The Evolution and Discovery of the Species of Equality in Euclid s Elements

Transcription

1 From the SelectedWorks of Lee T Nutini 2010 The Evolution and Discovery of the Species of Equality in Euclid s Elements Lee T Nutini Available at:

2 Nutini 1 The Evolution and Discovery of the Species of Equality in Euclid s Elements by Lee Thomas Nutini Mr. Jeffrey Smith Fall Tutorial November 1, 2010 The Graduate Institute St. John s College Annapolis, Maryland

3 Nutini 2 Introduction On its face, the term equality seems pretty absolute and static within our general usage: things that are equal are cleanly equal there s nothing more to be said about them. There don t seem to be degrees of equality, and most of us consider equality to be awfully easy to prove using the idea of number. In fact, political entities will often translate social inequalities numerically so that government statisticians may help to balance the scales, so to speak. This common example helps to illuminate the precise reason why Euclid s usage of equality is so intriguing. Equality seems extremely hard to achieve without some sort of numerical assistance, yet not one number appears in Euclid s first Book of the Elements. So, how does he accomplish such a feat, much less an entire evolution of the same idea, in what seems to be a series of logically sound propositions? Equality as presented in the Definitions The first thing to notice about Euclid s usage of equality is that he does in fact move it from the quantitative world to the qualitative world, or to mental conception exclusively. Interestingly enough, one can note how the equality of which Euclid speaks is something one must see with the mind s eye. To begin, let s take a look at Euclid s first usage of the term equal : Definition 10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. In this instance, the equality is of two angles, which references the inclinations of two lines to one another. For Definition 10, Euclid wants us to imagine the qualitative equivalencies in the inclinations: in other words, he wants us to see how the two inclinations in question are one and the same in their spatial presentation. The two equal angles could be drawn any

4 Nutini 3 size, as long as the quality of the space (i.e. the inclination) is exactly the same. This usage of equality is mimicked in fellow definitions 20 and 22. The only other instance within Book I s definitions occurs in 15, when Euclid provides that the circle s diameters (or distances) are all straight lines that are equal. Now, if we were to organize the variations of Euclidean equality into species (much as Darwin did), we might see how and why their differences matter. Definition 15 s example does not include angles, as we previously discussed, but it does include the supposed length of straight lines being equal. Lines are nothing more than their length, since they have no breadth, so a circle s diameters must cover the same length in space, regardless of size. If we were to attempt at a beginning of categorizing the various usages we find into distinct species of use, what might we say about these two types? Are they two examples of the same idea of equality? It seems that they might be. The angle example(s) deal with a sort of spatial inclination that would cause them to overlap or coincide with one another, provided they are actually adjacent angles, formed to be the same size. The lines that are so inclined will fall on one another if we were to superimpose them with our mind. Wouldn t the straight lines that make up a circle s diameter respond in the same way? If we imagine the circle as being a series of equal length radii fanned out, and then we collapsed that fan, the lengths would fall upon one another in the same coincidence as the angles in the previous example. Looking at them this way, we can clearly see that these two examples are both tokens of the species of equality we may call coincidence equality. What other types of equality exist in geometry? Equality as presented in the Postulates and Common Notions If we move beyond Euclid s definitions in Book I into his postulates, we find only one more usage. In Postulate 4, he posits that all right angles are equal, and this can be easily placed into our former category of coincidental equality. The postulate simply states that all

5 Nutini 4 angles of the same degree of inclination will be equal in their degree, which is basically a tautology. Now, if we take a look at Euclid s other given premises, the Common Notions, several characteristics of equality are enumerated. We find in CN 1 that things which equal the same thing also equal one another. This statement doesn t really tell us much about what equality is, except that it is transferable in the sense that the idea of being equal can be held by different placemarkers in the conceptual or actual world. Put simply, this Common Notion is telling us that an equivalency can have many faces, and those faces change nothing about their inherent equality. We may ask later what it is that is actually equal about two seemingly disparate things (e.g. two sides of an equation). Moreover, we find here that things can be equivalent based upon their equivalencies with other things: the equality of two or more things is a characteristic over and above what the things actually are (i.e. their appearances). In CN 2 and 3, we see that equal things maintain their equality through mathematical manipulation. Again, this tells us little about the question of varying species of equality existing within Euclid s geometrical world. In Common Notion 4, we are presented with the term coincide, which had previously designated as a type of equality: Things which coincide with one another equal one another. This seems to be precisely what we had discovered above that overlap or superposition gives us our most elementary form of two things being equal. We can now say with much assurance that, within the Definitions, Postulates, and Common Notions that are foundational to Euclid s geometry, we are presented with only one species of equality: coincidental equality. If we encounter new species of equality as we traverse the Propositions of Book I, it would truly be a phenomenal development due to the fact that a new type of equality would have to arise from foundational rules that seem to allow only for equality by coincidence.

6 Nutini 5 Equality within the Propositions While the term equal is used frequently throughout Book I s propositions, there are a few that ask us to question the nature of equality and how Euclid uses it to manipulate space. The first interesting use occurs within Proposition 4, when Euclid applies a triangle to another triangle in order to cause the points and sides to coincide. This is the superposition to which we referred above, in which we are asked to mentally move pieces of space onto one another in order to see how they align. What is most interesting about I.4 is that we are moving an entire figure, rather than just angles or lines. In this instance, we are using coincidence equality to prove that two entire figures are equal but in what sense are they equal? Euclid s proof allows us to know that two triangles sides and angles align, but how does that account for the entire space demarcated by the figure s boundaries? He is showing us, somehow, that the internal space that is bounded by the various inclinations of the equal angles will also be equal in the two triangles. Similarly, in I.8, Euclid proves that two triangles that are made up of three sides that are equal respective to one another will also have equal sides. The question remains whether this is nothing more than the equality of three angles or sides, or if something more can be said about the space they enclose. I feel it is important to notice that Euclid does not say the triangles are equal, but rather refers only to the angles and sides. Prior to Proposition 34, Euclid s usage of equality is mostly uninteresting to our current project: they deal mostly with manipulations of angles and lines that can coincide, without dealing much with enclosed space. However, I. 34 unveils an entirely new concept in reference to equality it is the first time that we are supposed to conceptualize a figure s area. Euclid proposes parallelogrammic figures that contain equal sides and angles (something we ve seen before), but also that the figure s area can be bisected by the

7 Nutini 6 diameter. It is extremely interesting that he changes his language here: Euclid hopes to say something about the space enclosed within a figure without referencing it in terms of the figure s angles and lines. This is a novel idea for his reader. What s more, within I.34 and I.40, Euclid begins saying that the figures themselves are the equal things, not just their constituent lines and angles. Both triangles and parallelograms are shown to have equality in themselves, and this is presumably because of the space that they enclose their area. In addition, the equality that is being manipulated in these propositions is within the heights of parallels, which provides a spatial area that may be enclosed in different ways by different figures (parallelograms or triangles). This series of propositions shows us that area is not the same as figure that different figures may even have the same area. Thus, a new species of equality has been discovered over and above the coincidence of mere angles and lines. In fact, we will come to see that coincidence need not ever be present for two figures to be equal in area. In Proposition 41, Euclid smartly utilizes the concept of area to make a remarkable shift in the power of equality. He begins to speak of triangles as whole figures being equal as whole figures due to what we previously called coincidental equality. What this means is that the coincidental equality referred to as early as I.4 and I.8 actually contained within them the necessity of two figures areas being equal as well. In I.41, we find that a triangle and a parallelogram may coincidentally share a base within the limits set forth by the height of parallels and have areas that we can define relative to one another (i.e. the parallelogram being double the triangle). In a much more sophisticated move, Euclid proves in I.42 that two types of figures may be equal in area, and moreover, that they have nothing coincidental about them! This truly breaks the mold for our conceptions of equality Euclid has demonstrated that it is not only coincidence that matters in the world of equality, but that

8 Nutini 7 space itself is something that can be contained in equivalent ways. For example, in I.42, we can be assured that the triangle and the parallelogram are equal only because they contain the same amount of space. Proposition 45 goes one step further in its power over manipulating areas: Euclid demonstrates how we might transform any rectilineal figure into a parallelogram with an equal area. This seems like the broadest reach we might need to understand two-dimensional space to convert any rectilineal area into other shapes with the same area. An important question for us to ask is whether or not this definition of equality via spatial area is any different from the coincidental equality to which we previously referred. Are they really two distinct species of equality? Is it a difference that really makes a difference? In the struggle to understand the two species of equality, we might want to explore how each operates on its equal geometrical entities. Within coincidental equality, we know that lines are equal because they have the same length; that angles are equal because they hold the same inclination; and that two figures are equal because they share both angles and lines. However, an interesting question can be asked in reference to the equality of angles: if an angle is formed by two lines inclination to one another, and it is this space that is equal, how long do the lines have to be that make up the angle? I would assume that the easy answer to this question is that they must be of the most minimal length, and that they do not have to be of equal length to hold an equal inclination. But what is the minimum length of a line? Focusing on this question makes it difficult to wrap your head around two angles being equal : the lines, which actually form the angle and present it to our senses, do not have to have anything in common except that their inclination be the same toward one another. The thing we are calling equal, then, is nothing more than an idea of the space in between the lines. Does this not seem like an idea of area, and not just coincidence? The area between

9 Nutini 8 two lines presents the degree of inclination (the angle) to our minds, but maybe this is not merely area that we perceive. We perceive quantitatively and qualitatively much more than contained area we perceive uncontained area that could exist infinitely as long as it holds its inclination. The only thing that limits the reach of the angle s area is a third line, which forms an enclosed figure a triangle. An angle rooted in an intersection of two lines does not have an ending unless it meets another linear intersection: thus, the angle qua angle can exist as a limitless extension of its defined inclination. So here in the case of the angle we have a species of equality by coincidence of inclination, but also a variant of the species of area that of undefined area. This area is quite unlike the area found later in Book I. That is to say, this area is not necessarily bounded or pinned to a certain size. So, two angles being coincidentally equal means that their inclination will always overlap, and that is all that is necessary. But it is when we begin to speak of enclosed figures like triangles, for example which actually convert coincidental equality into equality of area. It seems universally true that, in the case of enclosed figures, coincidental equality is always going to necessitate equality of area. This is why we do not need to question Euclid s shift in language later in Book I when he begins to say that two triangles are equal as figures (because of I.4 or I.8), rather than simply the constituent angles and lines are coincidentally equal. So, to summarize the cooperation of the species of equality in Book I, we might say that coincidental equality can stand alone as the most primal species because it is the only way to describe geometrical entities like lines and angles. However, when it comes to enclosed space, coincidental equality necessitates equality of area due to the inherent demarcations of those equal angles and lines. It is important to note how angles hold within them a variant of the species of area in addition to their coincidental inclination that of an undefined, possibly-infinite area being contained within the angle s reach. It is equally

10 Nutini 9 important to note that equality of area never necessitates coincidence which is why Euclid shows us that various rectilineal figures are translatable within parallels to parallelograms and triangles. Dealing with space solely in terms of area provides the geometer much greater reach in manipulating figures in that space. It seems that the difference between the species of equality is a difference that only becomes crucial when translating the idea of a particular area between two different types of figures that is, in the instances when equality of area can stand alone as sufficient proof of equality. The special case of Proposition 47 A new mutation? It seems as though a mutation occurs within the species of area near the end of Book I. Proposition 47 is unique in the way it treats its given figure. Unlike the rest of Book I, it does not refer to the coincidence of certain angles or lines, nor does it refer to the equality of the area of enclosed figures, per se. It begins to ask after the importance of as yet undefined space that surrounds a known enclosed figure! Because of I.47, we may know definitive information about the relation of areas that exist external to a figure (e.g. a right triangle, as in the proof). This proposition proposes a special case indeed because it wants to describe space that we could define, but currently has no definition or markings. The beauty of this special case lies in the fact that we are not constructing figures equal to given figures, but rather discovering areas that exist potentially around a given right-angled triangle. We come to know their area in relation to one another, and these areas have their foundation within the lengths of the triangle s sides. It is quite wondrous that we might know how to reach beyond our figures toward space, enclose it, and know its value in relation to the figures we have already come to know. Is this a new type of equality? If anything, it is quite similar to the variant we discovered within the formation of an angle. The things being equal in both instances are currently undefined and unbounded, and we may know them for certain by

11 Nutini 10 merely bounding them and setting them in relation to one another. Either way, we are given the power to reach beyond the figures we might see on paper in two-dimensional form. We are able to use our mind s eye to see and to know the space that surrounds what is already known, precisely due to what is already known. The angle sets up its inclination that we may extend to the horizon, and the right triangle contains within it the tools to know its surrounding spatial environment. Concluding Remarks Throughout Book I, Euclid displays a logical evolution of the idea of equality. Beginning with the Definitions, Postulates, and Common Notions, we find that coincidental equality is the most primal way to discover geometrical equality. However, it soon becomes clear that enclosed spaces require further attention. The space that becomes contained by a figure s angles and lines being coincidentally equal remains equal as well. For any figure, coincidental equality as a species will also necessitate an equality of area, though the converse is not true (interesting, because converses of true statements are generally also true in geometry). Equality of area can exist independently of coincidental equality because Euclid shows his reader that rectilinear figures can always be separated into triangles, which can then be placed into parallelograms. In instances like these, the idea of particular area can be shared by differing figures. Besides these two species, we discovered a fascinating variant or mutation to the species of area: equality of as yet undefined area. This mutation arises when one considers the particular space marked by an angle s inclination, or in the instance of I.47, in which we can access knowledge about a figure s surrounding geometrical environment. It seems especially important to note that such access could be helpful in discovering the nature of an entire spatial universe to defining an entire dimension. The way that coincidence equality assists us in knowing area equality, and that such equality of area can be

12 Nutini 11 translated to easily-accessible grids, like systems of parallels, seems invaluable to any effort to map our spatial universe.