MATHEMATICS-PLANE GEOMETRY. 



[BOOK v. PROPORTION: 



of a given line, and on tlie other side two equilateral tri- 

 angle*, one on each half of the line, then the two lines 

 drawn from the vertex of the larger triangle to the ver- 

 tices of the smaller triangles will trisect the given line : 

 rvquin-d the pn'f. 



Divide a given triangle into three equal ports by 

 lines drawn from the vertices of the triangle to a point 

 within it, 



~. I nscribe a circle in a rhombus. 



that tlio square circumscribing a circle is 

 double of the square inscribed in the same. 



9. If a circle be inscribed in a right-angled triangle, 

 and another be circumscribed about it, prove that the 

 sum of the sides containing the right angle will be equal 

 to the sum of the Uiauieton. 



10. From a given point in the arc of a circle, to draw 

 a tangent thereto without first finding the centre of {ho 

 circle. 



1 1. The straight lino bisecting any angle of a triangle 

 cuts tho circumference of the circumscribing circle in a 

 point which is equidistant from tho extremities of the 

 opposite side, and from the centre of the inscribed circle : 

 required the proof. 



12. If from any point within an equilateral and equi- 

 angular polygon perpendiculars be drawn to the several 

 sides, the sum of these perpendiculars will always be the 

 same, wherever the point from which they are drawn be 

 taken. 



13. Through a given point within, a circle it is required 

 to draw the shortest chord possible. 



CHAPTER V. 

 PROPORTION. 



A TREATISE ISTEJTDED AS A SUBSTITUTE FOB EUCLID'S 

 BOOK V. 



ISTBODUCTOBY. In the foregoing portion of elementary 

 geometry, we have given you the " Elements of Euclid," 

 substantially, in all their integrity : the modifications 

 we have introduced are for the most part merely of a 

 verbal character ; but while condensing tho language we 

 have been careful to preserve the spirit and rigour of the 

 original. In the few instances in which wo hare thought 

 an improvement might be introduced, or a defect sup- 

 plied, we have not hesitated to offer the suggestion, and 

 to propose the emendation : what little is done in this 

 way is sufficiently detailed in the Remarks appended to 

 the several books. You will, of course, submit these to 

 your own judgment always remembering that in mat- 

 ters connected with gepmetery, nothing is to be taken 

 upon trust ; mere opinion, unsupported by reasonings 

 reasonings which elevate it into proof must be regarded, 

 in thit subject, as of but little worth. 



We are now going to depart altogether from Euclid's 

 method of exposition, and to place before you a treatise 

 on Proportion, constructed on a different plan. We have 

 come to this determination only after mature delibe- 

 ration. It would, of course, be a much easier task to 

 transfer Euclid's fifth book into these pages. We could 

 find very little to remark upon in it, as the ancient Geo- 

 meter has displayed so much sagacity and penetration in 

 this, the moat elaborate of all his writings, that ho has 

 left to the moderns little or no room for improvement : 

 it must be studied just as it is (in Siinson's restoration), 

 or else be superseded in instruction by a treatise of equal 

 generality, but of geater simplicity. You will under- 

 stand, therefore, that we do not displace Euclid's fifth 

 book because of its imperfections, or because of its in- 

 adequacy to completely accomplish its objects ; but solely 

 because of its great difficulty to a beginner. Wo will 

 endeavour to give you here some notion of tho cause of 

 this difficulty. Tho subject of Euclid's fifth hook is 

 PROPORTION universal proportion : that is, not numerical 

 proportion merely, but proportion in reference to all 

 magnitudes and quantities whatever, whether numbers, 

 lines, surfaces, solids, or concrete quantities of any 

 kind. With proportion in numbers you are already 

 familiar : this will bo a help. You are also somewhat 

 acquainted with proportion in Algebra: this will be a 

 greater help ; for proportion in Geometry really accom- 

 plishes no more for things in general, than the same 

 doctrine in arithmetic and algebra accomplishes for what 

 tho notation of those sciences specially represents ; and 

 if this kind of proportion would do for geometry, the 

 fifth book of Euclid would become a very easy matter 

 indeed. Hut tho obstacle to this is, that geometrical 

 magnitudes, when compared together, are in many cases 

 found to be incommtiuiirable ; that is to say, two such 



magnitudes may be quite incapable of a common measure- 

 ment they may be of a nature not to admit of being 

 both measured by one and the same unit of measure- 

 ment, however minute the measuring unit be taken ; and, 

 consequently, both cannot be represented by numbers. 

 We have already adverted to an instance of this 

 kind* in the side and diagonal of a square, and to 

 another in the diameter and circumference of a circle. 

 You may divide the side of a square into as many equal 

 parts as you please from two parts to as many millions. 

 In every case each part is, of course, a measure of tho 

 side ; so that by applying such part, progressively, from 

 one extremity of the side onwards towards the other 

 extremity, that other extremity would at last be accu- 

 rately reached. 



But if tho same measure, however small it be taken, 

 be applied in like manner to the diagonal, the remote 

 extremity of it can never be accurately reached ; either 

 an unmeasured smaller portion will stiU be left, or elso 

 the applied measure will overlap and project beyond that 

 extremity. It is thus that these two lines are incom- 

 mensurable. That they are so, could not have been 

 found out by such practical or experimental tests as 

 those hero adverted to for illustration ; they must be 

 proved to be so by geometrical reasoning. 



It may be as well to caution you hero that you must 

 not speak of a line or quantity, by itself, as being incom- 

 mensurable ; this would be absurd. The diagonal of a 

 square is not itself incommensurable, since it has, of 

 course, its third part, fourth part, hundredth part, &c., 

 and is therefore mensurable by each of those parts ; but 

 as none of them will also measure the side, the two, con- 

 sidered together, are incommensurable ; there exists no 

 measure common to both. In tho same way in reference 

 to the circle the circumference itself is not incommen- 

 surable any more than tho diameter ; for each has its 

 fourth part, sixth part, (be. ; but it is incommensurable 

 with its diameter ; no length whatever can measure Imtli. 

 The circumference of a circle may bo ten feet and some 

 fraction of a foot ; the diameter will necessarily be 

 more than three feet ; but the exact fraction of a foot, 

 besides, it is not in the power of numbers to express. 



Now although Euclid makes no mention of iucommeii- 

 sur.-ilili! qu.intitie.s in his fifth book, he was well aware of 

 their existence ; and therefore, to render his theorems 

 on proportion general, he had to take care that this class 

 of i|iiantiti'M slmuld bo comprehended in his reasonings. 

 But proportion limited to numbers, or to the symbols 

 for numbers, would necessarily exclude incnmnicii- 

 surables ; ho therefore had to proceed quite independently 

 of arithmetic, and to secure to his propositions such a 

 universality that each theorem should rigorously apply, 

 whether tho quantities or magnitudes spoken of be 



Sec ante, p. 569. 



