NOTES AND ILLUSTRATIONS. NOTES TO BOOK I. DEFINITIONS. 1. THE primary objects which Geometry contemplates are, from their nature, incapable of decomposition. No wonder that ingenuity has only wasted its efforts to define such elementary notions. It appears more philosophical to invert the usual procedure, and endeavour to trace the successive steps by which the mind arrives at the principles of the science. Though no words can paint a simple sound, this may yet be rendered intelligible, by describing the mode of its articulation. The founders of mathematical learning among the Greeks were in general tinctured with a portion of mysticism, transmitted from Pythagoras, and cherished in the school of Plato. Geometry became thus infected at its source. By the later Platonists, who flourished in the Museum of Alexandria, it was regarded as a pure intellectual science, far sublimed above the grossness of material contact. Such visionary metaphysics could not impair the solidity of the superstructure, but did contribute to perpetuate some misconceptions, and to give a wrong turn to philosophical speculation. It is full time to restore the sobriety of reason. Geometry, like the other sciences which are not concerned about the operations of mind, must ultimately rest on external observation. But those ulti 1 mate facts are so few, so distinct, and obvious, that the subse quent train of reasoning is safely pursued to unlimited extent, without ever appealing again to the evidence of the senses, The science of Geometry, therefore, owes its perfection to the extreme simplicity of its basis, and derives no visible advantage from the artificial mode of its construction. The axioms are here rejected, as being totally useless, and rather apt to produce obscurity. 2. The term Surface, in Latin superficies, and in Greek επιφαyesa, conveys a very just idea, as marking the mere expansion which a body presents to our sense of sight. Line, or γραμμα, signifies a stroke; and, in reference to the operation of writing, it expresses the boundary or contour of a figure. A straight line has two radical properties, which are distinctly marked in different languages. It holds the same undeviating course-and it traces the shortest distance between its extreme points. The first property is expressed by the epithet recta in Latin, and droite in French; and the last seems intimated by the English term straight, which is evidently de rived from the verb to stretch. Accordingly Proclus defines a straight line as stretched between its extremities-ἡ επ' ακρων Τελαμενη. 3. The word Point in every language signifies a mark, thus indicating its essential character, of denoting position. In Greek, the term στιγμα was first used: but, this being degraded in its application, the diminutive σημειον, formed from σημα, a signal, came afterwards to be preferred. The neatest and most comprehensive description of a point was given by Pythagoras, who defined it to be " a monad having position." Plato represents the hypostasis, or constitution of a point, as adamantine; finely alluding to the opinion which then prevailed, that the diamond is absolutely indivisible, the art of cutting this refractory substance being the discovery of modern ages. 1. 4. The conception of an Angle is one of the most difficult 1 perhaps in the whole compass of Geometry. The term corresponds, in most languages, to corner, and therefore exhibits a most imperfect picture of the object intimated. Apollonius defined it to be " the collection of space about a point." Eu clid makes an angle to consist in " the mutual inclination, or κλισις, of its containing lines," -a definition which is obscure and altogether defective. In strictness, this can apply only to acute angles, nor does it give any idea of angular magnitude; though this really is as capable of augmentation as the magnitude of lines themselves. It is curious to observe the shifts to which the author of the Elements is hence obliged to have recourse. This remark is particularly exemplified in the 20th and 21st Propositions of his Third Book. Had Euclid been acquainted with Trigonometry, which was only begun to be cultivated in his time, he would certainly have taken a more enlarged view of the nature of an angle. 5. In the definition of Reverse Angle, I find that I have been anticipated by the famous mechanician Stevin of Bruges, who flourished about the end of the sixteenth century. It is satisfactory to have the countenance of an authority so highly respectable. 6. A Square is commonly described as having all its angles right. This definition errs however by excess, for it contains more than what is necessary. The original Greek, and even the Latin version, by employing the general terms ὁρθογωνιον, and rectanglum, dexterously, avoided that objection. The word Rhombus comes from ῥεμβῶν, to sling, as the figure represents only a quadrangular frame disjointed. The Lozenge, in heraldry and commerce, is that species of rhombus which is composed of two equilateral triangles placed on opposite sides of the same base. 7. It scarcely deserves notice, but I will anticipate the ob. jection which may be brought against me, for having changed the definition of Trapezium. The fact is, that I have only restricted the word to its appropriate meaning, from which Euclid had, according to Proclus, taken the liberty to depart. In the original, it signifies a table; and hence we learn the prevailing form of the tables used among the Greeks. Indeed the ancients would appear to have had some predilection ✔ for the figure of the trapez um, since the doors now seen in the ruins of the temples at Athens are not exactly oblong, but wider below than above. 8. Language is capable of more precision, in proportion as it becomes copious. As I have confined the epithet right to angles, and straight to lines, I have likewise appropriated the word diagonal to rectilineal figures, and diameter to the circle. In like manner, I have restricted the term arc to a portion of the circumference, its synonym arch being assigned to the use of architecture. For the same reason, I have adopted the term equivalent, from the celebrated Legendre, whose Elemens de Geometrie is one of the ablest works that has appeared in our times. These distinctions evidently tend to promote perspicuity, which is the great object of an elementary treatise.Euclid and all his successors define an isosceles triangle to have only two equal sides, which would absolutely exclude the equilateral triangle. Yet the equilateral triangle is afterwards assumed by them to be a species of isosceles triangle, since the equality of its angles is inferred at once as a corollary from the equality of the angles at the base of an isosceles triangle. This inadvertency, slight as it may appear, is now avoided. PROPOSITIONS. B A 9. The tenth Proposition may be very simply demonstrated, in the same manner as the next or its converse, by a direct appeal to superposition or mental experiment. For, suppose a copy of the triangle ABC were inverted and applied to it, the sides BA and BC being equal, if BA be laid on BC, the side BC again will evidently lie on BA, and the base AC coincide with CA. C Consequently the angle |