Elements of Geometry and Trigonometry from the Works of A.M. Legendre: Revised and Adapted to the Course of Mathematical Instruction in the United Sta Paperback – 25 February 2010

I'll speak first to the quality of the Forgotten Books reprint of this book.Except for four pages that were unfortunately missed in the scan (only one of any real importance), the content is the full text of Davies' 1834 edition and revision of Legendre's Geometry, in the A. S. Barnes and Co. printing of 1851, the last printing before Davies introduced a new edition that same year. I find blank pages at page 36, which would contain the first three propositions of Book II (a minor but real loss), and at the back of the book there are three pages missing that would contain tables of logarithmic sines and tangents for 1 degree, for 43 degrees, and for 45 degrees.The cover design of the book is as shown in the amazon photos. The print quality is equivalent to that of AMS Chelsea books. It is produced via "generation of a scan of the original" and there is no indication of previous ownership of the original text, such as library stamps or extraneous markings of any kind. It is as good a paperback as any other you might purchase, being glued, not sewn.Forgotten Books has the author on the cover as A. M. Legendre. This is not exactly inappropriate, as explained below. Charles Davies himself has printed on the title page: "Translated from the French of A. M. Legendre, By David Brewster, LL. D." The text was "revised and adapted" by Davies, not actually composed anew.I much prefer reading actual physical books to reading digital copies on a computer screen, so I plan to purchase more reprints of books that interest me. Forgotten Books is one reprint publisher I now trust (despite the four missing pages in Davies, it is still an excellent value), and I intend to purchase from them again.Now on to Charles Davies and his Elements of Geometry.A no-doubt valuable reference, which I haven't seen, is: Amy K. Ackerberg-Hastings, Mathematics is a Gentlemans Art: Analysis and Synthesis in American College Geometry Teaching, 1790-1840, Ph.D. dissertation, Iowa State University, 2000. There is an excerpt from this online, which is devoted to Davies: "Chapter Five. Two Circles Will Touch Each Other Internally: Charles Davies at the Art and Business of Teaching Geometry."Charles Davies taught at West Point Military Academy from 1823 to 1837, so researching him in relation to West Point may help. See, for example, the article from the Mathematical Association of America: "Mathematics Education at West Point: The First Hundred Years", by V. Frederick Rickey and Amy Shell-Gellasch.Ackerberg-Hastings' excerpt, along with pdf copies of Davies' books, shows that the main revision years of Davies' Elements of Geometry were 1828, 1834, 1851, 1857, 1862, and posthumously in 1885 edited by J. Howard Van Amringe incorporating revisions left by Davies in 1875. Ackerberg-Hastings claims 1882 for the posthumous edition, but I find it only with a preface dated 1885 by Van Amringe, although the copyright page lists 1862, 1882, 1885. Since 1862 was, what I'll call, the 5th edition, perhaps the 1882 edition was merely a reprint of that. Van Amringe's preface of 1885 gives the impression that it marks the first newly revised publication of Davies book since his death in 1876.V. Frederick Rickey and Amy Shell-Gellasch (mentioned above) write that: "The 1828 text, Elements of Geometry and Trigonometry, was his most popular book. In the period 1828-1895 it appeared in 33 editions/printings and some 300,000 copies." The book was, in fact, originally a translation and edited version of Legendre's Éléments de Géométrie of 1794. See the Ackerberg-Hastings excerpt for details. Of the 1834 edition (the edition reprinted by Forgotten Books), she writes:"Davies became a more significant editor with the 1834 printing of Elements of Geometry andTrigonometry. He substituted in material from Edinburgh Encyclopaedia in Book V and from Encyclopaedia Metropolitana in Book II [...]. Davies added eight axioms, several from Playfair's Elements or Simson's The Elements of Euclid: six statements on the relationships between equals and unequals, "Through the same point, only one straight line can be drawn which shall be parallel to a given line," and "All right angles are equal to each other," eliminating the first proposition from Carlyle's translation, which proved right angles are equal. Repeatedly, Davies elaborated on the reasoning process used to arrive at conclusions within proofs or in additional scholia. He also rearranged a very few definitions and propositions in the earlier books."The editions have interesting differences. The edition of 1851 seems to have been an attempt to simplify the text, and I prefer the 1834 edition over it. But the later editions of 1857, 1862, and 1885 return to the direction of 1834, while being true revisions and stylistic improvements. The trigonometry of the 1834 edition is presumably from Legendre. Starting with the edition of 1857, the trigonometry section expands considerably, and Davies tells us that it was written by "Professor Church of the Military Academy, for publication as a separate work; but, under the impression that it would be more useful as a component part of a regular course, he generously offered it as a substitute for the Trigonometry of the previous Editions."Davies distinguishes between axioms and postulates starting with the 1857 edition. An axiom "is a self-evident truth" and a postulate "is a self-evident problem." So his postulates include such statements such as: "Let it be granted, that a straight line may be drawn from one point to another point." His axioms include statements that Euclid considers postulates, such as: "All right angles are equal to each other." Davies lists what Euclid would call "common notions" under his axioms. So for Davies, it is an axiom that "Things which are equal to the same thing are equal to each other." What is odd is that Davies lists no postulates, in the meaning of his 1857 edition, in his editions prior to 1857. So he doesn't make explicit prior to 1857 that, for example, a straight line can be drawn from one point to another.[Edit: Axiom 11 states that "From one point to another only one straight line can be drawn." This axiom also occurs in the 1857 edition, where in the list of postulates he then asserts the possibility of drawing a straight line from one point to another. What is missing in 1834 are postulates granting this and other kinds of possibilities, such as a straight line bisecting a given angle or drawing a perpendicular from a point to a line. It is of interest to compare this with Euclid. To give one instance: Euclid I.9 derives the bisection of a rectilineal angle; Davies 1857 lists it as a postulate; Davies 1834 does neither.]Davies alters his definition of straight line over the course of his editions. In 1828 and 1834 he defines a straight line as being "the shortest distance from one point to another." Accounting for differences in translator's word-choice, this is probably Legendre's definition, if Farrar, in his 1825 translation of Éléments de Géométrie (a translation which Davies isn't using as his base text) is fairly true to Legendre's meaning when he writes: "A straight line or right line is the shortest way from one point to another." In 1851 Davies says "a straight line is one which lies in the same direction between any two of its points." Having altered the definition of straight line, in this edition he adds the axiom: "A straight line is the shortest distance between two points." In 1857 he defines a straight line as "one which lies in the same direction at every point." The corresponding axiom becomes: "A straight line measures the shortest distance between two points." I don't have current access to the 1862 edition. In the 1885 edition, "a straight line is one that doesn't change its direction at any point." The corresponding axiom states: "The shortest distance from one point to another is measured on the straight line which joins them."For an early use of direction in defining a straight line, see James Hayward 1829, Elements of Geometry upon the Inductive Method, where he defines a straight line as being "one which has the same direction throughout its whole extent." Hayward was Professor of Mathematics and Natural Philosophy at Harvard University. His text does not employ axioms. This was followed in 1837 by Benjamin Peirce, also of Harvard, in An Elementary Treatise on Plane and Solid Geometry, defining a straight line as being "one, the direction of which is the same throughout." Peirce does not mention Hayward (although as a student at Harvard from 1825 to 1829 he probably studied under him) and he speaks of his own use of direction as an undefined notion as an "innovation". Peirce uses the axiom: "A straight line is the shortest way from one point to another."Another interesting comparison among Davies' editions involves the definitions of geometry, point, and surface. For brevity, I'll compare the 1834 and 1857 editions only. In 1834, geometry is defined as "the science which has for its object the measurement of extension. Extension has three dimensions, length, breadth, and height or thickness." He defines a point in relation to a line. He says: "A line is length without breadth or thickness. The extremities of a line are called points: a point, therefore, has neither length, breadth, nor thickness, but position only." This definition of point binds points to lines, with his definition of straight line being "the shortest distance from one point to another" (i.e. from one extremity of a line to another). Lines, then, apparently, share extremities. (Euclid, after defining a point as "that which has no part", later says: "The extremities of a line are points." Saying that extremities are points is not to say that all points are extremities. Davies, however, gives no independent meaning beyond a point being an extremity of a line.)Davies does not later say in 1834 of a surface that it is composed of points, but only that a surface "is that which has length and breadth, without height or thickness" and that "a plane is a surface, in which, if two points be assumed at pleasure, and connected by a straight line, that line will lie wholly in the surface." This tells us that a plane is flat, the surface analogue of being straight. (Compare Euclid: "A plane surface is a surface which lies evenly with the straight lines on itself." I take this as meaning that no straight line in a plane extends beyond or outside the surface.)In 1857, Davies has definitions preliminary to his definition of geometry. He writes: "Space extends indefinitely in every direction. A Volume is a limited portion of space. It has three dimensions: length, breadth, and height or thickness. A Magnitude is anything that can be measured. A geometrical magnitude is a magnitude which possess one or more of the attributes of a volume. Geometry is the science which treats of the properties and relations of the geometrical magnitudes."He then gives us the definition of point: "A Point is that which has position, but not magnitude." Recall that in this edition he defines a straight line as being "one which lies in the same direction at every point." As in 1834, this binds the definition of a straight line (not line in general) to the definition of a point, although in a different way than in 1834. An axiom tells us: "A straight line measures the shortest distance between two points." Distance has no independent definition. Of a surface, he gives the definition: "A Surface is that which has length and breadth, without thickness. Surfaces are of two kinds, Plane and Curved." He defines a plane surface: "A Plane is a surface, such, that if any two of its points be joined by a straight line, that line will lie wholly in the surface."Because in 1857 he has defined points independently of lines, this definition of a plane, almost the same as in 1834, now suggests that points are an integral component of a plane. A surface, we might imagine, is comprised of points. The relation of a line to a surface isn't made explicit, but it appears that lines have an intimate relation to surfaces, since a line may have direction, and the "broken line" and "curved line" are defined also in terms of direction in this edition. Direction (an undefined notion) only makes (intuitive) sense in relation to length and breadth together (i.e. two dimensions), which a surface, but not a line, has.A curved line, being one "which changes its direction at every point" may certainly lie within a plane if it has the proper orientation, since a line has neither height nor breadth, but there is no indication of how lines, straight or curved, lie within a curved surface (what they look like), because a curved surface is defined simply as "a surface which is neither a plane nor composed of planes"; nor is there any indication what the shortest distance between points would be in a curved surface (although it would seem, by the axiom, to be a "straight line", a line which hasn't "changed direction" between those points). So what would "change of direction" mean in a curved surface, according to these definitions and the axiom? Presumably the same thing it means on a plane surface.In the introduction to the 1857 edition, we are corrected on some of these assumptions about the relations among points, lines, and surfaces:"A Volume being a limited portion of space, is necessarily divided from the indefinite space which surrounds it: that which so divides it, is called a Surface. Now, since that which bounds a volume is no part of the volume itself, if follows, that a surface has but two dimensions, length and breadth."If we consider a limited portion of a surface, then that which separates such portion from the other parts of the surface, is called a Line. This mark of division forms no part of the surface which it separates: hence, a line has length only, without breadth or thickness."If we regard a limited portion of a line, that which separates such a portion from the part beyond, at either extremity, is called a Point. But this mark of division forms no part of the line itself: hence a point has neither length, breadth, nor thickness, but place, or position, only."Although we use the term Volume to denote a given portion of space, the term surface to denote the boundary of a volume, the term line to denote the boundary of a surface, and the term point to designate the limit of a line, still, we may employ either of these terms, in an abstract sense, without any reference to the others."So a point, a line, and a surface are boundaries only. They are not components of space. A line and a surface are not composed of points. Points have no magnitude, and lines and surfaces do. They are all three, each within their dimensional limitations, boundaries within space, which "extends indefinitely in every direction", direction remaining undefined. (It seems that a surface and line, then, are not comprised of points of position, that extension or magnitude is not an arrangement of localities of position.) These remarks from the introduction of 1857 are absent from the edition of 1885.For those curious about the parallel axiom, in 1834 it's given as: "Through the same point, only one straight line can be drawn which shall be parallel to a given line." Parallel lines are defined as: "Two lines are said to be parallel, when being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced." In 1857 is added the additional remark: "They then have the same direction." The axiom in 1857 is the same as in 1834.[Edit: The edition of 1828 follows Legendre's text in having no axiom concerned with parallel lines. Legendre believed he had proven such an axiom as a theorem.]Charles Davies' Elements of Geometry and Trigonometry, known also simply as Davies' Legendre, was a popular and influential textbook in the United States during the 19th century. Looking into any one of his several editions is to see how geometry was taught in the past. This reprint of the 1834 edition is well worth the small expense to acquire it.

This (the Kessinger edition) is a perfectly formatted edition of this historical work. All the geometric figures are perfectly set next to the propositions.