Euclid’s Elements

"1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. 13. A boundary is that which is an extremity of anything. 14. A figure is A at which js contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; 16. And the point is called the centre of the circle. 17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute. 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction."

"1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part."

"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."

"In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."

"If a straight line [AB] be bisected and a straight line [BD] be added to it in a straight line, the rectangle contained by the whole [AD]... and the added straight line [BD] together with the square on the half [CB] is equal to the square on the straight line [CD] made up of the half and the added straight line. For let a straight line AB be bisected at the point C, and let a straight line BD be added to it in a straight line; I say that the rectangle contained by AD, DB together with the square on CB is equal to the square on CD. For let the square CEFD be described on CD, and let DE be joined; through the point B let BG be drawn parallel to either EC or DF, through the point H let KM be drawn parallel to either AB or EF, and further through A let AK be drawn parallel to either CL or DM. Then, since AC is equal to CB, AL is also equal to CH. But CH is equal to HF. Therefore AL is also equal to HF. Let CM be added to each; therefore the whole AM is equal to the gnomon NOP. But AM is the rectangle AD, DB, for DM is equal to DB; therefore the gnomon NOP is also equal to the rectangle AD, DB. Let LG, which is equal to the square on BC, be added to each; therefore the rectangle contained by AD, DB together with the square on CB is equal to the square on CD."

"To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. Let AB be the given straight line; thus it is required to cut AB so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. For let the square ABDC be described on AB; let AC be bisected at the point E, and let BE be joined; let CA be drawn through to F, and let EF be made equal to BE; let the square FH be described on AF, and let GH be drawn through to K. I say that AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on AH. For, since the straight line AC has been bisected at E, and FA is added to it, the rectangle contained by CF, FA together with the square on AE is equal to the square on EF. [Bk II. Prop 6] But EF is equal to EB; therefore the rectangle CF, FA together with the square on AE is equal to the square on EB. But the squares on BA, AE are equal to the square on EB, for the angle at A is right [Bk I. Prop 47, Pythagorean theorem]; therefore the rectangle CF, FA together with the square on AE is equal to the squares on BA, AE. Let the square on AE be subtracted from each; therefore the rectangle CF, FA which remains is equal to the square on AB. Now the rectangle CF, FA is FK, for AF is equal to FG; and the square on AB is AD; therefore FK is equal to AD. Let AK be subtracted from each; therefore FH which remains is equal to HD. And HD is the rectangle AB, BH for AB is equal to BD; and FH is the square on AH; therefore the rectangle contained by AB, BH is equal to the square on HA. therefore the given straight line AB has been cut at H so as to make the rectangle contained by AB, BH equal to the square on HA."

"Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order."

"Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out."

"Similar polygons inscribed in circles are to one another as the squares on the diameters."

"Circles are to one another as the squares on the diameters."

"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."

"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."

"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."

"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."

"Out of the pictures which are all that we can really see, we imagine a world of solid things; and... this world is constructed so as to fullfil a certain code of rules, some called axioms, and some called definitions, and some called postulates, and some assumed in the course of demonstration, but all laid down in one form or another in Euclid’s Elements of Geometry. ...This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one... 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—Mathematic 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 Pythagoreans, “the purifier of the reasonable soul.” Being thus in itself at once the inspiration and the aspiration of scientific thought, this book of Euclid has had a history as chequered as that of human progress itself."

"The discovery of incommensurable quantities threw an awful wrench in the machinery of geometry... The difficulty was finally overcome by Eudoxus' theory of proportion. But there was an indirect scare... In Euclid the theory of proportion and similar figures is postponed until the last possible moment, quite contrary to our present practice. Meanwhile, theorems which we prove by proportion were handled by the method of Application of Areas... The credit for discovering this seems to belong to the Pythagoreans:'According to the familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient, and belongs to the Pythagorean use.'What Proclus means...is... An area is applied to a given line segment, if we construct thereon a parallelogram of the given area and having a given angle. If the side of the parallelogram include not only the segment, but a prolongation, that part which is built on the extension is called the excess. On the other hand, if we use but a part of the segment, the parallelogram of the same height built on the unused part is called the defect. Let us see how the Greeks actually used the method. The deux ex machina was a simple figure called the 'gnomon'... Let us say that two plane figures are equivalent if they can be divide into the same number of figures... congruent, in pairs. They shall be called equivalent by completion if, by adding equivalent figures to them, the results are equivalent. ...we come to the actual use of the gnomon... in Euclid, II. 5... This is the identity\alpha \beta + (\frac{\alpha - \beta}{2})^2 = (\frac{\alpha + \beta}{2})^2Suppose... we wish to find the fourth proportional to... \alpha, \beta, \gamma. Euclid would reword this... apply to the line \alpha an area equal to that included by the lines \beta and \gamma. ...We construct a rectangle with non-parallel sides \beta and \gamma, extend the \beta side by the length \alpha, and complete the gnomon."

"The existence of incommensurable geometric magnitudes... necessitated a thorough reexamination and recasting of the foundations of mathematics, a task that occupied much of the fourth century B.C. During this period Greek algebra and geometry assumed the highly organized and rigorously deductive form that is set forth in the 13 books of the Elements that Euclid wrote about in 300 B.C. This systematic exposition of the Greek mathematical accomplishments of the preceding three centuries is the earliest major Greek mathematical text that is now available...(due perhaps to the extent to which the Elements subsumed previous expositions)."

"The discovery of incommensurability meant that there existed geometric magnitudes that could not be measured by numbers! ...It followed that geometric magnitudes could not be manipulated without hesitation in algebraic computations just as though they were numbers. ...The Greek answer ...is presented in the geometric algebra of Books II and VI... The product of two lengths a and b is not a third length, but rather the area of a rectangle with sides a and b. ... The principle Greek technique for the geometric solution of algebraic equations was based on the "application of areas." For example, given a segment... of length a, the construction in Proposition I.44 (Proposition 44 of Book I) ...a rectangle with base AB of length a and area equal to that of a given square of edge b... provides a solution of the equation ax = b^2. This corresponds to geometric division [x = \frac{b^2}{a}], and we say that the given area b^2 has been applied to the given segment AB. Proposition VI.28... shows how to apply a given area b^2 to a given segment of length a, but "deficient" by a square. ...This construction provides a geometric solution of the quadratic equation ax - x^2 = b^2."

"The Greeks... were well aware of geometric magnitudes that we call "irrational," but simply did not think of them as numbers. ...a direct result of their unyielding insistence on logical rigor. ...Book X ...devotes 115 propositions... to a comprehensive classification of irrational magnitudes of the forms a \pm \sqrt{b}, \sqrt{a} \pm \sqrt{b}, \sqrt{a \pm \sqrt{b}}, and \sqrt{\sqrt{a} \pm \sqrt{b}}, where a and b are commensurable lengths."

"I want now to glance for a moment at the development of the theoretical method, and while doing so especially to observe the relation of pure theory to the totality of the data of experience. Here is the eternal antithesis of the two inseparable constituents of human knowledge, Experience and Reason, within the sphere of physics. We honour ancient Greece as the cradle of western science. She for the first time created the intellectual miracle of a logical system, the assertions of which followed one from another with such rigor that not one of the demonstrated propositions admitted of the slightest doubt—Euclid's geometry]]. This marvellous accomplishment of reason gave to the human spirit the confidence it needed for its future achievements. ...But yet the time was not ripe for a science that could comprehend reality, was not ripe until a second elementary truth had been realized, which only became the common property of philosophers after Kepler and Galileo. Pure logical thinking can give us no knowledge whatsoever of the world of experience; all knowledge about reality begins with experience and terminates in it."

"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."

"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."

"The sacred writings excepted, no Greek has been so much read and so variously translated as Euclid."

"The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the “Principia” of Newton,"

"If we consider him [Euclid] as meaning to be what his commentators have taken him to be, a model of the most unscrupulous formal rigour, we can deny that he has altogether succeeded, though we admit that he made the nearest approach."

"Euclid himself is an elusive figure. We know nothing of him from contemporary references … Euclid’s Elements, which is believed to have been compiled around 300 BC, gives us an edited and anonymous treatment of some of the mathematics of Euclid’s predecessors, in a text that has passed through the hands of an unknown number of further scribes and editors before arriving at the versions that we now possess. ...Our evidence concerning pre-Euclidean mathematics is so indirect and fragmentary that the greatest part of the stories that are now told about the period must be invention."

"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."

"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."

"All our Greek texts of the Elements up to a century ago…purport in their titles to be either ‘from the edition of Theon’…or ‘from the lectures of Theon... [Greek commentaries] commonly speak of the writer of the Elements instead of using his name."

"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."

"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."

"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."

"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."

"Abraham now thinks that the aggregate of all his schooling did not amount to one year. He was never in a college or academy as a student, and never inside of a college or academy building till since he had a law license. What he has in the way of education he has picked up. After he was twenty-three and had separated from his father, he studied English grammar—imperfectly of course, but so as to speak and write as well as he now does. He studied and nearly mastered the six books of Euclid since he was a member of Congress. He regrets his want of education, and does what he can to supply the want. In his tenth year he was kicked by a horse, and apparently killed for a time."

"The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it."

"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 solve [Book I] Prop. XI algebraically, or to find the point H in AB such that the rectangle contained by the whole line AB and the part HB shall be equal to the square of the other part AH. Let AB contain a linear units, and AH one of the unknown parts contain x units, then the other part HB contains a - x units. And \therefore a(a - x) = x^2, by the problem, or x^2 + ax = a^2, a quadratic equation.Hence [by the ] x = \frac{\pm\, a \sqrt{5} - a}{2}.The former of these values of x determines the point H. So [substituting a = AB] that x = \frac{\sqrt{5}-1}{2} \cdot AB = AH, one part, and a - x = a - AH = \frac{3 - \sqrt{5}}{2} \cdot AB = HB, the other part. It may be observed that the parts AH and HB cannot be numerically expressed by any rational number. Approximation to their true values in terms of AB, may be made to any required degree of accuracy, by extending the extraction of the square root of 5 to any number of decimals."

"The property of similar triangles, and that contained in Prop. 47, Book I [the ], are most important theorems in Geometry."

"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 is a Platonist by persuasion and intimate with this system of philosophy, and so he has set before himself as goal of his system of Elements the constitution of the Platonic figures, as they are called."

"Proclus was obliged to defend his "pagan" religion, and especially the belief in political equity, against the communal mob violence, systematic book burning, and state repression targeted especially against his school by the new ruler-priests. It is in this context that Proclus turns to mathematics, for he regarded mathematics, and reason generally, as a key instrument in persuading those who violently advocated faith and inequity. It is not incidental that most theorems in the book are about equality (later reinterpreted as "congruence"). Read in the manner explained by Proclus, the Elements is a text which refutes point by subtle point all the key elements in the changed Christian doctrine of the 4th c. CE (reason vs faith, immanence vs transcendence, equity vs inequity, learning as reminiscence, hence past lives vs creation in the recent past, reincarnation vs resurrection, eternal truths hence an eternal cosmos vs apocalypse, images as aids to learning vs charges of idolatry)."

"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."

"Euclid's Elements is certainly one of the greatest books ever written, and one of the most perfect monuments of the Greek intellect."

"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."