THE word vcvcns, commonly inclinatio in Latin, is difficult to translate satisfactorily, but its meaningwill be gatheredfromsome general remarks by Pappus having reference to the two Books of Apollonius entitled veixrets (now lost). Pappus says*, "A line is
said to verge (veveiv)towards a pointif, being produced, itreach the point,"and he gives, among particular eases of the general form of the problem, thefollowing.
"Two lines being given in position, to place between them a straightlinegiveninlengthand verging towards agivenpoint."
"If there be given in position (1) a semicircle and a straight line at right angles to the base, or (2) two semicircles with their bases in a straight line, to place between the two lines a straight line given in length and verging towards a corner (ywviav) of a semicircle."
Thus a straight line has to be laid acrosstwo lin.esorcurvesso that itpassesthrough a given pointand theintercepton itbetween thelinesor curvesisequal toa given lengtht.
1. The following allusions to particular vcv'tms are found in Archimedes. The proofs of Props. 5, 6, 7 of the book OnSj)irals
use respectively three particular cases of the general theoremthat,
* Pappus(ed.Hultsch)vn.p.670.
t In the German translation of Zeuthen's work, Die Lehre von den KegeUchnittenim Altertum, i>eD<mistranslated by "
Einschiebung,"or as we mightsay"
insertion,"butthisfailstoexpress the condition that the required linemustpassthrougha givenpoint, justasinclinatio(andforthatmatterthe Greek term itself) failsto express the otherrequirementthat the intercepton thelinemustbeof given length.
ON THE PROBLEMS KNOWN AS NET2EI2. Cl ifA be anypoint on a circle and BCanydiameter, it ispossibleto
draw through A a straight line, meeting the circle again in P and
EGproduced in R, such that the intercept PRisequaltoanygiven
length. In each particular case the fact is merely stated as true withoutanyexplanationorproof, and
(1) Prop. 5 assumes thecase wherethetangentatA isparallel toEC,
(2) Prop. 6 the case where the points A, P in the figure are interchanged,
(3) Prop. 7 the case where A, P are in the relative positions shown inthefigure.
Again, (4) Props. 8and 9 eachassume (asbefore, withoutproof, and without giving any solution of the
implied problem) that, ifAE, BC be two chords of a circle intersecting at right angles in apoint D such tJtat ED > DC,
tfien it is possible to draw through A
another line ARP, meeting BC in R and
the circleagainin P, suchtlmtPR DE.
Lastly,with the assumptionsin Props.
5, 6, 7 shouldbe compared Prop. 8 of the Liber Assumptorum, which may well be
due to Archimedes,whatever may be said ofthecomposition ofthe whole book. This proposition proves that, ifin the first figure
APR is so drawn that PR is equal to the radius OP,tJien the arc
AB is three times the arc PC. In other words, if an arc AB of a circle be taken subtending any angle at the centre 0, anarc equal to one- third of the givenarc canbefound,i.e. t/iegiven anglecanbe trisected, ifonly APR can be drawn through A in such a manner
Cll INTRODUCTION.
that the intercept PR betweenthe circleand BOproducedisequalto theradius ofthe circle. Thusthe trisectionofan angleisreducedto a vcvcrts exactly similar to those assumed as possible in Props. 6, 7 ofthebook OnSpirals.
The vcvWs so referred to by Archimedes are not, in general, capable of solution bymeans of the straight line and circlealone, as may be easily shown. Suppose in the first figure that x
represents the unknown length OR, where is the middle point of JSC,and that k is the given length towhichPR isto be equal; also let OD =a, AD =6, BC =2c. Then,whetherEC bea diameter or (moregenerally)any chordofthecircle, we have
andtherefore kJb"+(x-
a)'
2= x2-r2.
The resulting equation, after rationalisation, is an equation of the fourth degreein #; or, ifwedenote thelength of ARbyy,we have,
for thedeterminationofxand ?/,thetwoequations
In other words, if we have a rectangular system of coordinate axes, the values of x and ysatisfying theconditions of theproblem can bedetermined asthe coordinates ofthepoints of intersection of acertain rectangular hyperbolaandacertain parabola.
In oneparticularcase, thatnamelyinwhich D coincides with the middle point of JtC, or in which A is one extremity of the diameter bisecting BC at right angles, a=0, and the equations reduce to the single equation
y*-ky =b*+c*i
which is a quadratic and can be geometrically solved by the traditional method of application of areas; for, if ube substituted for y k, sothat u = AP, the equation becomes
u(k+u)=b*+c8,
and we have simply "to apply to a straight line of length k a rectangle exceeding by a square figure and equal to a given area (&24-c2)."
The other vcOo-is referred to in Props. 8 and 9 can besolved in the more general form where k, the given length to which PR
is to be equal, has anyvalue within acertainmaximum andis not
ON THE PROBLEMS KNOWN AS NET2EI2. ciii necessarily equal to DE, in exactlythe same manner; andthetwo equations correspondingto (a) willbefor the second
figure
Here,again, the problem can be solvedby the ordinarymethod of application of areas in the particular case where AE is the diameter bisecting ECat rightangles; and it isinteresting to note that this
particular case appears to be assumed in a fragment of Hippocrates' Quadrature of lune* preserved in a quotation bySimplicius* from Eudemus' History of Geometry,while Hippo- crates flourished probably as early as 450 B.C.
Accordinglywe find that Pappus distinguishes differentclasses ofvvcrts correspondingtohis classification ofgeometrical problems in general. According tohim, the Greeksdistinguished threekinds of problems, some beingplane, others solid, andothers linear. He
proceedsthusf: "Those which canbe solvedbymeansofa straight line and a circumference of a circle may properly be called plane (cTrnrcSa); for the lines by means of which such problems are solved have their origin in a plane. Those however which are solved by using for their discovery (eupcni>) one or more of the sections of the cone have been called solid (o-Tcpea); for the construction requires the use of surfaces of solid figures, namely, those of cones. There remains a third kind of problem, that which is called linear (ypa/A/ufcdv); for other lines
[curves] besides those mentioned are assumed for the construction whose origin
is more complicated and less natural, as they are generated from more irregular surfaces and intricate movements." Among other
instances of the linearclassofcurves Pappus mentionsspirals, the curves known as quadratrices, conchoids and cissoids. He adds that "it seems to be a grave error which geometers fall into whenever any one discovers the solution of a plane problem by means of conies or linear curves, or generally solvesitby meansof a foreign kind, as is the case, forexample, (1)with theproblemin thefifth BookoftheConiesofApollonius relating to theparabolaJ,
*
Simplicius,Comment,in Aristot.Phys.pp.6168(ed.Diels). Thewhole quotationis reproduced byBretschueider,Die Geometric und die Geometervor Euklides, pp.109121. As regards theassumed constructionseeparticularly p.64andp.xxiv ofDiels1edition;cf.Bretschneider, pp.114, 115,andZeuthen, DiV?Lehrcvon denKeijelschnittenimAltertum, pp. 269, 270.
t Pappusiv.pp. 270272.
Cf.Apolloniusof Perya, pp.cxxviii. cxxix.
CIV INTRODUCTION.
and (2) when Archimedes assumes in his work on the spiral a
vevo-i? of a solid character with reference to a circle; for it is possible without calling in the aid of anything solid to lind the [proof of the] theoremgiven bythelatter [Archimedes], that is, to prove that the circumference of the circle arrived at in the tirst revolution is equal tothe straightlinedrawn at right angles tothe initiallinetomeet the tangentto thespiral."
The"solid veCo-is
"
referred toin thispassageis thatassumed to bepossiblein Props. 8and9 of thebook OnSpirals,andismentioned again by Pappus in another placewhere he showshow tosolvethe problem by meansof conies*. This solutionwill be givenlater,but,
whenPappusobjects tothe procedureof Archimedesasunorthodox, theobjectionappearsstrained ifweconsiderwhat preciselyit isthat Archimedesassumes. Itisnot theactual solutionwhich isassumed, butonly itspossibilityj and itspossibilitycan be perceived without any use of conies. For in the particular caseit is only necessary, as a condition of possibility, that DE in the second figure above should not be the maximum length which the intercept PR could have as APR revolves about A from the position ADK in the direction of the centre of the circle; and that DE is not the
maximum length which PR can have is almost self-evident. In
fact, if P, instead of moving along the circle, moved along the straightline throughEparallel to/?(7, andif ARP movedfrom the positionADEin thedirection ofthecentre,the lengthof PRwould continually increase, andafortiori, so longas Pison thearc ofthe circle cut off bythe parallel throughEtoBC, PRmustbe greater in length than DE; and on the other hand,as ARPmovesfurther in the direction of <#, it must sometime intercept a length PR
equal to DEbefore P reaches /?,when PR vanishes. Since, then, Archimedes*method merelydependsupon thetheoretical possibility of a solution of the vevo-is, and this possibility could be inferred from quite elementary considerations, he had no occasion to use conic sections for the purpose immediatelyin view, and he cannot fairlybesaid tohave solveda planeproblemby the use ofconies.
At the same time we may safely assume that Archimedes wasin possession ofasolution ofthe ycvcrts referred to. But there isno evidencetoshowhowhe solvedit,whether bymeansofconies, or otherwise. That he would have been able toeffectthesolution,
*
Pappusiv. p.298sq.