MATHEMATICS-PLANE GEOMETRY. 



[REMARKS on BOOK I. 



conception of the thing to win h it refers. If you 

 steadilv oontt-i a straight line, 



giving due consideration to its distinguishing peculiarity 

 uniformity of direction you mu-t M from 



the same point, cannot both be equidistant from a third ; 

 OM may be parallel to this third, or even-where equidis- 

 tant from it, since, as we have seen, parallels ixtpouible; 

 but one must of necessity meet 



Of the propositions passed over, the 22nd and 21th 

 are th only one* requiring any special notice here. 

 Both the**, as given in the text of Dr. Simson, are open 

 .in. In the first of them, it is taken for granted 

 that the two circles employed iu the construction must 

 cut one another ; and in the second, it U assumed that 

 the point F falls '*/" the lino E G. You ill find these 

 defects acknowledged iu the notes at the end of Simson's 

 Ewlul ; but they were first pointed out by Air. Thomas 

 Simpson, in his Element* of Geometry. The emen- 

 dations of the latter were, however, but ill received by 

 the " restorer of Euclid," who treated the " remarkcr,'' 

 a* he called him, with a good deal of contempt ; the 

 more to be reprobated, as the poor self-taught weaver 

 (for such Simpson in early life was) was very superior, 

 as a man of science, to his academical opponent, great as 

 were the merits of the hitter in the field of ancient 

 geometry. The biography of Thomas Simpson is full of 

 instruction and encouragement to the young and un- 

 aided student, who cannot fail to view with ink-rest the 

 steps by which a person in Simpson's position, without 

 books, money, or friends, plying his humble calling 

 among the lowest ranks of society, was conducted, by 

 the force of perseverance, to the proud eminence which 

 he eventually attained. In the annals of science he 

 ranks among the most distinguished mathematicians 

 of the last century ; and yet, at the age of nineteen, 

 he was ignorant of the first rudiments of common arith- 

 metic.* 



The defect above alluded to, in Dr. Simson's version 

 of the 24th Proposition, is removed in the present 

 edition ; and the objection mode to thu reasoning in 

 the 22nd may be disposed of as follows : After having 

 described the circles, as at page 538, reason thus : One 

 of thete circle* cannot be wholly without the other, for then 

 FG, the distance of their centres, would be either equal 

 to, or greater than the sum of the radii ; but, by hypo- 

 thesis, it islets. Neither can one of t!ie circles as, for 

 instance, that whose centre is G be wholly within tin 

 other ; for then the radius, F D, of the latter would be 

 equal to, or greater than F H ; but, by hypothesis, it is 

 leu ; hence, since one circle con be neither wholly without 

 the other, nor wholly within they must be partly wi'h- 

 ottf and parf'y within one another, . . they must cut in 

 some point K. This completion of the proof may be 

 introduced in a second reading of the first liook. 



Proposition XXXII. U among the most interesting 

 theorems of this first book ; but an objection to the 

 demonstration of it may be made, the occasion for which 

 had better be removed. You are directed to draw, 

 through the point C, a line C E parallel to .-V 15, by Pro- 

 position XXXI. ; and it is then inferred that the alter- 

 nate angles B A C. A C E are equal, by an appeal to 

 Proposition XXIX.; but to draw the parallel C E, 

 surely everybody would proceed by making the angle 

 ACE equal to B A C ; that is to say, wo should 

 make the alternate angles equal to get the parallels, and 

 should then make use of the parallels to prove the alter- 

 nate angles equal. You will at once nee that we should 

 avoid this circuitous method of proceeding, by making 

 the angle ACE equal to BAG by Proposition \ \ 1 1 1. ; 

 and then inferring the parallelism of A B, CE from 

 Proposition XXV1L, so that Proposition XXXI. need 

 not Ira called into operation at all, 



The corollaries to this proposition are remarkably 

 beautiful ; and the second, especially, cannot fuil to 

 excite, in a person who reads it for the tirat time, af.' 

 of surprise. It would indeed be a feeling of inrml- 

 U this were possible in geometry. That 'the sum of thu 



n of Simpwn. In T^> rn*lt >/ 

 ullon'l Vnf Ar.i'M- i.' Ihr'iann,,/. 



under IHffi- 



exterior angles, formed by prolonging the sides of a 



.ii'-al Ii,-ur0, should always lie exactly the some, 

 w hether the figure h.ive three sides or as many thon 

 is a truth so far beyond the reach of practical observ 

 and experiment, and apparently so improbable, tl. 

 the absence of geometry, it* existence could scarcely havo 

 been suspected, much less established ; and yet an orgu- 

 | meiit of half-a-dozen lines produces in every mind the 

 fullest conviction of the fact. 



But the corollary that precedes this, though less 

 striking, has, perhaps, the greater practical interest ; 

 among other things, we learn from it that the sum of 

 the angles of a four-side 1 ti.ure is twice as great as the 

 sum of the angles of a three-sided figure ; the sum of 

 the angles of a five-sided figure, tli'n- times as great ; of 

 a six-sided figure, four times as great, and so on ; but 

 the most noticeable practicable inference is, that only 

 three regular figuros.-t namely, the equilateral triangle, 

 the square, and the regular six-sided figure or h> -j-iiyoi, 

 can, by repetition, completely cover a surface ; in other 

 words, that, without leaving any blanks or interstices, 

 we may cover a surface with a mosaic work of equilateral 

 triangles, or of squares, or of regular hexagons, but not 

 with regular figures of any other kind. It would be im- 

 possible, for instance, to form a piece of tessella;e 1 ; 

 ment with slabs of any other legular figure but a 

 these three ; because the uniting together of any other 

 forms, by adjusting side to side, would not fill up the 

 space about the corners there would be cither left an 

 angular gap, or else the stones must overlap one another. 

 Y"'i will readily see the truth of this from the fol- 

 lowing considerations : 



Let us first consider the equilateral triangle : as the 

 three angles moke two ri_;lit angles, each must be J of 

 two right angles that is, J of Jour right angles ; cou- 

 ^equeutly, if tijc equilateral triangles were placed side by 

 side, a corner or vertex of each being at the same common 

 point, all the angular space about that point would lie 

 completely occupied ; and no one triangle would overlap 

 another, for the angles about a point amount to just four 

 right angles (Prop. XIII., Cor. 2). Let vis next con 

 the square ; and, as each angle of a square is a right 

 angle, it is plain thai/bill" squares, each with a vertex at 

 the same point, when placed in contact, will exactly till 

 the space about the point. 



The figure next in order is the regular /t<->i/rii:nn t or 

 five-sided figure. The corollary teaches us that tiie sum 

 of its five equal angles amounts to six right angles ; con- 

 sequently each angle is one right angle and ajijth. Now 

 you cannot multiply 1 \ by any whole uuinKer that will 

 make the product 4 no such number exists ; in other 

 words, you cannot arrange the angles of pentagons round 

 a point, as the common vertex of all, so as to fill up the 

 four right angles about that point ; the pentagon, ti: 

 fore, must be rejected. The next figure is the hexagon. 

 By the corollary wo loam t .at us six equal angles amount 

 to eight right angles: cm. '-.i -it an^'.e is equal 



to J of eight right angles ; or, which is- the sa'iio ti. 

 to J of four right angles. It follows, then-fort:, that 

 three regular hexagons, plac-'d hide dy ,si<le, round a 

 point, as a common vertex, will oxactly fill the space 

 about that point. 



Wo need jiot extend the cxaminstion any further ; for, 

 as the sides of regular figures increase iu number, the 

 angles inorea e in magnitude; and, as it has just him 

 seen that three angles of a six-sided i. ,-nlar figure, make 

 four right angles, more than two of a seven, or ei/ut, or 

 nine-siiled figure, could never be required ; but if 

 equal angles of any tignre could make /our right an 



uigle would equ - nrd. 



It thus follows, that if space about a point is to be filled 

 up by the juxtaposition of regular figures, these figures 

 must be either equilateral triangles, squares, or regular 

 hexagons. 



Of these three classes of figure*, it is demonstrable, by 

 more advanced principles, that tha hexagon will anoloM 

 a given amount of space, with less extent of outline or 



Hiitht-tinM flgnir* ire e*id to be regular, wbra they are both rj-ii- 

 Inlrtal und ryuianffiUar. 



