(listed in order as typed in)
His Motto
Mere Math'ns
His Power
His favorite pursuits
The first theoretical astronomers
The greatest of arithmeticians
The math giant
Greatness of
Lectures to three students
Lectures to three students
His style and method
His estimate of Newton
On the advantage of new calculi
M. and experiment
His Disquisitiones Arithmeticae (part one)
His Disquisitiones Arithmeticae (part two)
M., the queen of sciences
On number theory
On imaginaries
On the notation of sin^2
On infinite magnitude
On non-Euclidean Geometry (part one)
On non-Euclidean Geometry (part two)
On non-Euclidean Geometry (part three)
On non-Euclidean Geometry (part four)
On non-Euclidean Geometry (part five)
On non-Euclidean Geometry (part six)
On the nature of space
On his respect of the female courage
His Motto
"In other branches of science, where quick publication seems to be so much desired, there may possibility be some excuse for giving to the world slovenly or ill-digested work, but there is no excuse in mathematics. The form ought to be as perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. "pauca sed matura" was Gauss' motto."
-Glaisher, J. W. L.
Presidential Address British Association for the Advancement of Science, Section A, (1890); Nature, Vol.42, p. 467.
Mere Math'ns
"It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally trie of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings."
-Gauss
Gauss-Schumacher Briefwechsel, Bd. 4, (Altoma, 1862), p. 387.
His Power
"In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own.... With still superior methods this same problem was solved by the illustrious Gauss in one hour."
-Cajori, F.
History of Mathematics (New York, 1897), p. 248.
His favorite pursuits
"Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns."
-Gauss to Bolyai
Briefwechsel (Schmidt-Stakel), (1899), p. 55.
The first theoretical astronomers
"[Gauss calculated the elements of the planet Ceres] and his analysis proved him to be the first of theoretical astronomers no less than the greatest of "arithmeticians."
-Ball, W. W. R.
History of Mathematics (London, 1901), p. 458.
The math giant
"The mathematical giant {Gauss}, who from his lofty heights embraces in one view the stars and the abysses..."
-Bolyai, W.
Kurzer Grundriss eines Versuchs (Maros Vasarhely, 1851), p.44.
Greatness of
"Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss."
-Kronecker, L.
Zahlentheorie, Teil 1 (Leipsig, 1901), p.43
Lecture to three students
"I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession."
-Gauss to Bessel, 1810
Gauss-Bessel Briefwechsel (1880), p.107.
His style and method
"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."
-Ball, W. W. R.
History of Mathematics (London, 1901), p. 463.
His estimate of Newton
"For other great mathematicians or philosophers, he [Gauss] used the epithets magnus, or clarus, or clarissumus; for Newton alone he kept the prefix summus."
-Ball, W. W. R.
History of Mathematics (London, 1901), p. 362.
On the advantage of new calculi
"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea, to solve them mechanically in complicated cases in which, without aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Möbius's calculus. Such conceptions unite as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."
-Gauss, C. J.
Werke, Bd. 8, p. 298.
M. and experiment
"In very many cases the most obvious and direct experimental method of investigating a given problem is extremely difficult, or for some reason or other untrustworthy. In such cases the mathematician can often point out some other problem more accessible to experimental treatment, the solution of which involves the solution of the former one. For example, if we try to deduce from direct experiments the law according to which one pole of a magnet attracts or repels a pole of another magnet, the observed action is so much complicated with the effects of the mutual induction of the magnets and of the forces due to the second pole of each magnet, that it is next to impossible to obtain results of any great accuracy. Gauss, however, showed how the law which applied in the case mentioned can be deduced from the deflections undergone by a small suspended magnetic needle when it is acted upon by a small fixed magnet placed successively in two determinate positions relatively to the needle; and being an experimentalist as well as a mathematician, he showed likewise how these deflections can be measured very easily and with great precision."
-Foster G. C.
Presidential Address British Association for the Advancement of Science, Section A (1877)
His Disquisitiones Arithmeticae (part one)
"The 'Disquisitiones Arithmeticae' that great book with seven seals."
-Mertz J. T.
A History of European Thought in the Nineteenth century (Edinburgh and London, 1903), p. 721
His Disquisitiones Arithmeticae (part two)
"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory."
-Matthew G. B.
Theory of Numbers (Cambridge, 1892), Part 1 sect. 48)
M., the queen of sciences
"Mathematics is the queen of sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."
-Gauss.
Sartorius von Walterhausen: Gauss zum Gedächtniss. (Leipzig, 1856), p.79.
On number theory
"The higher arithmetic presents us with an inexhaustible store of interesting truths, - of thruths too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remained concealed.
-Gauss C. F.
Preface to Eisenstein's Mathematische, Abhandlungen (Berlin, 1847), [H. J. S. Smith]
On imaginaries
"That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, Ö1 had been called direct, inverse, and lateral units, instead of positive, negative and imaginary (or even impossible) such an obscurity would have been out of question."
-Gauss C. F.
Theoria residiorum biquadraticorum Commentatio secunda; Werke, Bd. 2 (Goettingen, 1863), p. 177.
On the notation of sin^2
"Sin^2 is odious to me, even though Laplace made use of it; should it be feared that sin
^2 might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(^2), well then, let us write (sin)^2, but not sin^2, which by analogy should signify sin (sin.<br>
-Gauss C. F.<br>
<i>Gauss-Schumacher Briefwechsel, Bd. 3, p. 292; Bd. 4, p.63.</i>
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On infinite magnitude
"I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without retriction."
-Gauss C. F.
Brief an Schumacher (1831); Werke; Bd. 8 p. 216
On non-Euclidean geometry (part one)
"I am exceedingly sorry that I have failed to avail myself of out former greater proximity to learn more of your work on the foundations of geometry; it surely would have saved me much useless effort and given me more peace, than one of my disposition can enjoy so long as so much is left to consider in a metter of this kind. I have myself made much process in this matter (though my other heterogeneous occupations have left me but little time for this purpose); though the course which you assure me you have reached, as much to the desired end, which you assure me you have reached as to the questioning of the truth of geometry. It is true that I have found much which many would accept as proof, but which in my estimation proves nothing, for instance, if it could be shown that a rectilinear triangle is possible, whose area is greater than that of any given surface, then I could rigorously establish the whole of geometry. Now most people, no doubt, would grant this as an axiom, but not I; it is conceivable that, however distant apart the vertices of the triangle might be chosen, this area might yet always be below a certain limit. I have found several other such theorems, but none of them satisfies me."
-Gauss C. F.
Letter to Bolyai (1799); Werke, Bd. 8 (Göttingen, 1900), p. 169.
On non-Euclidean geometry (part two)
"On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case, the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure."
-Gauss C. F.
Letter to Gerling (1816); Werke, Bd. 8 (Göttingen, 1900), p. 169.
On non-Euclidean geometry (part three)
"I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. perhaps in another world, we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics."
-Gauss C. F.
Letter to Olbers (1817); Werke, Bd. 8 (Göttingen, 1900), p. 177.
On non-Euclidean geometry (part four)
"There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180º. But it is otherwise with the statement that the sum of the angles cannot be less than 180º; this is the real Gordian knot, the rocks which cause the wreck of all.... I have been occupied with the problem over thirty years and I doubt if anyone has given it more serious attention, though I have never published anything concerning it. The assumption that the angle sum is less than 180º leads to a peculiar geometry, entirely different from Euclidean, but throughout consistent with itself. I have developed this geometry to my own satisfaction so that I can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori. The larger one assumes this constant the more nearly one approaches the Euclidean geometry, an infinitely large value makes the two coincide. The theorems of this geometry seem in part paradoxial, and to the unpracticed absurd; but on a closer and calm reflection it is found that in themselves they contain nothing impossible.... All my efforts to discover some contradiction, some inconsistency in this Non-Euclidean geometry have been fruitless, the one thing in it that seems contrary to reason is that space would have to contain a definitely determinate (though to us unknown) linear magnitude. However, it seems to me that notwithstanding the meaningless word-wisdom of the metaphysicians we know really too little, or nothing, concerning the true nature of space to confound what appears unnatural with the absolutely impossible. Should Non-Euclidean geometry be true, and this constant bear some relation to magnitudes which come within the domain of terrestrial or celestial measurement, it could be determined a posteriori."
-Gauss C. F.
Letter to taurinus (1824); Werke, Bd. 8 (Göttingen, 1900), p. 187.
On non-Euclidean geometry (part five)
"there is also another subject, which with me is nearly forty years old, to which I have again given some thought during leisure hours, I mean the foundations of geometry.... Here, too, I have consolidated many things, and my convictions has, if possible become more firm that geometry cannot be completely established on a priori grounds. in the mean time I shall probably not for along time yet put my very extended investigations concerning this matter in shape for publication, possibly not while I live, for I fear the cry of the Bœotians which would arise should I express my whole view on this matter. - It is curious too, that besides the known gap in Euclid's geometry, to fill which all efforts till now have been in vain, and which will never be filled, there exists another defect, which to my knowledge no one thus far has criticized and which (though possible) it is by no means easy to removed. This is the definition of a plane as a surface which wholly contains the line joining any two points. This definition contains more than is necessary to the determination of the surface, and tacity involves a theorem which demands proof."
-Gauss C. F.
Letter to Bessel (1829); Werke, Bd. 8 (Göttingen, 1900), p. 200.
On non-Euclidean geometry (part six)
"I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance.... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer v. Bolyai a genius of the first rank."
-Gauss C. F.
Letter to Gerling (1832); Werke, Bd. 8 (Göttingen, 1900), p. 221.
On the nature of space
"According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically."
-Sartorius, W. V. Waltershausen
Gauss zum Gedächtniss (Leipzig, 1856), p.81.
On his respect of the female courage
His Motto
"In other branches of science, where quick publication seems to be so much desired, there may possibility be some excuse for giving to the world slovenly or ill-digested work, but there is no excuse in mathematics. The form ought to be as perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. "pauca sed matura" was Gauss' motto."
-Glaisher, J. W. L.
Presidential Address British Association for the Advancement of Science, Section A, (1890); Nature, Vol.42, p. 467.
Mere Math'ns
"It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally trie of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings."
-Gauss
Gauss-Schumacher Briefwechsel, Bd. 4, (Altoma, 1862), p. 387.
His Power
"In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own.... With still superior methods this same problem was solved by the illustrious Gauss in one hour."
-Cajori, F.
History of Mathematics (New York, 1897), p. 248.
His favorite pursuits
"Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns."
-Gauss to Bolyai
Briefwechsel (Schmidt-Stakel), (1899), p. 55.
The first theoretical astronomers
"[Gauss calculated the elements of the planet Ceres] and his analysis proved him to be the first of theoretical astronomers no less than the greatest of "arithmeticians."
-Ball, W. W. R.
History of Mathematics (London, 1901), p. 458.
The math giant
"The mathematical giant {Gauss}, who from his lofty heights embraces in one view the stars and the abysses..."
-Bolyai, W.
Kurzer Grundriss eines Versuchs (Maros Vasarhely, 1851), p.44.
Greatness of
"Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss."
-Kronecker, L.
Zahlentheorie, Teil 1 (Leipsig, 1901), p.43
Lecture to three students
"I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession."
-Gauss to Bessel, 1810
Gauss-Bessel Briefwechsel (1880), p.107.
His style and method
"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."
-Ball, W. W. R.
History of Mathematics (London, 1901), p. 463.
His estimate of Newton
"For other great mathematicians or philosophers, he [Gauss] used the epithets magnus, or clarus, or clarissumus; for Newton alone he kept the prefix summus."
-Ball, W. W. R.
History of Mathematics (London, 1901), p. 362.
On the advantage of new calculi
"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea, to solve them mechanically in complicated cases in which, without aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Möbius's calculus. Such conceptions unite as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."
-Gauss, C. J.
Werke, Bd. 8, p. 298.
M. and experiment
"In very many cases the most obvious and direct experimental method of investigating a given problem is extremely difficult, or for some reason or other untrustworthy. In such cases the mathematician can often point out some other problem more accessible to experimental treatment, the solution of which involves the solution of the former one. For example, if we try to deduce from direct experiments the law according to which one pole of a magnet attracts or repels a pole of another magnet, the observed action is so much complicated with the effects of the mutual induction of the magnets and of the forces due to the second pole of each magnet, that it is next to impossible to obtain results of any great accuracy. Gauss, however, showed how the law which applied in the case mentioned can be deduced from the deflections undergone by a small suspended magnetic needle when it is acted upon by a small fixed magnet placed successively in two determinate positions relatively to the needle; and being an experimentalist as well as a mathematician, he showed likewise how these deflections can be measured very easily and with great precision."
-Foster G. C.
Presidential Address British Association for the Advancement of Science, Section A (1877)
His Disquisitiones Arithmeticae (part one)
"The 'Disquisitiones Arithmeticae' that great book with seven seals."
-Mertz J. T.
A History of European Thought in the Nineteenth century (Edinburgh and London, 1903), p. 721
His Disquisitiones Arithmeticae (part two)
"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory."
-Matthew G. B.
Theory of Numbers (Cambridge, 1892), Part 1 sect. 48)
M., the queen of sciences
"Mathematics is the queen of sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."
-Gauss.
Sartorius von Walterhausen: Gauss zum Gedächtniss. (Leipzig, 1856), p.79.
On number theory
"The higher arithmetic presents us with an inexhaustible store of interesting truths, - of thruths too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remained concealed.
-Gauss C. F.
Preface to Eisenstein's Mathematische, Abhandlungen (Berlin, 1847), [H. J. S. Smith]
On imaginaries
"That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, Ö1 had been called direct, inverse, and lateral units, instead of positive, negative and imaginary (or even impossible) such an obscurity would have been out of question."
-Gauss C. F.
Theoria residiorum biquadraticorum Commentatio secunda; Werke, Bd. 2 (Goettingen, 1863), p. 177.
On the notation of sin^2
"Sin^2 is odious to me, even though Laplace made use of it; should it be feared that sin^2 might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(^2), well then, let us write (sin)^2, but not sin^2, which by analogy should signify sin (sinBack to topOn infinite magnitude
"I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without retriction."
-Gauss C. F.
Brief an Schumacher (1831); Werke; Bd. 8 p. 216
On non-Euclidean geometry (part one)
"I am exceedingly sorry that I have failed to avail myself of out former greater proximity to learn more of your work on the foundations of geometry; it surely would have saved me much useless effort and given me more peace, than one of my disposition can enjoy so long as so much is left to consider in a metter of this kind. I have myself made much process in this matter (though my other heterogeneous occupations have left me but little time for this purpose); though the course which you assure me you have reached, as much to the desired end, which you assure me you have reached as to the questioning of the truth of geometry. It is true that I have found much which many would accept as proof, but which in my estimation proves nothing, for instance, if it could be shown that a rectilinear triangle is possible, whose area is greater than that of any given surface, then I could rigorously establish the whole of geometry. Now most people, no doubt, would grant this as an axiom, but not I; it is conceivable that, however distant apart the vertices of the triangle might be chosen, this area might yet always be below a certain limit. I have found several other such theorems, but none of them satisfies me."
-Gauss C. F.
Letter to Bolyai (1799); Werke, Bd. 8 (Göttingen, 1900), p. 169.
On non-Euclidean geometry (part two)
"On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case, the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure."
-Gauss C. F.
Letter to Gerling (1816); Werke, Bd. 8 (Göttingen, 1900), p. 169.
On non-Euclidean geometry (part three)
"I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. perhaps in another world, we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics."
-Gauss C. F.
Letter to Olbers (1817); Werke, Bd. 8 (Göttingen, 1900), p. 177.
On non-Euclidean geometry (part four)
"There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180º. But it is otherwise with the statement that the sum of the angles cannot be less than 180º; this is the real Gordian knot, the rocks which cause the wreck of all.... I have been occupied with the problem over thirty years and I doubt if anyone has given it more serious attention, though I have never published anything concerning it. The assumption that the angle sum is less than 180º leads to a peculiar geometry, entirely different from Euclidean, but throughout consistent with itself. I have developed this geometry to my own satisfaction so that I can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori. The larger one assumes this constant the more nearly one approaches the Euclidean geometry, an infinitely large value makes the two coincide. The theorems of this geometry seem in part paradoxial, and to the unpracticed absurd; but on a closer and calm reflection it is found that in themselves they contain nothing impossible.... All my efforts to discover some contradiction, some inconsistency in this Non-Euclidean geometry have been fruitless, the one thing in it that seems contrary to reason is that space would have to contain a definitely determinate (though to us unknown) linear magnitude. However, it seems to me that notwithstanding the meaningless word-wisdom of the metaphysicians we know really too little, or nothing, concerning the true nature of space to confound what appears unnatural with the absolutely impossible. Should Non-Euclidean geometry be true, and this constant bear some relation to magnitudes which come within the domain of terrestrial or celestial measurement, it could be determined a posteriori."
-Gauss C. F.
Letter to taurinus (1824); Werke, Bd. 8 (Göttingen, 1900), p. 187.
On non-Euclidean geometry (part five)
"there is also another subject, which with me is nearly forty years old, to which I have again given some thought during leisure hours, I mean the foundations of geometry.... Here, too, I have consolidated many things, and my convictions has, if possible become more firm that geometry cannot be completely established on a priori grounds. in the mean time I shall probably not for along time yet put my very extended investigations concerning this matter in shape for publication, possibly not while I live, for I fear the cry of the Bœotians which would arise should I express my whole view on this matter. - It is curious too, that besides the known gap in Euclid's geometry, to fill which all efforts till now have been in vain, and which will never be filled, there exists another defect, which to my knowledge no one thus far has criticized and which (though possible) it is by no means easy to removed. This is the definition of a plane as a surface which wholly contains the line joining any two points. This definition contains more than is necessary to the determination of the surface, and tacity involves a theorem which demands proof."
-Gauss C. F.
Letter to Bessel (1829); Werke, Bd. 8 (Göttingen, 1900), p. 200.
On non-Euclidean geometry (part six)
"I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance.... The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer v. Bolyai a genius of the first rank."
-Gauss C. F.
Letter to Gerling (1832); Werke, Bd. 8 (Göttingen, 1900), p. 221.
On the nature of space
"According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically."
-Sartorius, W. V. Waltershausen
Gauss zum Gedächtniss (Leipzig, 1856), p.81.
On his respect of the female courage
"But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur LeBlanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounters infinitely more obstacles that an man in familiarizing herself with complicated problems, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, without doubt she must have the noblest courage, quite extraordinary talents and superior genius."
-Carl Friedrich Gauss
Letter to Sophie Germain; circa April 1807)
Contributed by Jeni Rae Duschak.
