RAT 



RAT 



D A. 2dly. C F can never exceed C A ; therefore B F 

 can never exceed D A. ^dly. The arc coiitiiRially de- 

 creafing without limit, vanifhi'S in rcfpeft of the diameter, 

 wiiich is fixed ; therefore the chord A B, which is lefs than 

 the arc, likewife vanifhes in rcfpedt of the diameter A a. 

 But Afl:AB::AB:AF; therefore, if A B vanifhes 

 in refpe<5l of A a, A F will vanifli in rcfpeft of A B ; and 

 much more will A F vanifti in refpedt of A a ; but if A F 

 vanifh in refpeft of A a, it will alfo vanifh in rcfpedl of 

 i An, orC A. 



Now C A : C F :: A D : FB, and by divifion of pro- 

 portion CA:AF::AD:AD-FB. Therefore, 

 as A F vaniflies in refpeft of C A, fo does AD — F B 

 (the difference of the fine and tangent) vanifh in rcfpeft to 

 A D the tangent, aivd confequently in refpcft of F B the 

 fine. Whence we fee, that while the arc continually de- 

 creafes without limit, the fine approximates to the tangent ; 

 2dly, the fine never exceeds the tangent ; 3dly, although 

 the fine and tangent both vanifh in refpect of the radius, yet 

 their difference vanifh in refpetl of thefe quantities themfelves. 

 The/efore the ratio of equality is the limit of the varying 

 ratio which the fine and tangent have to each other, while 

 they both decreafe perpetually without limit, or it is their 

 ultimate ratio, or, as we may fay, they are ultimately equal. 

 The fine is lefs than the chord ; for in the right-angled tri- 

 angle A F B, the fide B F is lefs than the hypothenufe 

 A B ; the chord is lefs than the arc ; this is felf-evident, 

 the chord being a ftraight line, and tlie arc a curve, 

 both terminated by the fame points A and B. The arc is 

 lefs than the tangent ; for from the point D draw another 

 tangent to the circle in H ; and the lines A B H, A D H, 

 will be terminated by the fame points A and H ; and will 

 have the concavities turned the fame way ; therefore 

 the included arc A B H will be lefs than the fum of the 

 two equal tangents D A and D H ; confequently half that 

 arc, or A B, will be lefs than half the fum of the tangents 

 AD. 



Cor. I. — Hence the fine, chord, arc, and tangent, are all 

 ultimately in a ratio of equahty. This may appear becaufe 

 the chord and arc are included between the fine and tangent, 

 but perhaps more plainly thus. Of thefe four quantities, 

 •viz. the fine, chord, arc, and tangent, the fine is the lead, 

 and the tangent the greateft ; therefore, the difference be- 

 tween the fine and the tangent is greater than the difference 

 between any other two of thefe four quantities. If, there- 

 fore, the greatell of all thofe differences vanifh in refpett of 

 the leaft of all thofe quantities, much more wiD the differ. 

 ence between any other two of thefe four quantities vanifh 

 in refpeft of the quantities themfelves. 



Cor. 2. — Join B a, and in tlie right-angled triangle B F n, 

 the hypothenufe B a is greater than the fide F a ; therefore 

 A a — B rt is lefs than A a — Y a, or than A F ; but while 

 the arc A B decreafes continually without limit, A F va- 

 nifhes in refpett of A « ; much more then does A a — 'S>a, 

 the difference A a and B a, vanifh in refpeft oi A. a ; there- 

 fore the ultimate ratio of A a to B a is that of equality ; 

 and the ultimate ratio of i A «, or B C to B A, is that of 

 2 to I. See other applications of thefe principles in Lud- 

 lam's " Rudiments of Mathematics," from which the pre- 

 ceding article has been abflracted ; Newton's " Principia," 

 lib. i. ; Smith's " Fluxions ;" and Saunderfon's " Alge- 

 bra." 



Ratio, in our Law JVriters, is ufed for a judgment given 

 in a caufe. 



Hence, ponere ad rationem is to cite one to appear in judg- 

 ment. Walfingh. 88. 



Katio Status, Ragione di Jlato. See. R-E.\soti of State. 

 Vol. XXIX. 



Ratio ViBus. See Victus. 



RATIOCINATION, the aft of reafoniog. 



RATION, or Ratian, imthc jlrmy, a pittance or pro- 

 portion of ammunition, bread, drink, or forage, diflnbuted 

 to each foldier for his daily fubliftence. 



Some write the word radon, and borrow it from the 

 Spai'ih racion ; but they both come from the Latin ratio. 

 In fome parts they call it a remfon. 



The horfe have rations of hay and oats, when they cannot 

 go out to forrage. See Forrage. 



The rations of bread are regulated by weight. The or- 

 dinary ration of a foot foldier is a pound and a half of bread 

 per day. The officers have feveral rations, according td 

 their quality, and the number of attendants that they are 

 obliged to keep. 



When the ration is augmented on occafions of rejoicing, 

 it is called a double ration. 



The fhips' crews have alfo their rations or allowances of 

 bifcuit, water, &c. proportioned according to their flock. 

 The ufual ration at fea, particularly among the Portuguefc, 

 &c. is a pound and half of bifcuit, a pint of wine, and a 

 quart of frefh water per day ; and each month an arrobe, or 

 thirty-one pounds of fait meat, with fome dried fifh and 

 onions. 



RATIONABILES Expens.U, Reafonabk Expmces. 

 The commons in parliament, as well as the proftors of the 

 clergy in convocation, were anciently allowed rationabiles 

 expenfas ; that is, fuch allowance as the king, confidering 

 the prices of all things, fhaU judge meet to impofe on the 

 people to pay for the fubfiflcnce of their reprefentatives. 



This in the 17th of Edward II. was fettled at ten groats 

 per day for knights, and five for burgeiles : afterwards, 

 four fhillings a day for knights, and two fiiiUings for bur- 

 geffes ; which was then deemed an ample retribution, both 

 for expences, for labour, attendance, negleft of their own 

 affairs, &c. See BuiiGESs, a.n6. KtiiGHTS of the Shire. 



RATIONABILI Parte Bonorum, a writ which lies for 

 the wife, againfl the executors of her hufband, denying iier 

 the third part of her hufband's goods, after debts and fune- 

 ral expences paid. 



Fitzherbert quotes Magna Charta, and Glanville, to 

 prove, that, by the common law of England, the goods of 

 the deceafed, his debts firfl paid, fhould be divided into 

 three parts ; of which his wife is to have one, his children a 

 fecond, and the executors a third ; adding, that this writ 

 lies as well for the children, &c. as the wife. Such is the ge- 

 neral law of Scotland at this day. And whatever may hare 

 been the cullom of late years, in many parts of the king- 

 dom, or however it was introduced in derogation of the old 

 common law, the ancient method continued in ufe in the 

 province of York, the principality uf Walts, and the city 

 of London, till very modern times ; when in order to favour 

 the power of bequeathing, and to reduce the whole king- 

 dom to the fame flandard, three ftatutes have been provided; 

 the one 4 & 5 W. & M. c. 2. explained by 2 & 3 Ann. 

 c. 5, for the province of York ; another, 7 & 8 W. III. 

 c. 38. for Wales ; and a third, i I Geo. I. c. 18. for Lon- 

 don (fee Custom of London) ; whereby it is enafted, that 

 perfons within thofe diftrifts, and liable to thofe cufloms, 

 may, if they think proper, difpofe of all their perfonal 

 eftates by will ; and the claims of the widow, children, and 

 other relations, to the contrary', are totally barred. Thus 

 is the old common law now utterly abolifhed throughout all 

 the kingdom of England, and a man may devife the whole 

 of his chattels as freely as he formerly could his third part 

 or moiety. Blackft. Com. b. ii. 



ReBo de Rationabili Parte. See Recto. 



3 N RATIO- 



