KEMARKS ON BOOK I.] 



MATHEMATICS. PLANE GEOMETRY. 



555 



performance in the garb of fair reasoning. But Hume 

 knew nothing of mathematics ; and, in invading its pro- 

 vince, he trod upon slippery ground, and fell. Those who 

 did know something of the "exact sciences" betook 

 themselves to the investigation ; they started upon pre- 

 cisely the same principles as Hume did, without the 

 assumption of a single additional particular, and they 

 arrived, by a traiu of reasoning which it is impossible to 

 disturb, at a conclusion directly opposite to his Had 

 Hume known but a little of mathematics, and had he 

 revised his so-called argument, in the same sceptical spirit 

 which we recommend to you, a regard for his own literary 

 reputation merely, would have precluded him from ever 

 publishing to the world his Essay on Miracles.* 



You have already seen that the proposition.? of geo- 

 metry are of two kinds problems and theorems. A 

 proposition is called a problem, when the tiling proposed 

 is an operation to be performed a construction to be 

 effected : its object is a practical result, to be brought 

 about by a suitable disposal and combination of the ele- 

 mentary materials furnished by the postulates. A propo- 

 sition is called a theorem, when the thing proposed is a 

 truth to be demonstrated ; and for this demonstration 

 the elementary materials are furnished by the axioms. 

 It usually happens, however, that the proof of a theorem 

 requires the previous introduction of certain lines and 

 constructions ; and hence it is that Euclid commences his 

 fir.-t book with problems instead of theorems. The only 

 theorem in this book, that is quite independent of a pro- 

 i, is Proposition IV. Euclid probatily considered it 

 systematic to introduce the problems, which he 

 foresaw would be indispensable aa soon as the fourth 

 proposition was disposed of, at the beginning of the book, 

 than to interpose them between propositions so closely 

 related as the fourth and fifth. 



Of the three problems thus found to be necessary to 

 meet the demands of the subsequent theorems, the second 

 is the only one which seems to require any comment. 

 I ' .'liners in general find it difficult, and are apt to con- 

 sider that there is an unnecessary parade of geometrical 

 apparatus exhibited to effect so simple a matter as the 

 drawing of one straight line equal to another. There is 

 no doubt that a mere mechanic would pronounce the pro- 

 ceeding a very round-about one ; he would accomplish 

 the business at once, by drawing a straight line from A 

 at random ; and then, having taken the length B C in his 

 compasses, he would apply one foot at A, fixing it there 

 as a centre, and with the other foot would cut off the re- 

 quired length A L ; thus dispensing with all Euclid's 

 machinery the circles and the equilateral triangle. Now 

 this is all very well for the purposes of the practical 

 workman, who neither seeks nor expects rigid accuracy 

 in his constructions ; but you must remember that Euclid 

 ignores compasses, and that the instrumental transference 

 of one line to another is not warranted by any postulate. 

 No one can take accurately any stipulated length in a 

 pair of compasses ; the limitation of his vision precludes 

 ms pronouncing, with perfect certainty, that he has got 

 exactly the proposed length, neither more nor less. If 

 the minute error, whether in excess or in defect, be only 

 so small as to escape his senses, he cannot take cognizance 

 of it ; and he not only practically, but from necessity 

 disregards it. But without any additional postulate, 

 Euclid shows you how the thing proposed may be done 

 without the possibility of any error at all. In the or- 

 dinary editions of Euclid, we think justice is scarcely 

 done to the process indicated. You are directed, first, 

 to draw a line A E Imujrr than B C, and are then shown 

 how to cut off a part A L <-qnul to B C. Tne ingenuity 

 of Euclid's mode of proceeding would be more apparent, 

 if no superfluity of length were at first introduced. It 

 certainly seems a thing of much greater difficulty to draw 

 a line from a ]>oint A, till a certain prescribed length bo 

 attained, and then, but not till then, to stop. We would 

 therefore recommend you to leave the prolonging of D A 

 to the very la.st ; so that, having performed every other 



See Bahbage's /?riV/v'''<>r Treiitiie Appendix. Also Young's 

 Tkrte Ltclui ft on Mathftrt'iltcl. 



part of the construction, prolong D A, as a final step, till 



| the prolongation reaches to the circumference of the 



outer circle. A line A L will thus have been drawn equal 



to B C, and there will be no excess of length to throw 



I away. This is the mode of proceeding adopted in the 



present work. 



As to the other two problems, the first and third, but 

 little need be said ; the directions given by Euclid for 

 the construction of them are too clear and explicit to 

 render further explanation necessary. It may be well, 

 however, to invite your attention to two particulars in 

 connection with Proposition L, which instructs -us how 

 to describe an equilateral triangle upon a given finite 

 stmight line. Some commentators object to the word 

 finite as superfluous, considering the condition, " a given 

 straight line," to imply that the length, is fixed and de- 

 terminate. But a line may be given in position on\y, 

 without any limitation as to length, as in Proposition XI. ; 

 or it may be given in length merely, without any restric- 

 tion as to position : thus the line to be constructed in 

 Proposition II. is to have a given length, but is unre- 

 stricted as to position or direction. By " a given finite 

 straight line," Euclid means a line of given length, and 

 with given extremities ; and if any objection at all be 

 made, in reference to this ward finite, we think it should 

 be urged against the omission of it in Proposition II., 

 rather than against the introduction of it in Proposition 

 I. In the first proposition, we dare say Euclid thought 

 it prudent, to prevent all cavil, to state explicitly that 

 the extremities, A, B, are given ; and that lie did not 

 consider it necessary to repeat this in Proposition II. 

 Whenever a line is given in position merely, and restric- 

 tion as to its length forbidden, Euclid characterises it as 

 " a given straiglit line of unlimited length," as in Propo- 

 sition XII. ; but when it is a matter of perfect indifference 

 whereabouts the extremities are (position alone being all 

 that we are concerned with), the terms " a given straight 

 lino" are those always employed, as in Proposition 

 XXXI. A mere glance at this proposition will show you 

 that the given straight line of Proposition I. cannot be 

 so entirely free from restriction, as to length, as the given 

 straight line of Proposition XXXI ; and hence the pro- 

 priety of the restrictive term finite in the former. \Ve 

 should not have said so much about a mere word, had it 

 not been for the hypercriticism of others. You must 

 therefore regard these remarks, not as a comment upon 

 <l, butaa a comment upon his commentators. 



The other matter we should wish you to notice, in 

 connection witli this first proposition, is, that what is 

 called "the point C, in which the circles cut," is, in 

 fact, either of two points, one on each side of A B, so 

 that a second equilateral triangle may be described, on 

 the opposite side of the given line ; and it is this latter 

 position which the equilateral triangle, introduced into 

 the construction of Proposition IX., is to take. We 

 shall only further notice, that it would have been some- 

 what more explicit if Euclid had referred to the point 

 as where the circumferences cut, rather than as where the 

 circles cut ; though it is quite true that the circles them- 

 selves interpenetrate there ; but the distinction, very 

 properly made by Euclid in his definitions, between 

 circle and circumference, is in danger of being over- 

 looked by a beginner, in consequence of Euclid's mode 

 of expression, in reference to intersecting circumferences. 



Proposition IV. is the first of Euclid's theorems ; and, 

 being the first, and involving no construction, its proof 

 depends solely on the axioms. The demonstration 

 hinges upon what has been called the method of superposi- 

 tion ; that is, the imagining one figure to be placed upon 

 another, with a view to their perfect adaptation and coin- 

 cidence, and thence to the inference of their complet."! 

 equality (Axiom 8). If you wish to try the experiment 

 whether or not you have any taste or aptitude for geo- 

 metrical reasoning, you may, if you please, commence 

 with this theorem, and study the three preceding pn .- 

 bleins afterwards. It is a very beautiful specimen of 

 Euclid's mode of argumentation ; and is quite within 

 range of the powers of the merest beginner. But before 

 you address yourself to it, it may be as wall to reflect, 



