Euclid's Geometry as an Attempt to Solve 
              ________________________________________
                    a Problem with Plato's Forms
                    ____________________________

                       Thomas O'Neill 
                        Haverhill, MA 
                       TJPAO@aol.com 
                  (received: June 1, 1997)
 
 
	Following the Platonist tradition, which places a significant (even 
complete) emphasis on Universal Ideas (eidoi) as being what is ultimately 
real(1), Euclid created (or, more appropriately, collated) his system of 
geometry/ philosophy in a way that relied on the abstract, but in the process, 
he made the abstract tangible.  The shapes (and the relationships between 
them) which Euclid discusses in his Elements resemble, in nature, Plato's 
eidoi, but one can work with these abstractions on paper (or, traditionally, 
in the sand).  One can know what a perfect circle is by following the 
definitions provided by Euclid, but one cannot recreate a perfect circle in 
the apparent world.  The circle which one draws (for example, on paper or in 
the sand), of course, is flawed, given the Platonic presuppositions involved 
in Euclidean geometry.  The points, for example, may not be exactly 
equidistant from the center of the drawn circle(2). Nonetheless, by noting the 
imperfections in one's mind, one can work with the drawing of a circle as 
though it were perfect, knowing the entire time that it is not actually so.  
As a result of this possibility (working with the eidoi in the apparent world 
by means of Euclidean geometry), Plato's eidoi become more accessible(3).
 
	The significance of this similarity is that Euclid seems to have 
solved the most troublesome puzzle with which Platonic philosophy is plagued, 
namely how one can know and work with (discuss, write, etc.) that which is 
ultimately real (Form, Universal, Idea, eidos, etc.).  According to Plato, 
these eidoi are real, and human beings are capable of knowing them(4).  But, 
taking one's thoughts regarding the eidoi and expressing them verbally in a 
coherent manner is difficult, if not impossible(5).  Without a connection of 
some kind between the apparent world and that of the eidoi, one encounters 
hindrances imposed by the inability of language to accommodate such ideas 
before having the chance to make any significant progress toward understanding 
Plato's system.  In Plato's system, this connection does not exist; there is 
no way to work verbally (in an efficient manner) with the eidoi in the 
apparent world.  Euclidean geometry provides the necessary connection.  Euclid 
discusses certain eidoi (specifically, the eidoi involved in geometry) in such 
a way as to make them accessible to both the senses and language.  What one 
senses is not truly the eidos itself, but an example of the eidos, along with 
the verbal definition which  helps one to figure mentally what the ideal 
circle (for example) "looks" like(6).
 
	Euclid, regarding the problem of the eidoi which arises in Platonic 
philosophy, does not provide a flawless reconciliation, but this system does 
provide a means to understanding the notion of eidos in general.  With this 
general understanding of the notion of eidos, one can begin to understand the 
nature of the other eidoi, such as Truth and Beauty.  This is not to say that 
an understanding of Euclidean geometry will provide one with an understanding 
of the Good; such an assertion would be absurd.  But, the orientation which 
Euclidean geometry provides the reader is conducive to studying the eidoi.  
This type of geometry (since its foundations are in Platonic philosophy, as 
well as those from whom Plato borrowed ideas(7)) prepares one for the way in 
which one needs to think if one is to attempt to understand the nature of 
eidos itself, as well as the definitions which these eidoi constitute.
 
	In Euclidean geometry, one works with eidos in the apparent world.  In 
Platonic philosophy, by definition, this would be impossible(8).  But, in 
Euclidean geometry (as stated above), something similar to this is 
possible(9).  One is able to use objects in the apparent world to help sort 
out one's ideas regarding the geometric eidoi (for example, pencils and paper 
are often used in the study of geometric eidoi, even though both are in the 
apparent world).  Thus, despite the fact that these close facsimiles of eidoi 
(as the geometer works with them) are in the apparent world, one is really 
operating along a fine line, which actually permits one to operate among the 
eidoi and in the apparent world at the same time(10).
 
	Before entering into an in-depth approach to how Euclid addresses this 
puzzle produced by the Platonic notion of eidos, Plato's metaphysics (and as a 
result, the problem of being able to discuss eidoi in the apparent world; from 
here it will be called the "discussion puzzle") should be explained in a 
detailed manner.  In the Republic, Meno, Theaetetus, Sophist, and Parmenides, 
Plato provides all of the essential information to understanding the nature of 
the eidoi.  But, in none of these dialogues does he provide a clear and facile 
way to interpret definition or description.  Instead, metaphors and analogies 
(which are nearly impossible to decipher without leaving some degree of doubt) 
are employed, making numerous justifiable interpretations the result of 
repeated readings of these works.  Also, this type of writing style allows for 
the frequent misinterpretation of the message promulgated by Plato in these 
dialogues, resulting in the reader's not understanding the system as Plato had 
intended it to be comprehended.  A deep examination of Plato's philosophy and 
writing style, though, indicates that no other writing style could possibly be 
used in discussing some of the issues which his philosophical system 
generates.  Presenting his theory of the eidoi, in the Republic (or anywhere 
else), would have been impossible without the "analogy of the cave"(11).  
Further, this analogy was included only to clarify an earlier metaphor, 
specifically that of the "divided line"(12).
 
	Such figurative language is necessary in order for Plato to describe 
the foundation of his philosophical system:  the world of the eidoi.  
Introducing them at all, for Plato, must have been extraordinarily difficult.  
Although previous philosophers had dealt with the notion of eidos in some 
small way, none attacked it in the way that Plato did(13).  Subsequent 
philosophers were able to deal with eidos as easily as they did (without as 
many intricate metaphors as are present in Plato) only because they had 
Plato's foundation upon which to build (or, at least to which they could 
react).  Thus, Plato was doing something which nobody had done before.  This 
must have caused great difficulty for him in writing these ideas, for new 
ideas (especially those he expressed) often require new means of expression.  
Aristotle and subsequent philosophers (including Euclid) had Plato's ideas 
from which to work, and by then, language had become adapted to the needs of 
the ideas which Plato presented.
 
	The fact that Plato could not quite find the words to adequately 
express his metaphysical views opens the door for Euclid.  Plato's eidoi are 
not of the apparent world.  Consequently, discussing (and writing about) them 
is difficult, for such activity occurs in the apparent world.  Knowledge of 
the eidoi and of how to discuss them are different.  As Plato demonstrated 
(through Socrates in the Meno), even an intellectually uncultivated slave boy 
can (and does) know the eidoi.  But, being able to discuss these eidoi is more 
difficult(14).  This difficulty in explication is evident in the Sophist, 
Parmenides, and Theaetetus.  One is often led into cumbersome language and 
sometimes into seeming contradictions.  This, for example, occurs in the 
Sophist when the Eleatic stranger (The Eleatic stranger represents a follower 
of Parmenides, if not Parmenides himself.  Whichever he is, the Eleatic 
stranger's role is to represent one who adheres fervently to the philosophy of 
Parmenides, who actually did live in Elea) and Theodorus discuss motion and 
rest(15).  Perfect motion, Theodorus asserts, is needed to solve the problem 
at hand, and he and the Eleatic stranger come to the conclusion that such 
motion does not exist.  In this assertion, there lies no problem; it is simply 
the typical use of the so-called Socratic method.  But, the route which they 
take in arriving at this conclusion is circuitous and reliant on bizarrely 
stated propositions such as "not even white continues to flow white"(16).  
Such metaphors, even in context, are not at all clear, but they do reveal 
Plato's point, upon close examination.
 
	Plato's reliance on unclear figures of speech to explicate his 
philosophical system is not the sole difficulty with language which arises in 
his works.  There also exists a problem with the use of non-figurative 
language.  Some of the issues which Plato addresses are simply not conducive 
to either metaphor or analogy.  An attempt to deal with these issues using 
straightforward language is no clearer than such metaphor-laden examples as 
the "divided line."  In the Parmenides, for example, there is a portion of the 
dialogue (between Parmenides and Aristoteles) in which the literal use of 
words leads to "be," "is," and "exist" as having significantly different 
meanings(17).  These differences in meaning extend beyond the simple 
differences in nuance between "be" and "exist," and also beyond the 
differences in meaning which are imposed by the verb's tense (between, in this 
case, "be" and "is").  Even literal language, in Platonic philosophy, does not 
allow one to approach the eidoi directly.
 
     The problems stated above illustrate the difficulties which plague 
Platonic philosophy.  The problems do not necessarily lie within the system 
itself.  As a philosophical system, it is both consistent and plausible.  The 
puzzle instead lies in attempting to express one's thoughts about either 
Platonic philosophy itself or (if one accepts Plato's ideas regarding the 
nature of the universe) the eidoi themselves.  Neither can be done adequately 
given Plato's philosophical system.  A connection between the apparent world 
and the "world"(18) of the eidoi is needed.  There does not exist, in Platonic 
philosophy, an explicit segue between the apparent world and the eidoi. Yet, 
there is one which Plato does use without crediting it directly: geometry.  As 
stated earlier, Plato frequently uses examples from this discipline in 
explaining his philosophical system.  The most famous of these examples is 
that of the slave boy in the Meno.  During the time that Plato lived, though, 
a comprehensive body of geometric knowledge had not been established(19).  
There were the works of scattered philosophers and mathematicians (for 
example, the Pythagoreans), but nobody had brought it all together.  The 
result was that Platonic philosophy contained an underdeveloped example of the 
solution to its most troublesome puzzle. Euclid's Elements provided an 
explicated version of the "solution"(20) to the puzzle which arose in Platonic 
philosophy.
 
	In the Platonic system of philosophy (as well as that of Euclid), 
there is a group of eidoi which are conducive to operations in the apparent 
world; these eidoi are those of geometry.  Although a circle in the apparent 
world is really only a drawing of a circle, the process which led to the 
construction of that circle is what separates it from such eidoi as Truth and 
Beauty.  The key notion here is "constructed."  Despite the fact that 
triangles, circles, etc. can be found in the apparent world (only as examples 
of the actual eidos, though), they can nonetheless be constructed from an 
appropriate verbal definition(21).  The definition is such that one can 
understand it even in the apparent world.
 
	This notion is best explained with an example from Euclidean geometry. 
Euclid defines an equilateral triangle as a plane figure having three equal 
sides(22).  With this definition inmind, one can construct examples of 
equilateral triangles in the apparent world.  One can draw them on paper, for 
example, or (as the Greeks did) in the sand.  But, as this reproduction 
continues, one does not have to think of the equilateral triangle as being 
restricted by having a particular base-length or altitude.  One can look at it 
solely as an equilateral triangle, which though it has a certain altitude, 
etc., adheres to the definition.  For, the definition puts no restrictions on 
size.  There is no mention of size at all in Euclid's definition of an 
equilateral triangle.  This is because size is not an essential characteristic 
of equilateral triangularity.  Instead, the number of sides is essential (as 
is the congruence of the internal angles, but this is a function of the 
equality of the sides(23)).  In the drawing which one finds in the apparent 
world, one can disengage the eidos from the example. Although the observer 
sees an equilateral triangle, with each side being three inches long, one can 
just as easily disregard the length of the sides and work solely with the fact 
that there are three sides as opposed to any other number.  Even in the 
apparent world, one can recognize a presence of the eidoi.
 
	Euclidean geometry does not fit as crisply as the example above may 
imply.  This system operates under certain fundamental assumptions, as does 
Platonic philosophy.  The central assumption is that a point is that which has 
no part(24).  This definition does not provide a coherent definition of a 
point, at least not as it is used by Euclid later in the Elements.  This is of 
concern in Euclid's third definition, namely that of how a line is determined. 
 According to Euclid, a line is that which is determined by two points.  This 
raises a problem in that that which has no part determines that which has 
"part."  Despite the fact that a line is "breadthless length"(25), Euclid does 
not claim that it has no "part."  In fact, because it does have length, it 
must have "part."  But, to further complicate the issue, a line is composed of 
points (definition four)(26).  Thus, when grouped together, that which before 
had no part somehow does have "part."  The assumption which one must make is 
that this is possible.  Otherwise, none of the other eidoi (squareness, 
triangularity, circularity) would exist, for they all rely on the definition 
of "point."  With this assumption, though, Euclid's system is possible(27).
 
	After Euclid's fundamental assumptions and definitions are given, he 
begins to construct his geometric system.  Some of this was mentioned above, 
namely the construction of triangles and circles.  There is another 
significant step which occurs after Euclid constructs his basic shapes. 
Through deductive proofs, he shows the relationships which exist between these 
eidoi as a result of their common sources.  By common sources, the fundamental 
definitions on which everything in the system is built is meant. Everything in 
Euclidean geometry relies on the definition of a point.  Most of the system is 
tied in some way to the definition of a line.  These relationships deal with 
all facets of geometry, angle relationships, coplanarity, etc.  One such 
relationship is explored toward the end of the first book of the Elements.  
The notion of vertical angles is a relationship between intersecting lines.  
Non-adjacent angles, according to Euclid, will be equal in measure(28).
 
	The point of this is not merely to explain one of the relationships 
which exists in Euclidean geometry.  This relates directly to the issue of 
eidos in that Euclid's construction of the system from a single source allows 
for facile manipulation in the apparent world.  Euclid's system is explained 
in the apparent world (on paper or orally).  But, according to Plato, all that 
one gets in the apparent world is illusory; it is real only as "shadows"(29).  
This would mean that, according to Platonic philosophy(30), Euclidean geometry 
does not deal with any of the eidoi at all.  But, Euclid does deal with eidoi, 
and he approaches his geometric eidoi in the same way that Plato approaches 
geometric eidoi as he understands them.  A puzzle thus arises in that 
Euclidean geometry seems both impossible and necessary according to Platonic 
philosophy.
 
	This puzzle is not beyond solution.  The reason that it arises at all 
is a result of the unusual nature of geometric eidoi (which was discussed 
earlier).  Geometric eidoi are not inexpressible in the apparent world.  The 
representation of the eidos which one perceives in the apparent world may not 
itself be absolutely real (only the eidos is), but the nature of the 
representation is such that one can disengage the eidos from the example(31). 
In this way, the source of the puzzle is also that of the solution. According 
to a strict interpretation of Platonic philosophy, one cannot attain a 
knowledge of the eidoi by examining the apparent world(32). Instead, only 
deductive reasoning is possible.
 
	The same is true of Euclidean geometry.  Induction is not used as a 
method of reasoning about the nature of geometric eidoi, one can only reason 
deductively.  Although this similarity does exist between Euclidean geometry 
and Platonic philosophy, the reasoning behind each system's conclusion is 
different.  Plato relies on the notion of intrinsic knowledge.  One knows the 
eidoi solely by being a human being(33).  Thus, the foundations for deduction 
are established, and one can thereby attain a knowledge of what is ultimately 
real.  This requires the support of two substantive assumptions, namely that 
the soul is the essential characteristic of "human-ness" and that the soul can 
carry a knowledge of the eidoi into the apparent world(34).  Without these two 
assumptions, Plato's epistemology falls apart, and with it the rest of his 
system.
 
	Euclid does not have to make such bold assumptions in his system of 
geometry.  Since he is creating it (or at least collating what his 
predecessors had created or discovered), he does not need to assume that these 
geometric eidoi exist as ultimately real, nor does he have to explore their 
relationship to the apparent world.  Instead he can base his system on the 
assumption that a point exists, despite the fact that it has no "part." 
Whether it does or not is of little importance.  For, his system does not 
necessarily exist to explain the universe, despite the fact that it can be 
applied to the universe with a high degree of utility.  Euclid's primary goal 
was to construct a system which was internally consistent.  As a result of the 
nature of his first assumption (the nature of "point," definition 1(35)), 
namely that it does not necessarily have to be true in order for his system to 
be internally consistent, Euclid only has to establish that deduction is an 
effective means to laying out his system.  His proofs demonstrate that it is.
 
	Induction, as in Platonic philosophy, is useless in Euclid's system. 
Since he does not use the apparent world as the foundation for his system, he 
does not need to deal with that type of reasoning(36).  Euclid instead deals 
solely with universal principles, as opposed to particulars(37).  He details 
the laws of geometry in his Elements.  This does not mean that one cannot 
apply these laws to particular instances in the apparent world.  Euclid, 
though, does not provide numbers, lengths, and degree measures in his 
definitions (and also his proofs)(38).  This maintains the universal nature of 
these claims, while saving their applicability to the apparent world.  This 
applicability is the source of the importance of Euclid's system of geometry. 
It constitutes the means by which one can attain an understanding of the eidoi 
while working with entities in the apparent world.
 
	The fact that Euclid's universal laws can be applied to the apparent 
world constitutes a crucial similarity to Plato's eidoi.  Beauty, Truth, and 
Justice are all reflected in the apparent world.  Thus, it should not be 
unusual for Circle, Triangle and Line to exist in a manner similar to Beauty, 
Truth, and Justice.  They, too, are eidoi which are reflected in the apparent 
world, and it is the soul's knowledge of the eidoi (both geometric and 
otherwise) which provides human beings with the means to recognize the 
apparent world manifestations of the eidoi.  The soul is the source of all 
recognition.
 
	The essential difference between Platonic eidoi and Euclidean eidoi is 
that Platonic eidoi are similar to a one-way street.  One can only go from a 
knowledge of the eidoi to a manifestations of those eidoi in the apparent 
world; one must start with the eidoi (the converse would be induction).  One 
needs a knowledge of the eidoi before the apparent world will make any sense. 
A common argument against this view is the one on which inductive reasoning is 
built.  From a gathering of particulars, one can attain a knowledge of the 
general notion behind it, according to one who reasons inductively.  After 
looking at thirty tables, for example, one knows what a table is.  But, the 
flaw in this counter-argument to Platonic philosophy is that one may gather 
these particulars, but one cannot know of what they are particulars.  For, he 
will have no knowledge that there even are tables, let alone the knowledge to 
go look for them.
 
	Euclidean geometry does not exhibit this problem.  The inherently 
known eidoi are not necessary for Euclid's system.  With his definitions, he 
essentially invents his own fundamental notions.  But, when one explores the 
nature of the relationships shared by the various eidoi, in his system, one 
finds that they do correspond to the universe.  There are triangles in the 
universe, for example (one can verify this fact empirically), and the sum of 
the interior angles of each of these triangles is 180 degrees.  The reason 
that the sum of the interior angles is 180 degrees is that otherwise it would 
not be a triangle.  Unlike Plato's eidoi, those of Euclid do resemble a 
two-way street (though they do avoid the problems of induction).  One can go 
back and forth between the apparent world and that of the eidoi.  Knowing the 
definition of a rectangle (by means of print which is in the apparent world), 
one can identify rectangles in the apparent world.  The definitition itself is 
a universal claim (representative of an eidos), but one acquires a knowledge 
of it (in this case) by examining the apparent world (in this case, something 
in the apparent world which is shaped like a rectangle).
 
	More important than the above notions of identification and discovery 
is the actual process which occurs when one works with geometric eidoi.  Given 
the definition, one deals with examples in the apparent world; one applies the 
definition.  But, while working with the example of the eidos in the apparent 
world, this person must constantly refer to the essential qualities of (for 
example) a triangle in order to use the right rules and properties in regards 
to the apparent world and the eidoi.
 
	Euclid's "solution" to the puzzle which results from Plato's theory of 
the eidoi is, for the most part, an effective completion of a somewhat limited 
philosophical position.  But, Euclid's system does not constitute a complete 
solution to the given puzzle, even in regards to the parts which he appears to 
solve.  The primary concern is that although Euclid's system works, it works 
only for Euclid's system.  It is internally consistent, but it does not 
completely accommodate itself to the world of the eidoi.  Circles and 
triangles are not of exactly the same nature (from the perspective of human 
beings) as Justice and Truth(39).
 
	Geometric eidoi are defined in terms which are comprehensible to human 
beings because they do not rely on examples to provide a definition.  Most 
definitions of Justice and Beauty tend toward example more than toward actual 
definition.  Others fall into the problem of ambiguous terminology.  The 
definitions themselves contain terms which need to be explained, and what 
appears to be an infinite regress of definition results.  Euclid's geometric 
eidoi do not have this problem because the definitions are not concrete.  They 
are finite and dependent on fundamental definitions beyond which one cannot 
regress, for example the definition of a point(40).  The words used in this 
definition are so basic that they do not need to be defined (Euclid assumes). 
"Part," the essential term in this definition of "point," does not need 
further explanation, Euclid presupposes.  It is a commonly used and understood 
term, and Euclid (believes that he) does not deviate from the common usage.
 
	The other significant problem with Euclid's "solution" lies in the 
definitions themselves.  Many do not provide enough information to serve 
adequately as a definition.  The definition of a point will again serve as an 
effective example.  Euclid defines a point as that which has no part.  But, he 
does not say what it does have, or what it is at all.  By describing what 
something is not, one does not provide a definition of that entity.  Instead, 
one simply draws a distinction.  The result is that Euclid's system is not 
based on a definition at all.  An unusual fundamental assumption is first 
required,but this assumption has nothing to do with the first definition.  In 
fact, definition 2 is the first essential definition in the Elements(41).  It 
is here that the most demanding assumption occurs.
 
	One must assume in the second definition (without any indication from 
definition 1) that the nature of a point is such that when put together to 
form a line, length will result without also resulting in width.  The notion 
of breadthless length is not the problem.  The concern is "length" itself. For 
something to have length, it must have some type of "part."  A line is 
composed of and determined by points, and points (by definition) have no 
"part."  Length, it would see, relies on "part."  Thus, one is forced to 
assume that points, when gathered together to create a line, acquire or 
exhibit some properties which they normally do not possess or display.  This 
is a bold assumption to make, but the nature of Euclid's system demands it. 
Once this assumption is made, though, the rest of the system works flawlessly.
 
	It may seem that, throughout this article, the argument presented has 
come dangerously close to contradiction.  This concern must be addressed so 
that the reader does not misinterpret the argument and see a contradiction 
where one does not actually exist.  Euclid's geometric eidoi do not actually 
exist in the apparent world; they are defined in the apparent world, though, 
by means of language, and examples of these eidoi can be constructed in the 
apparent world if one uses the verbal definitions as guides.  But, despite the 
fact that they are defined verbally (human language being a facet of the 
apparent world, not the world of the eidoi),the perfect triangle, perfect 
circle, etc. do not and cannot exist in the apparent world.  But, as a result 
of the fact that they are defined in the apparent world, once an discuss and 
work with these eidoi in the apparent world.  By means of language and 
diagrams, one can provide examples of the eidoi which can be explained in such 
a way that the observer will be able to mentally disengage the eidos from the 
apparent world representation of the eidos.
 
	In order to completely understand what is meant by eidos (especially 
as Euclid discusses it in relation to Plato's philosophy), one must note the 
relationship between the verbal definition, the drawing, and the eidos itself. 
 There must, in fact, be a relationship, otherwise the system described by 
Plato (and subsequently by Euclid) is meaningless.  Geometric eidoi, as 
intended by Euclid, consist of verbal definitions, for example those provided 
by Euclid himself in the Elements which contain the essential characteristics 
of what it means (again, for example) to be a line.  This verbal definition 
can be employed to construct a visual (pictorial) example of a given eidos.  
The pictorial representation relies on the verbal definition for its 
foundation (that of which it constitutes an example), but it is not as 
accurate as the verbal definition.  The verbal definition does not constitute 
the eidos itself, though it is closer to it than the picture.  The eidos is an 
idea which is the foundation of the verbal definition. Essentially, a 
particular eidos is the starting point for all examples of that particular 
eidos.  It is that to which particulars can be traced in order to find out 
what exactly they are (for example, what it means to be a line).
 
	Plato's theory of the eidoi, though internally consistent, does have 
certain flaws.  The eidoi are not inconceivable, but they are difficult (if 
not impossible) to discuss and express in the apparent world.  One knows the 
eidoi through the soul (which is the "connection" between the apparent world 
and that of the eidoi), enabling one to think about the eidoi.  But, 
discussing the eidoi entails trying to carry them into the apparent world, 
where human language exists.  Since Plato considers the apparent world, to be 
tainted, and thus unable to accommodate the eidoi, language will always fail 
those who wish to discuss the eidoi.  To this problem, Euclid (though 
inadvertently) proposes a solution, which is helpful but not complete.
 
	Euclid is able to bring some eidoi into the apparent world, using a 
combination of language and image (drawing, picture, etc.).  Since, for 
example, a drawing of a circle is not perfect, one can note the imperfections 
by consulting the verbal definition.  By producing certain eidoi which are 
defined by language (in conjunction with the image, or imperfect picture), 
Euclid devises a new group of eidoi which extend into the apparent world 
(unlike those of Plato).  But, this does not completely solve Plato's problem. 
 Euclid's solution only works for some eidoi, namely geometric eidoi. Thus, 
one can see how one can understand the eidoi in general, but one is still left 
without a method of discussing particular eidoi such as Truth and Beauty.  
Essentially, Euclid's solution works for Euclid's system, but it does 
nonetheless provide significant insight into the Platonic notion of eidos.
 
Notes:
______
 
1.  See the diagram on the given page.  "Perfect beauty, justice, etc. are 
discussed.  Perfect, in this context, implies complete, universal. Plato.  
"The Republic."  Great Dialogues of Plato.  transl. W. H. D. Rouse (Mentor 
1984), p. 309.
 
2.  Euclid.  Elements, vol. 1.  transl. Sir Thomas L. Heath (Dover 1956), pp. 
183-5.
 
3.  When "Form" is capitalized in this article, it constitutes a reference 
specifically to its equivalent in ancient Greek, eidos.  Most often, though, 
the Greek itself will be used in order to preserve the full meaning of the 
term by Plato.
 
4.  "[T]here is nothing it [the soul] has not learnt." Plato.  "Meno."  Great 
Dialogues of Plato.  transl. W. H. D. Rouse (Mentor 1984), p. 47.
 
5.  This problem arises, for example, in both the Parmenides and the Sophist. 
The participants in the dialogues cannot adequately discuss "not-being" (for 
example, Parmenides) because, by discussing "not-being," they are giving it 
being.  Thus, it is no longer "not-being," since it has being. Plato.  
"Parmenides."  Plato:  Cratylus, Parmenides, Greater Hippias, Lesser Hippias.  
transl. H. N. Fowler (Loeb Classical Library 1992), p. 253.
 
6.  "Looks" is a misleading term.  What is meant is a comprehension 
(rationally) of what makes a circle a circle.  Although one actually sees 
nothing, people more readily associate with words which connote appearance or 
image as opposed to those that are grounded solely in the mind.
 
7.  Pythagoras and Parmenides are among them.
 
8.  Republic, pp. 313-5.
 
9.  It is not exact, but with one's knowledge of the definition, one can see 
that, when errors arise in calculation or comparison, it must be a result of 
the fact that one is working with an example of the eidos, not the eidos 
itself.
 
10.  Although the word "line" is used here metaphorically, one should note 
that this "fine line" along which one works is basically the same line which 
Plato uses in the Republic (p. 306), the divided line.
 
11.  Republic, pp. 313-5.
 
12.  Republic, p. 309.
 
13.  The only pre-Socratic philosopher who dealt with Form in a way which is 
at all similar to that of Plato is Pythagoras.  But, Pythagoras only dealt 
with one eidos, number.
 
14.  One should note that geometry was used in this incident (from the Meno), 
further emphasizing my point.  Discussion of, and dealings with, the eidoi 
require geometry as a medium.
 
15.  Plato.  "Sophist."  A Plato Reader.  ed. Ronald B. Levinson.  transl. 
Benjamin Jowett (Houghton Mifflin Company 1967), p. 428.
 
16.  Ibid.
 
17.  Parmenides, p. 253.
 
18.  "World" is put in quotation marks because the Forms are not actually in 
another world, even though that is a common misinterpretation.  The fact that 
one's ideas about the eidoi are expressed in the apparent world serves as a 
counterexample to the notion that the eidoi have a world of their own. 19.  
This, in fact, was done by Euclid.
 
20.  As stated earlier, hints as to the "solution" are contained in the 
dialogues (for example, the Meno).  But, Euclid explains this role of geometry 
in the universe.
 
21.  Definitions tell what the essential characteristics of something are, for 
example, a what makes a table a table (or a circle a circle).  Therefore, in 
geometry, one creates (things and ideas) in the apparent world based on one's 
knowledge of the pertinent eidoi.
 
22.  Euclid, p. 187.
 
23.  Euclid.  Elements, vol. 2.  transl. Sir Thomas L. Heath (Dover 1956).
 
24.  Euclid (vol. 1), p. 155.
 
25.  Euclid (vol. 1), p. 165.
 
26.  Euclid (vol. 1), p. 165.
 
27.  A demand like this (on the part of Euclid) is no different from the 
initial assumptions which are demanded in most philosophical systems.
 
28.  Euclid (vol. 1), p. 277.
 
29.  Republic, p. 315.
 
30.  It is important to remember that Euclidean geometry must be viewed in the 
context of Platonic philosophy, because both rely on many of the same 
fundamental presuppositions.  Also, to a certain extent, Euclidean geometry 
relies on Platonic metaphysics itself.
 
31.  The example is that which one encounters in the apparent world; it is an 
example of the eidos.
 
32.  This, of course, destroys the foundations of induction as a method of 
reasoning in the Platonic system.
 
33.  Since the soul is what makes a human being a human being, and knowledge 
of the eidoi is contained in the soul, it follows that a knowledge of the 
eidoi is part of what makes a human being a human being.
 
34.  This raises a problem similar to that raised by geometry.  There seems to 
be almost a dual existence to the soul.  It exists in both the world of the 
eidoi and in the apparent world.  But, this problem is not as easy to 
reconcile as that of geometry.
 
35.  Euclid (vol. 1), p. 155.
 
36.  This is because induction is generalizing from particulars, while 
deduction begins with universal claims.
 
37.  He does deal with particular examples in his proofs, but not in such a 
way as to rob them of their usefulness as universals (nearly synonymous with 
eidos).
 
38.  The only possible exception to this rule would be Euclid's dealings with 
right angles, which by definition, have a specific measure (although Euclid 
does not provide a numerical representation for this measure). Euclid (vol. 
1), p. 181.
 
39.  Geometric eidoi cannot be interpreted in a relative manner (i. e. 
relatively) in the apparent world, unlike Justice and Truth.
 
40.  This is Definition 1. Euclid (vol. 1), p. 155.
 
41.  Definition 2 reads as follows: "A line is breadthless length." Euclid 
(vol. 1), p. 158.