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Elements of Euclid's DataChristian Marinus Taisbak
1. Do we know intuitively what it means to be given?2.
Supplements, notations, and conventions
2a. Objects and ratios2b. Predicates and limitations
3. An example: theorem 41 of the Data4. Given in position5.
Given in magnitude6. Parallels and angles7. Given in form8.
Triangles9. Do we know the use of 'givens'?10. Conclusion11.
Epilogue
1. Do we know intuitively what it means tobe given?
If no one had written Euclid's Data, I would never have missed
it.Olaf Schmidt
Thus, the appropriate measure of the geometric researches
conductedby Euclid and his contemporaries is to be sought not in
the Elements,but in the Data
Wilbur Richard Knorr ((1986a), 102)
The confession by my teacher of Greek mathematics, Professor
OlafSchmidt of Copenhagen, one of the connoisseurs of Euclid's
Elementsof our century, conflicts conspicuously with the statement
of WilburKnorr as Schmidt must realize (should Knorr be right) that
he hasBrought to you by | St. Petersburg State University
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136 Christian Marinus Taisbak
not properly measured Euclid's achievement from his knowledge
ofthe Elements alone. To do so, he would have to have read the
Data, whichhe admits he has not done at any rate not so thoroughly
as Knorr,who gives the following short description of the work
((1986a), 109):
The Data is a complement to the Elements, recast in a form more
service-able for the analysis of problems. ... Each of its theorems
demonstratesthat a stated term will be given on the assumption that
certain otherterms are given. The subject matter overlaps that of
the Elements,dealing with ratios and with configurations of lines
and of plane fig-ures, both rectilinear and circular. Indeed, only
in rare instances doesthe Data present a result without a parallel
in the Elements.
I suppose that Knorr could bring this description into harmony
withhis statement, quoted above, by giving a suitable definition of
'parallelresults' (see the end of section 3 below). But whereas
Schmidt didn'tmiss the Data, in the sense of 'feeling a need for'
it, Knorr probablymisses its main point. Common to the
mathematician's view and thehistorian's is the belief that he knows
what it means to TJC given'.Everyone knows that, of course,
although anyone who has tried tofathom the 'common notions' of book
I of the Elements ought to havesuspicions about intuitively
intelligible words in Greek mathematics.
Many years ago, when first reading Ptolemy's Almagest, I
wonderedabout the very strange use of the predicate 'is given',
which is appliednot only to the input of a problem, but also to its
output (Alm I.38, seefigure 1):
B
Figure 1
AB and AC are both given in magnitude, measured in those units
ofwhich the diameter has 120; and let BC be joined. I say that BC
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Elements of Euclid's Data 137
This last assertion is translated by Manitius as '...dass
auch... BC sichbestimmen lsst', probably because he was convinced
that hardly anycontemporary mathematician would use the predicate
'is given' ofwhat has been found or proved to be true. But the
Greeks did so, andthat is one unfamiliar fact which must be
understood if Euclid's Data isto be properly assessed.
In this paper I will try to determine what the predicate 'is
given'means in the Data; my investigation will be based on the
followinghypothesis: that Euclid is trying to axiomatize its
meaning. I cannotcover all of the 94 theorems in the Data,1 but I
will point out some'elements' of the work, which were used for
geometric analysis in theData and elsewhere.
2. Supplements, notations, and conventions
'By their fruits ye shall know them.' (Matthew 7.20) I am not
suggestingthat Euclid (or whoever else wrote the Data) was one of
the falseprophets, but now and again he proves assertions with
axioms anddefinitions that nowhere are stated, whereas he fails to
use an explicitdefinition (3) where it is needed. Theorem 31 is
left in No Man's Land:if it is true, it is hard to see why; and if
it is false, why was it everpronounced and preserved? One way to
clear things up is to elicit thedefinitions and axioms from the
results obtained: to know by the fruitswhat the grain was. I shall
adopt this method in describing the Data,inserting my own
supplements whenever ideas are used without beingdefined in the
text. I am sure that I have not invented anything extra,but I may
have overlooked some necessary definition. I believe thatEuclid
would appreciate my approach, since most of the definitions inhis
works were probably written after the fact, as it were, when he
(orhis editor) felt a need for them.
1 The text of the Data edited by Heinrich Menge is printed with
a Latin translation involume 6 of Euclid's Opera Omnia (Leipzig:
Teubner). A translation of the Data withcommentary will be
forthcoming in 1992, if I live. Should anything in this paper
beless rigorous than it ought to be, I will have the opportunity to
amend it then. I wouldappreciate any comments, private or public,
that may contribute to a better under-standing of the Data. Brought
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138 Chnstian Mannus Taisbak
There is some variation in the terminology of the Data,
probablyreflecting different historical strata. In order not to
raise problems thatmay be irrelevant to the present investigation,
I have standardized thevocabulary when quoting propositions. For
example, I use one term'rectilinear figure' for chrion,
euthugrammon, and eidos in theorems49-55. A theorem of the Data is
identified by 'thm' followed by itsnumber; a definition is
identified by 'def followed by its number, e.g.,thm 25, def 4. My
own supplements are signalled by three digit num-bers and upper
case letters. THM 125 and THM 225 would be mysupplements to thm 25
of the Data. AXM 103 is one of my axioms.Propositions from Euclid's
Elements are identified as, e.g., VI.4, mean-ing the fourth
proposition of the sixth book.
2a. Objects and ratiosThe Data deals with objects in the plane.
Of course the theory can beextended to space, but that was not done
by Euclid. The objects are thewell-known ones from Euclid's
Elements. Their status is neither morenor less problematic in the
Data than in the Elements. They are:
Points.
Lines (grammai), which comprise three kinds: infinite
straightlines, line segments, (arcs of) circles, i.e., such lines
as can beproduced by permissible constructions with ruler and
com-pass. Normally Euclid does not distinguish between
infinitestraight lines and line segments, but the context never
leavesany doubt which he means. I allow myself to diverge
fromEuclid and speak of straights, segments, and arcs.
Angles.
Polygons, mostly triangles and parallelograms. Sometimes Iwill
refer to them as areas, though they should not be associ-ated with
any number, rational or real, but only with planefigures; more
often than not these figures cannot and will notbe measured in any
sense of the word. This is a crucial featureof Greek geometry, on
which one must insist.
(Segments of) circles. None will occur in this
presentation.Lines, angles, polygons and (segments of) circles are
called by thecommon name 'magnitudes' (megethe). I cannot now give
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Elements of Euclid's Data 139
ostensive definition of 'magnitude', but I shall have a little
more to sayin section 5.
Besides these objects, the Data involves ratios, logoi between
objectsof the same kind, most often line segments or polygons. The
ratio of Ato B will be written 'A:B'. A ratio is not an object, but
some kind ofconnection or relation between objects. The concept
itself is veryvaguely defined in the Elements V, def.3: 'Logos is a
sort of state orrelation (poia schesis) in respect of size between
two magnitudes of thesame kind.'2 With many reservations one might
compare Euclid'sconcept of ratio with the concept of field in
modern physics. A ratio isalways there whenever two magnitudes of
the same kind or twonumbers are present, but a ratio is not a
magnitude, nor a number.Rather, one might say that a ratio is a
pair of magnitudes or numbers.Two ratios (say A:B and C:D) can be
'the same', or one can be greaterthan the other, as defined in
Elements V, defs. 5 and 7. Euclid neverspeaks of 'equal'
ratios.
2b. Predicates and limitations
Any object and any ratio may serve as an argument for the
predicate'is given'. To suppress unwarranted connotations of the
term 'given' Ishall (sometimes) adopt a shorthand of putting the
name of the givenobject in brackets:
[P] means T is given';if ABC is a triangle, then [ABC] means
'the triangle ABC isgiven';
[A:B] means 'the ratio of A to B is given'.Concerning a
rectilinear figure in the plane, three questions suggestthemselves:
Where is it? How great is it? What is its shape? The Datadefines
three corresponding aspects of givenness: the figure may begiven
'in position', or given 'in magnitude', or given 'in form'. We
mightsay that its position, its magnitude, or its form is given,
but I prefer tofollow the Greek syntax, which uses the dative of
respect (thesei, megethei,eidei), to emphasize that these are three
different aspects of givenness.
2 On this definition see Euclid-Heath (1926), vol. 2,116.Brought
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140 Chnstian Marinus Taisbak
Clearly dimensions are relevant here. A point can be given in
posi-tion only; a straight can be given in position and in
magnitude; apolygon can be given in position, in magnitude, and in
form. Thebracket shorthand will be modified accordingly:
[AB]P means The segment AB is given in position'.[AB]m means
'The segment AB is given in magnitude'.[ABC]f means 'The triangle
ABC is given in form'.
Whenever in the Data the predicate 'given' is applied to an
objectwithout one of these modifications, it means 'given in
magnitude'. Insuch cases I too leave out the index. Seeing that a
point can only begiven in position, I shall always use the
shorthand [A] rather than [A]pto mean 'the point A is given'.
3. An example: theorem 41 of the Data
In order to acquaint the reader with the way one speaks about
givens,I present a typical theorem:
thm 41 If a triangle has one angle given, and the sides
contain-ing the given angle have a given ratio to one another,
thetriangle is given in form.
B C E
Figure 41
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Elements of Euclid''s Data 141
Let the triangle ABC (figure 41) have one given angle ZBAC;and
let the sides BA, AC containing the given angle have agiven ratio
to each other. I say that the triangle ABC is given inform.
Let a segment DE be set out, given in position and in
magni-tude. And let an angle ZEDF, equal to ZBAC, be constructedon
DE and at the point D on it.
The angle ZBAC is given; therefore ZEDF is also given. Now,since
at the straight DE given in position, and at the given pointD on
it, a straight DF is drawn, making the given angle ZEDF,DF is given
in position. And as the ratio AB:AC is given, letDE:DF be the same,
and let EF be joined; thus the ratio DE:DFis given. But DE is
given, and therefore DF is also given; butalso in position. The
point D is given; therefore also the pointF. And both E and D are
given; thus each of the lines DE, DF,and EF is given in position
and magnitude; therefore the trian-gle DEF is given in form.
And since two triangles ABC and DEF have one angle equal toone
angle, ZA = ZD, and the sides containing the anglesproportional,
the triangle ABC is similar to DEF, which is givenin form.
Therefore also ABC is given in form.
Although much in this theorem needs definition and explication,
thetheorem itself is easy to follow and intuitively acceptable. One
questionraises itself immediately, however: what does it mean to be
given in form?We are supposed to be able to prove this predicate of
the triangle whichis set before us, though not given. It is just
some triangle with a fewattributes which are given: an angle (in
whatever way such a thing canbe given to Euclid), and a ratio of
two of its sides (whatever ratio maymean, and however it is
possible to give it). Some things are given inposition, some in
magnitude, others are just given.
What status does the triangle ABC have? Since we can prove that
itis given in form, it must be given in form. But we do not know
that fromthe outset. It is not given in position. For if it were,
we would know eoipso that it is given in form, as we shall see. It
is definitely not given inmagnitude. But it is known to be a
triangle, with the properties of anytriangle. The sum of its angles
equals two right angles; any two sidestaken together are greater
than the third, etc. Thus, there is somegivenness masked in the
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142 Christian Marinus Taisbak
Which tools, which auxiliaries do we have at our disposal to
provethm 41? We have the Elements, more explicitly the first six
books,dealing with plane geometry. In this specific case, we have
used 1.23 toconstruct an angle equal to another, and VI. 12 to get
two segments inthe same ratio as the given ones. Obviously, thm 41
presupposes a fewtruths, which (you would think) must be procured
in the precedingtheorems. At the end of the proof we use Elements
VI.6:
If two triangles have one angle equal to one angle and the sides
aboutthe equal angles proportional, the triangles will be
equiangular andwill have those angles equal which the corresponding
sides subtend.
I suppose that something like theorem 41 was in Knorr's mind
when hewrote of 'parallel results'. In Elements VI.6 two triangles
TI and T2 havingone angle equal to one angle and the sides about
the angle in the sameratio are compared. A triangleT3 is
constructed equiangular with T] andcongruent with T2. Therefore, T,
is equiangular with T2. In Data 41, onetriangle T, is set out,
having one angle 'given' and the sides about theangle in a 'given'
ratio. A triangle T2 is constructed 'in position' with thesame
properties as Tlrand therefore equiangular with it. Since T2 is
'givenin position', it is also 'given in form', and therefore T] is
'given in form'.The Elements compare triangles. The Data deals with
individuals, andwith the 'knowledge' we may have of such individual
triangles withinthe language of givens. A superfluous discipline?
Rather, a useful one ingeometric analysis, where individual objects
prevail.
Why are objects said to be 'given' and not Tcnown'? I shall
leave thatquestion till the end of this paper; but I am sure that
the word waschosen deliberately at some time by some person as a
new term toprevent wrong or unwarranted ideas about knowledge.
4. Given in position
In the first theorems of the Data it is not evident that to be
given inposition is the most basic of the three kinds of givenness.
But thoroughanalysis shows that it is so, and so it must be, since
one important groupof objects, namely points, can only be given in
position. Points arethought of as separate entities, each with its
own identity; each point isdifferent from all other points. Now,
the only question one can mean-ingfully ask of a point's identity
is: Where is it? To describe thisindividuality, Euclid chose the
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Elements of Euclid's Data 143
def 4 'Given in position' is said of points and lines and
angleswhich always occupy the same place.
Heath (Euclid-Heath (1926), vol. 1, 132) says that 'given in
position'really needs no definition, and that we are not really the
wiser afterreading def 4. But I think we may argue from opposites,
and askourselves: If a point is not given (in position), what can
it be? Moving,if I am not mistaken. It might change its place,
metapiptein, exactly theterm introduced in thm 25 and known from
presocratic philosophers,particularly Democritus,3 who use it to
mean 'to change' (intransitive).Therefore, I supply two
biconditional definitions:
DEF 101 A point or a straight or an arc is given in position
ifand only if it does not move.
DEF 102 An angle is given in position if and only if its
vertexand legs are given in position.
In the Elements, points are not moving, nor are lines, but
elsewhere inGreek mathematics they accomplish useful movements. To
mentionone contemporary of Euclid, Autolycus4 has moving points;
and Ar-chimedes,5 when producing his helix, sends a point moving on
amoving straight.
We cannot help thinking of fixed coordinates in some
coordinatesystem when we are told that some point is given in
position and alwaysoccupies the same place (relatively to all the
others). But the time was notripe for Euclid to refer a point to
two straights ('which', to quote Petersent(1866),4], 'generally
have nothing to do with the problem considered'),if for no other
reason, then because the distance of the point from thoselines
could not always be measured by one and the same unit. After
all,real numbers were some two millenia from being invented. So
Euclidmeant just what he said by 'given in position'. If a point is
explicitlyclaimed to be given in position, it must remain where it
is, and keepdistinct from all other points. It may surprise a
modern reader that so'little' information is of any consequence,
but it is.
3 See Dieted 956), vol. 3,278 ff.4 On the Moving Sphere,
proposition 1.
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144 Christian Marimis Taisbak
How do we get started?AXM 103 Any point or straight in the plane
may be taken forgiven in position.
Whenever we need an item it is possible to 'let it be given'.
(See, forexample, thm 5). Apparently, we do not have to wait for
the Giver toget the idea but may demand it ourselves. After all,
the unprovedstatements of the Elements (and any mathematical
edifice) are just'demanded' (postulata in Latin, axiomata or
aitemata6 in Greek). Note thatthey are demanded, not taken.
It might be appropriate at this stage to introduce that
effectiveconstructor, The Helping Hand, the well-known factotum in
Greekgeometry, who sees that lines be drawn, points be taken,
perpendicu-lars dropped, etc. The perfect imperative passive is its
verbal mask. Noone who has read Euclid's Elements in Greek will
have missed it. Neveris there any of the commands or exhortations
so familiar from our ownclassrooms: 'Draw the median from vertex
A', or 'Cut the circle by thatsecant', or 'Let us add those squares
together'. The Helping Hand wasalways there to see that these
things were done. I wonder how Europeever inherited Greek
mathematics without the perfect imperative pas-sive.7
Thus, a point is given if it is taken for given, 'appointed' as
it were,or if it is proved to be given by thm 25, i.e., if it is
the point of intersectionof two lines that are given in position,
either two straights, or a straightand an arc, or two arcs:
thm 25 If two lines that are given in position cut one
another,their point of section is also given in position.
6 Another Pythagorean term?
7 Cf. Euclid-Heath (1926), vol. 1,242, note to line 8 of
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Elements of Euclid's Data 145
Figure 25
I reproduce the proof in my notation (figure 25):objects AB, CD:
lines (straights or arcs);
E: point;
hypotheses [AB]p/ [CD]p;E is point at which AB and CD
intersect;
assertion [E]
argument if not [E], then E would be moving (DEF101);ergo either
AB or CD would move; but they donot move (hyp).
What is proved in thm 25 is not that a point of section exists,
and it isirrelevant that two arcs or an arc and a straight may have
two points ofsection. What is proved is that if they do have a
point of section, thepoint is not a moving point, but a standing
point given in position. Theproof is a perfectly sound reductio ad
absurdum, if DEF 101 is supplied.But the application of thm 25 to
prove later propositions may causesome uneasiness.
Thm 25 presupposes that two lines are given. How can a line
begiven? According to AXM 103, it can just be taken for given; but
if not,thm 26 shows how, while at the same time opening new
horizons:8
8 Even without the supplements, which I have added because they
are used in someof the reasonings. The concept of direction is as
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146 Christian Marinus Taisbak
thm 26 If two points are given, the segment whoseendpoints are
the two points is given in position and in mag-nitude;
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Elements of Euclid''s Data 147
class of areas that are equal in the sense of the Elements 1.35.
In the Data,if a representative of such a class is given in
position, any member ofthe class is said to be given in magnitude
or to be a given magnitude.Let us define (or, perhaps, lay down as
axioms):
DBF 104 A segment or an area or an angle is given in magnitudeif
it is given in position.DEF105 A segment or an area or an angle is
given in magnitudeif it is equal to one that is given in
magnitude.
The latter definition is saved from circularity by the former.
They aremodelled on def 1 in the Data:
def 1 'Given in magnitude' is said of areas and lines and
anglesto which we can get equals.
Def 1 is always used as a combination of DEF 104 and 105. The
Greekterm for 'gef is porisasthai, which means 'procure', 'gef,
'provide', andindicates any kind of purchase without specification
of method. But itis evident from the use of def 1 that one cannot
procure 'impossible'things, but only objects which can be
constructed with ruler and com-pass. I take it that Euclid defines
an equivalence class of geometricobjects to be given if and only if
one of its representatives is given inposition. This turns out to
be one of the most effective definitions in theData.
6. Parallels and angles
Three (non-collinear) given points entail three straights, three
angles,and one triangle, all given in position. But also one given
straight andone given point not on the straight can ensure another
straight givenin position, namely the parallel through the
point:
thm 28 If through a given point a straight be drawn parallel
toone that is given in position, the straight drawn will be givenin
position.
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148 Chnstian Marinus Taisbak
D
Figure 28
The proof is instructive (figure 28):objects A: point;
BC, DAE: straights;hypotheses DAE parallel to BC, [A];assertion
[DAE]p.argument if not [DAE]p
then, with A standing, the straight DAE willmove and change its
position to, say, ZAH. But,by hypothesis, ZAH isparallel to BC,
which is also parallel to DAE. Therefore ZAH is parallel to DAE;but
they do meet each other in A, which isabsurd. Therefore DAE does
not move, and isgiven in position.
This proof makes free use of the theorems on parallels from the
Ele-ments, in casu, 1.30, from which it is deduced that there is
not more thanone parallel to a straight line through a point not on
the line. Thm 28proves that this one parallel does not move: while
being parallel it isgiven in position.
Theorems 31-38 deal with parallels. Their proofs depend on
twotheorems about angles, 29 and 30:
thm 29 If, at a straight given in position and at a given point
onit, a straight be drawn making a given angle, the straight
drawnis given in position.
thm 30 If, to a straight given in position from a given point ,
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Elements of Euclid's Data 149
Since the only difference between these theorems is the position
of thegiven point, the proofs run parallel and are perfectly sound
reductionesad absurdum: if the straight is not given in position,
it will move whilekeeping its properties, thereby changing the
angle; which immediatelyconflicts with 1.16.
We shall need a partial converse to thm 29 and 30 (in accordance
withDEF 102):
THM 125 If two lines that are given in position meet each
other,the angle which is contained by them is also given in
positionand in magnitude.9
The next theorem is a puzzle, resting totally on thm 25, and
neithermore nor less dubious than that one:
thm 31 If, from a given point, a segment given in magnitude
bedrawn to meet10 a straight given in position, the segment isgiven
in position.
B_
Figure 31
I give a very close paraphrase of the proof (figure 31). Let A
be thegiven point, and D the point (on the given line BC) where the
straightfrom A meets BC, AD being of given magnitude. Assertion: AD
is givenin position. With A as center and AD as radius let the
circleEDF have been drawn. The circle is given in position (def 6);
and BC is
9 The ancient definitions of angle are discussed thoroughly in
Euclid-Heath (1926), vol.1,176-181.1 know of nobody who has
reviewed the problem since Heath.
10 This is a rare instance of the verb prosballein, meaning 'to
draw towards so as to meetor reach'. There is no room for
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150 Christian Marinus Taisbak
given in position. Therefore their point of section D is given
in position(thm 25). And since A is given in position, AD is given
in position (thm26).
If we accept that this is a true mathematical theorem," a few
obviousinterpretations are ruled out:
1) It is not a problem of constructing a segment given
inmagnitude, from a point to a straight. The line is simply
as-sumed to have been drawn, the method being irrelevant to
theissue: that it is possible to have such a line drawn from A to
BC.Why, then, does Euclid have the circle drawn with A as centerand
AD as radius? Because analysis has shown that it is thelocus of
points at the given distance from A, so D must be onit. He will
need the circle to prove that D does not move, but isgiven in
position since it is also on BC.Constructions in the Data are
always secondary, i.e, auxiliaryto the proof, just commanded (via
The Helping Hand). Theymust be ensured by some problem in the
Elements, in casu,something like 1.12. The method of construction
is and must beirrelevant for the existence of the object. Therefore
a construc-tion in the Data may sometimes look superfluous or
misplaced,as in the present theorem. We would need the circle EDF
to beable to draw the segment AD. But then we should not need
todraw it again, in order to use it as the locus. Some
otherexamples of this kind of thing will be seen below.2) It is not
a uniqueness theorem, since, once the segment isdrawn, there may
exist another segment with the same prop-erties. Why didn't Euclid
prove that the segment drawn iseither the perpendicular on BC, or
one of two segments, sym-metrically situated on either side of the
perpendicular? Becausehe knew that already.12 Euclid does not
concern himself withuniqueness unless it has some specific
significance.13 Unique-
What else can we do? The text might have been better
transmitted. See Menge'sapparatus criticus at 52.20.
12 I shall not revive the discussion of 1.12. See Euclid-Heath
(1926), vol. 1,271 ff.13 As it does in X.42 ft.: a binomi(n)al line
(ek duo onomafon) can be split in only one way
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Elements of Euclid''s Data 151
ness has been discussed ever since the first proposition of
theElements. Should the equilateral triangle be above or below
thegiven line segment?3) What, then, is thm 31 about? As far as I
can see, it provesdeductively that if such a segment has been
drawn, it does notmove. Can such an insight be of any use?
Unfortunately, theData does not use thm 31 to prove any other
theorems.
Thm 32-33 concerns the correlation between angles and
transversallines in parallels:
thm 32 If a straight be drawn into parallels given in
positionmaking given angles, the straight drawn is given in
magnitude.thm 33 If a straight given in magnitude be drawn into
parallelsgiven in position, it will make given angles.
The proofs are straightforward applications of the theory of
parallelo-grams from book I of the Elements, combined with thm
25,26 and 29. Inthm 32, a segment is procured, given in position,
parallel and equal to thestraight drawn, which is therefore given
in magnitude. I shall quote thm33 in extenso with comments in curly
brackets (figure 33).
A H E B
Figure 33 c ^- p D
L
Let the straight EF given in magnitude have been drawn into
theparallels AB, CD, given in position. I say that it will produce
given anglesBEF, EFD. (One of them will suffice, since the
straights are parallel. Itmust be proved equal to one that is given
in position (DEF104,105).}
On AB let a given point H have been taken, and through H let HG
havebeen drawn parallel to EF. (If I were to draw it, I might mark
off FG = EH,using 1.33. However, the Helping Hand can draw it,
since it exists,without bothering about the method of drawing.}
Ergo FE = HG (I.34);and EFis given ; therefore HG is given (defl).
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152 Chnstian Marinus Taisbak
And H is given; thus the circle with center H and radius HG
will,when drawn, be given in position (def 6). Let it be drawn as
KGL; thenKGL is given in position. And CD is given in position;
thus G is givenin position (thm 25). (I would say, 'G is (a) point
of section between CDand the circle with center H and radius HG.
Therefore G is given inposition.' The ambiguity of thm 25 is not
reflected in this theorem, sincethe straight EF is supposed to have
been drawn beforehand, and thechoice between the two possibilities
made.)
And H is given: thus HG is given in position (thm 26). CD is
givenin position; ergo the angle HGD is given (DEF 102). And it is
equal toZEFD, which is therefore given. And the complement ZFEB is
alsogiven.
Theorems 34-38 are about parallels and the division of a
transversalline through a given point:
thm 34 If a straight be drawn from a given point into
parallelsgiven in position, it will be cut in a given ratio.
thm 35 If, from a given point to a straight given in position,
astraight be drawn and cut in a given ratio, and if a straight
bedrawn through the point of section parallel to the given one,the
straight drawn will be given in position.
thm 36 If, from a given point to a straight given in position,
astraight be drawn and a segment be added to it having a givenratio
to it, and if a straight be drawn through the endpoint ofthe
segment added parallel to the given, the straight drawn willbe
given in position.
thm 37 If, into parallels given in position, a straight be
drawnand cut in a given ratio, and through the point of section
astraight be drawn parallel to the given ones, the straight
drawnwill be given in position.
thm 38 If, into parallels given in position, a straight be
drawnand a segment be added to it having a given ratio to it, and
if astraight be drawn through the endpoint parallel to the
givenones, the straight drawn will be given in position.
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Elements of Euclid's Data 153
Figure 34
The five propositions may be summarized with small
divergencesfrom the text as follows (see figure 34). 37 and 38 are
substantially thesame as 36 and 35. In the latter is a given point,
in the former is anarbitrary point on a line given in position. The
four of them are partialconverse theorems to thm 34:
Common conditions for thm 34-38:
objects h, j, k: straights;t: transversal line;, , : points;
hypotheses [h]p;P on j and f, A on h and t, B on k and t.
Special conditions:
thm 3414hypotheses [P], k \ \ h, [k]p;assertion [PA:PB].
thm 35 (P external) and 36 (P internal)hypotheses [P], k II h,
[PA:PB];assertion [k]p.
14 A counterpart to 34 with internal P (between A and B) is
transmitted at the end ofthe proof, but relegated by Menge to the
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154 Christian Marinus Taisbak
thm 37 (analogue to 36) and 38 (analogue to 35)hypotheses lj]p,
k \ I h, [PA:PBJ;assertion [k]p.
7. Given in form
The main theorem in the Data, that is, the most general
assertion aboutgivenness in form and in magnitude is:
thm 55 If a rectilineal figure is given in form and in
magnitude,its sides will also be given in magnitude.
Intuitively, most people know what it means to say that a plane
figurehas a shape of its own, different from some and, perhaps,
similar toothers: it has something to do with angles and
proportion. So the Data'sdefinition of being 'given in form' is
hardly surprising, though perhapsa little too sophisticated to be
invented by Everyman:
def 3 A rectilineal figure is given in form if and only if15
each ofits angles is given and the ratios of its sides to each
other aregiven.
The statement looks like, and in fact is, a theorem. Although
Euclidapplies def 3 in many propositions, he does not use it where
he needsa definition, in thm 39, the starting point of the theory.
Instead he hasrecourse to what must be the more basic concept,
being given inposition. I offer some additional definitions,
elicited from thm 40 (seebelow), with which, by the way, def 3 can
be proved:
DEF 106 A rectilineal figure is given in position if its
verticesare given in position.
DEF 107 A rectilineal figure is given in form if it is given
inposition.DEF 108 A rectilineal figure is given in form if it is
similar toone which is given in form.16
15 The 'if and only if' clause is a defining relative clause in
Greek. From the use of def3 it is evident that it must be
interpreted as a biconditional.
16 Similar rectilineal figures are defined in VI, def 1. See
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Elements of Euclid''s Data 155
Figure 55
To understand thm 55, we may remark heuristically (see figure
55)that if a rectilineal figure P is given in form, its angles are
severallygiven, and its sides have given ratios to each other (def
3). This suggestsintuitively that (in modern idiom) the length of
one side (and thereforeof all sides) is a function of P's area. Or,
as the Data has it in thm 55, ifP is given in magnitude, its sides
will also be given in magnitude.
I now give an analysis for thm 55. P and Q are rectilineal
figuresgiven in form (in accordance with def 3), and a and b are
arbitrary sidesof P and Q respectively. It suffices to prove that
one side is given inmagnitude, since, if one is given, they all are
(def 3). Thus we want toprove that if [P]m then [a]m. Now, suppose
the conclusion to be true: anarbitrary side a of P is given in
magnitude. Let b be set out as a linesegment given in position and
therefore also in magnitude. Then a:b isa given ratio (thm 1). Let
Q be a rectilineal figure, constructed on theline segment b. If we
see to it that Q is similar and similarly situated toP, with b and
a homologous sides, Q will be given in form (def 3), andthe ratio
P:Q will be 'double' (diplasin; we should say 'square') of a:b,by
the corollary to VI.20. Therefore, since the 'double' of a given
ratiois itself given (thm 8), we have [P:Q]. And then, since P was
given inmagnitude, so is Q.
The analysis leaves us with an implication from one conjunction
toanother:
[P]m & [a]m & [&L -> [a:b] & [P:Q] &
[Q]m.In the proof of thm 55 this implication is partly
reversed:
[PL & [&L - [QL & [P:Q1 & fob] &
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156 Chnstian Marinus Taisbak
1st step [b]m > [Q]m (the converse of 55, proved in 52).2nd
step [P]m & [Q]m -> [P:Q] (thm 1).3rd step [P:Q] -> [a:b]
(thm 54, using thm 8 and the
important thm 24, which will be proved below)4th step [a:b]
-[fl]m (thm 2), and therefore (def 3) all the
sides of P are given in magnitude.
Here is a survey in reverse order of theorems 55-49 formulated
inmy notation:
thm 55 [P]m -> [A]m;thm 54 [P:Q] -> [a:b];thm 53 [a:fc]
for one pair a, b * [a:b] for any pair a, b;thm 52 [e]m -> [P]m
(converse of 55);thm 51 [a:b] -[P:Q] (converse of 54);thm 50 [a:b]
& P sim Q -[P:Q];17thm 49 a is the same as b [P:Q].
We shall need thm 1 and 2, but in this paper I shall not discuss
howsuch statements can be proved. Evidently the answer is either
trivial orvery subtle.
thm 1 The ratio of given magnitudes to each other isgiven.thm 2
If a given magnitude has a given ratio to another one,that
magnitude will be given.
A theorem of transitivity for given ratios is also needed. We
find it inthm 8, which is expressed in the well-known idiom of the
'commonnotions' of book I of the Elements: the word 'magnitude' is
absent fromthe text, which merely uses the definite article in the
plural: 'thosewhich have...':
thm 8 Magnitudes which have a given ratio to one and the
samemagnitude, also have a given ratio to one another.
17 In thm 50, an auxiliary to 51, P is similar and similarly
situated to Q; the sides a andb are, consequently, homologous sides
(in the sense of VI.4); but P and Q are not givenin form: the
theorem is about ratios between magnitudes (and might be put
beforethm 25), not about given forms. Knorr could legitimately
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Elements of Euclid's Data 157
[a:b] & lc:b] -> [a:c]The proof of thm 8, which I shall
not give, rests upon the 'di'isou'theorem, V.22, an important one
for the theory of proportions.18
I shall also not discuss the first part of the Data since it
does not dealwith positions. It ends with a solitary theorem
proving, to put it inmodern terms, 'the uniqueness of the square
root', though one wouldnot easily recognize this fact from the way
it is stated:
thm 24 If three line segments are proportional,and the first has
a given ratio to the third, it will also have agiven ratio to the
second.
objects a, b, c: line segmentshypotheses a:b = b:c,
[a:c]assertion [a:b]
To represent the proof of thm 24 I introduce the following
notation,which follows the tradition initiated by Dijksterhuis
(1929-1930), but ismore like that used by Herz-Fischler ((1987),
XIV):
S(e) for 'the geometric square on the line segment ;R(d.f) for
'the rectangle contained by (cf. II, def 1) the segmentsdandf.
Let d be a line segment given in position.19 Get/such that
d:f=a:c (VI.12).Then ld:b}, and, since Id}, [f\ (thm 2). Get e such
that d:e = e:f(VI.13); thenR(d.f) = S(e) (VI.17). Since [d] and
[/], [R(d.f)] (true, but unproved) andtherefore [S(e)]r and
therefore [e] (true, but unproved). Since [d], [d:e].The argument
continues:
ax = d:f,S(e):Rfo.c) = S(d):R(d.f) (VI.l, V.ll),S(a):S(b) =
S(d): S(e) (since a:b = b:c and d:e = e:f)a:b = d:e (VI.22)).
Hence, since [d:e], [a:b].
18 In Euclid's theory of numbers, book VII of the Elements, that
theorem is equivalentto the associative law for multiplication, as
I show in chapter 8 of Taisbak (1971).
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158 Christian Marinus Taisbak
As can be seen, the theorem amounts to proving thatif S(a):S(b)
= S(c):S(d) then a:b = c:d.
Normally, one is referred in this connection to VI.22, which is,
however,defective as transmitted in the Elements. To prove the case
for squares,and not for arbitrary rectilineal figures described
upon a and b, asimpler proof can be given, based on V.9.20
The proof of thm 24 makes use of statements about squares
andrectangles which have not been proved, namely,
If e and / are given, R(e.f) will be given,and its special
case,
If e is given, S(e) will be given.In the Data, these truths are
taken for granted (e.g., in thm 52, below).They can be proved using
thm 27, thm 29,1.46, and (perhaps) a supple-mentary theorem stating
that a rectilinear figure given in position isgiven in magnitude.
Thm 39 amounts to such a theorem, but, surpris-ingly, nothing is
said about magnitude in that proposition or in thetheorems that
follow it.
I noted that in the 3rd step of thm 55 a special case of the
followingproposition was used:
thm 54 If two rectilineal figures given in form have a given
ratioto each other, their sides will also have a given ratio to
eachother.
Euclid divides the proof into two cases in order to be able to
use thecorollary to VI.20:
Case 1. The form P is similar to the form Q. Get the
thirdproportional c to the similarly situated sides a and b. Then
a:c= P:Q (VI.19 cor.),and P:Q is a given ratio. Then (thm 24)
theratio a:b is also given. But since P and Q are given in form,
eachside in P has a given ratio to each side in Q (thm 53, below,
andthm 8).Case 2. The form P is not similar to the form Q. On one
side bof Q a rectilineal figure R similar and similarly situated to
P isconstructed. Then R is given in form, and so was Q.
Therefore
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Elements of Euclid''s Data 159
(thm 49) Q:R is a given ratio, and P:Q is given. Therefore P:R
isgiven (thm 8). The rest follows from case 1 of the theorem.
In fact, case 2 is not necessary for thm 55, and, as it stands,
thm 54 is amuch more general proposition than needed.
Thm 53 shows that the sides compared need not be homologous:thm
53 If two rectilinear figures are given in form, and one sideof one
figure has a given ratio to one side of the other, the othersides
will also have a given ratio to the other sides.
The assertion follows immediately from def 3 and thm 8.thm 52 If
on a segment given in magnitude a rectilinear figuregiven in form
is described, the figure is given in magnitude.21
This theorem is proved by drawing the square on the given
segmentand using thm 49 and the unproved special case of this
theorem: if asegment is given in magnitude, its square (which is a
given form) is alsogiven in magnitude. The converse of thm 54 is of
interest, though notnecessary for thm 55, since the figures in 55
are not arbitrary but similar:
thm 51 If two segments have a given ratio to each other
andarbitrary rectilineal figures given in form are described
uponthem, the figures will have a given ratio to each other.
With thm 50 as an intermediary the statement is a generalization
of thm49, in which the figures are drawn on the same segment:
thm 49 If on one and the same segment two arbitrary
rectilinealfigures given in form are described, they will have a
given ratioto each other.
49 is itself a generalization of thm 48, which deals with
triangles. Thuswe have reached the 'lower' level which is the basis
of the series ofassertions about polygons given in form, namely,
assertions abouttriangles. They are made to work through:
thm 47 Rectilinear figures that are given in form can be
dividedinto triangles that are given in form.
21 This is one of very few theorems where Euclid's figure has
more than four sides. Healways uses just enough to make the
reasoning general. In my diagrams for this paperI follow a modern
practice, so that the reader is not distracted by doubts as to
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160 Chnstian Mannus Taisbak
8. Triangles
The basic theorems concerning triangles given in form can be
surveyedin the following diagram (g in a column means that the item
above theline is given). Some of the theorems (40-44) are obvious,
while two ofthem (45-46) are surprising and may be considered as
applications ofthe obvious ones:
ZA ZB
g ggg(right)g
g
a:b b:c a+b : c b+c : a
gggg g
gg
thm
4041434442
4546
In all the theorems, except 42, one angle is given and something
more.In 40 it is one of the other angles. It then follows
immediately, from thesum of angles of a triangle, that the third
angle is also given. Thus it isnot necessary to have the third
angle given, though the statement ofthm 40 takes all three to be
given. In 41 the ratio of the sides containingthe given angle is
taken to be given, in 43 and 44 the ratio of the sidescontaining
another angle, the latter being dubious (under the
usualinterpretation). Finally in 45 and 46 the ratio of the sum of
two sides tothe third side is taken as given.
It may be instructive to begin with thm 42, which does not
involvea given angle; it is based on thm 39, which stands somewhat
apart fromthe others, having sides with given magnitudes, not only
given ratios:
thm 42 If a triangle's sides have given ratios to each other,
thetriangle is given in form.
thm 39 If a triangle has each of its sides given in magnitude,
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Elements of Euclid's Data 161
Note that thm 39 says nothing about the triangle being given in
mag-nitude, although this is actually proved. Only the form is
relevant forthm 42 and the following propositions.
In the proof of 39 a triangle ABC is set out with sides given
inmagnitude. Another triangle is constructed (not by Euclid, nor
you, norme, but by The Helping Hand) after the fashion of 1.22 (and
with,mutatis mutandis, the same words) on a segment given in
position andwith sides equal to the sides AB, BC, AC. The new
triangle will havevertices given in position and is therefore given
in form. And it is congruent with ABC, which is thereforegiven in
form (DEF106,107,108).
In thm 42 a triangle ABC is set out, with sides having given
ratios.Using 1.22 another triangle is constructed from segments
having thegiven ratios on a segment given in magnitude. The new
triangle will begiven in form (thm 39), since its sides are given
in magnitude. And it isproved to be similar to ABC, which is
therefore given in form.
For a long time while studying the Data I had the impression
that, inorder to be constructed, triangles (and figures in general)
had to havesome part given in position. Now I shall have to recant,
since thistriangle is not in any sense given in position, but only
in magnitude(which is irrelevant), and in form. But I am still
pondering over themeaning of ekkeisth, 'let it be laid out', from
the verb ektithemi, whichgives immediate associations to thesis,
'position'. If ekkeisth is inter-preted as 'let it be set out in
position', then the reference to thm 39 willbe unnecessary, and
that proposition will be superfluous in this con-text. The history
of the two theorems needs thorough investigation. Ishall leave the
question open, and go on to:
thm 40 If, in a triangle, each of its angles is given in
magnitude,the triangle is given in form.
Some might want a diorism to ensure that the sum of angles be
lessthan two right angles, but none is needed, since the object is
assumedto be a triangle. The proof also uses the triangle
inequality (1.20) as amatter of fact, as a 'masked given'. If def 3
were the basic definition ofbeing given in form, we might expect
Euclid to prove that the ratios ofthe sides are given. But what
happens? A line segment is taken to begiven in position and
therefore in magnitude. On the segment a triangleis constructed
with the given angles (1.23,1.32), and thus having verticesgiven in
position (thm 25). Euclid then says,
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162 Christian Marinus Taisbak
Therefore each side is given in position and in
magnitude.Therefore the triangle is given in form. And it is
similar to the triangle (VI.4); which, therefore, is given in
form.
I feel justified in adding the definitions DEF 106, 107, and
108, andmaintaining that def 3 should be a theorem.
Thm 41 was used as an example in section 3. In its proof
anothertriangle is constructed with the given angle, and with the
sides contain-ing it having the given ratio, and with vertices
given in position, so that(DEF 106,107) the new triangle is given
in form. It is proved (VI.6) tobe similar to the one that is set
out, which, therefore, is given in form,by DEF 108.
Thm 44 says,thm 44 If a triangle has one angle given and the
sides aboutanother angle have a given ratio to each other, the
triangle isgiven in form.
Roughly speaking, a figure ought (?) to be given in form, if it
is similarto figures having the same properties. On this
assumption, one caneasily verify that thm 44 is equivocal. If the
given angle is A, and thegiven ratio is ax, the theorem is true
only if a>c, or if ZC is a right angle(i.e., if a:c - sin A).
Otherwise, two very different triangles will bothhave the
properties in question (figure 44). But perhaps there are
ideasabout being given in form that I do not yet understand.
a c
Figure 44
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Elements of Euclid''s Data 163
In the Data, thm 44 is used only to prove thm 45, in which the
two'different7 triangles happen to be congruent. The 'proof of thm
44 restson:
thm 43 If in a right-angled triangle the sides about one of
theacute angles have a given ratio to each other, the triangle
isgiven in form.
D
objects ABC: triangle;hypotheses ZA right angle,
la:b];assertion [ABC]f.construction DHE right-angled
triangle,
ZH right,[DE]P/[DE:DH] = [a:bl
To prove thm 43 Euclid constructs (figure 43) a right-angled
triangleDHE with hypotenuse given in position and with the ratio
betweenhypotenuse and one of the legs given. Its vertices are
proved to be givenin position, whence the triangle is given in form
(DEF106,107). And itis similar to ABC (VI.17). Therefore the latter
is also given in form.
The proof of this proposition includes another example of a
'super-fluous' construction (see the discussion of thm 31 in
section 6). Afterfitting in the segment DH, Euclid goes on to draw
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164 Christian Marinus Taisbak
which is exactly the one he had to use to fit in DH, according
to IV.l.But then, the fitting of DH is just commanded (being
admitted fromIV.l), whereas the arc KHG is part of a new
construction to establishthat H is not a moving point (thm 25).
Since the sine of ZC is a:b, some moderns will want to interpret
thistheorem in trigonometric terms as indicating that the sine of
an acuteangle determines the angle uniquely. I would prefer to call
the wholetheory of triangles given in form Trigonosophy', Wisdom of
the Trian-gle. At a time when Ptolemy's trigonometry was still far
off, the use ofsuch wisdom was very limited. But wisdom it is.
For convenience I shall introduce some very useful auxiliary
trian-gles, the lesser and the greater isosceles adjuncts. The
lesser adjunct isknown from 1.20 (triangle inequality) and VI.3
(bisector of an angle ina triangle):
In a triangle ABC (figure 67) let the side BA be produced to
D,AD = AC, and let CD be joined. Then CAD is a lesser
isoscelesadjunct with respect to ZA. From the vertex B let BE be
drawnparallel to AC, to meet DC produced in E. Then EBD is
thegreater isosceles adjunct. The adjuncts are similar, and their
baseangles are equal to half of ZA. (By interchanging the letters
Band C, you get two others; while the greater adjuncts
arecongruent, the lesser are so only if AB = AC).
As I mentioned above, the following two propositions are not
elemen-tary. But they are applications of some of the elements of
the Data, andEuclid obviously regarded them as belonging among the
elementarytruths about triangles given in form. I therefore include
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Elements of Euclid's Data 165
thm 45 If a triangle has one angle given, and the sum of
thesides about the given angle has a given ratio to the third
side,the triangle is given in form.thm 46 If a triangle has one
angle given, and the sum of thesides about another angle has a
given ratio to the third side, thetriangle is given in form.
Both of these propositions can be (and in the appendix to
Menge'sedition are) proved by bringing in the lesser isosceles
adjunct; then 45follows directly from 44,46 from 41. Since 44 is
equivocal if applied tothis situation, 45 ought to be so, too; but
the two triangles which bothmeet the conditions are similar (see
figure 45). As this is the onlyinstance (that I know of) where thm
44 is used, the author may wellhave chosen to ignore its
ambiguity.
Figure 45 B
In the proofs which Menge edited as original (because they come
firstin the manuscripts) the given angle is halved, and VI.3 is
applied,followed by some appropriate manipulations with ratios from
book Vof the Elements. In the end the propositions follow from thm
44 and 41respectively; and the lesser adjunct has already done its
work in VI.3.
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166 Christian Marinus Taisbak
9. Do we know the use of 'givens'?
The following theorem illustrates the tricks that can be
performedwith isosceles adjuncts and one sort of proposition that
is found amongthe 94 in the Data:
thm 67 If a triangle has a given angle, then an area which is
thedifference between two squares, namely the square on the sumof
the sides containing the given angle and the square on thethird
side, will have a given ratio to the triangle.
Let the given angle be ZA. After constructing (figure 67) the
lesser andthe greater isosceles adjunct, Euclid asserts (out of the
blue, because henever mentioned the proposition in scholion 133)
that since BC is anarbitrary transversal line (through one vertex
of the isosceles triangleEBD),
S(BD) = R(DC.CE) + S(BC).But BD = BA + AD = BA + AC,so that
S(BD) = S(BA + AC) = R(DC.CE) + S(BC).
Now Euclid asserts and proves that R(DC.CE), the difference
betweenS(BA + AC) and S(BC),has a given ratio to the triangle ABC.
SinceZBAC is given, the lesser isosceles adjunct is given in form
(thm 40),whence
[DA:DC] (def3),and so [S(DA):S(DQ] (thm 50).Now, BA:AD = EC:CD
(VI.2),
R(BA.AD):S(AD) = R(EC.CD):S(CD) (VI.l),R(BA.AD):R(EC.CD) =
S(AD):S(CD) (enallax, V.16).
Since the ratio S(AD):S(CD) is given, so is R(BA.AD):R(EC.CD),
and,since AD = AC, so is R(BA.AC):R(EC.CD). But R(BA.AC) has a
givenratio to the triangle ABC (thm 66, see below); therefore also
R(EC.CD)has a given ratio to the triangle ABC (thm 8).
Scholion 133, which (as far as I know) is met in no other
source,obviously belongs to 'geometric algebra':
scholion 133 If, in an isosceles triangle, a straight is
drawnarbitrarily (from the apex) to the base, the square on the
seg-ment drawn plus the rectangle contained by the parts of thebase
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Elements of Euclid's Data 167
H
Figure 133D
The proof runs easily. Dropping the perpendicular from the apex
andapplying II.5 on the parts of the base (figure 133), and finally
addingthe square on the perpendicular, one has:
R(EC.DC) + S(CH) = S(HD)R(EC.DC) + S(CH) + S(BH) = S(HD) +
S(BH)K(EC.DC) + S(BC) = S(BD)
Thm 67 depends on:
thm 66 If a triangle has a given angle, the rectangle
containedby the sides that contain the given angle has a given
ratio to thetriangle.
Let the given angle be A, and let BD be the perpendicular on AC.
ThenR(BD.AC) is double the triangle; its ratio to R(BA.AC) is
BD:BA, whichis given according to thm 40 and def 3.
Thm 67 is enigmatic in several ways. What is its information
worth?What kind of analysis preceded it? In what mathematical
context? Weobserve that the vocabulary (meizon dunantai) is of the
sort that is foundprimarily in book X of the Elements. I feel sure,
although I cannot proveit, that this proposition was invented in a
context of geometric analysiswhere isosceles adjuncts were
frequently used and scholion 133 was auseful auxiliary. But at the
moment I fail to see the worth of theinformation that S(AC + AB) -
S(BC) has a given ratio to the triangleABC. I can see a vague (?)
connection with 11.12 and 13, but for me andyou the main difficulty
in appreciating the statement is that we do notget what we are
interested in, namely the value of the ratio, which isnot expressed
or expressible in any Greek measure, but amounts (intrigonometric
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168 Christian Mannus Taisbak
This is the sort of frustration that affects us everywhere in
the Data.We get very little information, hardly any 'knowledge' of
the givens.And why not? Probably because 'knowing' geometrical
objects wasproblematic at the time when the concept of given came
into being andthe consequences of incommensurability were just
being understood.There was next to nothing known about these
objects, and very littlethat is worth knowing: length, size,
distance, i.e., any of the rheta, theattributes that can be spoken
of with numbers. We look for coordinatesto pin down positions, for
lengths and areas to measure magnitudes.With forms we are at a
loss. Moreover, we do not need them any more.Mathematics lost
interest in them when the 'application of areas' wasrendered
obsolete by the theory of equations, and perhaps alreadywhen
trigonometry took over and codified the alliance between anglesand
ratios.
The situation is not much better in the case of those
propositionswhich ensure the existence of geometric objects. As an
illustration, let uslook at two famous proposition from the Data,
58 and 85 (see figure 58):
B D
Figure 58
thm 58 If a given area be applied to a given segment deficientby
a figure given in form, the latitudes of the deficiency
aregiven.thm 85 If two straights enclose a given area in a given
angle andif their sum is given, each of them will be given.
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Elements of Euclid's Data 169
objects AC, CD: parallelograms;hypotheses [AC]m, [AD]m,
[CD]f;assertion [BC]m, [BD]m.
At the beginning of the proof the parallelogram AC has been
appliedto the segment AD in such a way that the deficiency CD is
given in form.Then AD is bisected at E, and on ED a parallelogram
EF is drawn,similar and similarly situated to CD, and therefore
about the samediagonal; thus it is possible to draw the schema, the
gnomonlike figureEDFGCK.22 The rest follows from adding and
subtracting givens:
EF = gnomon EDFGCK + KG, given.AC = gnomon EDFGCK, given.EF = AC
+ KG, given.
KG is given, and similar to CD. Therefore the sides of KG are
given (thm55), whence EB is given, and therefore BD given. Ergo BC
is given.
Thm 85 follows immediately from thm 58 (figure 85):
B
Figure 85
Let AC be the given area enclosed by AB and BC, whose sum is
given.Let AD be equal to that sum, and let CD be a parallelogram on
BD. SinceCD is given in form (def 3), we have the conditions of 58:
a parallelo-gram AC is applied to a given segment AD such that a
given form CDis deficient. Therefore BD is given, and so BC.
22 Whenever the word schema occurs in the Elements and cognate
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170 Chnstian Mannus Taisbak
Evidently, the construction and proof of thm 58 presuppose
ananalysis along the same lines as VI.28. How do they differ?
VI.28constructs the parallelogram KG as the difference between EF
and thegiven area by means of some propositions in book VI of the
Elements.Thm 58 proves that KG is given in magnitude (being equal
to the saiddifference) and in form (being similar to the given
form). Therefore(thm 55) KC is given. No wonder that Olaf Schmidt
never missed theData for this problem. Its contribution is
definitely one of Knorr'sparallel results. And the analysis which
must have preceded the bisect-ing of the given segment is absent
from the Elements, as well as fromthe Data.
10. Conclusion
In conclusion, let me offer a thesis and an image. My thesis is
this. Itwill take us nowhere to 'translate' the 'givens' into
trigonometry or anykind of analytic geometry. Of course, thm 66 may
be said to prove theratio:
2 triangles : R(b.c) = sin A : 1,or even the formula:
2 triangles = be sin A.
But this interpretation is of no consequence as long as numbers
arerational only. Ptolemy allowed himself to express given chords
withsexagesimals, 'aiming at a continually closer approximation in
such amanner that the difference from the correct figure shall be
inapprecia-ble and imperceptible',23 but he did so simply because
they were givenand therefore ought to be determinable. But Euclid
did not think thatway: his 'givens' live in a world of their own
which we must masterwithout measure if we want to understand
it.
One important feature in a given is its character of being
thrust uponus not to be gotten rid of. In I.I we must use exactly
the given piece ofstraight line to construct the triangle. We learn
a Greek attitude to giftsfrom this, I am sure. And we are facing a
piece of Platonic ontology:
23 Almagest 1.10, translated by Thomas (1939), vol.
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Elements of Euclid's Data 171
what we construct is what already exists. We can do nothing by
way ofgenerating objects that are not already there. They are
given. By whom?I do not know if Euclid answered that question, but
he did set himselfto explain axiomatically what it means to be
given. The difficulties hereare not new. In his commentary on the
Data my namesake Marines,director of Plato's Academy starting in
485 C.E., was no less bewilderedthan I am about the meaning of
being given. He pretends at the endthat the theory of givens is
very useful, but I am afraid that some willstill not need it if
they know the Elements.
My image is this. The Data makes geometry into a play enacted on
astage we may call The Geometrical Plane. When the curtain rises to
eachact (theorem), some actors (geometrical objects) are on the
scene, a fewof which are presented to us in a certain fashion: they
are said to begiven. As the play goes on, it reveals that there are
more given actorson the scene than we believed at first sight,
which revelation turns outto be the point of the play. The
important feature of this play is thatthere are more things given
than meet the eye. Everything is latent onthe scene when the
curtain rises, but only a few objects and attributesare noticed by
the audience. As the play runs according to the rules ofthe author,
more and more facts are illuminated and emerge, until atlast the
scene is full of real entities, while several phantoms
(duplicatedcubes, squared circles, trisected angles) have been
exiled.
11. Epilogue
It is very much in line with the Data to end abruptly, leaving
manypoints unresolved. Consider common notion 2 of the Elements: if
equalsbe added to equals, the sums are equals. This is not
primarily a state-ment about magnitudes involved in an addition,
but about the relationof equality combined with the operation of
addition. I repeat: not themagnitudes, but the relation. Compare
now Data 3: if given magnitudesbe added together, their sum is
given. This is not primarily a statementabout magnitudes involved
in an addition, but about the property ofbeing given. That property
in its various manifestations is what Euclidis trying to elucidate.
Such an investigation can always take into ac-count one more
example. There is, then, no natural end to the Data.
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