Euclid's Elements

“ The 'Elements' of Euclid, which have withstood the action of so many centuries, will have at least the honor before ending of arousing a long series of works admirably enchained which will contribute, in the most effective way, to the progress of mathematics, at the same time that they furnish to the philosophers the points of departure the most precise and the most solid for the study of the origin and of the formation of our cognitions. ”

“ In teaching elementary geometry in England we have for a long time been accustomed to use the well-known Elements of Euclid. At the present moment, when we learn from the best testimony, namely, the admission of anti-Euclideans, that both in France and Italy dissatisfaction is felt with the system hitherto used, accompanied with more or less desire to adopt ours, we are urged by many persons to exchange our system for one which is falling out of favour on the continent. ”

“ We say so far as they are usually given, because these demonstrations usually imply that the sides of rectangles can always be expressed exactly in terms of some unit of length; whereas the student will find hereafter that this is not the case, owing to the existence of what are technically called incommensurable magnitudes. We do not enter on this subject, as it would lead us too far from Euclid's Elements of Geometry with which we are here occupied. ”

“ A remarkable feature of Euclid's, and of all Greek geometry before Archimedes is that it eschews mensuration. Thus the theorem that the area of a triangle equals half the product of its base and its altitude is foreign to Euclid.
Another extant book of Euclid is the Data. It seems to have been written for those who, having completed the Elements, wish to acquire the power of solving new problems proposed to them. The Data is a course of practice in analysis. It contains little or nothing that an intelligent student could not pick up from the Elements itself. ”

“ On the Continent, Euclid has for many years been abandoned, and his place supplied by numerous treatises, certainly not models of geometrical rigour and arrangement. The fact that for twenty centuries the Elements, or parts of them, have held their ground as an introduction to geometry is a proof that they are, at any rate, not unsuitable for such a purpose. They are, speaking generally, not too difficult for novices in the science ”

“ That a proposition may be false and yet furnish consequences which are true, can be seen from a simple example. Suppose a theorem were proposed for investigation in the following words; one angle of a triangle is to another as the side opposite to the first angle is to the side opposite to the other. If this be assumed to be true we can immediately deduce Euclid's result in I. 19; but from Euclid's result in I. 19 we cannot retrace our steps and establish the proposed theorem, and in fact the proposed theorem is false. ”

“ Your demonstration proceeds too rapidly and, it seems to me, you keep on assuming that all of Euclid’s theorems are [272] as familiar and available to me as his first axioms, which is far from true. And now this fact which you spring upon us, that four times the rectangle ea.ad is less than the square of de because the two portions ea and ad of the line de are not equal brings me little composure of mind, but rather leaves me in suspense. Salv.
Indeed, all real mathematicians assume on the part of the reader perfect familiarity with at least the elements of Euclid ”

“ He is not content with every where unfolding the full, and most accurate meaning of Euclid; but he continually rises in his discourse, and leads us into the depths of the Pythagoric and Platonic philosophy. We are surprised to find an use in Geometry, which at present it is by no means suspected to afford. For who would conceive that it is the genuine passage to true theology, and the vestibule of divinity? This, indeed, is by no means the case when it is studied for lucre, and applied to mechanical purposes; for then the soul is neither elevated nor enlightened ”

“ For if any one mingles principles, and things flowing from principles into one and the same, he disturbs the whole order of knowledge, and conglutinates things which can never mutually agree; since a principle, and its emanating consequent, are naturally distinct from each other. In the first place, therefore (as I have said) , principles in the geometric institution are to be distinguished from their consequents, which is performed by Euclid in each of his books; who, before every treatise, exhibits the common principles of this science ”

“ In the same manner every other point between A and B lies within the circle.
2. Take any point E in AB produced either way. Join CE. Then the angle ABC is greater than AEC [I. xvi.] ; therefore CAB is greater than AEC. Hence CE is greater than CA, and the point E is without the circle.
We have added the second part of this Proposition. The indirect proof given of the first part in several editions of Euclid is very inelegant; it is one of those absurd things which give many students a dislike to Geometry. ”

“ To the student of Euclid, for example, the impossibility of adequately defining any of the fundamental elements of the science of geometry—the point, the line, the surface—is a familiar fact. In so far as a science of geometry is possible at all, the exactness, which is its essential characteristic, is only {11} attainable by starting from data which are in themselves impossible, as of a point which has no magnitude, of a line which has no breadth, of a surface which has no thickness. ”

“ Euclid uses the reductio ad absurdum chiefly when he is demonstrating the converse of some former theorem; see I. 14, 19, 25, 40.
Some remarks on indirect demonstration by Professor Sylvester, Professor De Morgan, and Dr Adamson will be found in the volumes of the Philosophical Magazine for 1852 and 1853.
I. 6 is not required by Euclid before he reaches II. 4; so that I. 6 might be removed from its present place and demonstrated hereafter in other ways if we please. For example, I. 6 might be placed after I. 18 and demonstrated thus. ”

“ In Modern Geometry a curve is considered as made up of an infinite number of points, which are placed in order along the curve, and then the secant through two consecutive points is a tangent. Euclid’s definition for a tangent is quite inadequate for any curve but the circle, and those derived from it by projection (the conic sections) ; and even for these the modern definition is better.
iv. Circles are said to touch one another when they meet, but do not intersect. There are two species of contact:—
1. When each circle is external to the other.
2. When one is inside the other. ”

“ Proclus preserves a reply made by Euclid to King Ptolemy, who asked whether he could not learn geometry more easily than by studying the Elements—“There is no royal road to geometry.” Pappus of Alexandria, in his Mathematical Collection, says that Euclid was a man of mild and inoffensive temperament, unpretending, and kind to all genuine students of mathematics. This being all that is known of the life and character of Euclid, it only remains therefore to speak of his works.
Among those which have come down to us the most remarkable is the Elements (Στοιχεῖα) (see Geometry) . ”

“ The tenth book is the longest of the Elements. It is occupied with the consideration of commensurable and incommensurable magnitudes, and ends with the proposition that the diagonal and the side of a square are incommensurable. With regard to straight lines, Euclid distinguishes between those which are commensurable or incommensurable in length, and those which are so in power, the latter being defined to be straight lines the squares on which have or have not a common measure. ”

“ It is, however, on the excellencies of his Elements of Geometry that the durable reputation of Euclid depends; and though the hypercriticism of modern mathematicians has perhaps successfully maintained such objections against them as that they might have been more precise in their axioms, that they sometimes assume what might be proved, that they are occasionally redundant, and their arrangement sometimes imperfect, yet they still maintain their ground as a model of extreme accuracy, of perspicuity, and as a standard of exact demonstration. ”

“ Hence, an element is that whose consideration passes to the science of other things, and from which we derive a solution of the doubts incident to the particular science we investigate. For as there are certain first principles of speech, most simple and indivisible, which we denominate elements, and from which all discourse is composed; so there are certain principal theorems of the whole of geometry, denominated elements, which have the respect of principles to the following theorems ”

“ Euclid's first demonstration resembles his first demonstration of III. 9. He shews that the centre of each circle is on the straight line which joins K with the middle point of the straight line BG, and also on the straight line which joins K with the middle point of the straight line BH; therefore K must be the centre of each circle.
The demonstration which Simson has chosen requires some additions to make it complete. For the point K might be supposed to fall without the circle DEF, or on its circumference, or within it; and of these three suppositions Euclid only considers the last. ”

“ Amidst the great progress of algebra through the sixteenth century, the geometers, content with what the ancients had left them, seem to have had little care but to elucidate their remains. Euclid was the object of their idolatry; no fault could be acknowledged in his elements, and to write a verbose commentary upon a few propositions was enough to make the reputation of a geometer. Among the almost innumerable editions of Euclid that appeared, those of Commandin and Clavius, both of them in the first rank of mathematicians for that age, may be distinguished. ”

“ We have certainly seen nowhere so good an introduction to Modern Geometry, or so copious a collection of those elementary propositions not given by Euclid, but which are absolutely indispensable for every student who intends to proceed to the study of the Higher Mathematics. ”

“ It may be observed that the two triangles in I. 8 are equal in all respects; Euclid however does not assert more than the equality of the angles opposite to the bases, and when he requires more than this result he obtains it by using I. 4.
I. 9. Here the equilateral triangle DEF is to be described on the side remote from A, because if it were described on the same, side, its vertex, F, might coincide with A, and then the construction would fail. I. II. The corollary was added by Simson, It is liable to serious objection. For we do not know how the perpendicular BE is to be drawn. ”

“ 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. ”

“ As the Elements of Euclid are usually placed in the hands of young students, it is important to exhibit the work in such a form as will assist them in overcoming the difficulties which they experience on their first introduction to processes of continuous argument. ”

“ The cornicular angle is that which is made from the periphery of a circle and its tangent; that is, the angle comprehended by the arch L A, and the right line F A, which Euclid in (16. 3.) proves to be less than any right-lined angle. And from this admirable proposition it follows, by a legitimate consequence, that any quantity may be continually and infinitely increased, but another infinitely diminished ”

“ The converse of a true proposition is not necessarily true; the student however will see, as he proceeds, that Euclid shews that the converses of many geometrical propositions are true.
I. 6 is an example of the indirect mode of demonstration, in which a result is established by shewing that some absurdity follows from supposing the required result to be untrue. Hence this mode of demonstration is called the reductio ad absurdum. Indirect demonstrations are often less esteemed than direct demonstrations; they are said to shew that a theorem is true rather than to shew why it is true. ”

“ In offering to students and teachers a new edition of the Elements of Euclid, it will be proper to give some account of the plan on which it has been arranged, and of the advantages which it hopes to present.
Geometry may be considered to form the real foundation of mathematical instruction. It is true that some acquaintance with Arithmetic and Algebra usually precedes the study of Geometry ”

“ VI. Def. 1. The second definition is useless, for Euclid makes no mention of reciprocal figures. VI. Def. 4. The fourth definition is strictly only applicable to a triangle, because no other figure has a point which can be exclusively called its vertex. The altitude of a parallelogram is the perpendicular drawn to the base from any point in the opposite side.
VI. 2. The enunciation of this important proposition is open to objection, for the manner in which the sides may be cut is not sufficiently limited. ”

“ Join OC. Now because AB passes through the centre, and cuts the other chord CD, which does not pass through the centre at right angles, it bisects it [iii.] . Again, because AB is divided equally in O and unequally in E, the rectangle AE.EB, together with OE2, is equal to OB2—that is, to OC2 [II. v.] ; but OC2 is equal to OE2 + EC2 [I. xlvii.] Therefore
Reject OE2, which is common, and we have AE.EB = EC2; but CE2 is equal to the rectangle CE.ED, since CE is equal to ED. Therefore the rectangle AE.EB is equal to the rectangle CE.ED.
3. ”