T..M 



MATHEMATICS. PLANE GEOMETRY. 



[I.I-M U:KS UN u.i.'K t. 



Euclid's definitions that you<io "grant'' his postulatea, 

 and that you Jo t ully acquioace in the truth of hii axiom*. 

 all preparatory ground U cleared ; and you may proceed 

 at ouce t- -..! proposed. And here you are to 



observe, that both you and your author occupy different 

 positions. Kuclid at once, from this point, drop* his 

 authority as a master ; you withdraw vour submission 

 as a docile and obedient pupil. What he say*, you are 

 ,-:rd no lon::T only what he prorrt. You are to 

 exercise a vigilant watchfulness over every statement lio 

 mnk t receiving just so much of it as you cannot help 

 receiving, and no more. Faith, in anything lie advances, 

 is not to be thought of. Be as sceptical as you please 

 my, as sceptical as you can. Kuclid would not thank 

 you for any gratuitous concession whatever ; all he 

 demands is, that you will honestly respect the prelimi- 

 nary articles of agreement ; and, in spite of all your 

 opposition, and of all your scepticism, he will comptl you 

 however much against your wish to do unqualified 

 homage not to Aim but to the truth he propounds. 



Ana this is the attitude of mind you are to assume 

 in entering upon the propositions of geometry. There 

 must be no yielding to the dicta of a teacher no intel- 

 lectual obeisance to the authority of a great name. 

 Every truth you acquire, you must *o acquire as to feel 

 and know it to be a truth, from your own perfect indi- 

 vidual diiinV/i<i/i that it is so. Your conviction must be 

 to thoroughly inwrought and complete, that if a Newton, 

 or even a greater than Newton, should attempt to con- 

 trovert a truth, thus secured, the effect of such an at- 

 tempt upon your mind would be about the same as an 

 Mid.-avour to convince you that you are an inhabitant 

 of the moon. 



Now we think that, from these unqualified statements, 

 you may fairly make two inferences, well worthy of con- 

 sideration. The first is, that even in this, our frail and 

 erring state, there is offered to our notice a system of 

 unadulterated and incontrovertible TRUTH, built up by 

 pur.-ly human effort, and consolidated and rendered im- 

 perishable by purely human reason. The second infer- 

 ence is, that the reasoning process, by which such an 

 intellectual structure has been reared, must surely be of 

 the most faultless kind no logical error can have been 

 committed no conceivable objection unanticipated, and 

 no case of exception unprovided f >r. 



These considerations alone seem to us amply sufficient 

 to incline all who have the time and opportunity, to a 

 diligent study of geometry, apart from all regard to 

 practical applications. Only reflect for a moment upon 

 the habitt of mind which such study must necessarily 

 foster, where they in any way exist, or create where they 

 are wanting. The frequent contemplation of Truth has 

 a salutary and an ennobling influence. Next to Inspired 

 Truth, the truths of pure science furnish the most 

 exalted materials upon which the human mind can 

 exercise its powers. He who is earnestly and twxafnUy 

 engaged in this exercise, comes, at length, to love truth 

 for its own intrinsic excellence ; to be fascinated with 

 it* unadorned beauty ; and to entertain increased repug- 

 nance towards the deformities of falsehood. Habits of 

 mind, whether good or bad, are the fruits of seeds usually 

 own in youth ; they become formed and fixed from the 

 natural effects of those trains of thought in which we 

 most frequently indulge in early life ; and hence the 

 study of geometry, and of the sciences which carry out 

 it* pure principles, have an important influence, even in 

 a moral point of view. To secure the operation of this 

 influence is surely deserving an effort. The properties 

 of geometrical figures may be matters of perfect indif- 

 ference to us we may take but little direct interest 

 in what relates to triangles, parallelograms, and circles ; 

 but we cannot be indifferent to a truthful habit of mind; 

 and though all the theorems of Kuclid In- forgotten, yet 

 if Uiit remain as an abiding result, how great will be the 

 acquisition we shall have made ! 



Hut the iiit-IUrtiial advantage* connected with the 

 study of the "exact sciences, '"are even more certain 

 and palpable than the moral adi ntagi-s here alluded to. 

 * ou cannot read a proposition of Euclid u it ought to , 



be read and indeed as it must be read, in order to be 

 fully understood without a concentration of ,v 

 more intense than most other subjects, out of mathe- 

 matics, demand ; and since, as just noticed, there 

 must be no disposition to admit anything win- 

 without the most complete comict'oii of its truth, a 

 habit of scrutinising evidence, and of dkttniniiahm( 

 between plausibilities and proofs, is insensibly but 

 securely acquired. There is, perhaps, no faculty of the 

 mind which in early life stands in more need of culti- 

 vation than the reasoning faculty ; for that every one 

 reasons, or at least engages in what goes by the name 

 of reasoning, is so generally admitted as a distinguishing 

 peculiarity, that man has been even denned to be " a 

 reasoning animal." Locke says, " Would you have a 

 man reason well let him learn geometry ;" that is to 

 say, if to reason well be the only end in view all the 

 truths of Euclid being regarded as utterly valueless 

 still let geometry be studied. Yet geometry supplies no 

 rules ; it prescribes no directions for conducting a logical 

 process ; but, what is better, it places before us a col- 

 lection of the most exquisite models. It teaches by ex- 

 ample, not by precept ; and no one, with proper atten- 

 tion, can fail to profit by its lessons. 



You see, therefore, that Euclid is something more 

 than a mere problem-book for the use of architects and 

 surveyors : it is the most finished treatise on the " Art 

 of Reasoning" that the world possesses ; and it is chiefly 

 as such that we are anxious to recommend to our \ 

 friends a careful study of its contents. It may fail to 

 render you much direct professional service ; but the 

 mental discipline it furnishes will strengthen your judg- 

 ment, improve your logic, give additional acuteness to 

 your penetration, And, in fact, so enlarge and invigorate 

 all the faculties of your mind, that you will be enabled 

 to bring a higher degree of intellectual power to bear 

 upon any pursuit in which you may earnest ly engage. 

 It is not the properties of geometrical fignn s that can 

 do this : it is the reasonings by which they are established. 

 The several stages at which you arrive, in your progress 

 through Euclid, may present but few points of attracts >n ; 

 1'iit you mtut be benefited by the invigorating influence 

 of the journey. 



And here it may perhaps be as well, in order to prevent 

 misunderstanding, that we should offer a remark or two 

 in reference to a direction given you above namely, 

 that you should approach the demonstrations of Euclid 

 in a sceptical spirit. It has been foolishly, and most un- 

 justly affirmed by some, that the study of pure science 

 has a tendency to produce general scepticism. Now all 

 truths are harmonious ; they have common features and 

 common attractions, recommending themselves to our 

 homage by the same dignified aspect and bearing. How 

 can scientific truth ever be out of keeping with inspiivd 

 truth ? A religious sceptic is generally something moro 

 than a mere neutral as respects divine things : he is 

 usually a denier ; that is, he embraces a negative proposi- 

 tion, and acts upon it without proof ! How does geometry 

 sanction this .' In Euclid there are negative propositions 

 as well as affirmative ones. \Vo recommend you, anterior 

 to proof, to be equally sceptical as to both. In things 

 out of geometry, geometrical demonstration is, of course, 

 not to be hod ; it would be folly to look for it ; yet, if in 

 such things an affirmative be declared on the one hand, 

 and a negative on the other, do you not think that the 

 logic of geometry, as well as the logic of common s. 

 would incline us to tiwt, in support of which fume evi- 

 dence was offered, rather than to that which hod no such 

 support at all I 



But there are sceptics of a different stamp from tha 

 class noticed above ; men of literary and philosophical 

 habits, who do not content themselves with a "cold ne- 

 gation." They address themselves to the task of under- 

 mining the existing evidences of the truth of Christianity. 

 Such a man was David Hume, a distinguished writer 

 of the hist century. He wrote an essay to prove that u 



' never could have been performed, or, at 1 

 that we have no reliable evidence of its performance. Ho 

 aid down certain preliminary principles, and dressed his 



