RAT 



323 



RAT 



Ktio. the same ratio to the second that the third ha to the 



*"-* fourth. That in, if it !H> A + B : H : : C + I) : 1), then 



A : B : : ( ' : I). Fur it being a -|- b : b : : c -f- d : d, ly 

 article J'J, axd + bxd=f>x< +bx d. Consequently 

 a X d=b X f, and by article 13, a : b : : < : d, and there- 

 fore A : B: : (': I). 



XVIII. 1 1' tlie first of four magnitudes hns the same 

 ratio to the second that the third has to the fourth, 

 then the first and second together will have the same 

 ratio to the second that the third and fourth together 

 have to the fourth. That is, if it be A : B : : C : I), 

 thenA + B:B::C+D:D. Foritbeinga :h::c: rf.by 

 article 1 2, a X d=b x c ; and b X d being added to these 

 equals, we have axd+bxd=bxc + bxd. Conse- 

 quently, by article 13, a-f-o:6::c-f</:</, and there- 

 fore, A + B : B : : C + D : D. 



XIX. If a whole magnitude has the same ratio to a 

 whole that a magnitude taken from the first has to a 

 magnitude taken from the other; the remainder will 

 have the same ratio to the remainder that the whole 

 has to the whole. That is, if C be a part of A, and 1) 

 a part of B, and if it be A : B : : C : D, then A C : 

 B D : : A : B. For it being a : b : : c : d, by article 

 J2, a x d = b X c, and these equals being subtracted 

 from a x , we have a x b a. x </=o X b b x c. Hence, 

 byarticle 13,a c: b d: : a : b, that is A C : B D 

 : : A : B. 



If from the equals a X d, b X c, we take c x d, we 

 have axd c xd = b x c c x d, and then a c : 

 b d::c:d. That is A C : B D : : C : D. This 

 also follows from the above and article 15. 



XX. If four magnitudes be proportionals, the sum 

 of the first and second will have the same ratio to their 

 difference that the sum of the third and fourth has to 

 their difference. That is, if it be A : B : : C : D, then 

 A + I3 : A B : : C + D : C D. For it being a : b : : 

 f.d, by article 12, axd= bxc, and therefore a X d 

 -}- b x c=a xd bxc. To these equals add a x c 

 b X d, and then axe axd + bxc bx d=a x c -f- 

 axd bx c, bxd,or (a + b) x(c d)=(a 6) X 

 (c -f d.) Hence, by article ] 3, a + b : a b : : c + d : 

 c d, and therefore A + B : A B : : C 4- D:C 

 D. 



XXI. If there be any number of magnitudes, and as 

 many others, which, taken two and two in order, have 

 the same ratio; the first will have to the last of the 

 first magnitudes, the same ratio which the first of the 

 Others has to the last. First let there be three magni- 

 tudes, A, B, C, and other three, D, E, F, and let it be 

 A : B : D : ; E, and B : C : : E : F, and then it will be 

 A : C : : D : F. For, as a : b : : d : e, and b : c : : e :/, by 

 article 1 2, a x e b x d, and b xf=c X e, and therefore 

 axe bxd a d 



^x7 = bxf ~ = T' or a xf=c x d - Hencc by 



article IS, a : c : : d :/; that is, A : C : : D : F. 



Again, let there be four magnitudes, A, B, C, G, and 

 other four, D, E, F, II, and let it be A : B : : D : E, 

 B : C : : E : F, and C : G : : F : H, and then it will 

 be A : G : : D : H. For by the above A : C : : D : F, 

 and by what is now allowed C : G : : F : H. Conse- 

 quently, by the first case again, A : G : : D : II : and 

 in the same manner the demonstration may be extend- 

 ed to any number of magnitudes. 



XXII. If there be any number of magnitudes, and 

 as many others, which taken two and two in a cross 

 order, have the same ratio ; the first will have to the 

 last of the first magnitudes the same ratio which the 

 first of the others has to the last. First let there be 



three magnitude!, A. B, C, and other three, D, E, F, 

 and let it be A : B I : I , ,-md B : C : : D : E, and then 

 it will be A : C : : I) : F. For as it is a : b : : e :/, and 

 e: 6 : : c : d by article 12, a X/= b X t, and b x e = 

 c x d, and therefore a x /- r X'l. II ence, by article 1 3, 

 a: c: :d:f; that is A : C : : 1) : F. 



Again, let there be four magnitudes, A, B, C, G, and 

 other four, II, D, E, F; and let it be A : B : : E : F 

 H : C : : D : E, and C : G : : H : D, and then it will 

 be A : G : : H : F. For as in the preceding cae, A : C : : 

 I) : F, and therefore, again, by the first case, A : G : : 

 II : F, and in this way the demonstration may be ex- 

 tended to any number of magnitudes. 



By the method employed in the foregoing articles, 

 the doctrine of ratios may be easily extended. 



RATISBON or REGENSBWIIO, an ancient city of 

 Germany, in the kingdom of Bavaria, is situated on the 

 south bank of the Danube, opposite to the mouth of 

 the river Itegen. It is large and populous, and built 

 of stone, but the streets are crooked and narrow, and 

 the houses lofty and old-fashioned. The cathedral, 

 which is one of the best of its public buildings, is a 

 large Gothic edifice, built in 1400. The church and 

 abbey of St. Emmeran, which is like a small town, con- 

 tains some good paintings, and also a mathematical and 

 physical cabinet, The town-house is an old and un- 

 interesting building. Besides these buildings, we may 

 enumerate the palace of the Prince of Tourand Taxis, 

 in which there is a good library open to the public ; 

 the church of the Trinity, the Scottish church and 

 convent, and its library ; the building of the Jesuits' 

 College, the arsenal, and the Haidplatz. There are 

 also several hospitals in Ratisbon, two public libraries, 

 a botanical society, and a public drawing. school. A 

 bridge over the Danube, of fifteen arches and 1091 

 feet long, forms a communication between the city 

 and the suburb of Stadt-am-hof. 



Ratisbon has few manufactures, and very little trade. 

 The principal manufactures are those of linens, lace, 

 silk and worsted stockings, and needles. The fire- 

 arms of Kugelreuth, particularly his pistols, are great- 

 ly admired. Wood, provisions, corn, and salt are sent 

 down the Danube to Vienna. There are several brew- 

 eries and distilleries in the town, and dockyards for 

 building boats and small craft There are also here some 

 saw-mills driven by water. The hydromel of Ratis- 

 bon is in great request, and a considerable quantity 

 of it is exported. The two annual fairs of St, George 

 and St. Michael are well attended. The chief prome- 

 nades are, the alley of the Prince of Taxis, the high 

 and the low Wcorlh, and the Lime Trees. The inha- 

 bitants, who are principally Catholics, are computed 

 at 24,000. East Long. 12 4' 30". North Lat. 4<T 

 0' 53". 



RATZEBURG, a town of Denmark, in the duchy 

 of Lauenburg, and situated on an island in a lake of 

 the same name. The lake is about thirty miles long, 

 and nine broad, and communicates with the continent 

 on the east by a bridge, and on the west by a dike. 

 The streets of the town are regularly laid out, and the 

 houses are built in the Dutch fashion. The Regency 

 Ofiice, where the court of justice and the consistory 

 are held, stands in the market place ; and the cathe- 

 dral deserves to be visited. 



The principality of Ratzeburg, between Mecklen- 

 burg and Sax e- Lauenburg, contains about 136 square 

 miles, and 14,000 inhabitants, and is traversed by the 

 river Trave. The soil is fertile, producing much 

 wheat, and pasturing many cattle. It was once a 



