# Geometry

From Wikiquote

**Geometry** (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Quotes[edit]

*Quotes are arranged alphabetically by author*

### A - C[edit]

- The chemist smiles at the childish efforts of alchemists but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today.
- Florian Cajori (1893),
*A History of Mathematics*, p. 1.

- Florian Cajori (1893),

- When the value of mathematical training is called in question, quote the inscription over the entrance into the academy of Plato, the philosopher: "Let no one who is unacquainted with geometry enter here."
- Florian Cajori (1893),
*A History of Mathematics*, p. 3.

- Florian Cajori (1893),

- Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, Leibniz, and Lagrange played in creating that science.
- Florian Cajori (1893),
*A History of Mathematics*, p. 4.

- Florian Cajori (1893),

- The Egyptians carried geometry no further than was absolutely necessary for their practical wants. The Greeks, on the other hand, had within them a strong speculative tendency. They felt a craving to discover the reasons for things. They found pleasure in the contemplation of
*ideal*relations and loved science*as*science.- Florian Cajori (1893),
*A History of Mathematics*, p. 16.

- Florian Cajori (1893),

- The
*Eudemian Summary*says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression.- Florian Cajori (1893),
*A History of Mathematics*, p. 21.

- Florian Cajori (1893),

*Aristotle*(384-322 B.C.), the systematiser of deductive logic, though not a professed mathematician, promoted the science of geometry by improving some of the most difficult definitions. His*Physics*contains passages with suggestive hints of the principle of virtual velocities. About this time there appeared a work called*Mechanica*, of which he is regarded by some as the author. Mechanics was totally neglected by the Platonic school.- Florian Cajori (1893),
*A History of Mathematics*, p. 34.

- Florian Cajori (1893),

- When Ptolemy once asked Euclid if geometry could not be mastered by an easier process than by studying the
*Elements*, Euclid returned the answer, "There is no royal road to geometry."- Florian Cajori (1893),
*A History of Mathematics*, p. 35.

- Florian Cajori (1893),

- It is a remarkable fact in the history of geometry, that the
*Elements*of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences.- Florian Cajori (1893),
*A History of Mathematics*, p. 36.

- Florian Cajori (1893),

### D - L[edit]

- Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise.

- Since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art.
- Albrecht Dürer
*The Art of Measurement*(1525).

- Albrecht Dürer

- There is no royal road to geometry.
*μή εἶναι βασιλικήν ἀτραπόν ἐπί γεωμετρίαν**Non est regia [inquit Euclides] ad Geometriam via*- Reply given when the ruler Ptolemy I Soter asked Euclid if there was a shorter road to learning geometry than through Euclid's
*Elements*. - Attributed to Euclid by Proclus (412–485 AD) in
*Commentary on the First Book of Euclid's Elements*as translated by Glenn R. Morrow (1970), p. 57. ἀτραπός "road, trail, track" here takes the more specific sense of "short cut". The Latin translation is by Francesco Barozzi, 1560).

- I was informed by the priests at Thebes, that king Sesostris made a distribution of the territory of Egypt among all his subjects, assigning to each an equal portion of land in the form of a quadrangle, and that from these allotments he used to derive his revenue by exacting every year a certain tax. In cases however where a part of the land was washed away by the annual inundations of the Nile, the proprietor was permitted to present himself before the king, and signify what had happened. The king then used to send proper officers to examine and ascertain, by admeasurement, how much of the land had been washed away, in order that the amount of tax to be paid for the future, might be proportional to the land which remained. From this circumstance I am of opinion, that Geometry derived its origin; and from hence it was transmitted into Greece.

- He said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.
- H.P. Lovecraft
*The Call of Cthulu*, published in*Weird Tales*(1926)

- H.P. Lovecraft

### M - P[edit]

- I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."
- Benoît Mandelbrot As quoted in a review of
*The Fractal Geometry of Nature*by J. W. Cannon in*The American Mathematical Monthly*, Vol. 91, No. 9 (November 1984), p. 594.

- Benoît Mandelbrot As quoted in a review of

- I conceived, developed and applied in many areas a new [[geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it,
*fractal*geometry, from the Latin word for irregular and broken up,*fractus*. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit.- Benoît Mandelbrot As quoted in
*Encyclopedia of World Biography (1997) edited by Thomson Gale*

- Benoît Mandelbrot As quoted in

- If the Greeks had had a mind to reduce mathematics to one field... their only choice would have been to reduce arithmetic to geometry... it is hardly surprising that for nearly two millennia geometry took pride of place in mathematics. And it would have been obvious to any mathematician that a geometrical problem could not be stated or solved in the language of numbers, since the geometrical universe had more structure than the numerical universe.

If one desired to translate geometrical problems into the language of numbers,*one would have to invent (or discover) more numbers*.- Tim Maudlin,
*New Foundations for Physical Geometry: The Theory of Linear Structures*(2014)

- Tim Maudlin,

- O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all.
- Menaechmus (c. 350 BC) response to a request of Alexander the Great to be taught concisely, as quoted by Sir Thomas Little Heath,
*The Thirteen Books of Euclid's Elements*(1908) Vol.1*Introduction and Books I, II*p.1, citing Stobaeus,*Ecl.*(II. p. 228, 30, ed. Wachsmuth)

- Menaechmus (c. 350 BC) response to a request of Alexander the Great to be taught concisely, as quoted by Sir Thomas Little Heath,

- The doctrine of Proportion, in the Fifth Book of Euclid's
*Elements*, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's*Elements*, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it.

- Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration.
- Blaise Pascal,
*Œuvres*, Tom. I p. 66 as quoted by Robert Potts,*An Appendix to the larger edition of Euclid's Elements of Geometry*(1845)

- Blaise Pascal,

- Various relations being established in geometry between lines constituted under given conditions, as parts of geometrical figures,
*if*we choose to adopt the idea of expressing these lines by*numerical measures*, we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their*geometrical relations*however are absolutely general, and do not refer to any such distinction.- Rev. Baden Powell,
*On the Theory of Ratio and Proportion*(1836).

- Rev. Baden Powell,

### Q - Z[edit]

- It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. ...the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". ...If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry;
*the only exception is the formulation of the axiom of the parallels*. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry:*the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface*.- Hans Reichenbach (1928, tr. 1957)
*The Philosophy of Space and Time*§ 3.

- Hans Reichenbach (1928, tr. 1957)

- Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities.
- Hans Reichenbach (1928, tr. 1957)
*The Philosophy of Space and Time*§ 13.

- Hans Reichenbach (1928, tr. 1957)

- The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems which are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers... The eighteenth century doctrine of natural rights is a search for Euclidean axioms in politics. The form of Newton's
*Principia*, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source.- Bertrand Russell (1945)
*A History of Western Philosophy*Book One, Part I, Chapter III, Pythagoras, p. 36.

- Bertrand Russell (1945)

- The Greeks... discovered mathematics and the art of deductive reasoning. Geometry, in particular, is a Greek invention, without which modern science would have been impossible.
- Bertrand Russell (1945)
*A History of Western Philosophy*Book One, Part I, Chapter IV, Heraclitus, p. 39.

- Bertrand Russell (1945)