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lead summary, then a compact treatment of the topic, then a the more expansive related details. This argument would suggest that the history section for technical reference articles (and this would include essentially all the science and maths articles), a history section should almost never be near the top, since it is undoubtedly secondary (i.e. supportive) and not the primary content of the article. My own feeling is that it would generally best placed at the end of the article, at the start of the footer sections. In some sense it is natural to group it with the references and notes. To place it first is not a good idea to me: it makes accessing of the real meat of the article more clumsy, as one would have to skip over it every time the article is opened. â
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typical reader as the high school student or layperson who quickly wants to check facts about vectors, to learn or to refresh their memories. I do not see the bulk of those who go beyond the lead as wanting primarily the history section. Nevertheless, I am glad to see you are arguing on the basis of usage, not chronological order, and not still on typical section order. I have no strong opinion on the matter for this article, and was merely trying to suggest a "history first" general pattern as the norm would not be ideal. â
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emphasize quaternions, with vectors an afterthought at best. (I've now re-removed one of these paragraphs, a comparison of quaternion and complex multiplication.) Three such paragraphs remain. If you'd like to keep them, please rewrite them so that they are about vectors, not quaternions. (Or move content over to the quaternion article as appropriate.) Right now the section is extremely misfocused and misleading. --
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860:. When most people consider geometric vectors, e.g., in mechanics, they are usually not thinking of the "element of a Euclidean space" viewpoint, but rather are thinking of a vector in the sense described in this article: a directed line segment in a (naive) Euclidean space. It might be worth having more discussion somewhere to disambiguate the naive vectors described here and the elements of a
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It depends on whether most people come to technical articles to learn the subject, or to learn about the subject. A layperson who wants to learn about
Euclidean vectors will appreciate a little historical context. A student who wants to actually learn the rules of manipulating Euclidean vectors can
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That the definition of subtraction should be based on the definitions of opposite and of vector addition is indeed the case, in my opinion. And in that way a reader who is new to the topic will understand how subtraction of vectors is defined, though it would still be better if an illustration would
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I don't get your point. A more general tangent is not necessarily a vector. The tangent to a curved surface for example. But the two examples given, of a scalar valued function of position (a scalar field) and a curve parameterised by a scalar, both have well defined vector valued derivatives, even
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It is not that big a deal either way, because the history section, like any other section, is just a click away. But I doubt many people come here looking for "real meat". A layperson wants a general idea about a subject, without any technical details. And a mathematician, scientist, or engineer
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In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, like the co-ordinates, would reduce in an exactly
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Being placed at the start may simply reflect a liking for the sequence to match the chronology â after all, the history is what led up to what the current state of the discipline is. Yet, the start-at-the-top format of WP suggests that the content be in the order most useful for access: first the
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Yes, of course the history of a topic is important in an encyclopedia, and it's always nice to see historical information in math articles. But we have here a history section that is the history of a different topic than the subject of the article: there were 4 out of 5 or 6 paragraphs written to
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matrix and pointing out that the resulting automorphism group elegantly links all bases in a way that makes the concrete concept basis-independent. One can then ask whether there might be an even neater approach to basis independence, which then leads naturally to the notion of an abstract vector
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The history of vectors is quite convoluted. Many of the important features of vectors, like the dot and cross products and the del operator were arrived at through studying quaternions. There was also a parallel development following
Grassmann which is closer to what we would recognise as vector,
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The history of vectors focuses entirely on quaternions. A brief mention of quaternions is fine, but the section simply describes the history and properties of quaternions and leaves out the history of vectors entirely. The section obviously does not satisfy quality standards and if an experienced
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Even to the layperson, the history is still only context, not the core content, and such person can just as "easily skip to" the history section. It is the student who will be repeatedly accessing the article, not the layperson, so that "skipping" adds up. In this particular article, I see the
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means the opposite. The set of basis vectors is covariant (by definition), and the vector of coordinate scalars that is multiplied by the basis to get an invariant vector is contravariant. The wording in the article is confusing and should be fixed, but the meaning must not be switched around.
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Right now, this article has an excellent introduction, followed by an overview section that mostly repeats the same content, but with less clarity and a variety of issues (like the idea that "an arrow" is the definition). I suggest simply removing the "overview" part of the first section (I.e.,
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The problem with starting with the abstract definition is that it comes with no intuition. The point that is often lost is that concrete vector spaces are still vector spaces, despite not being defined equationally. It is enough for an object merely to satisfy the equations for a vector space,
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The same should apply to vector spaces, with the parallels being strikingly clear when one considers that a vector space over GF(2), when equipped with a second constant 1 as the complement of the origin 0, is equivalent to a
Boolean algebra via the evident translations in each direction between
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Consider the intelligent layperson who hears the word "vector" and wants to know what it means. To say that a vector is an element in a vector space is not helpful. I've been trying to find a good definition that will include vectors over an arbitrary field of scalars, and still be something a
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I disagree. We are an encyclopaedia not a text book so we need to cover a history of the subject. To properly discuss how the concept of vectors came about we need to discuss what came before, complex numbers and quaternions were important precursors. Also note this section shows where the word
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I agree with JBL. Your text would make sense in a context where operations on vectors would be defined geometrically (that is coordinate free). As the choice here is to start from coordinates, one has not to prove again the properties; one has only to show that the geometrical definition is
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The description given now is extremely simple. Your additions complicated it by tying the definition of subtraction to the definition of opposite and of vector addition. But that is not necessary, and doing so diluted the point of the paragraph (which is to define vector subtraction).
802:. In this paper, it is shown that it is necessary to separate the vectors into rectilinear and angular vectors. We introduce the concept of an inverse vector, which allows vector division operations. I hope that after reading, do not remain indifferent and help spread this article.
675:(necessarily plural to avoid confusion) initially ignores the axioms and begins with concrete Boolean algebras (a) because they arise naturally and (b) to make the point that one can speak about at least the concrete kind without reference to any axiomatic definition of the concept.
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In order to enable a larger group of readers to understand what is happening in the 'explanation' of the subtraction of vectors, I added extra explanations. Obviously I disagree with the removal of my additions. As it stands now, I think that too many readers won't understand it.
897:. Then the mathematicians took over and completely messed up with both the scope and the title of the article. :-) Now it's about any three-dimensional vector space over the real numbers with a positive-definite inner product, regardless of its relationships to physical space.
868:. A perusal of the archive shows that there is substantial confusion over what the scope of this article is, with formalists often trying to impose the "rigorous" definition (which is not even mathematically the same notion that the rest of the article is talking about).
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seem to have been written without reference to one another. A good goal would be to have all of these articles agree in terminology and style, and this article seems to be the place to start. There are probably other articles that should also be included in this project.
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The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the
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I don't have any problem with an article titled "Euclidean vector". My problem is with the lack of an article titled "vector (mathematics)". I haven't checked, but I suspect every mathematical encyclopedia has such an article. For example, this article at MathWorld
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I think there ought to be an article about vector spaces in general, and one about vectors as used in classical mechanics and engineering with a short mention about other kinds of vectors used in physics. The first is what is currently at
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is way too obvious and useless to be here. I don't mind if we remind readers that 1 + 1 = 2, but 0 + 1 = 1 is a little extreme. I mean I can't understand 90% of the mathematics on
Knowledge (XXG), and even I think this is too basic.
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had enabled analysis of two-dimensional space, but he arrived at a four-dimensional system. In 1846 Hamilton divided his quaternions into the sum of real and imaginary parts that he respectively called "scalar" and "vector":
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I disagree. The existence and behavior of the null vector is central to the notion of a vector space. Without it and it's admittedly trivial-seeming behavior that 0+a=a, you don't have a linear space, you have an
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layperson can understand. Something like "a vector is a mathematical object that has both magnitude and direction, though in abstract mathematics the concepts of magnitude and direction may also be abstract."
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Quite right. I've made everything "head"s and "tail"s in that section. (In general this article is something of a mishmash and needs someone to go through and sort it out. Not volunteering, though.)
695:. In both cases these are, up to isomorphism, the only finite/finite-dimensional such. (That the only non-free algebras here are some of the Boolean ones is an interesting but not central point.)
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would be a better title, and this article shouldnât imply that vectors can only describe
Euclidean situations. Geometric vectors need not even be metrical, e.g. displacement vectors in a generic
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The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions. The immediate predecessor of vectors were
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In general, a tangent vector is not the same thing as a vector. The section on viewing vectors as directional derivatives should clarify what is meant, in the context of
Euclidean space.
1773:, it is at the end, just before the footer sections. As such, I don't see that the placement in other articles should be used as a guide, but rather, a fresh motivations should be sought.
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of two vectors from the complete quaternion product. This approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth.
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I agree with
Quondum. Most people (both laypersons and others) are interested in the "real meat" and will be forced to skip the history section if it is placed at the beginning.
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It's hard to find an equationally defined class used in practice that wasn't originally motivated by its concrete instances. A Boolean algebra can be defined concretely as,
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For one thing, this is rather at odds with the way the article introduces vectors as directed line segments in the usual
Euclidean space (which is more properly speaking an
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This is absolutely correct. I've removed the section. Someone who wants to add a relevant history section is welcome to do so. The removed text is copied below. --
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The first thing to consider is whether the title "Euclidean vector" is the best title for this article, leaving no article on the more general subject "Vector".
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of a and b at the same point" or similar (this is what is illustrated). This subject is pretty fresh to me so I will leave it to somebody else to make the change.
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The benefit of the abstract definition, that it does not commit to a basis, can be had almost as well in the concrete case by defining an isomorphism of a concrete
590:. I think that's a good solution: After all, a vector (strictly speaking) is just an element of a vector space, so you can't really discuss one without the other.
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however at the time this was very marginal. It was not until the 1880's when Gibbs and
Heaviside both publish works which we would recognise as vector analysis.--
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in 1843 as a generalization of complex numbers. Initially, his search was for a formalism to enable the analysis of three-dimensional space in the same way that
2057:. In fact, in the current text there is not even the slightest attempt to explain the construction to the reader. The reader reads a statement and that's that.
1248:. Multiplication of two quaternions yields a third quaternion whose scalar part is the negative of the dot product and whose vector part is the cross product.
2037:. In the current version there is a drawing where a vector is constructed based on two other vectors, and then it is stated that this third vector is in fact
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learned the "real meat" in a college course, and isn't going to look for it in
Knowledge (XXG), except maybe as a reminder of something they have forgotten.
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the same. Also why, if they are both transformed by the forward transformation, is the need for an inverse to exist mentioned. Combined with the fact that
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all elements of all sets -- hence all vectors -- must be sets themselves, but the gravitational force acting on me right now in my frame of reference is
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I once even proposed to keep this article with its "new" scope and "new" title and to start another article which would then be the new counterpart to
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A further sampling of articles shows that the placement of the history section (when it is included) does not have a standard place. For example, in
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of b to the tip of a. That arrow represents the vector a â b, as illustrated below:". However the section on representations tells us that the
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the history section follows immediately after the Table of Contents. I've moved the history section in this article to follow that example.
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Why then does it say, just above that, that they "transform like the coordinates" and the give math transformation both the coordinates
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Acceptable, although I'd rather place it at the end of the article, because I assume people is more interested in everything else...
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538:, I think the scope and title of this article -- for physical vectors in 2 and 3 real-world dimensions -- are not badly chosen.
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Where should the history of a mathematical subject appear in a Knowledge (XXG) article? Checking three articles at random, in
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their respective languages. In particular, just as there is one finite concrete Boolean algebra 2 for each natural number
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The history, as we can see from the article, is only ONE LINE. Would someone help expand the histories? Thank you.
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Now the vector is not just one line. I suggest you look at the article "Angular vectors in the theory of vectors"
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953:. Plenty of times in advanced mathematics, seemingly trivial things are very important and need to be mentioned.
1307:, adapted from Gibb's lectures, and banishing any mention of quaternions in the development of vector calculus.
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has been demoted to "start class". Several people are trying to fix it. But this article and the articles
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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
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on Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
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It is worth noting that "direction" of a vector is sometimes "split" into "orientation" and "sense".
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easily skip to those rules (which are more technical than the non-student will care to read).
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I'm surprised this rather obvious mistake wasn't fixed sooner. I've gone ahead and changed it.
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1915:. For a non-singular curved surface the tangent plane will be spanned the tangent vectors.--
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seem to imply it should vary in the opposite way, but the description seems to say so to.
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The hatnote refers to vectors used in Physics, but in Relativity the vectors used are
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Does this section have contravariant and covariant backwards? Not only does the word
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I know Knowledge (XXG) is written for different readers, but this is just ridiculous.
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More formally, a Euclidean vector is any element of a Euclidean vector space, i.e. a
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I have removed the "formal definition" from the first paragraph of the article:
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Sometimes its easier to define what something is by defining what it isn't.
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equivalent with the algebraic definition. So, your addition is confusing.
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talk: Vector (mathematics and physics) #A CONCEPTDAB article is needed
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knowledgable editor doesn't revise it the section should be removed.
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then the set of tangent vectors is identical to the set of vector in
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begins with the finite axiomatization, but from the point of view of
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any model of the equational theory of the two-element Boolean algebra
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page give the contravariant transformation in terms of the inverse,
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The sum of the null vector with any vector a is a (that is, 0+a=a)
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whose square is negative one, quaternions have three independent
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of a and b at the same point, and then draw an arrow from the
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would be a better title. (Note also that physical vectors are
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1273:. Clifford simplified the quaternion study by isolating the
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are synonymous. I presume that it should read "place the
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carried the quaternion standard after Hamilton. His 1867
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Tangent space#Tangent vectors as directional derivatives
687:) is there one finite-dimensional concrete vector space
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It has been suggested that portions of this section be
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840:. A Euclidean vector space is automatically a type of
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Considering the length of the disambiguation pages at
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Knowledge (XXG) level-4 vital articles in Mathematics
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The article states: "to subtract b from a, place the
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before the subsection "examples in 1 dimension"). --
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London, Edinburgh & Dublin Philosophical Magazine
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Indeed: once upon a time this page was titled (IIRC)
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738:would be fine for it. For the latter, the current
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1659:WT: WikiProject Mathematics #âVectorâ redirects
1022:seems a good basis for extending the section.--
702:-dimensional vector space to be a non-singular
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1580:I've reverted this edit.
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333:Mathematics
324:mathematics
280:Mathematics
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70:Engineering
2099:Categories
1939:Lorentzian
1747:References
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1303:published
545:/Archive 5
492:linear map
1966:jacobolus
1956:I agree,
1767:Logarithm
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762:a set...
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610:(
594:(
571:(
551:(
516:(
475:.
364:.
259:.
154:.
53::
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