MATHEMATICS PLANE GEOMETRY. 



[REMARKS OK BOOK L 



circle, in fnct, as no human being ever formed. In like 

 manner, the straight line of Kuclid u rig.-n.twly what it 

 is affirmed to be /*//% (taught, and perfectly bnw.lt ii- 

 km Yon see, therefore, that we were fully warranted 

 in Mying, that Euclid c<mld not describe a circle, 

 nor draw a straight line; he has not, indeed, attempted 

 to do either ; the marks and diagrams which he exhibit! 

 to the eye, in connection with his reasoning*, are nothing 

 more than the outward symbols, of what actually exist* 

 in On mind alone. And the truths of geometry become 

 applicable to risible and tangible squares, circles, A-.-., 

 only on the supposition or assumption that they are per- 

 fect copies of our intellectual conceptions of these things. 

 A very able writer on Logic and mental Philosophy (Mr. 

 Stuart Mill), denies to the lines and figures of geometry, 

 the perfection here contended for. Assuming that all 

 our conceptions of form originate in our contemplation 

 of outward objects, which is no doubt true, he maintains 

 that our ideas of squares, circles, Ac., are only copies of 

 the confessedly imperfect forms presented to our eyes. 

 We would submit, however, that the mind can conceive 

 what it may surpass the powers of the hand to execute ; 

 and that we can imayine a perfection which art cannot 

 attain. A mere approximation to the perfect form 

 which is all that can be presented to the eye will sug- 

 gest the practically-unattainable perfection to the mind ; 

 and it may be safely asserted that the very infirmity f 

 our visual organs contributes to this perfection ; since 

 defects, too minute to be visible in the outward object, 

 cannot possibly accompany the mental impression of that 

 object. 



We have thought it right, in these introductory re- 

 marks, thus to state broadly, and unambiguously, what 

 the subject-matter of geometry really is. The objects 

 with which it deals, and to which its reasonings are 

 applied, are our perfect mental conceptions of figure, and 

 not the imperfect pictured forms wliich, to help these 

 conceptions, are traced upon paper. These are merely 

 the outward representations, or visible symbols of the 

 purely intellectual forms which they very conveniently 

 serve to suggest ; though, from the physical and instru- 

 mental imperfect ons which we know to be attached to 

 them, they are not accurately the things themselves ; 

 these latter, by an act of abstraction, being freed from 

 all material encumbrances ; so that, in fact, the forms 

 and figures of geometry are exclusively in the mind, and 

 not in matter. 



Tins is no bar to the practical applications of the sci- 

 ence. Whether a tangible square be perfect or imperfect 

 is of no moment, practically speaking, so long as its im- 

 perfections are undiscoverable by the senses ; inasmuch 

 as the rigorous conclusions of geometry may be applied 

 to it without practical or appreciable error. 



We now proceed to consider the basis upon which the 

 entire structure of geometry rests a basis so simple that 

 a child might lay the foundation-stones ; yet supporting 

 a fabric which, though so extensive, is, at the same time, 

 so secure, that the most powerful intellect cannot disturb 

 its stability. We need scarcely say that we allude to the 

 axioms and postulates of the science ; and, in connection 

 with the consideration of these, shall take occasion to 

 offer some suggestions as to the proper frame of mind in 

 which Euclid should be studied. 



Supposing, then, that you have your Euclid in your 

 hands, we commence by first directing your attention to 

 the fact, that the work is divided into distinct sections, 

 or Boo'.i; and that each book commences with an ex- 

 planation of the technical terms employed in it ; with 

 m concise but satisfactory description of the lines and 

 figures to be reasoned about, and a statement of the 

 elementary propositions to be admitted at possible, in the 

 practical constructions, and of the elementary proposi- 

 tions to be admitted at true in the reasonings. 



You will at once see, that in entering upon any doc- 

 trine which is to be established, not by the influence of 

 authority, but by the force of reasoning and sound argu- 

 ment, it is of much importance that preliminaries such 

 a* these should bo clearly and satisfactorily settled. If 

 * penon desire to communicate his own convictions to 



another, an<l, in undi-rtakin;; to do so, make reasoning 

 the only channel through which to convey them, there 

 must be first, a mutual concurrence an t<i the meaning 

 of the terms employed ; and secondly, a like concurrence 

 an to the fundamental principles to be assumed by the 

 one party, and admitted by the other. A good deal of 

 what goes by the name of reasoning and argument, in 

 the common affairs of life, is nothing but a sort of 

 wrangling disputation, solely from the neglect to estab- 

 lish a clear understanding on those points at the outset. 

 Euclid is careful to preclude this fertile source of am- 

 biguity, confusion, and error. He commences the seve- 

 ral portions of his subject with I><-jinitii>nt of the things 

 to be discussed, and of the peculiar terms to be employed 

 in the discussion ; he then tells you what he expects you 

 to admit as practically, or at least, as conceivably possible ; 

 and lastly, what he requires you to concede, without 

 demonstration, as necessarily true. 



You should not hurry over the definitions ; they have 

 been framed with great care. The character of a good 

 definition is this : that it is just sufficiently descriptive of 

 the thing defined to distinguish it from all other things, 

 but not more than sufficient for this purpose. If any- 

 thing more than what merely suffices to identify the 

 object defined be declared in a definition, that definition 

 is said to be redundant: it involves the assumption of 

 some property or peculiarity of the object, which it is 

 the province of reasoning to deduce from the properly- 

 restricted definition of it. All the properties of geometri- 

 cal figures are in this way deduced from, or as it were 

 drawn out of, the definitions of those figures ; for in the 

 definitions they are all virtually implied, and lie con- 

 cealed. If you were to define an equilateral triangle as 

 that which has three equal sides and three equal angles, 

 you would make a statement which is quite correct, as a 

 statement, but very faulty as a dffinitioii : the equality of 

 the three sides necessitates the equality of the three angle* 

 (Prop. V., Cor.), so that the equality of the angles is 

 virtually implied in the equality of the sides a truth 

 which must oe discovered to us by reasoning, not assumed 

 in a so-called definition. You will observe that Euclid 

 invariably constructs his figures solely in reference to the 

 descriptions of those figures embodied in the definitions, 

 quite regardless, at the time, of all other properties of 

 them ; and you will perceive that he has furnished 

 particulars just sufficient for this purpose, without one 

 superfluous item. 



NVo need scarcely state, that in speaking of Euclid 

 here, the emendations of Simson and other modern 

 editors are uniformly kept in view. It is much to be 

 regretted that, in the editions of Euclid most generally 

 studied, acknowledged blemishes are allowed to remain 

 in the text, while the proper emendations are given in 

 the form of notes at the end. Our veneration for a 

 writer on science should never be considered as ground 

 sufficient for us to endorse his errors and defects, nor 

 even to except to them only indirectly, and in the form 

 of supplementary annotation ; they ought, in justice to 

 him, as well as to those for whom he wrote, to be ex- 

 punged from the text of his instructions. In a work of 

 taste or imagination, the case would be different ; an 

 editor would have no right to replace the author's views 

 and peculiarities by his own ; but a book of science, so 

 extensively mud in education as Euclid is, should bo 

 rendered as perfect as possible ; and an editor of such a 

 book could incur little blame for expunging every ad- 

 mitted blemish from the text of his author. \Vu have 

 generally acted under this impression. 



Having thus previously establi.-he 1 the existence of his 

 geometrical forms, Euclid then proceeds, in his Theorems, 

 to deduce, by reasoning, all those properties necessarily, 

 though not obviously, implied in the definitions ; but, as 

 already noticed, before these existences can be proved, 

 .hat is to say, before the constructions employed by 

 Kuclid in his Problems can be actually effected, assent 

 uust bo given to the practicability, or, to be more explicit, 

 a the conceivability of certain fundamental operation*; 

 these are enumerated in the Postulates: and that before 

 the reasonings in his Theorems can be entered upon, 



