Celebrating 18 Years on the Web
TODAY IN SCIENCE HISTORY ®
Find science on or your birthday

Today in Science History - Quickie Quiz
Who said: “I have no satisfaction in formulas unless I feel their arithmetical magnitude.”
more quiz questions >>
Home > Category Index for Science Quotations > Category Index G > Category: Geometrical

Geometrical Quotes (10 quotes)

Astronomy was thus the cradle of the natural sciences and the starting point of geometrical theories. The stars themselves gave rise to the concept of a ‘point’; triangles, quadrangles and other geometrical figures appeared in the constellations; the circle was realized by the disc of the sun and the moon. Thus in an essentially intuitive fashion the elements of geometrical thinking came into existence.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 72.
Science quotes on:  |  Appear (115)  |  Astronomy (203)  |  Circle (55)  |  Concept (143)  |  Constellation (15)  |  Cradle (10)  |  Disk (3)  |  Element (162)  |  Essentially (14)  |  Existence (296)  |  Fashion (30)  |  Figure (68)  |  Give (200)  |  Intuitive (14)  |  Moon (199)  |  Natural Science (89)  |  Point (122)  |  Quadrangle (2)  |  Realize (90)  |  Rise (70)  |  Star (336)  |  Starting Point (13)  |  Sun (276)  |  Themselves (44)  |  Theory (690)  |  Think (341)  |  Triangle (10)

Geometrical axioms are neither synthetic a priori conclusions nor experimental facts. They are conventions: our choice, amongst all possible conventions, is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding all contradiction. ... In other words, axioms of geometry are only definitions in disguise.
That being so what ought one to think of this question: Is the Euclidean Geometry true?
The question is nonsense. One might as well ask whether the metric system is true and the old measures false; whether Cartesian co-ordinates are true and polar co-ordinates false.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 110.
Science quotes on:  |  A Priori (22)  |  Among (3)  |  Ask (157)  |  Avoid (52)  |  Axiom (52)  |  Cartesian (3)  |  Choice (79)  |  Conclusion (157)  |  Contradiction (54)  |  Convention (14)  |  Definition (191)  |  Disguise (10)  |  Euclidean (3)  |  Experimental (20)  |  Fact (725)  |  False (98)  |  Free (90)  |  Geometry (215)  |  Guide (62)  |  In Other Words (9)  |  Limit (123)  |  Measure (102)  |  Metric System (6)  |  Necessity (142)  |  Nonsense (40)  |  Old (147)  |  Polar (2)  |  Possible (155)  |  Question (404)  |  Remain (111)  |  Synthetic (16)  |  Think (341)  |  True (201)

Geometrical reasoning, and arithmetical process, have each its own office: to mix the two in elementary instruction, is injurious to the proper acquisition of both.
In Trigonometry and Double Algebra (1849), 92.
Science quotes on:  |  Acquisition (41)  |  Arithmetical (11)  |  Both (81)  |  Elementary (45)  |  Injurious (4)  |  Instruction (72)  |  Mix (19)  |  Office (22)  |  Process (261)  |  Proper (36)  |  Reason (454)  |  Teaching of Mathematics (31)

It has been the final aim of Lie from the beginning to make progress in the theory of differential equations; as subsidiary to this may be regarded both his geometrical developments and the theory of continuous groups.
In Lectures on Mathematics (1911), 24.
Science quotes on:  |  Aim (88)  |  Beginning (122)  |  Both (81)  |  Continuous (38)  |  Development (276)  |  Differential Equation (13)  |  Final (49)  |  Group (72)  |  Sophus Lie (5)  |  Mathematicians and Anecdotes (123)  |  Progress (362)  |  Regard (93)  |  Subsidiary (4)  |  Theory (690)

It needs scarcely be pointed out that in placing Mathematics at the head of Positive Philosophy, we are only extending the application of the principle which has governed our whole Classification. We are simply carrying back our principle to its first manifestation. Geometrical and Mechanical phenomena are the most general, the most simple, the most abstract of all,— the most irreducible to others, the most independent of them; serving, in fact, as a basis to all others. It follows that the study of them is an indispensable preliminary to that of all others. Therefore must Mathematics hold the first place in the hierarchy of the sciences, and be the point of departure of all Education whether general or special.
In Auguste Comte and Harriet Martineau (trans.), The Positive Philosophy (1858), Introduction, Chap. 2, 50.
Science quotes on:  |  Abstract (79)  |  Application (166)  |  Back (104)  |  Basis (89)  |  Carry (59)  |  Classification (85)  |  Departure (9)  |  Education (333)  |  Estimates of Mathematics (30)  |  Extend (41)  |  Fact (725)  |  First (313)  |  Follow (123)  |  General (156)  |  Govern (28)  |  Head (80)  |  Hierarchy (14)  |  Hold (92)  |  Independent (65)  |  Indispensable (27)  |  Irreducible (7)  |  Manifestation (33)  |  Mathematics (1149)  |  Mechanical (48)  |  Need (283)  |  Phenomenon (276)  |  Philosophy (257)  |  Place (174)  |  Point (122)  |  Positive (43)  |  Preliminary (5)  |  Principle (285)  |  Scarcely (13)  |  Science (2043)  |  Serve (57)  |  Simple (172)  |  Simply (52)  |  Special (74)  |  Study (461)  |  Whole (189)

That mathematics “do not cultivate the power of generalization,”; … will be admitted by no person of competent knowledge, except in a very qualified sense. The generalizations of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no contemptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards are products of a very high abstraction. … To perceive the mathematical laws common to the results of many mathematical operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler’s laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry—the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them—is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.
In An Examination of Sir William Hamilton’s Philosophy (1878), 612-13.
Science quotes on:  |  Abstract (79)  |  Abstraction (38)  |  Admit (44)  |  Agreement (39)  |  Aid (41)  |  Apprehension (15)  |  Arduous (3)  |  Binomial Theorem (3)  |  Call (127)  |  Case (98)  |  Class (83)  |  Common (118)  |  Competent (18)  |  Configuration (7)  |  Confuse (18)  |  Contemptible (8)  |  Creation (239)  |  Cultivate (19)  |  Demonstration (81)  |  Renι Descartes (81)  |  Diagram (13)  |  Difference (246)  |  Different (178)  |  Differential Calculus (8)  |  Difficulty (144)  |  Diversity (51)  |  Doubt (159)  |  Easy (98)  |  Effort (143)  |  Elementary (45)  |  Exercise (64)  |  Faculty (65)  |  Far (154)  |  Figure (68)  |  Fundamental (158)  |  Furnish (40)  |  Generalization (41)  |  Geometry (215)  |  Give (200)  |  Gravitation (38)  |  Great (524)  |  Group (72)  |  High (152)  |  Higher Mathematics (6)  |  Imagine (74)  |  Inductive (10)  |  Involve (47)  |  Johannes Kepler (90)  |  Knowledge (1293)  |  Large (130)  |  Law (513)  |  Lesson (41)  |  Line (89)  |  Management (12)  |  Mathematics (1149)  |  Mental (78)  |  Nature Of Mathematics (77)  |  Notion (57)  |  Object (169)  |  Operation (118)  |  Part (220)  |  Particular (75)  |  Perceive (40)  |  Person (153)  |  Physical Science (65)  |  Place (174)  |  Point (122)  |  Power (358)  |  Practical (129)  |  Preparation (41)  |  Problem (490)  |  Process (261)  |  Product (82)  |  Purely (28)  |  Qualify (4)  |  Reason (454)  |  Relative (39)  |  Require (79)  |  Result (376)  |  Rise (70)  |  Same (155)  |  Science (2043)  |  Scientific Mind (5)  |  Seize (14)  |  Sense (315)  |  Simple (172)  |  Single (119)  |  Situation (52)  |  Solve (76)  |  Strangely (5)  |  Successor (9)  |  Superior (40)  |  Tension (9)  |  Theory (690)  |  Train (45)  |  Triangle (10)  |  Universal (100)  |  Upwards (6)  |  Useful (98)  |  Vigorous (20)  |  Whole (189)  |  Wide (27)

The power of my [steam] engine rises in a geometrical proportion, while the consumption of fuel has only an arithmetical ratio; in such proportion that every time I added one fourth more to the consumption of fuel, the powers of the engine were doubled.
From 'On the Origin of Steam Boats and Steam Wagons', Thomas Cooper (ed.), The Emporium of Arts and Sciences (Feb 1814), 2, No. 2, 211-212.
Science quotes on:  |  Arithmetical (11)  |  Consumption (11)  |  Double (15)  |  Fuel (31)  |  Power (358)  |  Proportion (70)  |  Ratio (19)  |  Rise (70)  |  Steam Engine (42)

To characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.
From 'Geometry and Empirical Science', collected in Carl Hempel and James H. Fetzer (ed.), The Philosophy of Carl G. Hempel: Studies in Science, Explanation, and Rationality (2001), Chap. 2, 24. Also Carl Hempel, 'Geometry and Empirical Science', collected in J.R. Newman (ed.), The World of Mathematics (1956), Vol. 3, 1641.
Science quotes on:  |  Actual (47)  |  Body (243)  |  Characteristic (94)  |  Characterize (17)  |  Concept (143)  |  Connotation (2)  |  Customary (4)  |  Disclaimer (2)  |  Figure (68)  |  Geometry (215)  |  Importance (216)  |  Intend (16)  |  Movie (16)  |  Portrayal (2)  |  Primitive (41)  |  Property (123)  |  Pure (98)  |  Purely (28)  |  Relationship (71)  |  Similarity (20)  |  Spatial (8)

To emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations, ... let me take one more instance, by choosing a subject in which the purely mathematical interest is deemed supreme, the theory of functions of a complex variable. That at least is a theory in pure mathematics, initiated in that region, and developed in that region; it is built up in scores of papers, and its plan certainly has not been, and is not now, dominated or guided by considerations of applicability to natural phenomena. Yet what has turned out to be its relation to practical issues? The investigations of Lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation; which is merely the general geometrical method of regarding functional relations in that theory. Again, the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, made in the last twenty years, can all be, and now are all, I believe, deduced from similar considerations by interpreting functional relations between complex variables. In the dynamics of a rotating heavy body, the only substantial extension of our knowledge since the time of Lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion. Further, under the title of conjugate functions, the theory has been applied to various questions in electrostatics, particularly in connection with condensers and electrometers. And, lastly, in the domain of physical astronomy, some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas, the principles, the methods, and the results of the theory of functions. … the refined and extremely difficult work of Poincare and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects, developments which were made without a thought of such applications.
In Presidential Address British Association for the Advancement of Science, Section A, (1897), Nature, 56, 377.
Science quotes on:  |  Accept (65)  |  Accrue (3)  |  Achieve (63)  |  Advance (162)  |  Appear (115)  |  Applicability (6)  |  Application (166)  |  Apply (76)  |  Associate (14)  |  Astronomy (203)  |  Belief (503)  |  Body (243)  |  Build (117)  |  Certainly (31)  |  Chief (37)  |  Choose (59)  |  Complex (94)  |  Condenser (4)  |  Connection (107)  |  Consideration (85)  |  Conspicuous (7)  |  Construction (83)  |  Current (54)  |  Deduce (22)  |  Deem (6)  |  Develop (103)  |  Development (276)  |  Difficult (116)  |  Discontinuous (5)  |  Discussion (47)  |  Domain (40)  |  Dominate (19)  |  Dynamics (9)  |  Elaborate (20)  |  Electrostatic (5)  |  Emphasize (12)  |  Equation (93)  |  Extension (30)  |  Extremely (15)  |  Far (154)  |  Fluid Motion (2)  |  Function (128)  |  Functional (10)  |  General (156)  |  Guide (62)  |  Heavy (22)  |  Hydrodynamics (4)  |  Idea (577)  |  Important (202)  |  Initiate (6)  |  Instance (32)  |  Interest (235)  |  Interpret (18)  |  Introduce (41)  |  Investigation (175)  |  Issue (42)  |  Knowledge (1293)  |  Count Joseph-Louis de Lagrange (24)  |  Least (74)  |  Let (61)  |  Map (30)  |  Mathematician (364)  |  Mathematics (1149)  |  Merely (82)  |  Method (230)  |  Motion (158)  |  Natural (167)  |  Opinion (176)  |  Paper (82)  |  Particularly (21)  |  Phenomenon (276)  |  Physical (129)  |  Plan (87)  |  Henri Poincarι (93)  |  Portion (24)  |  Possible (155)  |  Practical (129)  |  Principle (285)  |  Property (123)  |  Pure Mathematics (63)  |  Purely (28)  |  Question (404)  |  Refine (4)  |  Regard (93)  |  Region (35)  |  Relation (149)  |  Representation (35)  |  Result (376)  |  Rotate (6)  |  Score (7)  |  Similar (35)  |  Sole (20)  |  Study And Research In Mathematics (59)  |  Subject (235)  |  Substantial (14)  |  Supreme (37)  |  Theory (690)  |  Thought (536)  |  Time (594)  |  Title (18)  |  Turned Out (4)  |  Unwise (4)  |  Variable (16)  |  Various (46)  |  Work (626)  |  Year (299)

[The famous attack of Sir William Hamilton on the tendency of mathematical studies] affords the most express evidence of those fatal lacunae in the circle of his knowledge, which unfitted him for taking a comprehensive or even an accurate view of the processes of the human mind in the establishment of truth. If there is any pre-requisite which all must see to be indispensable in one who attempts to give laws to the human intellect, it is a thorough acquaintance with the modes by which human intellect has proceeded, in the case where, by universal acknowledgment, grounded on subsequent direct verification, it has succeeded in ascertaining the greatest number of important and recondite truths. This requisite Sir W. Hamilton had not, in any tolerable degree, fulfilled. Even of pure mathematics he apparently knew little but the rudiments. Of mathematics as applied to investigating the laws of physical nature; of the mode in which the properties of number, extension, and figure, are made instrumental to the ascertainment of truths other than arithmetical or geometrical—it is too much to say that he had even a superficial knowledge: there is not a line in his works which shows him to have had any knowledge at all.
In Examination of Sir William Hamilton's Philosophy (1878), 607.
Science quotes on:  |  Accurate (32)  |  Acknowledgment (11)  |  Acquaintance (22)  |  Afford (16)  |  Apparently (19)  |  Apply (76)  |  Arithmetical (11)  |  Ascertain (15)  |  Ascertainment (2)  |  Attack (41)  |  Attempt (121)  |  Case (98)  |  Comprehensive (16)  |  Degree (81)  |  Direct (82)  |  Establishment (34)  |  Evidence (181)  |  Express (63)  |  Extension (30)  |  Famous (9)  |  Figure (68)  |  Fulfill (19)  |  Give (200)  |  Great (524)  |  Ground (90)  |  Hamilton (2)  |  Human Intellect (10)  |  Human Mind (80)  |  Important (202)  |  Indispensable (27)  |  Instrumental (5)  |  Investigate (65)  |  Know (547)  |  Knowledge (1293)  |  Law (513)  |  Line (89)  |  Little (184)  |  Mathematicians and Anecdotes (123)  |  Mathematics (1149)  |  Mode (40)  |  Nature (1211)  |  Number (276)  |  Physical (129)  |  Prerequisite (6)  |  Proceed (41)  |  Process (261)  |  Property (123)  |  Pure Mathematics (63)  |  Recondite (5)  |  Requisite (10)  |  Rudiment (4)  |  Say (228)  |  See (369)  |  Show (90)  |  Study (461)  |  Subsequent (19)  |  Succeed (26)  |  Superficial (11)  |  Tendency (54)  |  Thorough (17)  |  Tolerable (2)  |  Truth (914)  |  Unfitted (3)  |  Universal (100)  |  Verification (27)  |  View (171)  |  Work (626)


Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
Quotations by: • Albert Einstein • Isaac Newton • Lord Kelvin • Charles Darwin • Srinivasa Ramanujan • Carl Sagan • Florence Nightingale • Thomas Edison • Aristotle • Marie Curie • Benjamin Franklin • Winston Churchill • Galileo Galilei • Sigmund Freud • Robert Bunsen • Louis Pasteur • Theodore Roosevelt • Abraham Lincoln • Ronald Reagan • Leonardo DaVinci • Michio Kaku • Karl Popper • Johann Goethe • Robert Oppenheimer • Charles Kettering  ... (more people)

Quotations about: • Atomic  Bomb • Biology • Chemistry • Deforestation • Engineering • Anatomy • Astronomy • Bacteria • Biochemistry • Botany • Conservation • Dinosaur • Environment • Fractal • Genetics • Geology • History of Science • Invention • Jupiter • Knowledge • Love • Mathematics • Measurement • Medicine • Natural Resource • Organic Chemistry • Physics • Physician • Quantum Theory • Research • Science and Art • Teacher • Technology • Universe • Volcano • Virus • Wind Power • Women Scientists • X-Rays • Youth • Zoology  ... (more topics)
Sitewide search within all Today In Science History pages:
Visit our Science and Scientist Quotations index for more Science Quotes from archaeologists, biologists, chemists, geologists, inventors and inventions, mathematicians, physicists, pioneers in medicine, science events and technology.

Names index: | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

Categories index: | 1 | 2 | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

- 100 -
Sophie Germain
Gertrude Elion
Ernest Rutherford
James Chadwick
Marcel Proust
William Harvey
Johann Goethe
John Keynes
Carl Gauss
Paul Feyerabend
- 90 -
Antoine Lavoisier
Lise Meitner
Charles Babbage
Ibn Khaldun
Euclid
Ralph Emerson
Robert Bunsen
Frederick Banting
Andre Ampere
Winston Churchill
- 80 -
John Locke
Bronislaw Malinowski
Bible
Thomas Huxley
Alessandro Volta
Erwin Schrodinger
Wilhelm Roentgen
Louis Pasteur
Bertrand Russell
Jean Lamarck
- 70 -
Samuel Morse
John Wheeler
Nicolaus Copernicus
Robert Fulton
Pierre Laplace
Humphry Davy
Thomas Edison
Lord Kelvin
Theodore Roosevelt
Carolus Linnaeus
- 60 -
Francis Galton
Linus Pauling
Immanuel Kant
Martin Fischer
Robert Boyle
Karl Popper
Paul Dirac
Avicenna
James Watson
William Shakespeare
- 50 -
Stephen Hawking
Niels Bohr
Nikola Tesla
Rachel Carson
Max Planck
Henry Adams
Richard Dawkins
Werner Heisenberg
Alfred Wegener
John Dalton
- 40 -
Pierre Fermat
Edward Wilson
Johannes Kepler
Gustave Eiffel
Giordano Bruno
JJ Thomson
Thomas Kuhn
Leonardo DaVinci
Archimedes
David Hume
- 30 -
Andreas Vesalius
Rudolf Virchow
Richard Feynman
James Hutton
Alexander Fleming
Emile Durkheim
Benjamin Franklin
Robert Oppenheimer
Robert Hooke
Charles Kettering
- 20 -
Carl Sagan
James Maxwell
Marie Curie
Rene Descartes
Francis Crick
Hippocrates
Michael Faraday
Srinivasa Ramanujan
Francis Bacon
Galileo Galilei
- 10 -
Aristotle
John Watson
Rosalind Franklin
Michio Kaku
Isaac Asimov
Charles Darwin
Sigmund Freud
Albert Einstein
Florence Nightingale
Isaac Newton



who invites your feedback
Thank you for sharing.
Today in Science History
Sign up for Newsletter
with quiz, quotes and more.