Euclid’s Elements

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Sir Henry Billingsley's first English version of Euclid's Elements, 1570.

Euclid's Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems.

CONTENT : A - F , G - L , M - R , S - Z , See also , External links


  • And the whole [is] greater than the part.
    • Euclid (ca. 300 BC) Elements, Book I, Common Notions 5
  • Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery.
    • Euclid, The Thirteen Books of Euclid’s Elements T. L. Heath. CUP Archive, 1925, p. 133.
  • Euclid’s Postulate 5 [The Parallel Axiom].
    That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
    • Euclid, The Thirteen Books of Euclid’s Elements T. L. Heath Vol. 1 (Cambridge, 1908), p. 202. Reported in Moritz (1914)

Quotes about Euclid's Elements[edit]

Quotes are arranged alphabetically by author
Most quotes are reported in Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914.

A - F[edit]

  • The Definition in the Elements, according to Clavius, is this: Magnitudes are said to be in the same Reason [ratio], a first to a second, and a third to a fourth, when the Equimultiples of the first and third according to any Multiplication whatsoever are both together either short of, equal to, or exceed the Equimultiples of the second and fourth, if those be taken, which answer one another.... Such is Euclid’s Definition of Proportions; that scare-Crow at which the over modest or slothful Dispositions of Men are generally affrighted: they are modest, who distrust their own Ability, as soon as a Difficulty appears, but they are slothful that will not give some Attention for the learning of Sciences; as if while we are involved in Obscurity we could clear ourselves without Labour. Both of 300 which Sorts of Persons are to be admonished, that the former be not discouraged, nor the latter refuse a little Care and Diligence when a Thing requires some Study.
    • Isaac Barrow, Mathematical Lectures, (London, 1734), p. 388. Reported in Moritz (1914)
  • Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton.
    As it was, the very perfection... of Greek geometry retarded progress for centuries.
  • 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.
  • Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.
  • The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.
  • This book [Euclid] has been for nearly twenty-two centuries the encouragement and guide of that scientific thought 296 which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematics was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her. And hence she was called, in the dialect of the Phythagoreans, “the purifier of the reasonable soul”
  • [Euclid] at once the inspiration and aspiration of scientific thought.
  • In England the geometry studied is that of Euclid, and I hope it never will be any other; for this reason, that so much has been written on Euclid, and all the difficulties of geometry have so uniformly been considered with reference to the form in which they appear in Euclid, that the study of that author is a better key to a great quantity of useful reading than any other.
    • Augustus De Morgan, Elements of Algebra, (London, 1837), Introduction. Reported in Moritz (1914)
  • No one has ever given so easy and natural a chain of geometrical consequences [as Euclid]. There is a never-erring truth in the results.
    • Augustus De Morgan, Smith’s Dictionary of Greek and Roman Biography and Mythology, (London, 1902); Article “Eucleides”
  • The sacred writings excepted, no Greek has been so much read and so variously translated as Euclid.
    • Augustus De Morgan in: Smith’s Dictionary of Greek and Roman Biology and Mythology, (London, 1902), Article, “Eucleides.” Reported in Moritz (1914) : Euclid’s Elements compared with Newton's Principia
  • The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the “Principia” of Newton,
    • Augustus De Morgan in: Smith’s Dictionary of Greek and Roman Biology and Mythology, (London, 1902), Article, “Eucleides.” Reported in Moritz (1914): About the translations of Euclid’s Elements.
  • If we consider him [Euclid] as meaning to be what his commentators have taken him to be, a model of the most unscrupulous 297 formal rigour, we can deny that he has altogether succeeded, though we admit that he made the nearest approach.
    • Augustus De Morgan in: Smith’s Dictionary of Greek and Roman Biology and Mythology, (London, 1902), Article, “Eucleides.” Reported in Moritz (1914): About the translations of Euclid’s Elements.
  • The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
    • Leonhard Euler, Carl B. Boyer on Euler's Introduction to the Analysis of the Infinite in "The Foremost Textbook of Modern Times" (1950)

G - L[edit]

  • Professor Klein then speaks of "that artistic finish that we admire in Euclid's Elements," and mentions Allman's important historical work. I heartily concur in this estimate of Euclid, and desire to contrast it with the error of Charles S. Peirce, in the Nation, where he speaks of "Euclid's proof (Elements Bk. I., props. 16 and 17)" as "really quite fallacious, because it uses no premises not as true in the case of spherics." Our bright American seems to have forgotten Euclid's Postulate 6 (Axiom 12 in Gregory, Axiom 9 in Heiberg), "Two straight lines cannot enclose a space;" that is, two straights having crossed never recur.
  • The “elements” of the Great Alexandrian remain for all time the first, and one may venture to assert, the only perfect model of logical exactness of principles, and of rigorous development of theorems. 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. Die Entwickelung der Mathematik in den letzten Jahrhunderten, (Tübingen, 1884), p. 7. Reported in Moritz (1914)
  • There has been a rush of competitors anxious to be first in the field with a new text-book on the more "practical" lines which now find so much favour. The natural desire of each teacher who writes such a text-book is to give prominence to some special nostrum which he has found successful with pupils. One result is, too often, a loss of a due sense of proportion... It is, perhaps too early yet to prophesy what will be the ultimate outcome of the new order of things; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more.
  • As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into the vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous.
    • Oliver Heaviside, Electro-Magnetic Theory, (London, 1893), Vol. 1, p. 148. Reported in Moritz (1914)
  • It is certain that from its completeness, uniformity and faultlessness, from its arrangement and progressive character, and from the universal adoption of the completest and best line of argument, Euclid’s “Elements” stand pre-eminently at the head of all human productions. In no science, in no department of knowledge, has anything appeared like this work: for upward of 2000 years it has commanded the admiration of mankind, and that period has suggested little toward its improvement.
    • Philip Kelland, Lectures on the Principles of Demonstrative Mathematics, (London, 1843), p. 17. Reported in Moritz (1914)
  • The Elements of Euclid is as small a part of mathematics as the Iliad is of literature; or as the sculpture of Phidias is of the world’s total art.
    • Cassius Jackson Keyser, Lectures on Science, Philosophy and Art (New York, 1908), p. 8. Reported in Moritz (1914)
  • Let us for the present admit, that a new work were written on a plan different from that of Euclid, constructed upon different principles, built upon different data, and exhibiting the leading results of geometrical science of a different order. Let us wave also the great improbability, that even an experienced instructor should execute a work superior to that which has been stamped with the approbation of ages, and consecrated, as it were, by the collected suffrage of the whole civilised globe. Still it may be questioned whether, on the whole, any real advantage would be gained. It is certain that all would not agree in their decision on the merits of such a work. Euclid once superseded, every teacher would esteem his own work the best, and every school would have its own class book. All that rigour and exactitude, which have so long excited the admiration of men of science, would be at an end. These very words would lose all definite meaning. Every school would have a different standard; matter of assumption in one, being matter of demonstration in others; until, at length, Geometry, in the ancient sense of the word, would be altogether frittered away, or be only considered as a particular application of Arithmetic and Algebra.

M - R[edit]

  • Newton had so remarkable a talent for mathematics that Euclid’s Geometry seemed to him “a trifling book,” and he wondered that any man should have taken the trouble to demonstrate propositions, the truth of which was so obvious to him at the first glance. But, on attempting to read the more abstruse geometry of Descartes, without having mastered the elements of the science, he was baffled, and was glad to come back again to his Euclid.
  • There is a traditional story about Newton: as a young student, he began the study of geometry, as was usual in his time, with the reading of the Elements of Euclid. He read the theorems, saw that they were true, and omitted the proofs. He wondered why anybody should take pains to prove things so evident. Many years later, however, he changed his opinion and praised Euclid. The story may be authentic or not ...
  • To suppose that so perfect a system as that of Euclid’s Elements was produced by one man, without any preceding model or materials, would be to suppose that Euclid was more than man. We ascribe to him as much as the weakness of human understanding will permit, if we suppose that the inventions in geometry, which had been made in a tract of preceding ages, were by him not only carried much further, but digested into so admirable a system, that his work obscured all that went before it, and made them be forgot and lost.
    • Thomas Reid, Essay on the Powers of the Human Mind, (Edinburgh, 1812), Vol. 2, p. 368. Reported in Moritz (1914)

S - Z[edit]

  • Upon these accounts it appeared necessary and I hope will prove acceptable to all lovers of accurate reasoning and of mathematical learning to remove such blemishes and restore the the principal Books of the Elements to their original accuracy, as far as I was able; especially since these Elements are the foundation of a science by which the investigation and discovery of very useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewise is of the greatest use in the arts both of peace and war, to many of which geometry is absolutely necessary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unskilful editors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been justly celebrated as the most accurate of geometers, and by restoring to him those things which Theon or others have suppressed, and which have these many ages been buried in oblivion.
  • Geometry is nothing if it be not rigorous, and the whole educational value of the study is lost, if strictness of demonstration be trifled with. The methods of Euclid are, by almost universal consent, unexceptionable in point of rigour.
  • The doctrine of proportion, as laid down in the fifth book of Euclid, is, probably, still unsurpassed as a masterpiece of exact reasoning; although the cumbrousness of the forms of expression which were adopted in the old geometry has led to the total exclusion of this part of the elements from the ordinary course of geometrical education. A zealous defender of Euclid might add with truth that the gap thus created in the elementary teaching of mathematics has never been adequately supplied.
    • Henry John Stephen Smith, Presidential Address British Association for the Advancement of Science, (1873); Nature, Vol. 8, p. 451. Reported in Moritz (1914)
  • It is enough we have the Work. A Work! whose Propositions have such an admirable Connexion and Dependence, that take away but one, and the whole falls; whose Method is the most just, admitting nothing without a Demonstration, and no Demonstration but from what foregoes; and these so convincing, elegant and perspicuous, that it is beyond the Skill of Man to contrive better. Here the most potent and diligent Carpers have never been able to set Footing: This is the happy Empire wherein Truth has had an uninterrupted Reign for upward of 2000 Years, in profound Peace: No Disturbance at all. 'Tis from hence the Heroes of the geometrical World receive Force to vanquish the obstinate and ignorant, and extend their Dominions.
  • I should rejoice to see … Euclid honourably shelved or buried “deeper than did ever plummet sound” out of the schoolboys’ reach; morphology introduced into the elements of algebra; projection, correlation, and motion accepted as aids to geometry; the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity, continuity, infinity, and familiarization with the doctrines of the imaginary and inconceivable.
  • The early study of Euclid made me a hater of geometry, … and yet, in spite of this repugnance, which had become a second nature in me, whenever I went far enough into any mathematical question, I found I touched, at last, a geometrical bottom.
  • In comparing the performance in Euclid with that in Arithmetic and Algebra there could be no doubt that Euclid had made the deepest and most beneficial impression: in fact it might be asserted that this constituted by far the most valuable part of the whole training to which such persons [students, the majority of which were not distinguished for mathematical taste and power] were subjected.
    • Isaac Todhunter, Essay on Elementary Geometry; Conflict of Studies and other Essays, (London, 1873), p. 167. Reported in Moritz (1914)
  • All such reasonings [natural philosophy, chemistry, agriculture, political economy, etc.] are, in comparison with mathematics, very complex; requiring so much more than that does, beyond the process of merely deducing the conclusion logically from the premises: so that it is no wonder that the longest mathematical demonstration should be much more easily constructed and understood, than a much shorter train of just reasoning concerning real facts. The former has been aptly compared to a long and steep, but even and regular, flight of steps, which tries the breath, and the strength, and the perseverance only; while the latter resembles a short, but rugged and uneven, ascent up a precipice, which requires a quick eye, agile limbs, and a firm step; and in which we have to tread now on this side, now on that—ever considering as we proceed, whether this or that projection will afford room for our foot, or whether some loose stone may not slide from under us. There are probably as many steps of pure reasoning in one of the longer of Euclid’s demonstrations, as in the whole of an argumentative treatise on some other subject, occupying perhaps a considerable volume.
    • Richard Whately, Elements of Logic, Bk. 4, chap. 2, sect. 5. Reported in Moritz (1914)
  • To seek for proof of geometrical propositions by an appeal to observation proves nothing in reality, except that the person who has recourse to such grounds has no due apprehension of the nature of geometrical demonstration. We have heard of persons who convince themselves by measurement that the geometrical rule respecting the squares on the sides of a right-angles triangle was true: but these were persons whose minds had been engrossed by practical habits, and in whom speculative development of the idea of space had been stifled by other employments.
    • William Whewell, The Philosophy of the Inductive Sciences, (London, 1858), Part 1, Bk. 2, chap. 1, sect. 4. Reported in Moritz (1914)

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