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NOTES TO BOOK I.
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
mate facts are so few, so distinct, and obvious, that the subsequent 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 tripsveta, conveys a very just idea, as marking the mere expansion which a body presents to our sense of sight. Line, or yeagua, 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 zecta in Latin, and droite in French ; and the last seems intimated by the English term straight, which is evidently derived from the verb to stretch. Accordingly Proclus defines a straight line as stretched between its extremities— two axes,
Tilaksyn. 3. The word Point in every language signifies a mark, thus indicating its essential character, of denoting position. In Greek, the term a royuz was first used: but, this being degraded in its application, the diminutive anoetov, formed from roua, 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
4. The conception of an Angle is one of the most difficult
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.” Euclid makes an angle to consist in “the mutual inclination, or xxurts, 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. o
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 géoyanov, ,
and rectanglum, dexterously, avoided that objection. The word Rhombus comes from #23&n, to sling, as the figure repre
sents 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 objection 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