384- G E O M then a :e=e’ :a’, b :e=e’ :b', and so on. Prop. :24 comes to this : If a :b=c :d and e :b=_/'2 (I, then a+c :b=e+f:d. Some of the proportions which are considered in the above pro- positions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view. § 56. The last proposition in the fifth book is of a different character. Prop. ‘.25. If’ four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together. In symbols- If a, b, e, d be magnitudes of the same kind, and if a :b=c :d, and if a is the greatest, heuee d the least, then a+d>b+c. 5'57. 'e returu once again to the question, 'hat is a ratio? ‘»'e have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, Hov often is one line contained in another? But the answer to this question is given by a number, at least in some cases. aml in all cases if we admit incommensurable numbers. Considered from this point of view, vc may say the fifth book of the Elements shovs that some of the simpler algebraieal operations hold for ineommensurable numbers. 1n the ordinary algebraieal treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject. B001; VI. § 58. The sixth book eoutains the theory of similar figures. After a few definitions explaining terms, the tirst proposition gives the first application of the theor_v of proportion. Prop. 1. Triangles and parallelograms of the same altitude are to one another as their bases. The proof has already been considered in § 49. From this follows easily the important theorem Prop. 2. If a straight line be drawn parallel to one qf the sides of a triangle, it shall cut the other sides, or those sidesprodueed, pro- portionally ; and if the sides or the sides produced be cut proportion- ally, the straight line which joins the points of section, shall be parallel to the remaining side of the triangle. § 59. The next proposition, together with one added by Simson as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz., if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AL :CB ; but if C be taken in the line All produced, we shall say that AB is divided externally in the ratio AC : CE. The two propositions then come to this : TuF.or.E.t (Prop. 3).—Th-e biseetor of an angle in a triangle divides the opposite side -internally in a ratio equal to the ratio of the two sides including that angle; and eonversely, if a line through the rertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex. THEOREM (Simson's Prop. A).—Th-e line which bisects an exterior angle of a triangle di-rides the opposite side externally in the ratio of the other sides; and conversely, if a line through the 'certe.e of a triangle divide the base eaternally in the ratio of the sides, then it bisects an erterior angle at the rerteee of the triangle. lf we combine both we have— TnEor.I-:..——The two lines which bisect the interior and exterior angles at one 'i'erte.r of a triangle diride the opposite side internally and externally in the same ratio, ri:., in the ratio of the other two sides. § 60. The next fo11r propositions contain the theory of similar triantrles, of which four cases are considered. They may be stated toget ier. THEOREM.-—TlI:0 triangles are similar-,— 1. (Prop. 4). If the triangles are equiangular; 2. (Prop. 5). If the sides of the one are proportional to those of the other; 3. (Prop. 6). If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal ; 4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right, or both obtuse; homologous sides being in each case those which are opposite equal angles. An important application of these theorems is at once made to a right-angled triangle, viz. :— Prop. 8. In a right—an_r/lrrl triangle, if a. perpendicular be drawn from the right angle to the base, the tr-ianglrs on each side of it are similar to the whole triangle, and to one another. E T R. Y C'orollary.—From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the lam- is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side. §61. 'l‘here follow four propositions containing problems, viz., in language slightly different from l'Iu«-lid's:— Prop. 9. To diride a straight line into a giren number of equal parts. [EUCL1DIA.‘. Prop. 10. "o diride a straight line in (1 giren ratio. Prop. 11. To find a third proqmrtionul to two giren straight lin¢.-. Prop. 12. Tojind a fourth proportional to three gircn straight lines. Prop. 13. To jiml a mean proportional between two giren straight lines. The last three may be written as equations with one unknown qnant1ty,—v1z., if we call the given straight lines a, b, e, and the required line 1‘, we have to find a line .7: so that Prop. 11. a :b = b zae; Prop. 12. a:b=c:ac; Prop. 13. a : at = :2: : b. Ve shall see presently how these may be written without the signs of ratios. 3‘ 6'2. Euclid considers next proportions connected with parallelo- grams and triangles which are equal in area. Prop. 14. Equal parallclograms which hare one angle If/' the one equal to one angle of the other hare their sides about the equal ltuglezs‘ reciprocallyproportional; and parallelograms which hare one angle Qf the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another. Prop. 15. Equal triangles 1I'hieh hare one angle of the out. equal to one angle of the other, hare their sides about the equal ((ng.’:‘s reczproeall y proportional ; and triangles which hare one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another. The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure A l‘ belonging to the former the two equal parallelo- grams Ali and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter. It is worth noticing that the lines FE and DC‘: are parallel. Ve may state therefore the theorem- 'l‘HEOI—:E.I.—If two triangles are equal in area, and hare one angle in the one rertieally opposite to one angle in the other, then the two straight lines which join the remaining two rertiees of the one to tho.»-* of the other triangle are parallel. § 63. A most important theorem is Prop. 16. If four straight lines be proportionals, the rectangle contained by the e-.rtre2nes is equal to the rectangle contained by the means; and if the rectangle contained by the eastremes be equal to the rectangle contained by the means, the four straight lines are pr(q;or- tionals. In symbols, if a, b, c, d are the four lines, and C if a :b=c :d, then ad =bc ; and conversely, if ad=be, then a :b=c :d, where. ad and be denote (as in § 20), the areas of the rectangles contained by a and d and by b and e respeetivel_v. This allows us to transform every proportion bct'een four lines into an equation between two products. It shows further that the operation of forming a product _of two lines, and the operation of forming their ratio are each the ll1'L'l'>L‘ of the other. If we now define a quotient g of two lines as the number which multiplied into b gives a, so that (-5 b = a, we see that from the equality of two quotients E b d follows, if we multiply both sides by bd, %b.rl = %d. b,
ad = cb.