BOOKS 1v. v.] It will be well to show at once in an example how this _definition can be used, by p1'oving the first part of the first proposition 111 the sixth book. Trinnf/Irrs of the same altitude arc to one another as their buses, or if It and b are the bases, and a and B the areas, of two triangles which have the same altitude, thcn_tt_: b : : a : (3. 'I‘o prove this, we have, according to Definition 5, to show—— if ma>ub, then ma>n[-3, if ma=nb, then 'I)Ia=’)lfl, if ma<ab, then ’)Ila<nfl. That this is true is in our case easily seen. We may suppose that the triangles have a connnon vertex, and their bases in the same line. 'e sct oil‘ the base to along the line containing the bases in times; we then join the different parts of division to the vertex, and get m triangles all equal to a. The triangle on ma as base equals, therefore, ma. If we proceed in the same manner with the base b, setting it oil‘ 72. times, we find that the area of the triangle on the base 71b equals 111-], the vertex of all triangles being the same. hit if two triangles have the same altitude, then their areas are equal if the bases are equal ; hence 'ma=71)8 if mu=-nb, and if their bases are unequal, then that has the greater area which is on the greater base; in other words, -ma is greater than, equal to, or less than NB, according as mm is greater than, equal to, or less than 72b, which was to be proved. § 51). It will be seen that even in this example it does not become evident what a ratio really is. It is still an open question whether ratios are magnitudes which we can compare. We do not know whether the ratio of two lines is a magnitude of the same kind as the ratio of two areas. Though we might say that Def. 5 defines equal ratios, still we do not know whether they are equal in the sense of the axiom, that two things which are equal to a third are equal to one another. That this is the case requires a proof, and until this proof is given we shall use the -: instead of the sign =, which, however, we shall afterwards introduce. As soon as it has been cstablis-lied that all ratios are like magni- tudes, it becomes easy to show that, in some cases at least, they all‘o- nmnbcrs. This step was never made by Greek mathematicians. They distinguislied always most carefully between continuous magnitudes and the discrete series of nmnbers. In modern times it has become the custom to ignore this difference. I t‘, in determining the ratio of two lines, a common measure can be found, which is contained m. times in the first, and 9L times in the second, then the ratio of the two lines equals the ratio of the two numbers mm. This is shown by Euclid in Prop. 5, X. But the ratio of two numbers is, as a rule, a fraction, and the Greeks did not, as we do, consider fractions as numbers. Far less had they any notion of introducing irrational numbers, which are neither whole nor fractional, as we are obliged to do if we wish to say that all ratios are numbers. The incommensurable numbers which are thus introduced as ratios of ineommensurable quantities are now-a- days as familiar to us as fractions ; b11t a proof is generally omitted that we may apply to them the rules which have been established for rational numbers only. F.uclid’s treatment of ratios avoids this ditliculty. Ilis definitions holds for eommensurable as well as for incomnicnsurable quantities. liven the notion of incominensurable quantities is avoided in Book V. But he proves that the more elementary r11lcs of algebra hold for ratios. Ve shall state all his propositions in that algebmieal form to which we are now at-customed. This may, of course, be done without changing the character of Eur’-lid's method. § 51. Using the notation explained above we express the first propositions as follows :— Prop. 1. If a=ma’, b=mb’, c=9)zc', fl.»-n a+b+c=m(a'+b'+c’). Prop. :2. If (I =mb, and c=m(l, r=91b, and f = ml, then u +c is the same multiple of b as c+f is of d, viz. :— (r.+ c= (/n+ 7z)b, and c+f= (:n+7z)cl. Prop. 3. If a==mb, c=m(l, then is ’lL[t the same multiple of 17 th ~t .u- is of J, viz., 9m=mn7», 7zr.'=.z.ml. Prop. 4. If a :b: :c :(l, then mrl : nb : : mc ziul. Prop. 5. If a=mb, a.nd c=m(l til-:11 a -— c= m(b — (1). Prop. 6. If a=mb, c=nul, then are a — nb and c—ncl either equal to, or equirnultiples of, b and J, viz., a—7zb=(m—7z)b and c—ml=(m—71)(l, where m—n may be unity. All these propositions relate to cquimulttples. Now follow propositions about ratios which are compared as to their magnitude. § 5?. Prop. 7. If a=b, then a :c : :b :c and c :a. 2 zctb. The proof is simply this. As a=b we know that ma=mb; GEOMETRY therefore if ma>ac, then mb>1Lc, if 'ma=nc, then 7n.b=7zc, if mu<uc, then mb<nc, therefore the first proportion holds by Definition 5. Prop. 8. If a>b, then a :c>b:c, and c:a<c :b. The proof depends on Definition 7. Prop. 9 (converse to Prop. 7). If a :c : :b :c, orif czazzczb, then a=b. Prop. 10 (converse to Prop. 8). If It :c>b_:c, then a>b and if c : a<c :b, then a<b, Prop. 11. If a :b : 2c 2d, and a :b : :e :f, then c:cl::e:f. In words, if two ratios are equal to a third, they are equal to one another. After these propositions have been proved, we have a right to consider a ratio as a vnagnitzlde, for only now can we consider a ratio as something for which the axiom about magni- tudes holds: things which are equal to a third are equal to one another. We shall indicate this by writing in future the sign = instead of : : . The remaining propositions, which explain themselves, may then be stated as follows:— §53. Prop. 12. If a:b=c:d=e:f, then a+c+e:b+d+f=¢z :1). Prop. 13. If a:b=c:dandc:(l>c:f, then a:b>c : f. Prop. 14. If a :b=c : fl, and a>c, then b>(l. Prop. 15. Magnitudes have the same ratio to one another that their cquimultiples have— mar, :mb=a :1). Prop. 16. If 0-, b, c, cl are magnitudes of the same kind, and if It :b=c:(l, then It :c=b_:d. Prop. 17. If a+b :b=c+d zcl, then u :b=c : cl. Prop. 18 (converse to 17). If It : b=c :d then a+b :b=c+d :(l. Prop. 19. If a, b. 0, cl are quantities of the same kind, and if a :b=c :d, then a—c:b—(l=a:b. § 54. Prop. 20. If there be three onamzitudcs, and other three, u'Iu'ch have the same ratio, taken two and two, then if the first be greater titan the third, the fourth shall be greater than the stat]: ,- mzrl z_'f equal, equal ,- and if less, less. If we understand by azbzczclzcz . . . =a’:b’:c’:d':c’: . . . that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side, we may write this theorem in symbols, thus :— If It, b, c be quantities of one, and d, c, f magnitudes of the same or any other kind, such that n,:b:c=d:c:_f, and if «>0, then d> f, but if ' [t=c, then d= f, and if a<c, then cl< f. Prop. 21. If a:b=c:fandb:c=d:e, . 1 1 1 orif a:b:c=—:—:_, e (l and if a>c, then cl> f, but if a=c, then d= f, and if a<c, then cl< f. By aid of these two propositions the following two are proved. §55. Prop. 22. If t/uerc be any number of rnagnitzulcs, mid es many others, which hate the same ratio, taken two and two in order, the first shall Iuzrc to the last of the first 97zagm'tu(lcs the same ratio which the first of the others has to the last. Ve may state it more generally, thus :— If a:b:c:d:c:.. then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For mstancc—— ' . =a’:b’:c’:tl’:c':. . . , a :c=a’ :c’; b : e=b’:c’, &c. Prop. 23 we state only in symbols, viz. :- 1 1 .1 . 1 1 C,
If a:b:c:d:c:...=E,:5,_ _d,;E,...,