Geometry Quotes - page 3
You well know ... for which reason I began searching for a number of demonstrations proving a statement due to the ancient Greeks ... and which passion I felt for the subject ... so that you reproached me my preoccupation with these chapters of geometry, not knowing the true essence of these subjects, which consists precisely in going in each matter beyond what is necessary. ... Whatever way he [the geometer] may go, through exercise will he be lifted from the physical to the divine teachings, which are little accessible because of the difficulty to understand their meaning ... and because the circumstance that not everybody is able to have a conception of them, especially not the one who turns away from the art of demonstration.
Al-Biruni
J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise.
Florian Cajori
Paradoxical as it may seem, a Latin prose or a geometry problem, even though they are done wrong, may be of a great service one day, provided we devote the right kind of effort to them. Should the occasion arise, they can one day make us better able to give someone in affliction exactly the help required to save him, at the supreme moment of his need.
Simone Weil
The progression of a painter's work, as it travels in time from point to point, will be toward clarity: toward the elimination of all obstacles between the painter and the idea, and between the idea and the observer. As examples of such obstacles, I give (among others) memory, history or geometry, which are swamps of generalization from which one might pull out parodies of ideas (which are ghosts) but never an idea in itself. To achieve this clarity is, inevitably, to be understood.
Mark Rothko
The confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are cultivated. ... We cannot truly account for our acceptance of such theories without endorsing our acknowledgement of a beauty that exhilarates and a profundity that entrances us.
Michael Polanyi
Let us calculate the motion of bodies, but also consult the plans of the Intelligence that makes them move.
It seems that the ancient philosophers made the first attempts at this sort of science, in looking for metaphysical relationships between numbers and material bodies. When they said that God occupies himself with geometry, they surely meant that He unites in that science the works of His power with the perspectives of His wisdom.
From the all too few ancient geometers who undertook such studies, we have little that is intelligible or well-founded. The perfection which geometry has acquired since their time puts us in a better position to succeed, and may more than compensate for the advantages that those great minds had over us.
Pierre Louis Maupertuis
One of the most curious of these cases [geometrical paradoxers] was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness. He published a school geometry based on his views, which received the endorsement of a well-known New York school official and, on the basis of this, was actually endorsed, or came very near being endorsed, as a text-book hi the public schools of New York.
Simon Newcomb
Trained in a less severe school than that of geometry and physics, his reasonings are almost always loose and inconclusive. His generalizations seem to have been reached before he had obtained the materials upon which he rests them: His facts, though frequently new and interesting, are often little more than conjectures; and the grand phenomena of the world of life, and instinct, and reason, which other minds have woven into noble and elevating truths, have thus become in Mr. Darwin's hands the basis of a dangerous and degrading speculation.
David Brewster
If heat were the affecting force, direct indications of its presence could be found which would not make use of geometry as an indirect method. ...direct evidence for the presence of heat is based on the fact that it affects different materials in different ways. ...The forces... which we have introduced... have two properties: (a) They affect all materials in the same way. (b) There are no insulating [or isolating] walls. ...the definition of the insulating wall may be added here: it is a covering made of any kind of material which does not act upon the enclosed object with forces having property a. Let us call the forces which have the properties a and b universal forces; all other forces are called differential forces. Then it can be said that differential forces, but not universal forces, are directly demonstrable.
Hans Reichenbach
Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i.e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.
Hans Reichenbach
Arthropods and vertebrates share some broad features of general organization - elongated, bilaterally symmetrical bodies, with sensory organs up front, excretory structures in the back, and some form of segmentation along the major axis. But the geometry of major internal organs could hardly be more different... Arthropods concentrate their nervous system on their ventral (belly) side as two major cords running along the bottom surface of the animal. The mouth also opens on the ventral side, with the esophagus passing between the two nerve cords, and the stomach and remainder of the digestive tube running along the body above the nerve cords. In vertebrates, and with maximal contrast, the central nervous system runs along the dorsal (top) surface as a single tube culminating in a bulbous brain at the front end. The entire digestive system then runs along the body axis below the nerve cord.
Stephen Jay Gould
In the application of the ordinary principles of geometry and trigonometry to such Astronomical measures... it may sometimes be expedient to resolve the process into several successive steps, and these steps may perhaps require different kinds of treatment. But... all are simple and within ordinary comprehension, and the only complexity arises from the circumstance that the student may find it necessary to have a clear view of several such steps at once...
George Biddell Airy