R A S 



320 



RAT 



Rapin, 

 Rastadt. 



on the Whigs and Tories, and his great work, the His- 

 toire d' Angleterre appeared at the Hague in 9 vols. 8vo. 

 in J725, 1726, embracing the history of England from 

 the earliest periods down to the Proclamation of Wil- 

 liam and Mary. While he was collecting the materials 

 of that work, he made " An Abridgement of Ry- 

 mer's Fcedera," which was published in Le Clerc's Bib- 

 liotftcqiie Choisie. 



Rapin died of a fever at Wesel, on the l6th of May, 

 1725, in the 64th year of his age. 



RASTADT, a town of Germany, in the duchy of 

 Baden, and chief place of the district of the Murg, is 

 situated on the Murg not far from the Rhine. The 

 town is new and regularly built, and the principal 

 street is broad and handsome. It is surrounded by a 

 mound of earth, and contains the fine mansion of the 

 Princes of Baden. The principal manufactures are 

 those of carriages, fire-arms, silver, and plaited goods, 

 and mathematical and philosophical instruments. There 

 is here an institution for the education of young ladies. 

 The romantic valley of the Murg, which is considered 

 as Switzerland in miniature, deserves to be visited by 

 every traveller., Population about 4200. E^st long. 

 8 15', and north lat. 48 52'. 



RAT. See MAZOLOGY, Vol. XIII, p. 438, 440. 



RATAFIA, the name of a liquor made from the 

 kernels of cherries, apricots, c. In making the Rata- 

 fia of cherries, the cherries when bruised are put into a 

 vessel in which brandy has been long kept, and then 

 there is added the kernels of cherries, strawberries, su- 

 gar, cinnamon, white pepper, nutmegs, cloves, 10 quarts 

 of brandy being added for 20 Ibs. of cherries. 



RATHLIN, the name of an island situated between 

 the north coast of Ireland and Scotland. It is about six 

 miles long, and two and three-fourths of a mile broad, 

 having an indentation towards the middle, which forms 

 a large bay called Church Bay, which, except in wes- 

 terly winds, affords good anchorage and a safe harbour 

 for shipping. The number of plantation acres is about 

 200, which produce good barley. Kelp, however, is 

 the principal source of wealth in the island, about 100 

 tons being exported annually. Tumuli containing, 

 bones are found here, and also brazen swords and spear 

 heads. Rathlin is celebrated as the place where Ro- 

 bert the Bruce fled from his enemies, and the remains 

 of the fortress are still seen in which he is said to have 

 defended himself. Population about 1100. West 

 Long. 6 6\ North Lat. 55 20'. 



RATIO.* 



Ratio. UNDER this article the attainment of three important 

 V'""' objects in science has been attempted. The first aim- 

 ed at, was all the perspicuity of treating proportionali- 

 ty, or comparison of ratios, of which it is susceptible : f 

 the second was a satisfactory demonstration of the fifth 

 definition of the fifth book of Euclid ; | and the third, 

 an easy way of extending the subject by means of num- 

 bers, expressing the relative values of the magnitudes 

 under consideration. The consequences of these en- 

 deavours are now submitted to the perusal of the 

 reader. 



ARTICLE I. Ratio is the relation which one magni- 

 tude has to another, of the same kind, with respect to 

 quantity. 



II. If the first of four magnitudes has the same ra- 

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

 four magnitudes are said to be proportionals ; and, n 

 the contrary, if it be allowed that four magnitudes are 

 proportionals, it is understood that the first has the 

 same ratio to the second that the third has to the fourth. 

 In either of the two cases it is implied that the first is 

 exactly as great when compared to the second, as the 

 third is when compared to the fourth. 



III. If the first of four magnitudes be greater, when 

 compared to the second, than the third is when compar- 

 ed to the fourth, the first is said to have to the second 

 a greater ratio than the third has to the fourth. 



IV. If the first of four magnitudes be less, when 

 compared to the second, than the third is^when com- 

 pared to the fourth, the first is said to have to the se- 

 cond a less ratio than the third has to the fourth. 



V. If the first of four magnitudes has to the second 

 the same ratio which the third has to the fourth ; then 

 if the first be equal to the second, the third is equal to 

 the fourth ; if greater, greater, if less, less. For let A 

 be the first, B the second, C the third, and D the fourth 

 of the four magnitudes, and first, let A be equal to B. 

 Then as, by hypothesis, it is exactly as great when 

 compared to B, as C is when compared to D ; and as A 

 is equal to B, it is evident that C must be equal to D. 

 Secondly, let A be greater than B. Then, as A is ex- 

 actly as great when compared to B, by hypothesis, as 

 C is when compared to 1), and as A is greater than B, 

 C must be greater than D. Lastly, let A be less than 

 B. Then, as A, by hypothesis, is exactly as great when 

 compared to B, as C is when compared to D, and as A 

 is less than B, C must be less than D. 



VI. If the first of four magnitudes has the same ra- 

 tio to the second which the third has to the fourth, and 

 if any equimultiples whatever be taken of the first and 

 third, and also any whatever of the second and fourth ; 

 the multiple of the first will have the same ratio to the 

 multiple of the second, that the multiple of the third 

 has to the multiple of the fourth. 



Let A the first have to B the second, the same ratio 

 that C the third has to D the fourth ; 

 and let EG be any equimultiples 

 whatever of A and C, and FH any 

 whatever of B and D ; and then E 

 will have the same ratio to F, that 

 G has to H. A B D 



Ratio. 



* The Editor has been indebted for this valuable article to the Rev. Abraham Robertson, D.D. F.R.S. and Savilian Professor of As- 

 tronomy, Oxford. 



f See Dr. Hutton's Mathematical and Philosophical Dictionary, under the word RATIO. 



I Dr. Barrow, in the 21st, 22d, and 23d of his Mathematical Lectures, gives a statement of opinions concerning this fifth definition, 

 and endeavours to defend it against all objections. 



