RAT
Ratios greater than the true. Ratios lefs than the true.
I
2
0X2
I
2
1
3
8
I X2
3
2 6
IXI
2
ii
76
4x1
28 32x1
8 11
3x1
4
87
'9
7x4
106
39
87
32
193 1264
71x6
465
106 1 158
1264 '457
39x1 426
1457
21768
536x1 8008
465x1 536
23225 25946
8544x1 9545
2721 23225
1001 x8 8544
49171 fcSV.
18089x10 EsV.
25946
9545X1
S3c.
Divide the greater term 2.71828, &c. by the IefTer 1, or the greater I by the Iefler 0,367879, E3V. and again the Iefler by the remainder, and this again by the laft remainder, and fo on : the quotients arifing will be 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, ij 12, i, 1, 14, 1, 1, 16, 1, 1, &c. Thefe being found, two rows or columns of ratios muft be made, the one containing the ratios greater than the true, and the other fuch as are lefs ; beginning the computation from the ratio's 1 to 0, o to 1, which are moil remote from the truth,' and from thence proceeding to fuch as approach continually nearer. Let then the terms 1 and o be mul- tiplied by the firfl: quotient 2, and write the produces 2 and o, under the terms o and 1 ; then adding, there will arife the ratio 2-fo to o-|-i, or 2 to I. Multiply the terms of this ratio by the. fecond quotient 1, and add the produces 2 and 1 to the terms 1 and o in the firfl: column, there wil! arife the ratio 2+1 to i-fo or" 3 to 1. The terms of this ratio being multiplied by the third quotient 2, and the produces 6 and 2 added to the terms 2 and I of the fecond column, will give the ratio 8 to 3. Thefe multi- plied by the fourth quotient 1, and the products 8 and 3 added to the preceding terms 3 and 1 in the firfl; column, will give the ratio 11 to 4. Whole terms multiplied by the fifth quotient 1, and the produces 11 and 4 added to the ratio 8 to 3 give the ratio 19 to 7 ; whofe terms being mul- tiplied by the flxth quotient 4, and the products 76 and 28 be- ing added to 11 and 4, give the ratio 87 to 32. And thus we may go as far as we think fit ; proceeding alternately from one column to the other. This being done, we fhall find ratios greater than the true, to be 3 to 1, 11 to 4, 87 32, 193 to 71, 1457 to 536, 23225 to 8544, 49171 to 18089, &c. And the ratios lefs than the true will be, 2 to 1, 8 to 3, 19 to 7, 106 to 39, 1264 10465, 2721 to 1001, 25946 to 9545, &c. And thefe are the principal and primary ratios^ which continually approximate to the ratio propofed.
But if the whole feries of ratios greater than the true, be required, fo that no ratios greater than the true, and ex- prefled in fmaller terms, fhall come nearer the truth ; and If alfo the whole feries of ratios lefs than the true, and fuch, as that no ratios lefs than the true, and preffed in fmaller terms, fhall approach nearer to the truth, be defired, then other fecondary ratios mud be in- ferted between the primary already found. And thefe take place where the quotients furpafs unity. They may be found by changing the multiplication by the quotient as above directed, into a continual addition of the terms, as often as there are units in the quotient. Thus, the firft quotient being 2, the terms 1 and o are to be twice added to the terms O and 1. The funis will give the ratios 1 to I and 2 to 1. Thefe laft terms 2 and 1, die fecond quo- tient being 1, muft be once added to the terms 1 and o, and the fum will give the ratio 3 to 1. Alfo the terms 3 and 1, the third quotient being 2, are to be twice added to the terms 2 and 1 ; and the funis will give the ratios 5 to 2, 8 to 3. Thefe laft terms 8 and 3, the fourth quotient be- ing 1 muft be once added to the terms 3 and 1, and the fums will give the ratio 11 to 4. Thefe terms 1 1 and 4, the fifth quotient being x, muft be once added to the terms 8 and 3, and the funis will give the ratio 19 to 7. Laftly, thefe terms 19 and 7, the lixth quotient being 4 muft be four times added to the terms 1 1 and 4. The fums will give the ratios 30 to 11, 49 to 1 8, 68 to 25, 87 to 32. And fo we may proceed as far as we think fit. The operation be- ing performed, we ihall find the whofe feries of ratios greater than the truth to be 1 to o, 3 to 1, 1 1 to 4, 30 to Suppl. Vol. II.
RAT
to ir, 49 to 18, 68 to 25, 87 to 32, &f f , and in like
manner, the whole feries of ratios lefs than the truth wil' be to 1, 1 to i, 5 to 2, 8 to 3, 19 to 7, &c.
Example of the operation.
Ratios greater than the true. 1 0x2
Ratios lefs than the true»
3
8
1x2 3
11 19
4X 7
3° 19
11
7
49 19
18 7
68 19
25 7
87 fcfe.
32x1
(Sc.
IXI
I
3x1
4
- 9
87
7X4 3 2
106
39x1
&c.
See Cotes harmonia menfurarum, p. 7, &c. The ingenious author has not given the demonftration of his method ; but Dr. Saunderfon has fhewn the reafon of it, in ' the fifth book of his algebra, to which we refer the reader. By means of this method, the approximated values of the ratio of the circumference, to the diameter of a circle, may be found in fmall terms. Thus the proportion of the cir- cumference of a circle to its diameter, being according to Van Ceulen's numbers, when abridged, 3 141 59265359 to ioooooooopoo ; dividing the greater by the IefTer, &c. as before directed, we fhall find the quotients 3, 7, 15 and 1, which will give (he following ratios, 3 to 1, 22 to 7, 333 to 106, and 355 to 113; the fecond is that of Archi- medes, and the fourth that of Adrian Metius. This proportion of 113 to 355 approaches very near the truth, only erring by 2 in the feventh decimal, when reduced to that form, for 355 : 1 13 : : 3, 1415129 : 1 now it ought to be 3,1415927 to 1. To remember this proportion the bet- ter, we may make ufe of the following artifice. Take the three firfl: odd numbers i, 3, 5, and write each twice, thus 113355, then will the three firfl: figures of this number 1I3, be the diameter, and the three laft, 355 the circumference. Thefe approximations are of ufe in many practical parts of mathematics. See inftances thereof in Huygen's chjeriptio automati planetarii among his poflhumous works, torn. 2. p. 174. Edit. Amft. 1728, where he defcribes his method, and demonftrates it. In mufic, it is the foundation of the different febemes of geometiical temperatures. See the ar- ticle Temperament.
Ratio modidaris. See Logarithm.
RATIONIS os, a term ufed by fome anatomical writers to ex- prefs the osfmcipitis.
RATIONARIUM, among the Romans, a book which con- tained the accounts of the empire. Pitifc. Lex. Ant. in voc. It was otherwife called breviarium. See Breviary.
RATSBANE. See the article Arsenic.
RATTLE fnake, a very dreadful fpeeies of fjrpent, whofe bite is fatal, if not timely remedied, and which is diftin- guiflied from all other ferpents by the rattles in its tail. This is compofed of feveral fcaly fubflances, and is faid to encreafe by the creature's age ; every year adding one fcale to it. It moves over the rocks and mountains with pro- digious fwiftnefs, but is lefs nimble on even ground, than many other fnakes.
It grows to four or five feet long, and fometimes, though rarely, more ; and one of more than four feet long, having been diflected, and accurately defcribed by Dr. Tyfon, the account that gentleman gives cf it may not be un- acceptable to the reader. The body where largcft, which was near the middle, meafured fix inches and a half round. Its neck only three inches. Its head was fiat, as that of the viper ; and as the jaws are very broad and protuberant, and the nofe fharp, it fomewhat refembles the head of an arrow. At the end of the nofe are phiced the noflrils, and between thefe and the eyes there are two other holes which may be miftaken for ears ; but they only go into a hollow of the hone of the fkull, without any perforation into the brain. The viper Jus nothing of thefe holes. The eyes are round, and-wholly refemble thofe of the viper. The whole body of the creature greatly alfo refembles the viper, but for the Angularity of the rattle ; and over the eyes thvre are two large fcales, looking like eye-brows. The fcalcs which cover the head are very fmall, thence they become gradually larger as they reach toward the middle D d d of
