"The so-called 'Treasury of Analysis' [also Heuristic] is, to put it shortly, a special body of doctrine for the use of those who, after having studied the ordinary Elements, are desirous of acquiring the ability to solve mathematical problems, and it is useful for this alone. It is the work of three men, Euclid, the author of the Elements, Apollonius of Perga, and Aristaeus the elder. It teaches the procedures of analysis and synthesis." (Pappus of Alexandria, cca. 4th century BC)
"My father, in my earliest childhood, taught me the rudiments of arithmetic, and about that time made me acquainted with the arcana; whence he had come by this learning I know not. This was about my ninth year. Shortly after, he instructed me in the elements of the astronomy of Arabia, meanwhile trying to instill in me some system of theory for memorizing, for I had been poorly endowed with the ability to remember. After I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself. This is the knowledge I was able to acquire and learn without any elementary schooling [...]" (Girolamo Cardano, "De Vita Propria Liber" ["The Book of My Life"], 1576)
"Arithmetic [...] teaches all the various operations of numbers and demonstrates their properties. [...] The Greeks are said have received it from the Phoenicians. The ancients, who have treated arithmetic most exactness, are Euclid, Nicomachus of Alexandria, and Theon of Smyrna. It was difficult either for the Greeks or the Romans to succeed much in arithmetic, as both used only letters of the alphabet for numbers, the multiplication of which, in great calculations, necessarily occasioned abundance of trouble. The Arabic ciphers [...] are infinitely more commodious, and have contributed very much to the improvement of arithmetic." (Charles Rollin, "The Ancient History of the Egyptians, Carthaginians, Assyrians, Babylonians, Medes and Persians, Grecians and Macedonians", 1754)
"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." (Nikolai I Lobachevsky," Geometric researches on the theory of parallels", 1840)
"If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)
"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent." (Florian Cajori, "A History of Mathematics", 1893)
"Every one knows there are mathematical axioms. Mathematicians have, from the days of Euclid, very wisely laid down the axioms or first principles on which they reason. And the effect which this appears to have had upon the stability and happy progress of this science, gives no small encouragement to attempt to lay the foundation of other sciences in a similar manner, as far as we are able." (William K Clifford et al, "Scottish Philosophy of Common Sense", 1915)
"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." (Godfrey H Hardy, "A Mathematician's Apology", 1940)
"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions." (Eric T Bell, Mathematics Magazine, 1949)
"We shall now compare the non-Euclidean geometry of Lobachevski-Bolyai with the geometry of Euclid studied in high school. One is immediately struck by how much the two geometries have in common. In both geometries two points determine a unique line, and two lines can have at most one point in common (this follows from the fact that lines of hyperbolic geometry are segments of lines in the plane). Further, in both geometries it is possible to carry a point and a ray issuing from it, by a motion, into any other point and a preassigned ray issuing from the latter point. The hyperbolic length of a segment and magnitude of an angle share many properties with their Euclidean counterparts; for example, in both geometries the length of the sum of two segments is the sum of their lengths, and the measure of the sum of two angles is equal to the sum of their measures." (Isaak Yaglom, "Geometric Transformations", 1973)
"That is, the physicist likes to learn from particular illustrations of a general abstract concept. The mathematician, on the other hand, often eschews the particular in pursuit of the most abstract and general formulation possible. Although the mathematician may think from, or through, particular concrete examples in coming to appreciate the likely truth of very general statements, he will hide all those intuitive steps when he comes to present the conclusions of his thinking to outsiders. It presents the results of research as a hierarchy of definitions, theorems and proofs after the manner of Euclid; this minimizes unnecessary words but very effectively disguises the natural train of thought that led to the original results." (John D Barrow, "New Theories of Everything: The Quest for Ultimate Explanation", 1991)
"Calculation with numbers is the obvious model for calculation with letters, but a geometric model is also conceivable, since numbers can be interpreted as lengths. Indeed, the coordinate geometry of Fermat and Descartes was based on algebra. They found that the curves studied by the Greeks can be represented by equations, and that algebra unlocks their secrets more easily and systematically than classical geometry. But to apply algebra in the first place, Fermat and Descartes assumed classical geometry. In particular, they used Euclid’s parallel axiom and the concept of length to derive the equation of a straight line,"(John Stillwell, "The Four Pillars of Geometry", 2000)
"The many faces of geometry are not only a source of amazement and delight. They are also a great help to the learner and teacher. We all know that some students prefer to visualize, whereas others prefer to reason or to calculate. Geometry has something for everybody, and all students will find themselves building on their strengths at some times, and working to overcome weaknesses at other times. We also know that Euclid has some beautiful proofs, whereas other theorems are more beautifully proved by algebra. In the multifaceted approach, every theorem can be given an elegant proof, and theorems with radically different proofs can be viewed from different sides." (John Stillwell, "The Four Pillars of Geometry", 2000)
"Mathematicians often get bored by a problem after they have fully understood it and have given proofs of their conjectures. Sometimes they even forget the precise details of what they have done after the lapse of years, having refocused their interest in another area. The common notion of the mathematician contemplating timeless truths, thinking over the same proof again and again - Euclid looking on beauty bare - is rarely true in any static sense." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)
"There are many ways to use unique prime factorization, and it is rightly regarded as a powerful idea in number theory. In fact, it is more powerful than Euclid could have imagined. There are complex numbers that behave like 'integers' and 'primes', and unique prime factorization holds for them as well. Complex integers were first used around 1770 by Euler, who found they have almost magical powers to unlock secrets of ordinary integers. For example, by using numbers of the form a + b√ -2. where a and b are integers, he was able to prove a claim of Fermat that 27 is the only cube that exceeds a square by 2. Euler's results were correct, but partly by good luck. He did not really understand complex 'primes' and their behavior." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"Euclid sharply distinguished between number and magnitude, associating the former with the operation of counting and the latter with a line segment. So, for Euclid, only integers were numbers; even the notion of fractions as numbers had not yet been conceived of. He represented a whole number n by a line segment that was n times the chosen unit line segment. However, the opposite procedure of distinguishing all line segments by labeling them with numerals representing counting numbers was not possible. Obviously, this one-way correspondence of counting number with magnitude implies that the latter concept was more general than the former. The sharp distinction between counting number and magnitude, made by Euclid, was an impediment to the development of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"In 1884, just 40 years after the publication of Grassmann’s 'Algebra of Extension', Gibbs developed his vector algebra following the ideas of Grassmann by replacing the concept of the outer product by a new kind of product known as vector product and interpreted as an axial vector in an ad-hoc manner. This, in fact, went against the run of natural development of directed numbers started by Grassmann and completely changed the course of its development in the other direction. Grassmann’s outer product reveals the fact that the Greek distinction between number and magnitude has real geometric significance. Greek magnitudes, in fact, added like scalars but multiplied like vectors, asserting the geometric notions of direction and dimension to multiplication of Greek magnitudes. This revealing feature is a reminiscence of the distinction, carefully made by Euclid, between multiplication of magnitudes and that of numbers. Thus, Herman Grassmann fully accomplished the algebraic formulation of the basic ideas of Greek geometry begun by Renė Descartes." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often. especially in secondary and collegiate mathematics. the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubti mportant, they do not possess the richness and variety that one finds with inequalities." (Claudi Alsina & Roger B Nelsen, "When Less is More: Visualizing Basic Inequalities", 2009)
"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît B Mandelbrot, "The Fractalist", 2012)
"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
See also the quotes on Euclidean Geometry, respectively Non-Euclidean Geometry
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