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Plato (427 BC to 437 BC)

The ludicrous state of solid geometry made me pass over this branch. *Republic*, VII, 528.

The knowledge of which geometry aims is the knowledge of the eternal.

*Republic*, VII, 52.

Archimedes of Syracuse (287 BC to 212 BC)

Archimedes to Eratosthenes greeting. ... certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

*The Method *

Nichomachus of Gerasa (about 100 AD)

... arithmetic ... which is the mother of geometry ...

*Introduction to Arithmetic*

Proclus Diadochus (411 to 485)

According to most accounts, geometry was first discovered among the Egyptians, taking its origin from the measurement of areas. For they found it necessary by reason of the flooding of the Nile, which wiped out everybody's proper boundaries. Nor is there anything surprising in that the discovery both of this and of the other sciences should have had its origin in a practical need, since everything which is in process of becoming progresses from the imperfect to the perfect.

*On Euclid*

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantity as such, music the relations between quantities, geometry magnitude at rest, spherics magnitude inherently moving.

*A Commentary on the First Book of Euclid's Elements*

Al-Biruni (973 to 1048)

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 teaachings, 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.

*Book on the Finding of Chords *

Abraham bar Hiyya (1070 to 1136)

Who wishes correctly to learn the ways to measure surfaces and to divide them, must necessarily thoroughly understand the general theorems of geometry and arithmetic, on which the teaching of measurement ... rests. If he has completely mastered these ideas, he ... can never deviate from the truth.

*Treatise on Mensuration *

Albrecht Dürer (1471 to 1528)

But when great and ingenious artists behold their so inept performances, not undeservedly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.

*The Art of Measurement.* 1525.

And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art...

*Course in the Art of Measurement*

Johannes Kepler (1571 to 1630)

Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God.

Conversation with the Sidereal Messenger [an open letter to Galileo Galilei]: *Dissertatio cum Nuncio Sidereo* (Prague, 1610)

George Berkeley (1685 to 1753)

The method of Fluxions is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature.

Colin Maclaurin (1698 to 1746)

The supposition of an infinitely little magnitude [is] too bold a Postulate for such a Science as Geometry.

*Treatise on Fluxions *

Jean D'Alembert (1717 to 1783)

Geometry, which should only obey Physics, when united with it sometimes commands it. If it happens that the question which we wish to examine is too complicated for all the elements to be able to enter into the analytical comparison we wish to make, we separate the more inconvenient [elements], we substitute others for them, less troublesome but also less real, and we are surprised to arrive, notwithstanding a painful labour, only at a result contradicted by nature; as if after having disguised it, cut it short or altered it, a purely mechanical combination could give it back to us.

*Essai d'une nouvelle théorie de la résistance des fluides* (1752)

Nikolai Lobachevsky (1792 to 1856)

In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it came to us from Euclid.

As belonging to these imperfections, I consider the obscurity in the fundamental concepts of the geometrical magnitudes and in the manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory of parallels, to fill which all efforts of mathematicians have so far been in vain.

*Geometric researches on the theory of parallels* (1840)

Bernhard Riemann (1826 to 1866)

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible.

From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

*On the hypotheses which lie at the foundation of geometry* (1854)

Peter Guthrie Tait (1831 to 1901)

Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical geometry.

Article on Quaternions *Encyclopaedia Britannica * (1911)

Henri Poincaré (1854 to 1912)

If geometry were an experimental science, it would not be an exact science. it would be subject to continual revision ... the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.

Alfred N Whitehead (1861 to 1947)

I regret that it has been necessary for me in this lecture to administer a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are ...

*The Concept of Nature*

William Osgood (1864 to 1947)

Geometry is the noblest branch of physics.

Maurice Böcher (1867 to 1918)

... there is what may perhaps be called the method of optimism which leads us either wilfully or instinctively to shut our eyes to the possibility of evil. Thus the optimist who treats a problem in algebra or analytic geometry will say, if he stops to reflect on what he is doing: "I know that I have no right to divide by zero; but there are so many other values which the expression by which I am dividing might have that I will assume that the Evil One has not thrown a zero in my denominator this time."

*Bulletin of the American Mathematical Society* **11** 1904, 134.

George Polya (1887 to 1985)

Geometry is the science of correct reasoning on incorrect figures.

Ludwig Wittgenstein (1889 to 1951)

We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.

*Tractatus Logico Philosophicus* (New York 1922).

William Edge (1904 to 1997)

Algebra exists only for the elucidation of geometry.

Jean Dieudonné (1906 to 1992)

Analytical geometry has never existed. There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry. Out with them!

For example, it is well known that Euclidean geometry is a special case of the theory of Hermitian operators in Hilbert spaces.

It is indubitable that a 50-year-old mathematician knows the mathematics he learned at 20 or 30, but has only notions, often rather vague, of the mathematics of his epoch, i.e. the period of time when he is 50.

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