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Nineteenth Century Geometry
In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm. During this period, the content of geometry and its internal diversity increased almost beyond recognition; the axiomatic method, vaunted since antiquity by the admirers of geometry, finally attained true logical sufficiency, and the ground was laid for replacing, in the description of physical phenomena, the standard geometry of Euclid by Riemann's wonderfully pliable system. Modern philosophers of all tendencies — Descartes and Hobbes, Spinoza and Locke, Hume and Kant — had regarded Euclidean geometry as a paradigm of epistemic certainty. The sudden shrinking of Euclidean geometry to a subspecies of the vast family of mathematical theories of space shattered some illusions and prompted important changes in the philosophical conception of human knowledge. Thus, for instance, after these nineteenth-century developments, philosophers who dream of a completely certain knowledge of right and wrong secured by logical inference from self-evident principles can no longer propose Euclidean geometry as an instance in which a similar goal has proved attainable. The present article reviews the aspects of nineteenth century geometry that are of major interest for philosophy and hints in passing, at their philosophical significance.
- 1. Lobachevskian geometry
- 2. Projective geometry
- 3. Klein's Erlangen program
- 4. Axiomatics perfected
- 5. The differential geometry of Riemann
- 6. Lie Groups
- Bibliography
- Academic Tools
- Other Internet Resources
- Related Entries
1. Lobachevskian geometry
Euclid (fl. 300 BCE) placed at the head of his Elements a series of 'definitions' (e.g., "A point is that which has no part") and 'common notions' (e.g., "If equals be added to equals, the sums are equal"), and five 'requests'. Supposedly these items conveyed all of the information needed for inferring the theorems and solving the problems of geometry, but as a matter of fact they do not. However, the requests (aitemata)—usually called 'postulates' in English—must at any rate be granted or Euclid's proofs will not go through. Some of them are plainly practical:
1. To draw a straight line from any point to any point. 3. To draw a circle with any center and any radius.
However, the fifth one sounds more like a statement of fact. Euclid's text can be rendered in English as follows: "If a straight line [c] falling on two straight lines [a and b] makes the interior angles on the same side less than two right angles, the two straight lines [a and b], if produced indefinitely, meet on that side on which are the angles less than the two right angles" (terms in brackets added for clarity). This sounds far-fetched. Still, it can be readily paraphrased as a recipe for constructing triangles, (See Figure 1.) Every triangle is formed by three coplanar straight lines that meet, by pairs, at three points. Given any segment PQ, draw a straight line a through P and a straight line b through Q, so that a and b lie on the same plane; verify that the angles that a and b make with PQ on one of the two sides of PQ add up to less than two right angles; if this condition is satisfied, it should be granted that a and b meet at a point R on that same side of PQ, thus forming the triangle PQR. This request is known as "Euclid's Postulate". If the request is rejected—say, because we believe that the world is finite and there is no room in it to accommodate vertex R if the interior angles in question add up to very little less than two right angles—then much of Euclid's system of geometry will not go through.
Figure 1
In the darker ages that followed, Euclid's sense of mathematical freedom was lost and philosophers and mathematicians expected geometry to rest on self-evident grounds. Now, if a is perpendicular and b is almost perpendicular to PQ, a and b approach each other very slowly on one side of PQ and it is not self-evident that they must eventually meet somewhere on that side. After all, the hyperbole indefinitely approaches its asymptotes and yet, demonstrably, never meets them. Through the centuries, several authors demanded—and attempted—a proof of Euclid's Postulate. John Wallis (b. 1616, d. 1703) derived it from the assumption that there are polygons of different sizes that have the same shape. But then this assumption needs proof in turn. Girolamo Saccheri (b. 1667, d. 1733) tried reductio. He inferred a long series of propositions from the negation of Euclid's Postulate, until he reached one which he pronounced "repugnant to the nature of the straight line". But Saccheri's understanding of this "nature" was rooted in Euclidean geometry and his conclusion begged the question.
In the 1820's, Nikolai I. Lobachevsky (b. 1793, d. 1856) and Janos Bolyai (b. 1802, d. 1860) independently tackled this question in a radically new way. Lobachevsky built on the negation of Euclid's Postulate an alternative system of geometry, which he dubbed "imaginary" and tried inconclusively to test for validity at the astronomical scale by calculating the sum of the internal angles of triangles formed by stars on the sky. Bolyai excised the postulate from Euclid's system; the remaining rump is the "absolute geometry", which can be further specified by adding to it either Euclid's Postulate or its negation. From the 1790's Carl Friedrich Gauss (b. 1777, d. 1855) had been working on the subject in the same direction, but he refrained from publishing for fear of scandal. Since Lobachevsky was the first to publish, the system of geometry based on the said "absolute geometry" plus the negation of Euclid's Postulate is properly called Lobachevskian geometry.
The construction introduced above to explain Euclid's Postulate can also be used for elucidating its negation. Draw the straight line a through point P at right angles with the segment PQ. If Euclid's Postulate is denied, there are countless straight lines through Q, coplanar with a, that make acute angles with PQ but never meet a. Consider the set of real numbers which are the magnitudes of these acute angles. Let the greatest lower bound of this set be μ. Evidently, μ > 0. There are exactly two straight lines through Q, coplanar with a, that make an angle of size μ with PQ. (See Figure 2.) Call them b1 and b2. Neither b1 nor b2 meets a, but a meets every line through Q that is coplanar with a and makes with PQ an angle less than μ. Gauss, Lobachevsky and Bolyai—unbeknownst to each other—coincided in calling b1 and b2 the parallels to a through Q. μ is called the angle of parallellism for segment PQ. Its size depends on the length of PQ, and decreases as the latter increases.
Figure 2
Suppose that the angle of parallellism for PQ is one half a right angle. In this case, b1 and b2 make a right angle at Q and we thus have two mutually perpendicular straight lines on the same plane as a, which fail to meet a.
Lobachevsky's geometry abounds in surprising theorems (many of which had already been found by Saccheri). Here are a few: The three interior angles of a triangle add up to less than two right angles. The difference or "defect" is proportional to the triangle's area. Hence, in Lobachevskian geometry, similar triangles are congruent. Moreover, if a triangle is divided into smaller triangles, the defect of the whole equals the sum of the defects of the parts. Since the defect cannot be greater than two right angles, the area of triangles has a finite maximum. If a quadrilateral, by construction, has three right angles, the fourth angle is necessarily acute. Thus, in Lobachevskian geometry there are no rectangles.
There is a simple formal correspondence between the equations of Lobachevskian trigonometry and those of standard spherical trigonometry. Based on it, Lobachevsky argued that any contradiction arising in his geometry would inevitably be matched by a contradiction in Euclidean geometry. This appears to be the earliest example of a purported proof of relative consistency, by which a theory is shown to be consistent lest another theory — whose consistency is typically taken for granted — be inconsistent.
Lobachevskian geometry received little attention before the late 1860s. When philosophers finally took notice of it, their opinions were divided. Some regarded it as a formal exercise in logical deduction, with no physical or philosophical significance, which employed ordinary words—such as 'straight' and 'plane'—with a covertly changed meaning. Others welcomed it as sufficient proof that, contrary to the influential thesis of Kant, Euclidean geometry does not convey any prerequisites of human experience and that the geometrical structure of physical space is open to experimental inquiry. Still others agreed that Non-Euclidean geometries were legitimate alternatives, but pointed out that the design and interpretation of physical experiments generally presupposes a definite geometry and that this role has been preempted by Euclid's system.
No matter what philosophers might say, for mathematicians Lobachevskian geometry would probably have been no more than an odd curiosity, if a niche had not been found for it within both projective and differential geometry, the two main currents of nineteenth-century geometrical research (§§ 2 and 5).
2. Projective geometry
Today projective geometry does not play a big role in mathematics, but in the late nineteenth century it came to be synonymous with modern geometry. Projective methods had been employed by Desargues (b. 1591, d. 1661) and Pascal (b. 1623, d. 1662), but were later eclipsed by Descartes's method of coordinates. They prospered, however, after Jean-Victor Poncelet (b. 1788, d. 1867) showed that the projective properties of figures furnished grounds of proof that were at least as powerful as, and certainly more intuitive and ostensibly compelling than the Cartesian procedure of setting up and solving equations between numbers representing points.
Projective properties are those preserved by projections. Take, for example, two planes Γ and H and a point P outside them. Let Φ be any figure on Γ. Draw straight lines from P through each point of Φ. The figure formed by the points where these lines meet H is the projection of Φ on H from P. Generally this figure will differ from Φ in size and shape. But the projection of any number of straight lines on Γ meeting each other at certain points generally consists of an equal number of straight lines on H meeting respectively at the projection of those points. What happens, however, if the straight line joining P with some point Q of Γ never meets H, because PQ happens to lie on a plane parallel to H? (See Figure 3.)
Figure 3
To obviate such irksome exceptions, projective geometry added to each straight line in space an ideal point, shared by every line parallel to it. Continuity requires then that all ideal points lie on a single ideal plane, which meets each family of parallel planes along a different ideal line. Fundamentalists may shudder at this seemingly wanton multiplication of entities. However, it had been practised in arithmetic for centuries, as the initial stock of natural numbers 1, 2, 3, … , was supplemented with zero, the negative integers, the non-integral rationals, the irrationals, and the so-called imaginary numbers.
The points of a straight line stand in mutual relations of neighborhood and order. To see how the ideal point fits into these relations let H rotate continually about the straight line m where it intersects Γ. (See Figure 4.) When H is parallel to PQ—say, at time t—the projection of Q on H from P is the ideal point of the straight line through P and Q. Right before t the said projection is an ordinary point of H, very far from m. Right after t the projection is again an ordinary point of H, very far from m, but at the opposite end of the plane. Studying the continuous displacement of the projection during a short time interval surrounding t, one concludes that if A and B are any two points of H that stand, respectively, on either side of m, the ideal point of the straight line through A and B must be placed between A and B. Thus, in projective geometry, the points of a straight line are ordered cyclically, i.e., like the points of a circle. As a result of this, the neighborhood relations among points in projective space and on projective planes differ drastically from those familiar from standard geometry, and are highly counterintuitive. It is fair to say that projective geometry signified a much deeper and far-reaching revolution in human thought than did the mere denial of Euclid's Postulate.
Figure 4
In the new setting, the projective properties of figures can be defined unexceptionably. a one-one mapping f of projective space onto itself is a collineation if it sends any three collinear points A, B, and C, to three points (A), (B), and (C), which are collinear too. Projective properties (and relations) are those which are preserved by collineations. Here are a few examples of projective properties. Of three or more points: to lie on the same straight line; to lie on the same plane. Of three or more straight lines: to meet at the same point; to lie on the same plane. Of three or more planes: to intersect along the same straight line; to share the same point. Of curves: to be a conic. Of surfaces: to be a quadric.
3. Klein's Erlangen program
In a booklet issued when he joined the faculty at Erlangen (1872), Felix Klein (b. 1849, d. 1925) took stock of the enormous growth and diversification of geometry and proposed a standpoint from which its many branches could be organized into a system. From this standpoint, the task of a branch of geometry can be stated thus:
Given a manifold and a group of transformations of the manifold, to study the manifold configurations with respect to those features which are not altered by the transformations of the group. (Klein 1893, p. 67)
In nineteenth-century mathematics, 'manifold' often designated what we now call a set, but Klein apparently had something more specific in mind:
If n variables x1, … , xn are given, the … value systems we obtain if we let the variables x independently take the real values from −∞ to +∞ constitute what we shall call … a manifold of n dimensions. Each particular value system (x1, … ,xn) is called an element of the manifold. (Klein 1873, p. 116)
If S is a manifold in either sense, by a transformation of S we mean a one-one mapping of S onto itself. It is clear that
- If T1 and T2 are transformations of S, the composite mapping T2 ○ T1, which consists of T1 followed by T2, is also a transformation of S;
- the composition of transformations is associative, so that, if T1, T2 and T3 are transformations of S, (T3 ○ T2) ○ T1 = T3 ○ (T2 ○ T1);
- the identity mapping I that sends each point of S to itself is a transformation of S such that, for any transformation T, T ○ I = I ○ T = T;
- for every transformation T there is a transformation T−1, the inverse of T, such that T−1 ○ T = I (T−1 sends each point of S back to where it was brought from by T).
By virtue of conditions (i)-(iv), the transformations of S form a group GS in the precise sense that this term has in algebra. GS includes subgroups, i.e., subsets which contain I and satisfy conditions (i) and (iv). If H is a subgroup of GS and Φ is a feature of S, or of its elements or parts, which is not affected by the transformations of Φ, we say that Φ is H-invariant. The only GS-invariant is the cardinality of S (i.e., the number of elements in the manifold). On the other hand, the group {I}, consisting of the identity alone, trivially preserves every conceivable feature. Between these two extremes there can be many different subgroups with all sorts of interesting invariants, depending on the respective group structure. If S is not an arbitrary (structureless) set, but a numerical manifold as described by Klein, it inherits structure from the real number field, which contributes to characterize the different subgroups of GS and their invariants. Thus, the group of continuous transformations preserves the topological properties (neighborhood relations), and the group of linear transformations preserves the projective properties.
Can metric properties be fixed in this way? Traditionally one defines the distance between two points (x1, … ,xn) and (y1, … ,yn) of a numerical manifold as the positive square root of (x1 − y1) 2 + … + (xn − y n)2. The group of isometries consists of the transformations that preserve this function. However, this is just a convention, adopted to ensure that the geometry is Euclidean. Using projective geometry, Klein thought of something better. No real-valued function of point pairs, defined on all projective space, is an invariant of the projective group, but there is a function of collinear point quadruples, called the cross-ratio, which is such an invariant. Drawing on work by Arthur Cayley (b. 1821, d. 1895), Klein (1871, 1873) considered the cross-ratio of point quadruples <P1,P2,P3,P4>. such that P3 and P4 belong to a given conic κ on a projective plane, while P1 and P2 range over a region R that is bounded by or otherwise fixed by κ. Since P3 and P4 must be the points where the straight line through P1 and P2 meets κ, the said cross-ratio may be regarded as a function of the point pair <P1,P2>. The collineations that map a given conic onto itself form a group, and the said function is clearly an invariant of this group. Klein showed that a certain function of this function behaves like an ordinary distance function on R. According to the nature of the conic κ, the structure determined by this function satisfies either (i) all the theorems of Euclidian plane geometry, or (ii) all those of Lobachevskian plane geometry, or (iii) those of a third geometry which Klein himself discovered and dubbed 'elliptic'. (In elliptic geometry every straight line meets every other, and the three internal angles of a triangle always add up to more than two right angles. Klein's names for the geometries of Euclid and Lobachevsky were 'parabolic' and 'hyperbolic', respectively.)
This is how Klein's approach works for Lobachevskian geometry on the plane. Let κ be a real conic—a conic comprising only real points—on the projective plane. Let Gκ be the set of all collineations that map κ onto itself. Gκ is a subgroup of the projective group. Consider now the cross-ratio of point quadruples <P1,P2,P3,P4> such that P3 and P4 belong to κ, while P1 and P2 range over the interior Int(κ) of the region of the real plane bounded by κ. (P ∈ Int(κ) if and only if P is a real point and no real tangent to κ passes through P.) As noted above, the choice of points P1 and P2 fixes P3 and P4, so the said cross-ratio may be regarded as a function of the first pair of points only, say, fκ(P1,P2). The function fκ is clearly Gκ-invariant. Put dκ(P1,P2) = c log fκ(P1,P2), where c is an arbitrary real-valued constant, different from 0, and log x denotes the principal value of the natural logarithm of x. Klein was able to show that dκ behaves precisely like a Lobachevskian distance function on Int(κ). In other words, every theorem of Lobachevskian geometry holds for suitable figures formed from points of Int(κ), if the distance between any two of these points is given by the function dκ. Consider, for instance, four points P1,P2,P3, and P4 in Int(κ), such that dκ(P1,P2) = dκ(P2,P3) = dκ(P3,P4) = dκ(P4,P1). They are the vertices of a Lobachevskian equilateral quadrilateral Q, which can have at most three right angles, in which case the fourth interior angle of Q must be acute. (Where 'right angle' means, as usual, an angle equal to its adjacent angle, and two angles in Int(κ) are said to be equal if one is the image of the other by a transformation of group Gκ).
If κ stands for a different sort of conic, not an ordinary real one, the function dκ obtained by the above procedure behaves on suitably defined regions of the projective plane like a Euclidian distance function or like the distance function of elliptic geometry (this depends on the nature of the conic κ). Thus, depending on whether κ belongs to one or the other of three kinds of conic, the group of collineations that map κ onto itself is structurally identical with one of the three groups of Lobachevskian, Euclidean, or elliptic isometries. Similar results hold for the three-dimensional case, with κ a quadric surface.
Klein's result led Bertrand Russell (b. 1873, d. 1970) to assert, in his neo-Kantian book on the foundations of geometry (1897), that the general "form of externality" is disclosed to us a priori in projective geometry, but its metric structure—which can only be Lobachevskian, Euclidean or elliptic—must be determined a posteriori by experiment. Henri Poincaré (b. 1854, d. 1912) took a more radical stance: If geometry is nothing but the study of a group,
one may say that the truth of the geometry of Euclid is not incompatible with the truth of the geometry of Lobachevsky, for the existence of a group is not incompatible with that of another group. (Poincaré 1887, p. 290)
The application to physics is immediate: "Among all possible groups we have chosen one in particular, in order to refer to it all physical phenomena, just as we choose three coordinate axes in order to refer to them a geometrical figure" (ibid., p. 291). The choice of this particular group is motivated by its mathematical simplicity, but also by the fact that "there exist in nature some remarkable bodies which are called solids, and experience tells us that the different possible movements of these bodies are related to one another much in the same way as the different operations of the chosen group" (ibid.). These remarks of Poincaré signalled the beginning of conventionalism in the philosophy of science and provided its initial motivation.
Klein's group-theoretical view of geometry enjoyed much favor among mathematicians and philosophers. It achieved a major success when Minkowski (1909) showed that the gist of Einstein's special theory of relativity was the (spacetime) geometry of the Lorentz group, an essential result that Klein (1911) lived to enjoy. It implies that the recent debate on the priority of Minkowski chronogeometry over Lorentz invariance or vice-versa is utterly idle, for these are logically equivalent and thus, in effect, two sides of the same coin (as explained by Acuña (2016)). However, Klein's Erlangen program failed to cover the differential geometry of Riemann (§5), which Einstein (1915, 1916) placed at the core of his general theory of relativity.
4. Axiomatics perfected
According to Aristotle, scientific knowledge (episteme) must be expressed in statements that follow deductively from a finite list of self-evident statements (axioms) and only employ terms defined from a finite list of self-understood terms (primitives). For over two millennia it was generally assumed that Aristotle's ideal is actually realized in Euclid's Elements. As a matter of fact, there is a logical gap already in Euclid I.1 (the solution of this problem rests on an unstated assumption of continuity) and it is not clear that Euclid regarded his postulates as self-evident (by calling them 'requests' he suggested he did not). The idea of securing knowledge by logical deduction from unquestionable principles had a powerful fascination for modern scientists such as Galileo and Newton, both of whom fondly practised axiomatics, at any rate as a literary form, like Spinoza in his Ethics. Still, a truly satisfactory and, if one may say so, serious instance of axiomatization of a branch of knowledge was not available in print until 1882, when Moritz Pasch (b. 1843, d. 1930) published his Lectures on Modern Geometry.
Pasch viewed geometry as a natural science, whose successful utilization by other sciences and in practical life rests "exclusively on the fact that geometrical concepts originally agreed exactly with empirical objects" (Pasch 1882, p. iii). Geometry distinguishes itself from other natural sciences because it obtains only very few concepts and laws directly from experience, and aims at obtaining from them the laws of more complex phenomena by purely deductive means. The empirical foundation of geometry was encapsulated by Pasch in a core of basic concepts and basic statements or axioms. The basic concepts refer to the shape and size of bodies and their positions relative to one another. They are not defined, for no definition could replace the "exhibition of appropriate natural objects," which is the only road to understanding such simple, irreducible notions (ibid., p. 16). All other geometric concepts must be ultimately defined in terms of the basic ones. The basic concepts are connected to one another by the axioms, which "state what has been observed in certain very simple diagrams" (p. 43). All other geometric statements must be proved from the axioms by the strictest deductive methods. Everything that is needed to prove them must be recorded, without exception, in the axioms. These must therefore embody the whole empirical material elaborated by geometry, so that "after they are established it is no longer necessary to resort to sense perceptions" (p. 17). "Every conclusion which occurs in a proof must find its confirmation in the diagram, but it is not justified by the diagram, but by a definite earlier statement (or definition)" (p. 43). Pasch understood clearly the implications of his method. He writes (p. 98):
If geometry is to be truly deductive, the process of inference must be independent in all its parts from the meaning of the geometric concepts, just as it must be independent from the diagrams. All that need be considered are the relations between the geometric concepts, recorded in the statements and definitions. In the course of deduction it is both permitted and useful to bear in mind the meaning of the geometric concepts that occur in it, but it is not at all necessary. Indeed, when it actually beomes necessary, this shows that there is a gap in the proof, and—if the gap cannot be eliminated by modifying the argument—that the premises are too weak to support it.
Pasch's Lectures on Modern Geometry dealt with projective geometry. The first axiomatization of Euclidean geometry that was up to Pasch's standards—Foundations of Geometry by David Hilbert (b. 1862, d. 1943)—appeared in 1899 and exercised enormous influence on twentieth century mathematics and philosophy. Hilbert invites the reader to consider three arbitrary collections of objects, which he calls 'points', 'straight lines' and 'planes', and five undefined relations between (i) a point and a straight line, (ii) a straight line and a plane, (iii) three points, (iv) two pairs of points ('segments') and (v) two equivalence classes of point triples ('angles'). The conditions prescribed in Hilbert's 20 axioms—including the Axiom of Completeness added in the second edition—are sufficient to characterize the said objects and relations up to isomorphism. Isomorphism—i.e., structural equivalence—can hold, however, between different, intuitively disparate, systems of objects. Hilbert availed himself of this feature of axiomatic theories for studying the independence of some axioms from the rest. To prove it he proposed actual instances (models) of the structure determined by all axioms but one, plus the negation of the omitted one. Frege complained that the geometric axioms retained in these exercises could be applied to Hilbert's far-fetched models only by tampering with the natural meaning of words (cf. Alice's conversation with Humpty Dumpty). Hilbert replied on 29 December 1899:
Every theory is only a scaffolding or schema of concepts together with their necessary mutual relations, and the basic elements can be conceived in any way you wish. If I take for my points any system of things, for example, the system love, law, chimney-sweep, … and I just assume all my axioms as relations between these things, my theorems—for example, the theorem of Pythagoras—also hold of these things. … This feature of theories can never be a shortcoming and is in any case inevitable.
All this follows, of course, from the very nature of axiomatics, as explained in the passage quoted from Pasch. Indeed, such truth-preserving semantic permutations were no news in geometry after Gergonne (1771–1859) drew attention in 1825 to the following principle of duality: Any true statement of projective plane geometry gives rise to another, equally true, dual statement obtained by substituting 'point' for 'line', 'collinear' for 'concurrent', 'meet' for 'join', and vice versa, wherever these words occur in the former. (In projective space geometry, duality holds for points and planes.) The same result is secured, of course, by exchanging not the words, but their meanings.
5. The differential geometry of Riemann
In a lecture "On the hypotheses that lie at the foundation of geometry", delivered to the Faculty of Philosophy at Göttingen in 1854 and posthumously published in 1867, Bernhard Riemann (b. 1826, d. 1866) presented some radically innovative views on this matter. He noted that the measurable properties of a discrete manifold can be readily determined by counting. (Think of the population of a country, and the proportion of born-again Christians, or of couples who divorced within the first year of their marriage.) But continuous manifolds do not admit this approach. In particular, the measurable properties of physical space, which are the subject of geometry, depend on the binding forces that act on it. The distance between two points in space can be ascertained with a rod, or a tape, or by optical means, and the result depends essentially on the physical behavior of the instruments used. Up to now, the measurable properties of space have been successfully described in accordance with Euclidean geometry. However, "the empirical concepts on which the metric determinations of space are based—the concepts of a rigid body and a light ray—lose their validity in the infinitely small; it is therefore quite likely that the metric relations of space in the infinitely small do not agree with the assumptions of geometry, and in fact one would have to accept this as soon as the phenomena can thereby be explained in a simpler way" (Riemann 1854, p. 149). To prepare physicists for this eventuality, Riemann proposed a more general conception of geometry. Riemann's basic scheme makes allowance for much greater generality than he actually reaches for; but, in his judgment, it should be enough for the time being to characterize the geometry of continuous manifolds in such a way that it agrees optimally with Euclidean geometry on a small neighborhood of each point.
Riemann extends to n dimensions the methods employed by Gauss (1828) in his study of the intrinsic geometry of curved surfaces embedded in Euclidean space (called 'intrinsic' because it describes the metric properties that the surfaces display by themselves, independently of the way they lie in space). Looking back at Gauss's work one gets a better intuitive feel for Riemann's concepts (see Torretti 1978, pp. 68–82). However, for the sake of conciseness and perspicuity, it is advisable to look forward and to avail oneself of certain concepts introduced by later mathematicians as they tried to make sense of Riemann's proposal. Consider the modern formulation of Riemann's theory in the supplement A Modern Formulation of Riemann's Theory.
In his study of curved surfaces, Gauss introduced a real-valued function, the Gaussian curvature, which measures a surface's local deviation from flatness in terms of the surface's intrinsic geometry. Riemann extended this concept of curvature to Riemannian n-manifolds. By using his extended concept of curvature, he was able to characterize with great elegance the metric manifolds in which all figures can freely move around without changing their size and shape. They are the Riemannian manifolds of constant curvature. This idea can be nicely combined with Klein's classification of metric geometries. Regarded as Riemannian 3-manifolds, Euclidean space has constant zero curvature, Lobachevskian space has constant negative curvature, and elliptic space has constant positive curvature. Pursuant to the Erlangen Program, each of these geometries of constant curvature is characterized by its own group of isometries. But Klein's conception is too narrow to embrace all Riemannian geometries, which include spaces of variable curvature. Indeed, in the general case, the group of isometries of a Riemannian n-manifold is the trivial group consisting of the identity alone, whose structure conveys no information at all about the respective geometry.
6. Lie Groups
For a philosopher, the most satisfying feature of the tremendous complication attained by 19th-century mathematics was perhaps the promptness with which the newly created (or discovered?) mathematical structures found their way into empirical science, enabling the intellectual grasp and handling of actual phenomena. We shall wind up this survey of 19th-century geometry with a few light remarks on a particularly rich and fruitful structure that has pride of place in current physics, namely, Lie groups, so called after Sophus Lie (1842–1899), the Norwegian mathematician who studied them in depth after 1870. A Lie group is, of course, a group in the algebraic sense we met in §3, that is, a set G such that (i) every ordered pair <x,y> ∈ G is associated to a unique element x · y ∈ G (known as the product or the sum of x and y); (ii) the product operation is associative, i.e., (x · y) · z = x · (y · z), for every x, y, z ∈ G; (iii) there is a one and only one element 0 ∈ G such that for every x ∈ G, x · 0 = 0 · x = x (0 is the identity or neutral element of G); (iv) for every x ∈ G, there is one and only one element x−1 ∈ G such that x · x−1 = 0 (x−1 is known as the inverse of x). But a Lie group is also a smooth manifold, as described in the supplement A Modern Formulation of Riemann's Theory: the set G can be represented patchwise by systems of real-valued (or alternatively complex-valued) coordinates, mutually linked by well-defined, differentiable coordinate transformations wherever their respective patches overlap. The group and manifold structures of G are meshed together by the condition that the product operation is a differentiable mapping of G × G into G.
A simple, yet important, example of a Lie group is the group SO(2), instantiated by the rotations of the plane about an arbitrary fixed point. The manifold is topologically compact and therefore cannot be covered by a single coordinate patch, but three will suffice: one including, say, all counterclockwise rotations by more than three radians and less than four, which can be naturally coordinatized using the real numbers in the open interval (3,4); another patch comprising the inverses of the former, which can be mapped onto the open interval (−4,−3), and a third one covering all counterclockwise rotations by less than two right angles plus their clockwise inverses, which can be mapped onto the open interval (−π,π). Indeed, all the groups we encountered in §3, which Klein used for characterizing the Euclidean geometry of space and the classical Non-Euclidean geometries, are Lie groups, and their respective smooth manifold structures allow for topological quirks. Thus the Euclidean isometries constitute a disconnected manifold, with mirror reflection not included in the same component as the subgroup of Euclidean motions.
Like all smooth manifolds, a Lie group G has a tangent vector space attached to each element. In particular, the tangent space at the neutral element 0 of G becomes the Lie algebra of G by the definition of the so-called Lie bracket, a bilinear mapping of T0G × T0G into T0G, which, for all u,v,w in T0G satisfies the condition [u,u] = 0 and the Jacobi identity: [u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0. The Lie algebra of G throws much light on the structure of G through the homeomorphic ("exponential") mapping of a neighborhood of 0 ∈ T0G into a neighborhood of 0 ∈ G.
In the supplement A Modern Formulation of Riemann's Theory we touch on the idea of a fiber bundle, formed by two smooth manifolds F and M, bound together by a "projection" mapping π of F onto M, which partitions the manifold F into "fibers", mapped by π to the different points of M. A fiber bundle <F,M,π> becomes a principal fiber bundle <F,M,π,G> if a Lie group G, known as the bundle's structure group, acts on F in such a way that each fiber of F is an orbit of the action and a few other conditions are met. For instance, the Lorentz group is the structure group of the principal fiber bundle of tetrads (orthonormal 4-tuples of tangent vectors at each point) on any relativistic spacetime, no matter how bizarre. In such ways, Lie groups provide a means of unifying the many models allowed by a physical theory and of introducing some degree of homogeneity among them.
During the last third of the 20th-century, fibre bundles and their Lie groups have virtually taken over fundamental physics. This is not the place to explain how or why, but the unstoppable evolution of physics towards ever more mathematically sophisticated, prima facie less straightforward representations of its subject-matter, deserves the attention of philosophers. It is clear that the concept of a definite stable thing out there, which might, at least in principle, be held and manipulated, is no longer so serviceable to us as it was once to our flint-carving ancestors.
Primary Sources
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- Cayley, Arthur, 1859. "A sixth memoir upon quantics," Philosophical Transactions of the Royal Society of London, 149: 61–90.
- Ehresmann, Ch., 1957. "Les connexions infinitésimales dans un espace fibré différentiable," in Colloque de Topologie (Espaces Fibrés), Bruxelles 1950, Paris: Masson, pp. 29–55.
- Einstein, A., 1915. "Die Feldgleichungen der Gravitation," Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1915), pp. 844–847.
- Einstein, A., 1916. "Die Grundlagen der allgemeinen Relativitätstheorie," Annalen der Physik, 49: 769–822.
- Euclides, Elementa, I. L. Heiberg (ed.), Leipzig: B. G. Teubner, 5 volumes., 1883–88. (For English translation, see below under Heath).
- Gauss, C. F., 1828. Disquisitiones generales circa superficies curvas, Göttingen: Dieterich. (English translation by A. Hiltebietel and J. Morehead: Hewlett, NY, Raven Press, 1965.)
- Hilbert, D., 1899. "Die Grundlagen der Geometrie," in Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals, Leipzig: B.G. Teubner, pp. 3–92.
- Hilbert, D., 1968. Grundlagen der Geometrie, mit Supplementen von P. Bernays. Zehnte Auflage. Stuttgart: Teubner. (Tenth, revised edition of Hilbert 1899.)
- Klein, F., 1871. "Über die sogenannte Nicht-Euklidische Geometrie," Mathematische Annalen, 4: 573–625.
- Klein, F., 1872. Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen: A. Duchert.
- Klein, F., 1873. "Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz)," Mathematische Annalen, 6: 112–145.
- Klein, F., 1893. "Vergleichende Betrachtungen über neuere geometrische Forschungen," Mathematische Annalen, 43: 63–100. (Revised version of Klein 1872).
- Klein, F., 1911. "Über die geometrischen Grundlagen der Lorentz-Gruppe," Physikalische Zeitschrift, 12: 17–27.
- Lie, S., 1888–1893. Theorie der Transformationsgruppen (3 volumes), Unter Mitwirkung von F. Engel, Leipzig: Teubner.
- Lobachevsky, N. I., 1837. "Géométrie imaginaire," Journal für die reine und angewandte Mathematik, 17: 295–320.
- Lobachevsky, N. I., 1840. Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin: F. Fincke. (English translation by G. B. Halsted printed as a supplement to Bonola 1955.)
- Lobachevsky, N. I., 1856. Pangéométrie ou précis de géométrie fondée sur une théorie générale et rigoureuse des parallèles, Kazan: Universitet.
- Locke, J., 1690. An Essay concerning Humane Understanding (in four books), London: Printed for Thomas Basset and sold by Edward Mory. (Published anonymously; the author's name was added in the second edition).
- Minkowski, H., 1909. "Raum und Zeit," Physikalische Zeitschrift, 10: 104–111.
- Pasch, M., 1882. Vorlesungen über neueren Geometrie, Leipzig: Teubner.
- Poincaré, H., 1887. "Sur les hipothèses fondamentales de la géométrie," Bulletin de la Société mathématique de France, 15: 203–216.
- Poncelet, J. V., 1822. Traité des propriétés projectives des figures, Paris: Bachelier.
- Ricci, G. and T. Levi-Cività, 1901. "Méthodes de calcul différentiel absolu et leurs applications," Mathematische Annalen, 54: 125–201.
- Riemann, B., 1854. "Über die Hypothesen, welche der Geometrie zugrunde liegen," Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1867): 133–152. (For English translation, see below under Spivak.)
- Riemann, B., 1861. "Commentatio mathematica, qua respondere tentatur quaestioni ab illustrissima Acad. Parisiensi propositae," in Bernhard Riemanns gesammelte mathematische Werke und wissenschaftlicher Nachlass, Leipzig: Teubner, 1876, pp. 391–404.
- Russell, B., 1897. An Essay on the Foundations of Geometry, Cambridge: Cambridge University Press. (Unaltered reprint: New York, Dover, 1956.)
- Saccheri, G. 1733. Euclides ab omni nævo vindicatus sive conatus geometricus quo stabiliuntur prima ipsa universæ geometriæ principia, Mediolani: Ex Typographia Pauli Antonii Montani. (Reprint, with facing English translation by G. B. Halsted: New York, Chelsea, 1986.)
Secondary Literature
- Acuña, Pablo, 2016. "Minkowski spacetime and Lorentz invariance: The cart and the horse or two sides of a single coin?," Studies in History and Philosophy of Science (Part B: Studies in History and Philosophy of Modern Physics), 55: 1–12.
- Blumenthal, L. M., 1961. A Modern View of Geometry, San Francisco: Freeman.
- Boi, Luciano, 1995. Le problème mathématique de l'espace: Une quête de l'intelligible, Berlin: Springer.
- Bonola, R., 1955. Non-Euclidean Geometry: A critical and historical study of its development. English translation with additional appendices by H.S. Carslaw. New York: Dover.
- Freudenthal, H., 1957. "Zur Geschichte der Grundlagen der Geometrie," Nieuw Archief vor Wiskunde, 5: 105–142.
- Freudenthal, H., 1960. "Die Grundlagen der Geometrie um die Wende des 19. Jahrhunderts," Mathematisch-physikalische Semesterbericht, 7: 2–25.
- Gallot, S., D. Hulin, and J. Lafontaine, 2004. Riemannian Geometry, Berlin: Springer, 3rd edition. (An up-to-date textbook, with solutions to odd-numbered exercises. A section is devoted to the "pseudo"-Riemannian geometry employed in Relativity Theory.)
- Giedymin, J., 1982. Science and Convention: Essays on Henri Poincaré's Philosophy of Science and the Conventionalist Tradition, Oxford: Pergamon.
- Greenberg, M. J., 2008. Euclidean & Non-Euclidean Geometry: Development and History, New York: Freeman, 4th edition. (An excellent tool for self-study at the high-school senior or college freshman level.)
- Heath, T. L., 1956. The Thirteen Books of Euclid's Elements, translated from the text of Heiberg with introduction and commentary, New York: Dover, 3 volumes, 2nd edition, revised with additions.
- Magnani, L., 2001. Philosophy and Geometry: Theoretical and Historical Issues, Dordrecht: Kluwer.
- Nagel, E., 1939. "The formation of modern conceptions of formal logic in the development of geometry," Osiris, 7: 142–224.
- O'Neill, B., 1983. Semi-Riemannian Geometry with Applications to Relativity, New York: Academic Press.
- Nomizu, K., 1956. Lie Groups and Differential Geometry, Tokyo: The Mathematical Society of Japan.
- Ronan, M., 2008. "Lie Theory," in T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton, NJ: Princeton University Press, pp. 229–234.
- Rosenfeld, B. A., 1988. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, translated by Abe Shenitzer, New York: Springer.
- Spivak, M., 1979. A Comprehensive Introduction to Differential Geometry (5 volumes), Berkeley: Publish or Perish, 2nd edition. (Contains an excellent English translation, with mathematical commentary, of Riemann's lecture "On the hypotheses that lie at the foundation of geometry"; see Vol. 2, pp. 135ff.)
- Torretti, R., 1978. Philosophy of Geometry from Riemann to Poincaré, Dordrecht: Reidel. (Corrected reprint: Dordrecht, Reidel, 1984).
- Trudeau, R. J., 1987. The Non-Euclidean Revolution, Boston: Birkhäuser.
- Winnie, J. W., 1986. "Invariants and objectivity: A theory with applications to relativity and geometry," in R. G. Colodny (ed.), From Quarks to Quasars, Pittsburgh: Pittsburgh University Press, pp. 71–180.
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I thank John Norton for the illustrations and for ideas leading to a better presentation of some mathematical concepts. I am also very grateful to Edward Zalta for his painstaking editorial work and for having identified and firmly rejected a murky passage in the first version of this article. [Added in 2003:] I thank John Corcoran for his very apt critical comments, which prompted changes in several passages of the 1999 text.


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These recipes were entered by customers, growers, and market managers at the many locallygrown.net markets. Account-holders at those markets can see what recipes are in season, buy ingredients from their local growers while looking at the recipe itself, add comments and photographs, mark favorites, and more. Buying and cooking with locally grown food has never been easier!
Apple Cider-Glazed Turnips
From Lawrenceville, GA

Apple cider's sweetness is a perfect foil for turnip's earthy character. Serve these alongside roasted chicken, pork, or lamb.
Source: Whole Foods (Entered by James Bouchard)
Serves: 4-6
3 Table Spoon
2 lbs
Turnips, peeled and cut into sticks
2/3 cup
Apple Cider
3 Table Spoon
Brown Sugar
1 pinch
Sea Salt
1 pinch
Fresh Ground Black Pepper
2 Table Spoon
Chopped Fresh Parsley
Step by Step Instructions
- Preheat over to 400F. Grease a 9x13 inch baking dish with 1 tablespoon butter.
- Arrange turnips in dish in a single layer
- Combine remaining 2 tablespoons butter, cider, sugar, salt and pepper in a small pot and cook over medium heat until melted and well combined. Pour this mixture over turnips.
- Cover dish with foil and bake for 20 minutes. Stir turnips and continue to bake, uncovered, stirring occasionally, until fork tender and liquid has reduced, 20-30 minutes more.
- Sprinkle with parsley and serve.


text:
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"Looking back, and knowing what you know now… what would you have done differently, and why?"
This question (and many of its variations), is really popular among those interviewing the rich and famous. It's meant to elicit golden nuggets of priceless information, acquired over a long and illustrious career. It's an old trick, and it still works.
As an interviewer I've probably used it dozens of times, and I could only get away with my lack of originality by editing myself out. I usually kept the answer until the end of the conversation. After a short musical interlude, the celebrity I was speaking with would "spontaneously" get philosophical, and come up with this profound life lesson that resonated long after the interview was over.
Mission accomplished!
Last week, the tables were turned when a young colleague asked me same question: "Looking back, and knowing what you know now… what would you have done differently, and why?"
At that moment one realizes that it's much easier to ask than to answer, but I knew pretty quickly what I was going to say. It brought me back to the beginning of my American career, some sixteen years ago. Here's what I came up with:
"I wish I would have listened to my heart, instead of to my mind, when I thought of becoming a voice-over."
I realize that this is not an eye-opening, Zen-like insight, but I know I'm not the only one struggling with the battle between warm feelings and cold logic.
At that time my analytical, practical mind came up with all these brilliant rationalizations as to why a VO-career would never work for me. This was at the beginning of a new millennium, and I had just arrived in the United States.
I had very little money, no contacts in the industry, and I didn't know where to begin. How would I promote myself in a country with over 300 million people? Who would hire this nobody from Holland with his funny accent?
I felt overwhelmed, unprepared, and insecure.
Of course there was no Facebook or LinkedIn group where aspiring voice-overs could ask questions. There were no books about the business, and the concept of home studios did not exist. It was much easier to find a job waiting tables, and as someone who needed to make money, that's exactly what I did.
My first job was at The Fish House in Lambertville, NJ, and even though I was a vegetarian, I knew how to sell sardines, swordfish, and Chilean sea bass. Because I didn't know anybody, the so-called celebs who frequented this restaurant didn't impress me.
One day, a colleague took me aside and said: "Do you know who you just served?"
I had no idea.
"The coach of the Eagles!" he replied enthusiastically. "You know… THE EAGLES!!"
I looked at him with a straight face, and said: "What Eagles?"
In hindsight I think coach Andy Reid appreciated that I treated him like a regular customer. He even laughed at one of my wine jokes. His wife Tammy wanted to know why the Jersey Chard she was drinking had such a distinctive yellow glow. I told her the vineyard was next to a nuclear power plant.
Fortunately she though it was funny.
Meanwhile, I didn't know that I had just taken the first step in becoming a real actor: I was waiting tables!
The restaurant was also where people began commenting on my voice, my accent, and my ability to speak several languages. To me it was kind of a party trick to help my tip jar, but kind customers asked: "No offense, but why are you a waiter? You should really do something with that voice of yours!"
Encouraged, I signed up for an open casting call at Mike Lemon Casting in Philadelphia. My heart told me that's where I should go, but my mind was skeptical. Once again it came up with a million reasons as to why I wouldn't make the cut. All those reasons made perfect sense, but they were all wrong.
That day, voice casting director Joanne Joella signed me on the spot, and my American adventure in voice-overs officially began.
Well, not quite.
Even though I was booking some decent jobs here and there, my mind told me this wasn't going to last, and that I really needed a serious position doing serious work. I was doing well on tips as a waiter, but recommending Jersey wine and pan-seared scallops did not make a career.
That's how I ended up in a call center, surveying European hard- and software specialists by telephone. Of course these overworked, stressed out professionals had nothing better to do than talk to me, and they all loved telling me about their satisfaction with the latest network servers.
This job had two amazing perks. One: Because we called businesses in Germany and in the Netherlands, I lived on European time, getting up at 2:00 AM, making my first call at 3:00 AM (9:00 AM in Amsterdam and Munich). Two: I had to use a script from which I was not allowed to deviate.
That was my second step in becoming a real actor: I got to use scripts!
A year or two into that pathetic call center job, something wonderful happened. All the interviewers were mercifully replaced by an automated voice response system that was much better at taking verbal abuse from German software specialists who were sick of revealing their satisfaction with product X on a scale of zero to ten, zero meaning completely dissatisfied, and ten meaning completely satisfied.
It was time for me to move up the ladder!
Did I listen to my heart this time, and would I be pursuing a full-time voice-over career?
No, my friends. My mind talked me into accepting a job as a customer service trainer at Wachovia Bank. As we all know, banks are a secure place to work. Some of them even offer benefits.
Yea for me!
Luckily, I knew nothing about the financial industry or balancing books, and I suffer from dyscalculia. That's like being dyslexic but with numbers instead of words. It's particularly useful when you have to stare at bank accounts all day long, and figure out why this infuriated client got slammed with five overdraft fees after buying a burger with money he didn't have.
Here's what I loved about this job. Since I was the lead trainer, I was in front of a class of sleepy, unmotivated students all day long.
Looking back, Wachovia was my third step in becoming a real actor: I got to perform in front of a live audience!
By the way, if you can't remember the name Wachovia, that's perfectly understandable. Wachovia was eventually overrun by the Wells Fargo wagon, and they brought in their own training team to cultivate a new corporate culture.
Good for them. Great for me!
After three pointless, mind numbing, soul crushing, dream dashing jobs, I finally got the message:
"Follow Your Heart, you idiot! Become a full-time voice talent, and conquer the world."
And that's exactly what I did.
I strongly believe that living is learning, and that every job helped prepare me for the future I created for myself. Yet, when I look back at all those years of doing things for money while my heart wasn't in it… When I think of how miserable I used to be, and how happy I am now… I often wonder:
If only I had known…
If only I would have taken the risk, and had followed my dreams from the get-go. Where would I be now?
Would I be a household name? Would obnoxious fans ask for my autograph at crazy comicons and conventions? Would agents fight to represent me? Would I be rich and famous?
Well, if that were me, I'm pretty sure that one day, a young reporter would knock on my door. After an in-depth, hour-long interview, he would pause and get ready for that very last killer-question:
"Looking back, and knowing what you know now… what would you have done differently, and why?"
Paul Strikwerda ©nethervoice
PS Be sweet. Please retweet!
photo credit: you probably don't wanna know via photopin (license)


text:
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text:
Teaching You To Become A Good Steward
"I can plod." The man who said those words was William Carey (1761-1834). Perhaps you recognize the name? He is remembered as the father of modern missions, a man who by God's grace pioneered an awakening in the country of India. He faced severe obstacles and devastating setbacks, yet his ministry was used mightily by God. And he could plod. But is plodding—honorable as it sounds—really what we ought to be doing for God? Is it not more honorable to engage in bold advances for the Kingdom? Shouldn't we pray for miracles and expect God to do great things? Why plod? Shouldn't we expect great things from God? Shouldn't we do something more than just plod?
William Carey was obviously a courageous missionary, and by all accounts, a success. But lest we adopt his plodding mantra without careful thought, let's think about some of the negative ramifications of plodding.
Perils of Plodding
- Plodding can result in discouragement. The dictionary definition of plodding is "slow-moving and unexciting." To illustrate, let's say a man is a pastor of a small church. He's a plodder. He's a faithful guy. But as his ministry rolls on, month after month, year after year, he realizes that not a whole lot is happening. Revivals are not forthcoming. Droves of people are not getting saved. In fact, some of the families that he thought he "put back together," have come totally unglued. The man plods, but discouragement sets in.
- Plodding can result in burnout. When a person pours intense labor into something for which they are not gifted, burnout will result. For example, consider a pastor whose gift is Bible exposition. He is not gifted at administration. However, his church of 200 people and lots of programs demands a great deal of administration. Soon, the pastor discovers that the lion's share of his time is consumed by administrative details. Despite his lack of administrative skill, he tries to organize programs and oversee events. He feels like he's banging his head against a wall. And eventually, he gives in. He is exhausted. He is fatigued. He plods. And burns out.


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Slow Cooker Sweet & Tangy Chicken
This time of year is always super busy for me. I don't know how y'all who have kids do it, because I don't even have to prepare for back-to-school! It just seems like every year my schedule explodes in August & early September. I think it's because everyone is trying to cram in as much "end of Summer" fun as they can before the weather gets cooler.
I generally do a lot of cooking on the weekends, but with all the weekend trips and visitors lately I haven't had a ton of time to spend in the kitchen. That's where my slow cooker comes in! It's so nice to toss a few ingredients in a pot in the morning and then come home from work to a hot and ready meal. This particular Slow Cooker Sweet & Tangy Chicken recipe fits that scenario perfectly. There's next to no morning prep time and when you get home the meat literally falls apart for shredding and you just have to thicken the sauce for 2-3 minutes. Serve it over some brown minute rice and steam a side of broccoli, snow peas or an Asian veggie mix and voila! Dinner is served.
In addition to being unbelievably simple, this recipe is really tasty. It's sweet and tangy with a garlicky Asian flair. It pairs really well with rice, but you could certainly serve it over vegetables or pasta, stuff it in a baked potato or add a tortilla and some mix-ins and make it a wrap. At only 209 calories or 7 Green, 5 Blue or 5 Purple Weight Watchers SmartPoints per serving, the possibilities are endless! To view your WW PersonalPoints for this recipe and track it in the WW app or site, click here!
Looking for more simple slow cooker recipes? Try Meatballs & Marinara, Mexican Pulled Pork, Beef Braciole, Chicken Parmesan, Barbecue Beef, Apple Bacon BBQ Chicken, Taco Chili, Beef & Broccoli, Buffalo Chicken or Balsamic Beef.
Slow Cooker Sweet & Tangy Chicken
- ¾ cup packed brown sugar
- 2/3 cup apple cider vinegar
- ¼ cup diet lemon-lime soda
- 2 tablespoons minced garlic
- 2 tablespoons low sodium soy sauce
- ½ teaspoon black pepper
- 2 lbs raw boneless skinless chicken breasts
- 2 tablespoons corn starch
- 2 tablespoons water
- Crushed red pepper flakes, to taste
- Combine the brown sugar, cider vinegar, soda, garlic, soy sauce and black pepper and pour into your slow cooker. Lay the chicken breasts across the bottom and stir sauce to cover chicken (if chicken peeks up over sauce that's okay). Cook on low for 6-8 hours.
- Remove chicken from sauce and shred using forks (it should fall apart easily).
- Pour the sauce from the slow cooker into a small saucepan (or if you have a slow cooker with a removable pot specifically meant to be used on the stove top you can just use the same pot - this is the slow cooker I use). In a small bowl, stir together the corn starch and water. Add the corn starch mixture to the sauce and bring to a boil over high heat. Boil for 2-3 minutes until sauce has thickened into a glaze. Remove from heat and let stand for another 1-2 minutes (sauce will continue to thicken while standing).
- You can either mix the chicken back into the sauce or serve the sauce poured over the chicken (as pictured). Sprinkle each serving with a pinch of crushed red pepper flakes to taste before serving.
5 per servings (P+ calculated using the recipe builder on weightwatchers.com) adapted from Six Sisters' Stuff
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Posted on Thursday, December 20, 2012
NEW WILMINGTON, Pa. - Westminster College broadcast and digital communications students and Westminster's Cable Network (WCN) received several awards from College Broadcasters, Inc. (CBI) for projects featured on WCN.
"I'm so proud of our students for receiving these CBI awards," said Brad Weaver, communications studies lecturer. "It says so much about their talent and skill. They are the heart of Westminster Cable and Titan Radio. The students make it a joy to teach at Westminster and serve as an inspiration for their peers. This kind of recognition really shows why our students are placed so quickly after graduation."
"Spring Break with a Purpose" was named Best TV Special Broadcast. The video followed nearly 60 Westminster College students during spring break as they participated in three work trips in the Carolina's through Westminster's Chapter of Habitat for Humanity and Campus Ministries.
Students who worked on the project as reporters and producers include: Corey Benedict, Courtney Crown, Josh Dunn, Ashley Durham, Ryan Johnston, Connor Kobis, Joe Ligo and Hannah Paczkowski. Alyssa Hanna served as executive producer.
"Winning the CBI for 'Spring Break with a Purpose' was a huge accomplishment," Durham said. "Not only did we justify ourselves as journalists, but we did it by showing the wonderful things Westminster does beyond its community, and that is just truly amazing."
"I am very proud of the 'Spring Break with a Purpose' crew for sticking with the class and accepting the challenge of working when other students were on vacation," Weaver said.
"The Truth About Natural Gas Fracking in Lawrence and Mercer Counties" was named Best TV News Reporting. The video covered a meeting held at Wilmington High School by the activist group The Fracking Truth Alliance. The group held a panel discussion featuring professionals from various fields involved with natural gas drilling. The piece was researched, written, edited and produced by Ligo. Jessie Klousnitzer and Johnston assisted with interviews and filming.
"Titan Radio Never Stops" was named Best TV Promo. Ligo, the program director for the Westminster's student-run Titan Radio, came up with the concept and produced the video.
"This 30-second promo for Titan Radio features stop-motion animation made up of hundreds of still photographs," Ligo said. "After the success of the music video 'Easy' for J.D. Eicher and the Goodnights, I wanted to try producing more stop-motion animation projects. The idea came to me in the radio station one day, so I decided to follow through with it. This promo was shot and edited in one morning."
"Get Out" was named Runner Up for Best Public Service Announcement. The video promoted the services offered by Titan Traverse, Westminster College's Adventure Education and Leadership Development program that includes a rope course, trust building and problem-solving activities, and a variety of wilderness based adventure activities. Ligo and Andrea Tice worked on this project.
Weaver's students said their accomplishments would not be possible without his support.
"Mr. Weaver challenges us constantly to get better at our profession," Corey Benedict said. "During the 'Spring Break with a Purpose' trip, he helped us out getting the shots we needed and helped us put the documentary all together. It was a pleasure working with him."
The broadcast and digital communications program is one of the largest and fastest growing at Westminster College, offering students interested in radio and television an education with hands on experience, including: remote broadcasts of sports and other special events, in-studio production of news, magazine and interview shows, announcing and managing Titan Radio, special feature productions, and web and internet journalism on www.wcn247.com.
The objective of the broadcast and digital communications program is to prepare students for entry-level and subsequent management positions in digital media: radio, television, cable and a range of other organizations.
Contact Weaver at (724) 946-7238 or email for additional information.


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LRQA has presented STX with approval in principle for a 50k dwt MR tanker design capable of using LNG as a fuel.
LRQA has presented STX with approval in principle for a 50k dwt MR tanker design capable of using LNG as a fuel. This development is the result of close collaboration between STX and LR.
STX incorporated an LNG-fuelled propulsion system into their existing conventional MR tanker design with technical assistance and support from LRQA, ensuring that the design is in compliance with the new IGF code as well as LRQA's Gas Fuelled Rules, covering other applicable requirements. LRQA also supported STX in carrying out a risk assessment of the LNG-fuelled propulsion system for the vessel design.
Mr. EK Choi, Chief Technical Officer of STX, commented: "STX has an extensive reference list for designing and building MR tankers. The experience and know-how that has been built to date has been utilised to develop this new LNG-powered MR tanker. The design enables the lowest possible fuel consumption and allow owners to select a design that will meet the future SOx and NOx emission challenges."
He added, "We are very happy to cooperate with one of the leading classification societies, LRQA, for the development of this design and to create a viable solution to meet the market demand for environmentally-friendly and technology-enabled products."
Presenting the AiP, Young-Doo Kim, LRQA's North Asia Technical Support Office Manager, said: "It's been an exciting project where we are delighted to have been able to support STX in incorporating LNG as a fuel option."


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Book Blurb:
Within her walls are stories. Love. Laughter. Tears. Lives lived to the fullest, and sometimes cut short before they really get started. She's seen it all, from homelessness to super-stardom, boys dressed as girls, girls dressed as lamps, and boy, does she have secrets she could tell. Her one truth is that everyone who darkens her doors is accepted with open arms and a song.
Her name is Queers, and for one of L.A.'s few dueling piano bars, she has a lot of heart. On her stage, the beautiful voice of Duff McKinley rings so true he captures the shattered heart of his boss, Brad Mosely, who checked out of life three years ago. But even lifelong dreams have untold consequences.
Duff came to L.A. to chase his musical aspirations, but his best friend Garrett Slater accompanied him to chase a wish of his own: finding The One, his white knight. He believes Landon Kennedy fits the bill. Suave, handsome, and powerful in the music industry, Landon can show Garrett a life he's only seen in movies. But even as Landon helps Duff realize his dreams, he brings Garrett into a nightmare he may never leave, one from which Garrett's roommate, Jackson Moriarty, is determined to rescue him. If Garrett repays Jackson by seeing his own charms, so much the better.
Yes, within her walls, Queers has long history. Unrequited love, sinister intentions, fame, fortune, and a group of friends and lovers who would do anything for each other no matter the cost.
AJ Rose once again will blow your mind with his talent. This book will take you on a journey that will pull on every emotion you have! Do not miss this ride!
There is also a Book Trailer available to watch... If only to wet your whistle even more, check it out...
It is available on Amazon, and soon (If not already by the time you read this) iTunes, B&N and ARe.
MAKE SURE YOU BUY THIS BOOK! It's worth every penny! I can't wait to hear what you think.


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Skirting Orac SX137
For more than 40 years, the Belgian company Orac Decor has been investing all our love in the development and production of decorative finishing elements for your interior. Orac Decor ornaments are made from advanced materials, so they are resistant, lightweight and easy to install. These unique product features distinguish Orac Decor profiles from products made of wood, MDF or plaster. The majority of our products are made from a high-quality polyurethane that allows for sharp details (Puretouch). Part of the collection is made from an extruded, impact-resistant polymer (Duropolymer), while the basic collection uses Durofoam. A large number of cornice mouldings, panel mouldings and skirting are available in a flexible variant: Orac Flex. Naturally, all of these products can be painted immediately.


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Daily mass readings : Solemnity of the Assumption of the Blessed Virgin Mary
Mass during the Day
Reading 1 : RV 11:19A; 12:1-6A, 10AB
God's temple in heaven was opened,
and the ark of his covenant could be seen in the temple.
A great sign appeared in the sky, a woman clothed with the sun,
with the moon under her feet,
and on her head a crown of twelve stars.
She was with child and wailed aloud in pain as she labored to give birth.
Then another sign appeared in the sky;
it was a huge red dragon, with seven heads and ten horns,
and on its heads were seven diadems.
Its tail swept away a third of the stars in the sky
and hurled them down to the earth.
Then the dragon stood before the woman about to give birth,
to devour her child when she gave birth.
She gave birth to a son, a male child,
destined to rule all the nations with an iron rod.
Her child was caught up to God and his throne.
The woman herself fled into the desert
where she had a place prepared by God.
Then I heard a loud voice in heaven say:
"Now have salvation and power come,
and the Kingdom of our God
and the authority of his Anointed One."
Responsorial Psalm : PS 45:10, 11, 12, 16
R. (10bc) The queen stands at your right hand, arrayed in gold.
The queen takes her place at your right hand in gold of Ophir.
R. The queen stands at your right hand, arrayed in gold.
Hear, O daughter, and see; turn your ear,
forget your people and your father's house.
R. The queen stands at your right hand, arrayed in gold.
So shall the king desire your beauty;
for he is your lord.
R. The queen stands at your right hand, arrayed in gold.
They are borne in with gladness and joy;
they enter the palace of the king.
R. The queen stands at your right hand, arrayed in gold.
Reading 2 : 1 COR 15:20-27
Brothers and sisters:
Christ has been raised from the dead,
the firstfruits of those who have fallen asleep.
For since death came through man,
the resurrection of the dead came also through man.
For just as in Adam all die,
so too in Christ shall all be brought to life,
but each one in proper order:
Christ the firstfruits;
then, at his coming, those who belong to Christ;
then comes the end,
when he hands over the Kingdom to his God and Father,
when he has destroyed every sovereignty
and every authority and power.
For he must reign until he has put all his enemies under his feet.
The last enemy to be destroyed is death,
for "he subjected everything under his feet."
R. Alleluia, alleluia.
Mary is taken up to heaven;
a chorus of angels exults.
R. Alleluia, alleluia.
Gospel : LK 1:39-56
Mary set out
and traveled to the hill country in haste
to a town of Judah,
where she entered the house of Zechariah
and greeted Elizabeth.
When Elizabeth heard Mary's greeting,
the infant leaped in her womb,
and Elizabeth, filled with the Holy Spirit,
cried out in a loud voice and said,
"Blessed are you among women,
and blessed is the fruit of your womb.
And how does this happen to me,
that the mother of my Lord should come to me?
For at the moment the sound of your greeting reached my ears,
the infant in my womb leaped for joy.
Blessed are you who believed
that what was spoken to you by the Lord
would be fulfilled."
And Mary said:
"My soul proclaims the greatness of the Lord;
my spirit rejoices in God my Savior
for he has looked with favor on his lowly servant.
From this day all generations will call me blessed:
the Almighty has done great things for me
and holy is his Name.
He has mercy on those who fear him
in every generation.
He has shown the strength of his arm,
and has scattered the proud in their conceit.
He has cast down the mighty from their thrones,
and has lifted up the lowly.
He has filled the hungry with good things,
and the rich he has sent away empty.
He has come to the help of his servant Israel
for he has remembered his promise of mercy,
the promise he made to our fathers,
to Abraham and his children forever."
Mary remained with her about three months
and then returned to her home.
Views: 7


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World Beyond War
Building a Global Movement to End All War
by David Swanson
I've been involved in starting enough activist campaigns and coalitions to know when one has more potential than any other I've seen. When hundreds of people and organizations are signing up on the website before you've announced it anywhere, and nine months before you plan to officially launch, and when a large percentage of the people signing on ask how they can donate funding, and when people from other countries volunteer to translate your declaration into other languages, and when committees form of volunteer women and men to work on a dozen different aspects of the planning — and they actually get to work in a serious way, and when none of this is due to anything in the news or any statement from anyone in government or any contrast between one political party and another, then it's time to start thinking about what you're going to help build as a movement.
In this case I'm talking about a movement to end, not this war or that war, but the institution of war as an acceptable enterprise for the human species. The declaration of peace that people and groups are signing reads, in its entirety:
"I understand that wars and militarism make us less safe rather than protect us, that they kill, injure and traumatize adults, children and infants, severely damage the natural environment, erode civil liberties, and drain our economies, siphoning resources from life-affirming activities. I commit to engage in and support nonviolent efforts to end all war and preparations for war and to create a sustainable and just peace."
This can be signed at http://WorldBeyondWar.org — and we fully expect a million people to sign it in short order. There's a great weariness in resisting militarism piecemeal, in reforming or refining war, in banning a weapon or exposing a tactic. All of that is a necessary part of the work. This will be a campaign of numerous partial victories, and we'll be directing our efforts toward various strategic weaknesses in the military-industrial complex. But there is enthusiasm right now for stopping not just missile strikes into Syria, not just deadly sanctions and threats to Iran, but stopping also — as part of these actions — the thinking that assumes war must always be with us, the casual discussions of how "the next war" will be fought.
So, we've set up an online center for addressing the concerns of the anyone who thinks we might need to keep war around or who thinks war will stay around regardless of what we do. We address a number of myths, including the myths that war is inevitable, and war is necessary, and war is beneficial. Then we provide a number of reasons for ending war, including these:
War threatens our environment.
We need $2 trillion/year for other things.
We've also provided an explanation of how nonviolent tools are more effective in resisting tyranny and oppression and resolving conflicts and achieving security than violence is, in other words how we can be more secure without war and without preparations for war.
This movement to abolish war, will be a movement to create a better world in which we are better able to address real crises, such as those in the earth's natural environment, rather than manufactured crises, such as the urgent need to drop missiles on Syria — which vanishes the moment we block that proposal.
Our plan is to announce on the International Day of Peace, September 21, 2014, a broader, wider, more mainstream and more international movement for peace and nonviolence than we've seen in a while, and a coalition capable of better uniting those doing good work toward that end in various corners of the globe and of our societies.
But we've only just begun to work out our plans, and we'd like everyone's input. If you go to http://WorldBeyondWar.org and sign the declaration, it will ask you to indicate how you might like to be involved beyond that. You can check any of a number of ways or invent your own. You can get involved in shaping our thinking and our plans and activities. You can also enter a brief statement of your own. Here are a few of the many entered already:
"I support this proposal and agree with this great and important initiative to abolish militarism and war. I will continue to speak out for an end to the institution of militarism and war and for institutions built on international law and human rights and nonviolent conflict resolution." — Mairead Maguire, Nobel Peace Laureate
"As a 29 year veteran of the US Army/Army Reserves, retiring as a Colonel and having served as a U.S. diplomat for 16 years and resigning in 2003 in opposition to the Iraq war, I firmly believe war does not resolve political issues. We must work diligently to force the governments of our nations to use diplomacy, not weapons." — Ann Wright
"Everybody's worried about stopping terrorism. Well, there's a really easy way: stop participating in it." — Noam Chomsky
"It is so inspiring to see a new group coming together not to focus on a particular war or weapons system, but on all war–everywhere. And it's great to have such beautifully crafted arguments about why war is not inevitable and how war contributes to so many other global ills. This coalition is worthy of Martin Luther King's call to end violence and instead put our energies and resources into 'life-affirming activities.' Bravo!" — Medea Benjamin
"We must work to end all war because: 1. In war there are no winners, only losers. 2. To thrive, humans need peace, which cannot be created by war. 3. We need all our ingenuity, creativity, technology and will to find a solution to runaway climate change. We cannot afford the military-industrial complex." — Sally Reynolds, Abingdon Peace Group
"The abolition of war is an idea whose time has come. We are at a transformative moment in history. Our Mother Earth is under siege from destructive global warming and industrialization. It is essential that we mobilize to save our planet. War is a cruel and untenable distraction, draining trillions of dollars and incalculable losses of intellectual firepower away from the essential work that needs to be done to create a livable future for humanity." — Alice Slater, Global Council of Abolition 2000
"War is a crime against humanity. When 90% of the casualties of war are civilians including children, its time to End ALL WARS! The world badly needs the resources to meet human and environmental needs. Wars are not making us more secure, but creating more enemies. There are more effective means of achieving security than war and killing other people's children. As former President Eisenhower said, 'I like to believe that the people of the world will want peace so much that governments will have to get out of the way and let them have it.' When the people of the world decide to end war, we can end it. At least 99% of the world's people do not benefit at all from all the wars our governments are waging. The time is NOW. Please join us." — David Hartsough
David Swanson is the author of War Is a Lie and Daybreak: Undoing the Imperial Presidency and Forming a More Perfect Union. He blogs at davidswanson.org and warisacrime.org, works as Campaign Coordinator for the online activist organization rootsaction.org, and hosts Talk Nation Radio. Among his many publishing venues, Swanson is a Contributing Author for New Clear Vision.


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Local integration - A success story in Lesotho
Local integration - A success story in Lesotho
MASERU, Lesotho (UNHCR) - Refugees are not a big issue in the Southern African kingdom of Lesotho. Not because there are none - the country hosts about 100 refugees - but because they have been well-integrated into local society and are practically Basotho, as Lesotho nationals are called.
Many of the refugees are living and working alongside their Basotho hosts. Some are married to Basotho women and have set up families in Lesotho. Like their hosts, they make their living by running business like hair salons or offering services like car battery-charging, mechanics and other small trading. There are also doctors, nurses, teachers and engineers among them.
Starting a new life in Lesotho can be tough and challenging. But Congolese Mvibudulu Menakuntima, like many of his fellow refugees, has managed to cope with the situation. He is gradually gaining acceptance in his host community, to the extent that his in-laws accepted his lobola [dowry] for his Basotho wife, a primary school teacher. The couple and their two children live in Upper Thamae, a densely populated and popular residential area in Lesotho's capital of Maseru.
Menakuntima runs a hair salon in a shabby room near his rented two-bedroom home. In order to expand his sources of revenue, he also sells second-hand spare parts in the same shop. Outside the shop, he has installed a car battery-charging device. He runs another car battery-charging service outside his home.
Like many Congolese, Menakuntima is very enterprising. He confided that he has constructed a house nearby, which he is renting to a Basotho national. He spends part of the rent he collects to pay for his own rented accommodation, and the rest to meet the costs of keeping his young family, including the payment of school fees. Gloria, his eldest daughter, goes to a privately-run English-medium primary school.
Originally from the Democratic Republic of the Congo (DRC), Menakuntima is among the hundreds of thousands who fled the recurring civil wars in his country to seek asylum abroad. But why Lesotho? "I wanted to get as far as my imagination could take me because I was sick and tired of continuous insecurity in my country," he said.
Menakuntima, whose first name means "patience" in Kikongo, knows all about perseverance. Leaving the DRC in 1992, he spent two months travelling through Angola, Namibia and South Africa before arriving in Lesotho.
His parents knew of his intention to leave and supported his flight. He started his journey alone from his home in Matadi, south-west of Kinshasa, and except for the few people he met on the road, including other asylum seekers, he kept to himself until his final destination.
"I took some few clothes with me and $100 I had so painfully saved," he recalled.
He said the easiest part of his journey was the stretch between Matadi and the Angolan border, where, due to ongoing tension and a lack of working government machinery, he did not have to undergo immigration formalities. Once inside Angola, he first walked and hitchhiked to Nzeto, then travelled by canoe to Luanda. From there, he took buses and lorries to Windhoek in Namibia.
"Up to this moment I still had my $100 bill intact. I was obliged, however, to sell some of my clothes between Luanda and Windhoek in order to buy food and drink."
He encountered linguistic problems in Namibia, but luck was on his side. He came across another Congolese who spoke some English, enough to locate a long-distance truck driver who agreed to sneak him into South Africa. Having to pay for his way this time round, he changed his dollars into South African rands, a currency also acceptable in Namibia.
The truck driver dropped him off near a church around the Central Park in Johannesburg, where he met other asylum seekers from different countries, including his compatriots. They briefed him on the problems faced by asylum seekers in South Africa, and some advised him to move on to Swaziland or Lesotho.
By that time Menakuntima had neither money nor clothes to sell, having spent his money on transport and food. But again a Good Samaritan appeared and approached a priest at a church in downtown Johannesburg. He explained to the priest that Menakuntima was an asylum seeker from the DRC who, with no means of assistance in South Africa, would like to move on to a nearby country. The priest gave him 60 rands, which enabled him to board a bus to Maseru, where he is now happily in the process of integration.
Many - if not all - of the refugees hosted by Lesotho have been in the country for at least 10 years, and have gradually integrated into the Basotho community and way of life.
Local integration is one of the best long-term solutions to any refugee problem, the other options being voluntary repatriation to one's home country and resettlement to a third country.
In Southern Africa, UNHCR works closely with national governments to encourage local integration. Lesotho has signed and ratified all the international conventions on refugees. The kingdom's openness towards integrating refugees is an enviable example for many, not only for its partners in the Southern African Development Community (SADC), but also for countries outside the region.
The UN refugee agency pulled out of Lesotho in the mid-1990s, and currently has no assistance programme in the country. Refugee issues are handled by the UN Development Programme (UNDP) Representative in consultation with UNHCR in South Africa and the Lesotho National Commissioner of Refugees.
By Fidellis Swai
UNHCR Regional Office for Southern Africa