogram*, which are equal and |rallel, two and 

 two. It i* in fact a quadrangular prism. 



r, 



When all the parallelograms are rectangles, we have one of the 

 figura to which our eye* are most accustomed, as in the case of a die, 

 box, a plank, a room, 4c. Ac. Persons not acquainted with mathe- 

 mrtf~ would hardly believe that English mathematicians seldom 

 expre** thi* most limple and elementary of all solids in less than ten 

 y liable*, a* follow* : 



rect-an-gul-ar par-all-el-o-pi-ped. 



A more *imple term might easily be obtained, and one perfectly con- 

 rictent with analogy, namely rigU tolitt. Thus a right line might be 

 conceived u generated by the most simple motion of a point ; a right 

 turface (or rectangle) by the most simple motion of a right line ; nml a 

 right solid (or rectangular parallelepiped) by the most simple motion 

 of a right surface. We shall consider the properties of a right solid 

 in the article RECTANGLE. 



When the adjacent rectangles of a right solid are squares, the solid 

 U a Ci'BK, for which fortunately there is a shorter term than equi- 

 lateral rectangular parallelepiped. 



The number of cubic unite in a parallelepiped is found by multiply- 

 ing the number of square units in either base by the number of linear 

 unit* in the perpendicular distance between that base and the opposite 

 one. The diagonals AO, BH, CE, DF, meet in the same point, which 

 buecU them all, and the sum of their squares is equal to the sum 

 of the squares of all the twelve sides of the solid. 



PARALLELS (ra^x(AAi)\a, 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 



. _D 



to mtisfy ounelve* of the existence of such pairs as A n, r n, which 

 neither diverge nor converge, and to which common perpendiculars, such 

 a* M S and r q. all of the same length, can be drawn through any point 

 of either. Moreover the angle* R s B and K T i> 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- 

 ne [DiHEOTiosJ. and also the notion of differing directions, which is 

 uggwted by two line* which make an angle, we may readily see that 

 the relation of situation which, adopting Euclid's term, may be called 

 parmlUitm, b really that which would be also conveyed by the words 

 mm*** of direction ; *o that if two line* A and B be parallel, A may 

 be wbeUtated for or B for A, in any proposition which involves 

 relation* of direction only, without affecting the truth of that proposi- 

 tion, if true, or iU falsehood, if falre. 



Geometry, a* every beginner knows, depend* upon a small number 

 of *elf-evident truth*, or rather of propositions the truth of which 

 (it* ont utrpfum) i* *o soon and an easily perceived, that no one 

 doubt* of them when *UUd with ordinary attention to clearness of 

 ezpreesion. The exception alluded to appear* for the first time in 

 Kuclid, and ha* been the occasion of a controversy which ha* lasted 

 from hi* time to the preeant. 



It will be remarked that the definition of parallel line* i* purely 

 negative : it describe* what they are not, not what they are : if lines 



wliii-h meet, or which will meet if produced, be called interaeotora, 

 |ralleU are Hun-intersectors. Those who would fom.d omctry upon 

 definitions entirely, may think that the difficulty of the theory of 

 iiarallels arise* from insufficient definition : but those who believe it to 

 be deducible from real and positive conceptions, having nothing mili- 

 tary about them, must suspect that, in this purely negative definition 

 of parallels, we have not sufficiently described that very obvious 

 rd.it i> in of intuition which distinguishes parallelism from convergence, 



however short the lines we image to ourselves, or however lit tin 



""" we think of what will take place if they are produced. Euclid, 



proceeding upon axioms the admission of which is not coii- 



' to be a question connected with the present difficulty, establishes 



the following proposition : If the two lines s B and T o make the angles 

 r s T and 8 T u equal, or us B and STD equal, or BSTandsTO together 

 equal to two right angles (all which amount to the same thing), then g B 

 and T D are non-iuterscctors. 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 B 8 T and s T D be 

 together less than two right angles, 8 B and T D will meet, if produced, 

 and on that side on which they make with B T the angles less than two 

 right angles. The last clause is not a necessary part of the i 

 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. 



Euclid obviously puts the wlw/e difficulty into an assumption ; 

 which, though the most direct course, is not that which is best calcu- 

 lated to give the highest degree of evidence to geometrical truth - 

 it is a more obvious proposition that two lines which intersect one 

 another cannot both be parallel to a third line, and this being granted, 

 Euclid's axiom readily follows. If it should be objected that this is 

 merely Euclid's axiom in another form, it is replied that the form is a 

 more easy one, and therefore preferable: just as it would be wiser 

 to assume " Every A is B and every Bis A, "than the identical but miv 

 complicated proposition " Every A is B, and everything which is not \ 

 is not B." 



It is known then that the difficulty is entirely removed if we grant 

 that "two lines which intersect are not both parallel to any third line," 

 or, which is the same thing, that " through a given point not more 

 than one line can be drawn parallel to a given line." The tin 

 Euclid being thus improved so for as it is capable of being done by a 

 mere difference of statement, it remains to ask, 1. Whether assump- 

 tion can be dispensed with altogether, and a direct proof of Euclid's 

 axiom, or something equivalent, to it, given * 2. if the preceding 

 question be answered in the negative, can any more simple assumption 

 be made the foundation of the theory ? 



The attempts to answer one or the other question in the affirmative 

 have been very numerous, and have (without any exception but < .m- in 

 which new axioms of another sort are introduced) tacitly contained the 

 defect which their authors were desirous of avoiding. The author of 

 '.Geometry without Axioms' (General Perronet Thompson) has collected 

 and commented ou thirty instances, of which we here make a brief 

 abstract, adding one or two more. 



1. The axiom of Euclid in question. 2. Ptolemy; his proof 

 assumes the symmetrically of parallel lines on one side and the other 

 of any line which cuts both. 3. Proclus assumes that iutersectors 

 diverge infinitely from the point of intersection, and that parallels do 

 not. 4. Clavius assumes that a line which is everywhere equidistant 

 from a straight line, is itself straight. 5 and 6. Two demonstrations 

 of Dr. Thomas Oliver (1604) assume Clavius's axiom. 7. Wolf, 

 Boscovich, T. Simpson (in the first edition of his Elements), and 

 Bonnycastle, define parallel lines as those which always preserve the 

 same distance, which is Clavius's axiom in the disguise of a definition. 

 8. D'Alembert would define parallels as lines one of which has two 

 poults equidistant from the other, but acknowledged that he could not 

 complete the proof of the axiom of Clavius. 9. T. Simpson (Ele- 

 ments, 2nd edition) proposed to assume that two lines, one of which 

 has two points unequally distant from the other, must meet. 10. 

 Robert Simpson proposed to assume that a straight line canni 

 approach to and then recede from another, without cutting it. 11, 

 V HI igtion. Be/out, &c., would define parallels as lines which make the 

 same angle with a third line : if a third line mean tome one third line, 

 the difficulty remains just as before ; if any third line, the difficulty is 

 tacitly removed by an assumption. 12. Ludlani, Playfair, &c., 

 recommend the axiom which we have also recommended, namely, 

 that two intersecting straight lines cannot be both parallel 

 to a third. 13. Leslie proposes to attain the same axiom in a sort 

 of experimental manner, by making a line revolve about a point. 

 II. Playfair (in his Notes) proposes to assume that a straight line 

 which turns completely round, and thus regains its first position, must 

 i ui n through four right angles, whether it constantly revolves about 

 one point, or whether the pivot of revolution changes. 15 and 10. 

 Franceschini (1787) proposes to assume that the projections of a 

 straight line on a line making on acute angle with it, increase without 

 limit with the projected line. 17. Some have proposed to define 

 parallel!) a* "lines having the same direction," assuming it to be ob- 

 viously contained in the conception of direction, that two similar 



